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\centerline {{\bf TOWARDS A MACROSTATISTICAL MECANICS}\footnote
*{Based on a talk given at the Workshop on "Mathematical Physics
Towards the 21st Century", held at Beersheva, Israel, March
14-19, 1993}}
\vskip 1cm
\centerline {{\bf by Geoffrey L. Sewell}\footnote{**}{Partially
supported 
by European Capital and Mobility Contract No. CHRX-CT92-0007}}
\vskip 0.5cm
\centerline {\bf Department of Physics, Queen Mary and Westfield
College}
\vskip 0.5cm
\centerline {\bf Mile End Road, London E1 4NS}
\vskip 1cm
\centerline {\bf ABSTRACT}
\vskip 0.5cm
I discuss the general question of the derivation of the
statistical mechanics of macroscopic variables from the quantum
structures of many-particle systems, as represented in the
thermodynamic limit. I then present a number of strands of such
a macrostatistical mechanics, including (a) a derivation of the
electrodynamics of superconductors from their order structure and
gauge covariance; (b) a hydrodynamics, with transition from
deterministic to stochastic flow, of a quantum plasma model; and
(c) a non-linear generalisation of Onsager's irreversible
thermodynamics.
\vskip 1cm
{\bf 1. Introduction.} The 'miracle' of statistical physics is
that the microscopically chaotic dynamics of many-particle
systems conspires to generate macroscopic laws of relatively
simple structure, such as those of thermodynamics and
hydrodynamics. In view of the complexity of the microscopic
picture and the simplicity of the macroscopic one, it is natural
to seek an approach to the subject, that is centred on macro-
observables, with microscopic imput limited to very general
principles, e.g. conservation laws, ergodicity, etc. Such an
approach would evidently be at the opposite pole from the
conventional microscopically based 'many-body theory'. 
\vskip 0.2cm
In fact, there are already important macroscopically-based 
theories in statistical physics, most notably Onsager's [On]
linear irreversible thermodynamics and Landau's [LL, \ Ch.17]] 
fluctuating hydrodynamics. However, these theories are essentially 
heuristic because, apart from questions of rigour, they lack the 
structures needed for a precise characterisation of their purportedly 
key ingredient of macroscopicality. Moreover, the same thing can be 
said about all works devised within the traditional framework of the 
statistical mechanics of strictly finite systems. 
\vskip 0.2cm
On the other hand, the 'revolution' in statistical mechanics,
based on
the formulation of many-particle systems in the thermodynamic
limit [Ru, HHW, He], have provided us with just the framework
required for a sharp characterisation of macroscopicality, and
even of different levels thereof [Se1, GVV]. My objective here
is to discuss the project of extracting a 'macrostatistical
mechanics' (MSM) from the quantum structures of many-particle
systems, within this framework. In fact, we already have some
strands of such a discipline in the works of Hepp and Lieb [HL] 
on the derivation of the macroscopic dynamics, with non-equibrium
phase transition, of a laser mode; of myself [Se1,Ch.4]
on the formulation of an extended classical thermodynamics with
phase structure; and of the Leuven school [GVV] on macroscopic
fluctuation theory.
\vskip 0.2cm
In this article, I shall bring together further contributions, 
from three different areas, towards an MSM. The first of these, 
$({\S}2),$ consists of a derivation of the electromagnetic 
properties of superconductors from their order structure and 
gauge covariance [Se2,3]. The second, $({\S}3),$ is an extraction
of the hydrodynamics, with non-equilibrium phase transition, of
a 
quantum plasma model [Se4];  and the third, $({\S}4),$ consists
of 
a quantum-statistical derivation of a non-linear generalisation
of 
Onsager's irreversible thermodynamics [Se5]. I shall conclude,
in 
${\S}5,$ with some further brief comments about the project of
a  
macroscopically-centred statistical mechanics.
\vskip 0.5cm
{\bf 2. Macroscopic Quantum Theory of Superconductivity.} At the
{\it phenomenological} level, the principal electrodynamic
properties of superconductors are their capacity to support
persistent electric currents (superconductivity) and their
perfect diagmagnetism (Meissner effect). These two properties are
intimately related, since the Meissner effect is the mechanism
whereby the supercurrents screen the magnetic field they generate
from the interior of the body [Lo]. Thus, superconductivity
arises from the combination of the Meissner effect with the
thermodynamic metastability of the supercurrents and their
magnetic fields. 
\vskip 0.2cm
Although it appears to be widely accepted that the microscopic
theory of Bardeen-Cooper-Schrieffer [BCS] leads to the
electrodynamics of metallic superconductors, the arguments
employed both there and in related works [An, Ri] are radically
flawed in that (a) they are based on totally uncontrolled
approximations, and (b) they violate (exact) gauge covariance of
the second kind {\it at the Hamiltonian level}, and thus do not
even admit precise definition of a local current density. As
regards ceramic, i.e.
high $T_{c},$ superconductivity, the microscopic theory is 
less developed than in the metallic case, and has not yet led to
an electrodynamics.
\vskip 0.2cm
On the other hand, the BCS characterisation of the structure of
the superconductive phase by electron pairing, first proposed by
Schafroth [Sc], has been amply substantiated by experiments on
the Josephson effect [Jo] and the quantisation of trapped
magnetic flux in multiply-connected superconductors [DF].
Moreover, Yang [Ya], generalising ideas of O. Penrose [Pe, PO],
pointed out that this characterisation is captured by the
hypothesis of {\it off-diagonal long range order} (ODLRO). This
is a macroscopic quantum property, representing a well-defined
order structure in a gauge covariant way. Furthermore, it is a
property also possessed by certain ans\"atze, e.g. [ZA], for the
high $T_{c}$ superconductive phase of ceramics.
\vskip 0.2cm
I shall now sketch an approach [Se2,3] I have made to the
electrodynamics of superconductors, based on the assumption of
ODLRO. This is designed to relate the electromagnetic properties
to the order structure of these systems in purely macroscopic 
quantum terms (cf. eqno. (2.8) below). For brevity, I shall confine 
myself here to the derivation of the Meissner effect from ODLRO.
\vskip 0.2cm
{\bf The Model.} We take the quantum model, ${\Sigma},$ to be an
infinitely extended system of electrons, and possibly also of
ions or phonons, in a Euclidean space $X:$ lattice systems may
be formulated analogously. Points in $X$ will generally be
denoted by $x,$ (sometimes by $y,a$ or $b$) and the Lebesgue
measure by $dx.$  It will be assumed that the model enjoys the
properties of gauge covariance of the second kind, and that its
interactions are translationally invariant.
\vskip 0.2cm
The electronic part of ${\Sigma}$ is formulated in terms of a 
quantised field ${\psi}=({\psi}_{\uparrow},{\psi}_{\downarrow}),$
satisfying the canonical anticommutation relations. Thus, in a 
standard way, the $C^{\star}-$algebra ${\cal F}_{el}$ 
of the CAR over the Hilbert space ${\cal H}:=L^{2}(X,dx)$ is
defined by the specifictions that
\vskip 0.2cm
(1) there are linear maps
${\psi}_{\uparrow},{\psi}_{\downarrow},$ 
from ${\cal H}$ into ${\cal F}_{el}$ satisfying the CAR
$${\lbrack}{\psi}_{\alpha}(f),{\psi}_{\beta}(g)^{\star}
{\rbrack}_{+}= {\delta}_{{\alpha},{\beta}}
{\langle}g,f{\rangle}_{\cal H}; \
{\lbrack}{\psi}_{\alpha}(f),{\psi}_{\beta}(g){\rbrack}_{+}
=0\leqno(2.1)$$
\vskip 0.2cm
(2) ${\cal F}_{el}$ is generated by ${\lbrace}{\psi}_{\alpha}(f),
{\psi}_{\alpha}(f)^{\star}{\vert}f{\in}{\cal H}; \ {\alpha}=
{\uparrow},{\downarrow}{\rbrace}.$
\vskip 0.2cm
The algebra ${\cal F}_{el}$ is then the {\it  field algebra} of
the electrons. Space translations and gauge transformations are 
represented by the homomorphisms ${\sigma}_{0}, \ {\gamma}$ of
the additive  groups $X, \ C_{R}^{\infty}(X),$ respectively, into
$Aut({\cal F}_{el}),$ defined by the formulae
$${\sigma}_{0}(a){\psi}(f)={\psi}(f_{a}) \ {\forall}a{\in}X, \ 
with \ f_{a}(x):=f(x-a)\leqno(2.2)$$
and
$${\gamma}({\chi}){\psi}({\phi})={\psi}({\phi}{\exp}(i{\chi}))
\ {\forall}{\chi}{\in}C_{R}^{\infty}(X)\leqno(2.3)$$
The {\it global} gauge automorphisms are those for which ${\chi}$
is constant. 
\vskip 0.2cm
Let ${\cal K}:={\lbrace}f_{1}{\otimes}f_{2}
{\vert}f_{1},f_{2}{\in}{\cal H}{\rbrace}.$ We define the {\it
pair field}
${\Psi}$ to be the mapping of ${\cal K}$ into ${\cal F}_{el}$
given by
$${\Psi}(F)={\psi}_{\uparrow}(f_{1}){\psi}_{\downarrow}(f_{2})
 \ for \ F=f_{1}{\otimes}f_{2}\leqno(2.4)$$
Hence, by (2.2)-(2.4),
$${\sigma}_{0}(a){\Psi}(F)={\Psi}(F_{a}), \ with \
F_{a}(x_{1},x_{2})
=F(x_{1}-a,x_{2}-a)\leqno(2.5)$$
and
$${\gamma}({\chi}){\Psi}(F)={\Psi}(g({\chi})F);\leqno(2.6)$$
$$ \ with \  (g({\chi})F)(x_{1},x_{2}):=F(x_{1},x_{2})
{\exp}i({\chi}(x_{1})+{\chi}(x_{2}))$$
In a standard way, we define the {\it observable algebra} of the
electrons to be the subalgebra ${\cal A}_{el}$ of ${\cal F}_{el}$
that is elementwise invariant under the global gauge
automorphisms
${\psi}{\rightarrow}{\psi}{\exp}(i{\alpha}),$ with ${\alpha}$
constant. Space translations and gauge transformations of the
electronic observables are then given by the 
restrictions to ${\cal A}_{el}$ of the automorphism groups
${\sigma}_{0}(X)$ and ${\gamma}(C_{R}^{\infty}(X)),$
respectively. 
\vskip 0.2cm
The construction of the $C^{\star}-$algebra ${\cal A}_{0}$ of the
observables of the other species of particles of ${\Sigma}$ is 
effected in a similar way. We take the $C^{\star}-$algebra of 
observables of the system to be ${\cal A}:={\cal A}_{el}
{\otimes}{\cal A}_{0}$ and canonically identify ${\cal A}_{el}$
with its subalgebra ${\cal A}_{el}{\otimes}I.$ It is assumed that
the automorphism groups ${\sigma}_{0}(X)$ and
${\gamma}(C_{R}^{\infty}(X))$ extend from ${\cal A}_{el}$ to
${\cal A}.$ 
\vskip 0.2cm
We assume, for simplicity,\footnote *{The more generally valid
assumption of a $W^{\star}-$dynamical system [Se6], employed in
${\S4}$ 
of this article, leads to
precisely the same results as the present one.} that the dynamics
of ${\Sigma},$ as given by a canonical limiting form of that of
finite versions of the system, corresponds to a one-parameter
group ${\lbrace}{\alpha}(t){\vert}t{\in}{\bf R}{\rbrace}$ of
automorphisms of ${\cal A}.$
\vskip 0.2cm
An equilibrium state ${\omega}$ at inverse
temperature ${\beta}$ may then be characterised by the
Kubo-Martin-Schwinger (KMS) conditions [HHW], namely that, for
arbitrary $A,B{\in}{\cal A},$ there is a function $F$ on {\bf C}
that is analytic in the interior of the strip
$Im(z){\in}[0,{\hbar}{\beta}]$ and continuous on its boundaries,
and satisfies the relations
$$F(t)={\omega}(A{\alpha}(t)B); \ F(t+i{\hbar}{\beta})=
{\omega}(({\alpha}(t)A)B)\leqno(2.7)$$
\vskip 0.3cm
{\bf Off-Diagonal Long Range Order.} A state ${\omega}$ is said
to possess the property of off-diagonal long-range order (ODLRO)
if there is a mapping ${\Phi}:{\cal K}{\rightarrow}{\bf C},$ such
that
$${\lim}_{{\vert}y{\vert}\to\infty}[{\omega}({\Psi}(F)^{\star}
{\Psi}(G_{y}))-{\overline {{\Phi}(F)}}{\Phi}(G_{y})]=0
 \ {\forall}F,G{\in}{\cal K}\leqno(2.8)$$
where the bar denotes complex conjugation; and ${\Phi}(G_{y})$
does not 
tend to zero, for all $G{\in}{\cal K}$ as
$y{\rightarrow}{\infty}.$
\vskip 0.2cm
${\Phi}$ is then termed the {\it macroscopic wave function} for
the state ${\omega}.$
\vskip 0.2cm
${\bf Note}$ that, although ${\Psi}(F)$ does not belong to the
observable algebra ${\cal A},$ the argument of ${\omega}$ in
(2.8) does.
\vskip 0.3cm
${\bf Lemma \ 2.1.}$ {\it The ODLRO conditions 
define the macroscopic wave function up to a constant phase 
factor; i.e., if ${\Phi}_{1},{\Phi}_{2}$ are two such functions
satisfying these conditions, for the same state ${\omega},$ then
${\Phi}_{2}={\Phi}_{1}{\exp}(i{\eta}),$ where ${\eta}$ is a 
real-valued constant.}
\vskip 0.3cm
{\bf Proof.} Assuming that ${\Phi}_{1},{\Phi}_{2}$ both 
satisfy the ODLRO conditions with respect to the same state
${\omega},$ it follows from (2.8) that
$${\lim}_{{\vert}y{\vert}\to\infty}[{\overline {{\Phi}_{1}(F)}}
{\Phi}_{1}(G_{y})-{\overline {{\Phi}_{2}(F)}}
{\Phi}_{2}(G_{y})]=0$$
Since this is valid for all $F,G{\in}{\cal K},$ we may
replace $F$ by $F^{\prime}({\in}{\cal K}),$ thereby obtaining 
$${\lim}_{{\vert}y{\vert}\to\infty}
[{\overline {{\Phi}_{1}(F^{\prime})}}
{\Phi}_{1}(G_{y})-{\overline {{\Phi}_{2}(F^{\prime})}}
{\Phi}_{2}(G_{y})]=0$$ 
On multiplying the complex conjugate of the first equation by
${\Phi}_{2}(F^{\prime})$ and that of the second one by
${\Phi}_{2}(F),$ and then taking the difference, we see that
$${\lim}_{{\vert}y{\vert}\to\infty}
{\overline {{\Phi}_{1}(G_{y})}}
[{\Phi}_{1}(F){\Phi}_{2}(F^{\prime}))-{\Phi}_{1}(F^{\prime})
{\Phi}_{2}(F)]=0\leqno(2.9)$$
Since, by the above definition of ODLRO, there are elements $G$
of ${\cal K}$ for which ${\Phi}(G_{y})$ does not tend to zero,
as ${\vert}y{\vert}{\rightarrow}{\infty},$ it follows that the
quantity in the square brackets of (2.9) vanishes. Consequently,
as ${\Phi}_{1,2}$ are non-zero, by the same stipulation, 
$${\Phi}_{2}(F)=c{\Phi}_{1}(F) \ {\forall}F{\in}{\cal K}$$
where c is a complex-valued constant, and since
${\Phi}_{1},{\Phi}_{2}$ both satisfy (2.8), it follows
immediately that this is just a constant phase factor
${\exp}(i{\eta}).$
\vskip 0.3cm
{\bf Gauge and Space Translational Covariance.} Assume now that 
the system is subjected to a classical, static magnetic field,
of induction 
$$B=curlA\leqno(2.10)$$
The time-translational automorphisms then become $A-$dependent,
and we denote them by ${\alpha}_{A}({\bf R}).$ We assume that
they are covariant w.r.t. space translations and gauge
transformations, i.e. that 
$${\sigma}_{0}(a){\alpha}_{A}(t){\sigma}_{0}(-a)=
{\alpha}_{A_{a}}(t); \ with \ A_{a}(x):=A(x-a)\leqno(2.11)$$
and
$${\gamma}({e{\chi}\over {\hbar}c}){\alpha}_{A}(t){\gamma}
({-e{\chi}\over
{\hbar}c})={\alpha}_{A+{\nabla}{\chi}}(t)\leqno(2.12)$$
We note now that, in the particular case where $B$ is uniform,
it may be represented by the vector potential 
$$A={1\over 2}B{\times}x\leqno(2.13)$$
Thus, choosing 
$${\chi}(x)=-{1\over 2}(B{\times}a).x,\leqno(2.14)$$
we see that                                   
$$A+{\nabla}{\chi}=A_{a}\leqno(2.15)$$ 
Hence, defining ${\sigma}:X{\rightarrow}Aut({\cal A})$ by the
formula 
$${\sigma}(a)={\gamma}({e{\chi}\over {\hbar}c}){\sigma}_{0}(a),
\leqno(2.16)$$
it follows from (2.11),(2.12) and (2.16) and the definition of
${\cal A}$ that ${\sigma}$ is a representation of $X$ in
$Aut{\cal A}$ which commutes with the time translations  
${\alpha}_{A}({\bf R}),$ with the gauge fixed by equn. (2.13). 
In view of these properties, we take
${\sigma}$ to be the space translational automorphisms in the
presence of the uniform magnetic field $B.$
\vskip 0.2cm
{\bf Note.} By (2.5),(2.6) and (2.16), the canonical extension
of ${\sigma}$ to ${\cal F}_{el},$ and in particular to the pair
field ${\Psi},$ yields
$${\sigma}(a){\Psi}(F)={\Psi}(s(a)F)\leqno(2.17)$$
where  
$$(s(a)F)(x_{1},x_{2})=F(x_{1}-a,x_{2}-a){\exp}
({ie(B{\times}a).(x_{1}+x_{2})\over 2{\hbar}c})\leqno(2.18)$$
Hence, 
$${\lbrack}s(a),s(b){\rbrack}_{-}=2isin
({eB.(a{\times}b)\over {\hbar}c})s(a+b) \ and \ s(a)s(-a)=
I \ {\forall}a,b{\in}X \leqno(2.19)$$ 
where ${\lbrack},{\rbrack}_{-}$ denotes commutator. Thus, as
${\sigma}(a)$ and ${\sigma}(b)$ do not, in
general intercommute, it follows from (2.17) that the extension
of ${\sigma}(X)$ to the field algebra ${\cal F}_{el}$ is non-
abelian. This is crucial for our derivation of the Meissner
effect below.
\vskip 0.3cm 
{\bf ODLRO and the Meissner Effect.} The essential distinction 
between normal diamagnetism and the Meissner effect is that the 
former can support a uniform, static, non-zero magnetic induction
and 
the latter can not. Thus, we base our derivation of the Meissner 
effect on considerations of the response of an state with ODLRO 
to the action of a uniform magnetic field.
\vskip 0.3cm
${\bf Proposition \ 2.2.}$ {\it The system cannot support
uniform,
non-zero magnetic induction in translationally invariant states
with ODLRO.}
\vskip 0.3cm      
The following Corollary follows immediately from this Propostion
and elementary thermodynamics.
\vskip 0.3cm
${\bf Corollary \ 2.3.}$ {\it Assuming that translational
symmetry 
is preserved in the equilibrium state ${\omega}_{A},$ then either
\vskip 0.2cm
(1) ${\omega}_{A}$ possesses the property of ODLRO and $B=0$
\vskip 0.2cm
or
\vskip 0.2cm
(2) ${\omega}_{A}$ does not possess ODLRO and is normally
diamagnetic.}
\vskip 0.2cm
{\it Further, assuming that, in the absence of a magnetic field,
the free energy density of the ODLRO phase is lower, by
${\Delta},$ than that of the normal one, the former phase will
prevail, and thus the system will exhibit the Meissner effect,
provided that the applied field, $H,$ satisfies the 
condition that} ${\vert}H{\vert}<H_{c}:=(8{\pi}{\Delta})^{1/2}$ 
\vskip 0.3cm
{\bf Comment.} This result signifies that the Meissner effect
arises 
from a rigidity of the macroscopic wave-function in the face of 
an applied magnetic field. This corresponds precisely to 
London's [Lo] idea of rigidity, transferred to the macroscopic
level.
\vskip 0.3cm
${\bf Proof \ of \ Proposition \ 2.2.}$ By the translational
invariance of ${\omega}_{A}$ and the formula (2.17) for
${\sigma},$ 
$${\omega}_{A}({\Psi}(s(a)F)^{\star}{\Psi}(s(a)G)){\equiv}
{\omega}_{A}({\Psi}(F)^{\star}{\Psi}(G))\leqno(2.20)$$
Further, it follows from the ODLRO condition (2.8) and the
definition (2.18) of $s$ that
$${\lim}_{y\to\infty}[{\omega}_{A}({\Psi}(s(a)F)^{\star}
{\Psi}(s(a)G_{y}))-{\overline {{\Phi}(s(a)F)}}
{\Phi}(s(a)G_{y})]=0$$
and hence, by (2.20), that
$${\lim}_{y\to\infty}[{\omega}_{A}({\Psi}(F)^{\star}{\Psi}
(G_{y}))-{\overline {{\Phi}_{a}(F)}}
{\Phi}_{a}(G_{y})]=0\leqno(2.21)$$
where
$${\Phi}_{a}={\Phi}{\circ}s(a)\leqno(2.22)$$
Equation (2.21) implies that ${\Phi}_{a},$ as well as ${\Phi},$
serves as a macroscopic wave function for the state
${\omega}_{A},$ and therefore, by Lemma 2.1 and equn. (2.22),
that 
$${\Phi}{\circ}s(a)=c(a){\Phi}$$
where $c(a)$ is a complex number of unit modulus. Similarly, if
$b$ is another element of $X,$ then
$${\Phi}{\circ}s(b)=c(b){\Phi}$$
where $c(b)$ is also a complex number of unit modulus. Thus, the
last two equations imply that
$${\Phi}{\circ}{\lbrack}s(a),s(b){\rbrack}_{-}=0$$
On taking the composition product of this equation with 
$s(-a-b)$ and using (2.19), we see that, 
$$sin({eB.(a{\times}b)\over {\hbar}c}){\Phi}=0 \ {\forall}
a,b{\in}X$$
from which it follows that {\it either} $B=0$ {\it or}
${\Phi}=0,$ which is the required result.
\vskip 0.5cm
{\bf 3. Hydrodynamical Phase Transition in a Quantum Plasma.} 
\vskip 0.2cm
The quantum Jellium model is a system of electrons, interacting
via
Coulomb forces both with one another and with a uniform,
positively charged, neutralising background. It is thus a model
of a many-particle system with realistic interactions. Its
thermodynamic properties have been established at the level of
mathematical physics [LN]. I shall now sketch a treatment [Se4,7]
leading from its many-particle Schr\"odinger equation to an
Eulerian hydrodynamics with a non-equilibrium phase transition.
The three principal stages of the derivation of this result are
the following. Firstly, we derive a classical Vlasov equation,
governing a {\it large scale} description of its dynamics, from
the microscopic Schr\"odinger equation, together with certain
initial and regularity conditions. Secondly, we observe that the
Vlasov dynamics corresponds precisely to the Liouville equation
for a {\it Lagrangian} hydrodynamics, governing the evolution of
the time (t)-dependent position, $X_{t}(x),$ of a 'fluid
particle'located initially at the point $x.$  Then, finally, we
show that this Lagrangian hydrodynamics reduces to a
deterministic {\it Eulerian} form, provided that a certain
parameter, governing the non-uniformity in the initial density
profile, lies below a certain critical value, and that otherwise
it corresponds to a {\it stochastic} flow.
\vskip 0.3cm
{\bf The Jellium Model,} ${\Sigma}^{(N,L)},$ consists of $N$
electrons 
in a cube, $K^{(L)},$ of side $L,$ with uniform neutralising
positive charge background. We assume periodic boundary
conditions. Our objective is to obtain a quantum theoretical
derivation of the hydrodynamics of ${\Sigma}^{(N,L)}$ in a limit
where $N$ and $L$ tend to infinity and the mean particle density,
$${\overline n}=N/L^{3}\leqno(3.1)$$
remains fixed and finite. 
\vskip 0.3cm
Denoting the gradient operator in $K^{(L)}$ by ${\nabla}^{(L)},$
we represent the position vectors and momenta of the electrons
by the standard multiplicative and differential operators
${\lbrace}X_{j},P_{j}=-i{\hbar}{\nabla}_{j}^{(L)}{\vert}j=1,..
\,N{\rbrace}$ in the Hilbert space ${\cal H}^{(N,L)}$ of square
integrable antisymmetric functions on $(K^{(L)})^{N}.$ The
Hamiltonian is then the self-adjoint operator in ${\cal
H}^{(N,L)}$ given by
$$H^{(N,L)}={-{\hbar}^{2}\over 2m}{\sum}_{j=1}^{N}
{\Delta}_{j}^{(L)}+e^{2}{\sum}_{j,k(>j)=1}^{N}U^{(L)}
(X_{j}-X_{k})\leqno(3.2)$$
where $-e,m$ are the electronic charge and mass, respectively,
${\Delta}^{(L)}$ is the Laplacian for $K^{(L)},$ and $U^{(L)}(X)$
is the difference between ${\vert}X{\vert}^{-1},$ periodicised
w.r.t. $K^{(L)},$ and its space average over that cube, i.e.
$$U^{(L)}(X)={4{\pi}\over L^{3}}{\sum}^{(L)}
{{\exp}(iQ.X)\over Q^{2}}\leqno(3.3)$$
the superscript $(L)$ over ${\Sigma}$ signifying that summation
is taken over the non-zero vectors $Q=(2{\pi}/L)(n_{1},n_{2},
n_{3}),$ with the $n$'s integers. The pure states of
${\Sigma}^{(N,L)}$ are represented by the normalised vectors,
${\Psi}^{(N)}$ in ${\cal H}^{(N,L)}$ and their evolution is
governed by the time-dependent Schr\"odinger equation
$$i{\hbar}{{\partial}{\Psi}_{T}^{(N)}\over {\partial}T}=
H^{(N,L)}{\Psi}_{T}^{(N)}\leqno(3.4)$$
with $T$ the time variable. We shall assume the following initial
kinetic and potential energy bounds for ${\Sigma}^{(N,L)}$-more
precisely, for the family of systems
${\lbrace}{\Sigma}^{(N,L)}{\rbrace},$ with
$N,L$ satisfying (3.1).
\vskip 0.2cm
$(I.1)^{(L)}$ The expectation value of the total kinetic energy
per particle, for the initial state ${\Psi}_{0}^{(N)},$ is less
than some finite $N-$independent constant $B/2m,$ i.e.
$$({\Psi}_{0}^{(N)},P_{1}^{2}{\Psi}_{0}^{(N)})<B\leqno(3.5)$$
\vskip 0.2cm
$(I.2)^{(L)}$ The expectation value of the total potential
energy, for the the initial state ${\Psi}_{0}^{(N)},$ is less
than some finite $N-$independent constant, $e^{2}C/2,$ times
$N^{5/3}.$ This bound corresponds to the electrostatic energy of
a continuous distribution of charge, whose density is a smooth
function of $X/L,$ and signifies that
$$({\Psi}_{0}^{(N)},U^{(L)}(X_{1}-X_{2}){\Psi}_{0}^{(N)})<
CN^{2/3}\leqno(3.6)$$
\vskip 0.2cm
We base our macroscopic description of the model on scales of
length, time and particle momentum given by $L,  \
{\omega}^{-1}$ and $mL{\omega},$ where ${\omega}$ is the
classical plasma frequency, i.e. 
$${\omega}=(4{\pi}{\overline n}e^{2}/m)^{1/2}\leqno(3.7)$$
For this description, we employ the rescaled space and time
coordinates, $x=X/L, \ t={\omega}T,$ respectively. Under this
rescaling, ${\Sigma}^{(N,L)}$ is mapped onto a system
${\Sigma}^{(N)}$ of particles in a unit cube,
$K,$ with periodic boundaries. This rescaling induces a unitary
mapping of ${\cal H}^{(N,L)}$ onto the Hilbert space ${\cal
H}^{(N)}$ of antisymmetric square integrable functions on
$K^{N},$ under which ${\Psi}_{T}^{(N)}$ is transformed to the
state of ${\Sigma}^{(N)}$ given by
$${\psi}_{t}^{(N)}(x_{1},.. \ .,x_{N})=L^{3N/2}
{\Psi}_{{\omega}^{-1}t}^{(N)}(Lx_{1},.. \ .,Lx_{N})\leqno(3.8)$$
The Schr\"odinger equation (3.4) thus transforms to 
$$i{\hbar}_{N}{{\partial}{\psi}_{t}^{(N)}\over {\partial}t}=
H^{(N)}{\psi}_{t}^{(N)}\leqno(3.9)$$
where
$$H^{(N)}={1\over 2}{\sum}_{j=1}^{N}p_{j}^{2} \ + \
N^{-1}{\sum}_{j,k(>j)=1}^{N}U(x_{j}-x_{k})\leqno(3.10)$$
$$p_{j}=-i{\hbar}_{N}{\nabla}_{j}\leqno(3.11)$$
$${\hbar}_{N}={{\hbar}\over mL^{2}{\omega}}=
{{\hbar}\over m{\omega}}({{\overline n}\over
N})^{2/3}\leqno(3.12)$$
is a dimensionless effective 'Planck constant', ${\nabla}$ is the
gradient operator for $K,$ and 
$$U(x)={\sum}_{q}^{(1)}{\exp}(iq.x)/q^{2}\leqno(3.13)$$
the superscript $(1)$ over ${\Sigma}$ signifying that summation 
is taken over the non-zero vectors $2{\pi}(n_{1},n_{2},n_{3}),$ 
with the $n$'s integers. It follows from (2.13) that
$${\Delta}U(x)=1-{\delta}(x)\leqno(3.14)$$
where ${\Delta}$ is the Laplacian and ${\delta}$ the Dirac 
distribution for $K.$
\vskip 0.2cm
We formulate the dynamics of ${\Sigma}^{(N)}$  in
terms of its characteristic functions 
$${\mu}_{t}^{(N,n)}({\xi}_{1},.. \ .,{\xi}_{n};{\eta}_{1},.. \
.,{\eta}_{n}):=\leqno(3.15)$$
$$({\psi}_{t}^{(N)},{\Pi}_{j=1}^{n}
({\exp}(i{\xi}_{j}.p_{j}/2){\exp}(i{\eta}_{j}.x_{j})
{\exp}(i{\xi}_{j}.p_{j}/2)){\psi}_{t}^{(N)})$$
where the ${\xi}'$s and ${\eta}'$s run over the ranges ${\bf
R}^{3}$ and $(2{\pi}{\bf Z})^{3},$ respectively. 
\vskip 0.2cm
The initial condition for ${\Sigma}^{(N)},$  corresponding to
$(I.1)^{(L)}$ for ${\Sigma}^{(N,L)},$ is
\vskip 0.2cm
$(I.1)$
$$({\psi}_{0}^{(N)},p_{1}^{2}{\psi}_{0}^{(N)})
<N^{-2/3}b \ {\rightarrow}0 \ as \
N{\rightarrow}{\infty}\leqno(3.16)$$
{\it where $b$ is a finite constant.}
\vskip 0.2cm
The initial condition $(II.2)^{(L)}$ transforms to the following
form for ${\Sigma}^{(N)},$ which is unaffected by the above
rephasing of the initial state.
\vskip 0.2cm
$(I.2)$
$$({\psi}_{0}^{(N)},U(x_{1}-x_{2}){\psi}_{0}^{(N)})<c
\leqno(3.17)$$
{\it where $c$ is a finite constant. We further assume that}
\vskip 0.2cm
$(I.3)$
$${\lim}_{N\to\infty}{\rho}_{0}^{(N,1)}(x)={\sigma}_{0}(x)
\ {\forall}x{\in}K\leqno(3.18)$$ 
{\it where the function ${\sigma}_{0}$ is continuous; and that}
\vskip 0.2cm
$(I.4)$
$${\lim}_{N\to\infty}({\mu}_{0}^{(N,n)}({\xi}_{1},.. \
.,{\xi}_{n};{\eta}_{1},.. \
.,{\eta}_{n})-{\Pi}_{1}^{n}{\mu}_{0}^{(N,1)}
({\xi}_{j},{\eta}_{j})){\equiv}0\leqno(3.19)$$
This last condition signifies that, in the limit
$N{\rightarrow}{\infty},$ the initial correlations of
${\Sigma}^{(N)}$ are of zero range: the assumption behind
this is that the initial state of the microscopic model
${\Sigma}^{(N,L)},$ carries only short range correlations, as in
a pure phase [Ru, Se1].
\vskip 0.2cm
We note here that it was shown by explicit construction, in the 
Appendix of Ref. [Se7], that the initial conditions (I.1)-(I.4)
are perfectly viable. 
\vskip 0.2cm
The macroscopic dynamics of the Jellium model, then, is
represented by the time-dependence of the characteristic
functions ${\mu}_{t}^{(N,n)},$ in the limit
$N{\rightarrow}{\infty},$ subject to the initial conditions
(I.1-4). 
\vskip 0.2cm
{\bf The Vlasov Dynamics.} We note now that the dynamics of the
model ${\Sigma}^{(N)},$ as given by (3.9) and (3.10), has the
simplifying features that (a) the effective, dimensionless Planck
constant, ${\hbar}_{N},$ governing its quantum behaviour, tends
to zero as $N{\rightarrow}{\infty};$ and (b) the pair interaction
potential scales as $N^{-1}.$ The properties (a) and (b) are
generally the hallmarks of a classical and of a mean field
theory, respectively, in the limit $N{\rightarrow}{\infty}.$ 
In fact, were it not for the Coulomb singularity in $U,$ on e
could infer immediately from [NS, Sp] that the Schr\"odinger
equation (3.9), subject to the initial conditions (I.1-4), led
to a classical (mean field theoretic) Vlasov dynamics in the
limit $N{\rightarrow}{\infty}.$ Specifically, if the two-body
potential $U$ in (3.10) were a twice continuously differentiable
function, then the following results would prevail.
\vskip 0.2cm
(A) [NS, Sp] Pointwise convergence of ${\mu}_{t}^{(N,1)}$,
as $N{\rightarrow}{\infty},$ to the characteristic function,
${\mu}_{t},$ of a classical probability measure $m_{t}$ on
$(K{\times}{\bf R}^{3}),$ that satisfies the
weak Vlasov equation
$${d\over dt}{\int}f(x,v)dm_{t}(x,v)=
{\int}v.{{\partial}\over
{\partial}x}f(x,v)dm_{t}(x,v)\leqno(3.20)$$
$$ \ \ -{\int}{\nabla}U(x-y).
{{\partial}\over {\partial}v}f(x,v)
dm_{t}(x,v)dm_{t}(y,w) \
{\forall}f{\in}C_{0}^{2}(K{\times}{\bf R}^{3})$$
\vskip 0.2cm
(B) [NS, Sp] Pointwise convergence of ${\mu}_{t}^{(N,n)}$ to the
characteristic function of the classical probability measure
$m_{t}{\otimes}m_{t}.. \ .{\otimes}m_{t} \ (n \ factors)$ on
$(K{\times}{\bf R}^{3})^{n}.$
\vskip 0.2cm
(C) [Se7] The initial condition for $m,$ ensuing from (I.1) and
(I.3), that
$${\int}dm_{0}(x,v)f(x,v)={\int}dx{\sigma}_{0}(x)
f(x,0)\leqno(3.21)$$ 
\vskip 0.2cm
(D)[Ne, Se7] The unique determination of the solution of the
Vlasov equation (3.20), subject to the initial condition (3.21),
by that of the Newtonian mean field theoretic problem
$${dX_{t}(x)\over dt}=V_{t}(x); \ {dV_{t}(x)\over dt}=
-{\int}dy{\sigma}_{0}(y){\nabla}U^{\prime}(X_{t}(x)-X_{t}(y));
\leqno(3.22)$$
$$ \ with \ X_{0}(x)=x; \ V_{0}(x)=0$$
according to the formula
$${\int}dm_{t}(x,v)f(x,v)={\int}dx{\sigma}_{0}(x)
f(X_{t}(x),V_{t}(x))\leqno(3.23)$$
\vskip 0.2cm
In order to extend these results to the Jellium model,
${\Sigma}^{(N)},$ we require certain regularity conditions on the
dynamics of the system, that serve to tame the Coulomb
singularity. Accordingly, we invoke certain supplementary
assumptions (R), specified in [Se4], whose essential import is
that the Coulomb repulsion keeps the electrons apart,
sufficiently to ensure that the magnitude of the internal
electric field remains bounded. Under these assumptions, we have
the following result.
\vskip 0.3cm
{\bf Proposition 3.1.}[Se4] {\it The Schr\"odinger equation
(3.10) for ${\Sigma}^{(N)},$ together with the initial conditions
(I.1-4) and the regularity conditions R, leads to the Vlasov
dynamics specified by (A)-(D).}
\vskip 0.3cm
{\bf Eulerian Versus Stochastic Hydrodynamics.} The Newtonian
mean field theory, given by (3.22), corresponds to a {\it
Lagrangian} hydrodynamics, in which $X_{t}(x)$ and $V_{t}(x)$ are
the position and velocity, respectively, of a 'fluid particle';
and the Vlasov equation (3.20) is just the Liouville equation
representing its probabilistic description. Our aim now is to
show that the reducibility of the Vlasov dynamics to a
deterministic Eulerian hydrodynamics depends on the invertibility
of $X_{t}.$ 
\vskip 0.2cm
{\bf Case (a):} $X_{t}$ {\bf Invertible.} In this case, the
Jacobean 
$$J_{t}(x)={{\partial}(X_{t}^{(1)},X_{t}^{(2)},X_{t}^{(3)})\over 
 {\partial}(x^{(1)},x^{(2)},x^{(3)})}\leqno(3.24)$$ 
with $x^{(j)}$ (resp. $X_{t}^{(j)}$) the j'th component $x$
(resp. $X_{t}$), is strictly positive, and so we can define
$${\sigma}_{t}(x)={\sigma}_{0}(X_{t}^{-1}(x))/J_{t}(x);
 \ and \ u_{t}(x)=V_{t}(X_{t}^{-1}(x))\leqno(3.25)$$ 
Thus, since, by (3.23) and (3.25),
$${\int}dm_{t}(x,v)f(x,v)={\int}dx{\sigma}_{t}(x)
f(x,u_{t}(x)) \ {\forall}f{\in}C(K)\leqno(3.26)$$
i.e., formally,
$$dm_{t}(x,v)={\sigma}_{t}(x){\delta}(v-u_{t}(x))dxdv$$
it follows that $u_{t}(x)$ and ${\sigma}_{t}(x)$ are the drift
velocity and normalised particle density, respectively, at
position $x$ and time $t.$ Moreover, by (3.14), (3.22) and
(3.25), $u_{t},{\sigma}_{t}$ evolve according to the 
Euler-cum-Maxwell hydrodynamical equations
$${{\partial}{\sigma}_{t}\over {\partial}t}+{\nabla}.
({\sigma}_{t}u_{t})=0; \ {{\partial}u_{t}\over
{\partial}t}+(u_{t}.{\nabla})u_{t}=E_{t}; \
{\nabla}.E_{t}=({\sigma}_{t}-1)\leqno(3.27)$$
\vskip 0.3cm
{\bf Case (b): $X_{t}$ Non-Invertible.} In this case, we cannot
employ the formulae (3.25) to define the time-dependent
density and drift velocity. Instead, we have to consider the
situation where the equation
$$X_{t}(y)=x$$
has several solutions, labelled by an index set $S,$ for $y$ as
a function of $x$ and $t,$ i.e.
$$y={\lbrace}Y_{t}^{(s)}(x){\vert}s{\in}S{\rbrace}$$
In this case, equation (3.26) implies that
$${\int}dm_{t}(x,v)f(x,v)={\sum}_{s{\in}S}{\int}dx
{\sigma}_{t}^{(s)}(x)f(x,u_{t}^{(s)}(x))\leqno(3.28)$$
where
$${\sigma}_{t}^{(s)}(x)={\sigma}_{0}(Y_{t}^{(s)}(x)){\vert}
K_{t}^{(s)}(x){\vert}; \ \ u_{t}^{(s)}(x)=
V_{t}(Y_{t}^{(s)}(x))\leqno(3.29)$$
and
$$K_{t}^{(s)}(x)={{\partial}(Y_{t}^{(s,1)},Y_{t}^{(s,2)}),
Y_{t}^{(s,3)})\over
{\partial}(x^{(1)},x^{(2)},x^{(3)})}\leqno(3.30)$$
Thus, (3.28) signifies that the macroscopic state of the system
at time t corresponds to a statistical mixture of different
streams, indexed by $S,$ with dnsities ${\sigma}_{t}^{(s)}$ and
drift velocities $u_{t}^{(s)}.$ This implies that the local
density and drift velocity of the system are now {\it
stochastic} variables of a hydrodynamics still governed by the
Vlasov equation.
\vskip 0.2cm
We may summarise the above observations in the following form.
\vskip 0.3cm
{\bf Proposition 3.2.} {\it If $X_{t}$ is invertible, then the
macroscopic dynamics of the system corresponds to a deterministic
hydrodynamics given by the Euler-cum-Maxwell equations (3.27).
Otherwise, it is a stochastic flow, still governed by the Vlasov
equation.}
\vskip 0.3cm
{\bf Example of Hydrodynamic Phase Transition.} Let us now
specialise to the situation where ${\sigma}_{0}$ is a function
of just a single coordinate, $x^{(1)}.$ In this case, as the
regularity conditions (R) imply uniqueness of the solution of
(3.22) [Se4], the flow becomes effectively one-dimensional, with
$X_{1,t}$ dependent on the coordinate $x^{(1)}$ only, and
$X_{t}^{(2)},X_{t}^{(3)}$ remaining fixed at $x^{(2)},x^{(3)},$
respectively. For notational convenience, we shall henceforth
drop the superscript $(1)$ from $X_{t}^{(1)}$ and $x^{(1)},$ and
denote by $X_{t}(x)$ the first component of the vector hitherto
represented by this symbol. Thus, by equations (3.22),
$$X_{t}(x)=x+{\int}_{0}^{t}ds(t-s){\int}_{0}^{1}dy
{\sigma}_{0}(y)F(X_{s}(x)-X_{s}(y))\leqno(3.31)$$
where
$$F(x)=-{\int}_{0}^{1}dx^{(2)}{\int}_{0}^{1}dx^{(3)}
{{\partial}U\over {\partial}x}(x,x^{(2)},x^{(3)})\leqno(3.32)$$
Let
$$J_{t}(x)={{\partial}X_{t}(x)\over {\partial}x}\leqno(3.33)$$
Then, by Prop. 3.2 and the implicit function theorem, the
neccessary and sufficient condition for deterministic
hydrodynamics is that $J_{t}$ has no zeroes. We note also that
the definition (3.33) permits us to re-express (3.31) in the form
$$X_{t}(x)=x+{\int}_{0}^{t}ds(t-s){\int}_{0}^{1}dy
{\sigma}_{0}(y)F({\int}_{y}^{x}dzJ_{s}(z)))
\leqno(3.31)^{\prime}$$
\vskip 0.3cm
{\bf Proposition 3.3.} {\it Under the specified assumptions, the
hydrodynamics of the model is deterministic and given by the
Euler-cum-Maxwell equations (3.27), provided that}
$${\sigma}_{0}(x)>{1\over 2}{\forall}x{\in}[0,1]\leqno(3.34)$$
{\it Otherwise there is a transition to a stochastic flow at a
certain time ${\tau},$ given by the least positive value of $t$
for which} 
$${\sigma}_{0}(x)+(1-{\sigma}_{0}(x)){\cos}(t)=0\leqno(3.35)$$
{\it for some $x{\in}[0,1].$}
\vskip 0.3cm
{\bf Proof.} By (3.13), (3.14) and (3.31)-(3.33),
$$J_{t}(x)=1+{\int}_{0}^{t}ds(t-s)
J_{s}(x)(-1+{\int}_{0}^{1}dy{\sigma}_{0}(y)
{\delta}(X_{s}(x)-X_{s}(y)))\leqno(3.36)$$
where now ${\delta}$ is the Dirac distribution on $[0,1],$
subject to periodic boundary conditions. 
\vskip 0.2cm
Let us first suppose that $X_{t}$ is invertible, i.e. that
$J_{t}$ is strictly positive, over a time interval
$0{\le}t<{\tau}_{0},$ for some positive ${\tau}_{0}.$ In this
case, 
$$J_{s}(x){\delta}(X_{s}(x)-X_{s}(y)){\equiv}{\delta}(x-y) \
{\forall}s{\in}[0,{\tau}_{0})$$
and therefore (3.36) reduces to
$$J_{t}(x)=1+{\int}_{0}^{t}ds(t-s)
({\sigma}_{0}(x)-J_{s}(x))$$
i.e.
$$({d^{2}\over dt^{2}}+1)J_{t}(x)={\sigma}_{0}(x);
 \ with \ J_{0}(x)=1; \ {\dot J}_{0}(x)=0$$
where ${\dot J}_{t}=dJ_{t}/dt.$ Hence,
$$J_{t}(x)={\sigma}_{0}(x)+(1-{\sigma}_{0}(x)){\cos}(t)$$
In view of the non-negativity of ${\sigma}_{0},$ this equation
implies that $J_{t}$ is strictly positive for all $t{\ge}0$
if and only if the condition (3.34) is fulfilled. Otherwise,
$J_{t}$ changes sign at some point $x{\in}[0,1]$ when $t$ reaches
the value ${\tau}$ specified in the statement of the Proposition.
Hence, our assumption of the invertibility of $X_{t}$ is
untenable if (3.34) is violated; and therefore, by Prop. 3.2, the
flow becomes stochastic in this case.
\vskip 0.2cm
The proof (cf. [Se4]) of the converse, i.e. that (3.34) implies
the invertibility of $X_{t}$ and thus the deterministic Euler-
cum-Maxwell flow, stems from the explicit form of our regularity
condition (R).
\vskip 0.5cm
{\bf 4. Macrostatistics and Non-Equilibrium Thermodynamics.}
Here, I present an approach to the general problem of formulating
the structures imposed on non-equilibrium thermodynamics by the
underlying quantum mechanics of many-particle systems.
This is based on a combination of macroscopic and
microscopic treatments of a generic system, ${\Sigma},$ of
particles occupying a Euclidean space $X={\bf R}^{d}.$ The
argument consists of four parts. The first (${\S}4.1$) is a
formulation of a mathematical framework for non-equilibrium
thermodynamics, based on a macroscopic, continuum mechanical
model, $M,$ of ${\Sigma}.$ The second (${\S}4.2$) is an algebraic
quantum statistical formulation of a microscopic model, ${\cal
Q}$ of the same system. The third (${\S}4.3$) consists of a
treatment of the connection between $M$ and ${\cal Q},$ leading
to a {\it classical macrostatistical mechanics} $(CMSM)$ of the
observables of ${\cal Q}$ that correspond to the hydrodynamical
variables of $M.$ This admits a clear formulation of {\it local
equilibrium} conditions and of a generalised version of Onsager's
regression hypothesis [On], namely that the macroscopic fluctuations
about the flow given by $M$ is governed by the same dynamics as
the externally induced 'weak' perturbations to that flow. In
${\S}4.4,$ I derive a generalised form of the Onsager reciprocity
relations for $M$ from the structure of $CMSM.$ The key elements
of this derivation are the symmetry properties of the
model, stemming from the microscopic reversibility of ${\cal Q},$
and the assumption of local equilibrium.
\vskip 0.3cm
{\bf 4.1. The Macroscopic Model, $M$.}  In the
classical thermodynamical description, the {\it equilibrium
states} of ${\Sigma}$ are represented by a certain set
$q=(q_{1},.. \ .,q_{n})$ of intensive variables, corresponding
to global densities of extensive conserved quantities, which we
shall characterise in ${\S}4.2.$ Its entropy density, $s,$ is
then a function of these variables, and its equilibrium
properties of governed by the form of $s.$ The thermodynamic
variables ${\theta}=({\theta}_{1},.. \ .,{\theta}_{n})$ conjugate
to $q$ are defined by the formula
$${\theta}={{\partial}s(q)\over {\partial}q}, \ i.e. \ 
{\theta}_{k}={{\partial}s(q)\over {\partial}q_{k}}, \ 
k=1,2.. \ ,n\leqno(4.1)$$
and the pressure, $p,$ is the function of ${\theta}$ given by
$$p({\theta})=sup_{q}(s(q)-q.{\theta})\leqno(4.2)$$
the dot representing the ${\bf R}^{n}$ inner product. Thus, the
inverse of the formula (4.1) (in the pure phase region) is
$$q=-{{\partial p}({\theta})\over
{\partial}{\theta}}\leqno(4.3)$$
\vskip 0.2cm
In non-equilibrium thermodynamics, the state of the system is
generally not translationally invariant, and the macroscopic
variables $q$ become functions of position, $x,$ and of time,
$t.$ In other words, the macrostate is an n-component {\it
classical field} 
$$q(x,t){\equiv}q_{t}(x){\equiv}
(q_{1,t}(x),.. \ .,q_{n,t}(x)).$$ 
The laws of continuum mechanics, such as those of
hydrodynamics or heat conduction, are then of the {\it
deterministic} form
$${\dot q}_{t}={\cal F}(q_{t})\leqno(4.4)$$ 
where ${\cal F}$ is some functional of the fields $q_{t}.$
Equivalently, defining the local thermodynamic conjugates
${\theta}_{t}$ of the fields $q_{t}$ by
$${\theta}_{t}(x){\equiv}{\theta}(x,t):={{\partial}s(q(x,t))\over 
{\partial q}},\leqno(4.5)$$ 
it follows from (4.3) that the equation of motion (4.4) may be
re-expressed as
$${\dot q}_{t}{\equiv}{d{\dot q}_{t}\over dt}=
{\Phi}({\theta}_{t}); \ {\Phi}:={\cal F}{\circ}
(-{{\partial}p\over {\partial}{\theta}})\leqno(4.6)$$
\vskip 0.2cm
{\bf Note.} The definition (4.5) of ${\theta}_{t}$ does not
require the microstate of ${\Sigma}$ to simulate one of
equilibrium at a local level. On the other hand, some assumption
of local equilibrium will be needed for the determination of
properties of the functional ${\cal F}.$
\vskip 0.2cm
{\bf Example.} An example of a law of the above form is the non-
linear, n-component diffusion given by
$${\dot q}_{k,t}={\sum}_{l=1}^{n}{\nabla}.
(L_{k,l}({\theta}_{t}){\nabla}{\theta}_{l,t})
\leqno(4.6a)$$
\vskip 0.2cm
Returning to the general structure, we impose the following
assumptions on the dynamical law (4.6). 
\vskip 0.2cm
(M.1) {\it The system is confined to a single phase region, i.e.
the values of $q_{t}(x), \ {\theta}_{t}(x)$ always lie in domains
where the functions $s$ and $p$ are $C^{({\infty})}.$}
\vskip 0.3cm
(M.2) {\it ${\cal F}(q)=0$ if $q(x)$ is constant, i.e. the
system is in equilibrium if its local thermodynamic variables
$q(x)$ (or equivalently ${\theta}(x)$) are spatially uniform.}
\vskip 0.3cm
(M.3) {\it The equation of motion (4.6) is covariant w.r.t.
space-time translations and scale transformations
$x{\rightarrow}{\lambda}x,t{\rightarrow}{\lambda}^{r}t,$ with r
a positive integer (=2 in the case of
(4.6a)).}\footnote *{Note that this assumption is not
always fulfilled: for example, the Navier-Stokes equation is not
scale invariant.}  
\vskip 0.3cm
Let $y_{t}, \ {\eta}_{t}$ be 'small' perturbations of $q_{t}, \
{\theta}_{t},$ respectively, and let 
\vskip 0.3cm
$$[{\Lambda}({\theta}){\eta}_{t}](x):=
{d\over dz}{\Phi}({\theta}_{t}(x)+z{\eta}_{t}(x))
{\vert}_{z=0}\leqno(4.7)$$
in the pointwise sense, for the class of functions ${\eta}_{t}$
for which the r.h.s. of this formula is well-defined. Then
it follows from (4.6) and (4.7) that the {\it linearised}
equation of motion for $y_{t}$ is
$${\dot y}_{t}={\cal L}({\theta}_{t})y_{t}\leqno(4.8)$$
where
$${\cal L}({\theta}_{t})={\Lambda}({\theta}_{t})
B({\theta}_{t})\leqno(4.9)$$
and $B({\theta})$ is the Hessian $s^{{\prime}{\prime}}(q),$ i.e.
$$(B({\theta}))y)_{k}:={\sum}_{l=1}^{n}
{{\partial}^{2}s(q)\over {\partial}q_{k}{\partial}q_{l}}
y_{l}\leqno(4.10)$$
The next assumption permits us to formulate the perturbed
dynamics as the evolution of a tempered distribution. Thus,
defining ${\cal S}^{(n)}(X)$ to be the space of the Schwartz
${\cal S}-$class ${\bf R}^{n}-$ valued functions on $X,$ and
${\cal S}^{(n){\prime}}(X)$ to be its dual space, we assume that
\vskip 0.3cm
(M.4) {\it the linear operator ${\cal L}({\theta}_{t})$ extends,
by continuity, to a transformation of ${\cal S}^{(n){\prime}}(X),$
and the formula (4.8), considered now as an equation of motion
in that space, has a unique solution}
$$y_{t}=T({\theta}_{.}{\vert}t,t^{\prime})y_{t^{\prime}} \ 
{\forall}t{\ge}t^{\prime}{\ge}0\leqno(4.11)$$
with
$$T({\theta}_{.}{\vert}t,t_{0}){\equiv}
T({\theta}_{.}{\vert}t,t_{1})T({\theta}_{.}{\vert}t_{1},t_{0})
 \ and \ T({\theta}_{.}{\vert}t,t)=1 \
{\forall}t{\ge}t_{1}{\ge}t_{0}{\ge}0$$ 
\vskip 0.3cm
{\bf Comments.} In the equilibrium case, where
${\theta}_{t}={\overline {\theta}},$ a constant, the
two-parameter
family $T({\theta}{\vert},.,)$ of transformations of ${\cal
S}^{(n){\prime}(X)}$ reduces to a one-parameter semi-group
${\lbrace}{\tilde T}({\overline {\theta}}{\vert}t){\vert}t{\in}
{\bf R}_{+}{\rbrace},$ where
$${\tilde T}({\overline {\theta}}{\vert}(t-t_{0}){\equiv}
T({\overline {\theta}}{\vert}(t,t_{0})=
{\exp}({\cal L}({\overline {\theta}})(t-t_{0}))\leqno(4.12)$$
Further, in the case of the non-linear diffusion (4.6a), ${\cal
L}({\theta}^{(0)})$ reduces to the form
$${\cal L}({\overline {\theta}})=L({\overline {\theta}})
B({\overline {\theta}}){\Delta}; \ L({\overline {\theta}})
=[L_{kl}({\overline {\theta}})];\ B=[B_{kl}({\overline
{\theta}})]
\leqno(4.12a)$$
where ${\Delta}$ is the Laplacian.
\vskip 0.3cm 
The following Proposition follows immediately from
(M.1) and (M.4). 
\vskip 0.2cm
{\bf Proposition 4.1.} {\it The process $y.$ is covariant w.r.t. 
space translations and space-time scale transformations, i.e.
defining}
$${\phi}_{t}^{(a,b,{\lambda})}(x){\equiv}
{\phi}^{(a,b,{\lambda})}(x,t):=
{\phi}(a+{\lambda}x,b+{\lambda}^{r}t), \ for 
\ {\phi}={\theta},q,y\leqno(4.13)$$
and for arbitrary $(a,b,{\lambda}){\in}X{\times}{\bf R}
{\times}{\bf R}_{+}),$ 
$$(T({\theta}_{.}{\vert}t,t^{\prime})
y_{{t}^{\prime}})^{(a,b,{\lambda})}{\equiv}
T({\theta}_{.}^{(a,b,{\lambda})}{\vert}t,t^{\prime})
y_{{t}^{\prime}}^{(a,b,{\lambda})}\leqno(4.14)$$
\vskip 0.3cm 
{\bf 4.2. The Quantum Model,} ${\cal Q}.$ In order to
accommodate the dynamics of both the microscopic and macroscopic
observables of ${\Sigma},$ we construct its quantum model,
${\cal Q},$ as a $W^{\star}-$dynamical system [Se6] $({\cal
A},{\alpha},{\cal N}({\cal A})),$ where ${\cal A}$ is a
$W^{\star}-$algebra of observables, $t{\rightarrow}{\alpha}_{t}$
is a representation of ${\bf R}$ in $Aut({\cal A}),$
corresponding to the dynamics of the system, and ${\cal N}({\cal
A}))$ is the folium of normal states on ${\cal A}.$  In fact,
this algebra is constructed as the weak closure of the largest
locally normal representation, ${\pi},$ of the standard [HHW]
quasi-local $C^{\star}-$algebra, ${\tilde {\cal A}},$ that can
support the dynamical and thermodynamical structures we require.
Thus, we characterise ${\cal A},{\alpha}$ and ${\pi}$ according
to the following prescription.
\vskip 0.2cm
(1) Let $L$ be the set of bounded open regions of X. Then for
each ${\Lambda}{\in}L,$ there is a type-I factor, ${\cal
A}({\Lambda})({\subset}{\cal A}),$ representing the observables
of that region and satisfying the conditions of isotony and local
commutativity; and ${\cal A}$ is the weak closure of ${\cal
A}_{L}:=U_{{\Lambda}{\in}L}{\cal A}({\Lambda}).$ 
We extend automorphisms ${\gamma}$ of ${\cal A}$ to unbounded
observables, $Q,$ affiliated\footnote *{Recall that, if
${\cal M}$ is a $W^{\star}-$algebra of operators in a Hilbert
space ${\cal H},$ then the unbounded operators, in ${\cal H},$
affiliated to ${\cal M}$ are the densely defined ones that
commute with ${\cal M}^{\prime}.$} to this algebra, according to
the formula
$${\exp}(i{\lambda}{\gamma}(Q))={\gamma}({\exp}(i{\lambda}Q))
 \ {\forall}{\lambda}{\in}{\bf R}\leqno(4.15)$$
We similarly extend anti-automorphisms of ${\cal A}$ to its
unbounded affiliates.
\vskip 0.2cm
(2) The local energy observable (Hamiltonian) $H({\Lambda})$ for
the region ${\Lambda}({\in}L)$ is, in general, an unbounded one,
affiliated to ${\cal A}({\Lambda}).$ We assume that ${\cal A}$
is equipped with a Wigner time-reversal anti-automorphism,
${\rho},$ which leaves this observable invariant, i.e.
$${\rho}H({\Lambda})=H({\Lambda})\leqno(4.16)$$ 
\vskip 0.2cm
(3) Space translations are represented by a homomorphism
${\sigma}$ of $X$ into $Aut({\cal A}),$ such that
${\sigma}(x){\cal
A}_{0}({\Lambda}){\equiv}{\cal A}_{0}({\Lambda}+x)$ and
${\sigma}(x)H({\Lambda})=H({\Lambda}+x)),$ this
last condition representing an assumption of translationally
invariant interactions. 
\vskip 0.2cm
(4) The dynamical automorphisms ${\alpha}$ are given by infinite
volume limits of those of finite versions of ${\Sigma},$
according to the formula
$${\alpha}_{t}A=s-{\lim}_{{\Lambda}{\uparrow}}
({\exp}(iH({\Lambda})t/{\hbar})A{\exp}(-iH({\Lambda})t/{\hbar}))
\ {\forall}t{\in}{\bf R}, \ A{\in}
{\cal A}_{L}\leqno(4.17)$$ 
Thus, by the translational covariance of $H({\Lambda}), \
{\alpha}_{t}$ commutes with the space translational
automorphisms, ${\sigma},$ and, by (4.16), satisfies the {\it
microscopic reversibiliy} condition
$${\rho}{\alpha}_{t}{\rho}={\alpha}_{-t} \ {\forall}t
{\in}{\bf R}\leqno(4.18)$$
\vskip 0.2cm
(5) We formulate the thermodynamics of the system in terms a
Hermitian quantum field ${\hat q}=({\hat q}_{1},.. \ .,{\hat
q}_{n}),$ the ${\hat q}_{k}'s$ being densities of
locally conserved quantities. We assume that ${\hat q}$ is a
tempered distribution, affiliated to ${\cal A},$ that transforms
covariantly w.r.t space translations. Thus, ${\hat q}$ is a
mapping of ${\cal S}^{(n)}(X)$ into the
self-adjoint affiliates of ${\cal A}.$ We assume that the
function ${\exp}(i{\hat q}(.))$ is strongly continuous, and
that ${\hat q}$ transforms covariantly w.r.t. space translations,
i.e.
$${\sigma}(x)[{\hat q}(f)]={\hat q}(f_{x}), \ where \
f_{x}(y)=f(x-y) \ {\forall}x,y{\in}X,f{\in}{\cal S}^{(n)}(X)
\leqno(4.19)$$
We assume, for simplicity, that ${\hat q}$ is invariant under
time reversals, i.e. 
$${\rho}{\hat q}(f){\equiv}{\hat q}(f)\leqno(4.20)$$
and that its components satisfy the following commutation
relations, which signify that their space integrals over finite
volumes intercommute, up to 'surface effects'.
$${\lbrack}{\hat q}_{k}(g),q_{l}(h){\rbrack}_{-}=
i{\hbar}j_{k,l}({\nabla}(gh)) \ {\forall}g,h{\in}
{\cal S}(X),\leqno(4.21)$$
where $(gh)(x):=g(x)h(x)$ and $j_{k,l}$ a tempered distribution.
Further, denoting ${\alpha}_{t}[{\hat q}(f)]$ by ${\hat
q}_{t}(f),$ we assume local
conservation law of the form
$${{\partial}{\hat q}_{t}(f)\over
{\partial}t}=j_{t}({\nabla}f)\leqno(4.22)$$
where $j_{t}$ is a tempered distribution.
\vskip 0.2cm
(6) We take the equilibrium thermodynamic variables of ${\cal Q}$
to be the 'observables at infinity' [LR], given by the global
spatial average ${\hat q}^{\infty}$ of ${\hat q}$ over $X.$
Further, denoting by ${\hat s}$ the standard [Ru] entropy density
functional on the translationally invariant states on ${\cal
A}_{0},$ we assume that these observables form a complete
thermodynamic set, in the sense that [Se1, Ch.4] 
\vskip 0.2cm
(a) for each expectation value, $q,$ of ${\tilde q},$ there is
precisely one translationally invariant state $({\in}{\cal
N}({\cal A}))$ that maximises ${\hat s};$ and 
\vskip 0.2cm
(b) no proper subset of ${\hat q}$ possesses this
property. 
\vskip 0.2cm
The equilibrium thermodynamics of the system is thus given by the
form of the resultant entropy density, $s(q).$ We identify $q,s$
with the objects denoted by these symbols in the macroscopic
model $M;$ and, defining ${\theta}$ according to (4.1), we denote
by ${\omega}_{\theta}$ the maximising state of condition (a). We
assume that this state is stationary,\footnote *{The proof of
this is straightforward for lattice systems, since one can show
within the framework of [Se1,Ch.4], that, for these,
${\omega}_{\theta}$ satisfies the KMS conditions, and is
therefore stationary.} i.e. ${\alpha}-$invariant, in view of the
fact that ${\hat q}_{\infty}$ is a globally conserved quantity;
and we designate it as the equilibrium state corresponding to
${\theta},$ i.e. to $q.$ We note that, by (4.18) and the 
thermodynamic completeness condition (a),
$${\omega}_{\theta}{\circ}{\rho}={\omega}_{\theta}\leqno(4.23)$$ 
\vskip 0.2cm
(7) We define ${\pi}$ to be the largest locally normal
representation of ${\tilde {\cal A}}$ that supports the dynamical
group ${\alpha}$ and the quantum field ${\hat q},$ as defined
above.
\vskip 0.3cm
Thus, the quantum model, ${\cal Q},$ is given
by $({\cal A},{\alpha},{\sigma},{\hat q},{\cal N}({\cal A})),$
as specified by the conditions (1)-(7).
\vskip 0.3cm
{\bf 4.3. Relationship between $M$ and ${\cal Q}.$} Our
formulation of this relationship is based on the idea that the
phenomenological law (4.6) corresponds to the dynamics of the
quantum field ${\hat q}$ in a large-scale limit. Thus,
in view of the scale-invariance assumption (M.1) for
$M,$ we introduce a length parameter $L$ and reformulate ${\cal
Q}$ on a length scale $L$ and a time scale $L^{r},$ defining the
quantum field ${\hat q}^{(L)}$ on these scales by the formula
$${\hat q}_{t}^{(L)}(f){\equiv}{\hat q}^{(L)}(f,t):=
{\hat q}(f^{(L)},L^{r}t)\leqno(4.24)$$
where
$$f^{(L)}(x):=L^{-d}f(x/L)\leqno(4.25)$$
We assume that $L$ is also the length scale of spatial variations
of the initial state, ${\omega}^{(L)},$ of ${\cal Q},$ i.e.
that there is a map $A{\rightarrow}{\overline A}$ of ${\cal A}$
into $C(X)$ and a tempered distribution $q_{0}({\in}{\cal
S}^{(n){\prime}}),$ such that
$${\lim}_{L\to\infty}{\omega}^{(L)}({\sigma}(Lx)[A])=
{\overline A}(x) \ {\forall}x{\in}X\leqno(4.26)$$
and
$${\lim}_{L\to\infty}{\omega}^{(L)}
({\hat q}_{0}^{(L)}(f))=q_{0}(f) \ {\forall}
f{\in}{\cal S}^{(n)}\leqno(4.27)$$
We define the fluctuation field ${\hat {\xi}}^{(L)}$ by the
formula
$${\hat {\xi}}_{t}^{(L)}(f):=L^{d/2}({\hat q}_{t}^{(L)}(f)-
{\omega}^{(L)}({\hat q}_{t}^{(L)}(f))), \ {\forall}f{\in}
{\cal S}^{(n)}(X), \ t{\in}{\bf R}\leqno(4.28)$$
Our basic assumptions for the large-scale dynamics of the model
are the following.
\vskip 0.3cm
(I) {\it For each $M-$ process $q_{.},$ there is an equivalence
class of initial states, ${\lbrace}{\omega}^{(L)}{\rbrace},$
parametrised by $L,$ such that
\vskip 0.2cm
(a) the mapping 
$$f^{(1)},.. \ .,f^{(m)};t^{(1)},. \ .,t^{(m)} \
{\rightarrow}{\omega}^{(L)}({\hat{\xi}}_{t^{(1)}}^{(L)}(f^{(1)})
.. \ .({\hat {\xi}}_{t^{(m)}}^{(L)}(f^{(m)}))$$ 
of $({\cal S}^{(n)})^{m}{\times}{\bf R}^{m})$ into ${\bf C}$ is
continuous, for all positive integers $m.$
\vskip 0.2cm
(b) The expectation value of the quantum field ${\hat
q}_{t}^{(L)}$ reduces to that of the classical one, $q_{t},$
of $M$ in the limit $L{\rightarrow}{\infty},$ i.e.}
$${\lim}_{L\to\infty}{\omega}^{(L)}({\hat q}_{t}^{(L)}(f)))=
q_{t}(f) \ {\forall}f{\in}{\cal S}^{(n)}(X)\leqno(4.29)$$
{\it where $q_{t}$ is the solution of (4.4), with initial value
given by (4.27).}
\vskip 0.2cm
{\it (c) The quantum stochastic process ${\hat {\xi}}^{(L)}$
converges to a classical one, ${\xi},$ as
$L{\rightarrow}{\infty},$ i.e.}
$${\lim}_{L\to\infty}{\omega}^{(L)}
({\hat {\xi}}_{t^{(1)}}^{(L)}(f^{(1)})
.. \ .({\hat {\xi}}_{t^{(m)}}^{(L)}(f^{(m)}))=\leqno(4.30)$$
$${\bf E}[{\theta}_{.}{\vert}({\xi}_{t^{(1)}}
(f^{(1)}).. \ .{\xi}_{t^{(m)}}(f^{(m)})]$$
$$ \ {\forall}t^{(1)},. \
.,t^{(m)}{\in}{\bf R}_{+}, \ f^{(1)},.. \ .,f^{(m)}{\in}
{\cal S}^{(n)}(X), \ r{\in}{\bf N}$$
{\it where the expectation functional ${\bf E}[{\theta}_{.}{\vert}.]$
is governed by the restriction of ${\theta}_{.}$ to the close
interval between the minimum and maximum of} ${\lbrace}t^{(1)},.
\ .,t^{(m)}{\rbrace}.$
\vskip 0.3cm
{\bf Comments.} (1) The classicality of the limits of $q^{(L)}$
and ${\xi}^{(L)}$ here are assumed to arise from the commutation
rules (4.21), together with asymptotic abelian properties of
${\cal Q}$ with respect to time.
\vskip 0.2cm
(2) Since (c) implies that the dispersion in ${\hat
q}_{t}^{(L)},$ for the state ${\omega}^{(L)},$ tends to zero as
$L{\rightarrow}{\infty},$ it follows from (b) that the quantum
process ${\hat q}_{t}^{(L)}$ reduces to the classical one,
$q_{t},$ in this limit.
\vskip 0.2cm
(3) It follows from (b) and (c) that ${\xi}_{t}$ is an ${\cal
S}^{(n){\prime}}-$valued random variable.
\vskip 0.3cm
Let 
$${\omega}_{\tau}^{(L)}:={\omega}^{(L)}{\circ}
{\alpha}(L^{r}{\tau}) \
{\forall}{\tau}{\in}{\bf R}_{+}\leqno(4.31)$$
Then ${\lbrace}{\omega}_{t}^{(L)}{\rbrace}$ satisfies the
conditions of (I), and the replacement of ${\omega}^{(L)}$ by
${\omega}_{\tau}^{(L)}$ corresponds to that of ${\theta}_{.}$ by
${\theta}^{({\tau})}$ in (4.30), where 
$${\theta}_{t}^{({\tau})}={\theta}_{t+{\tau}}, \
{\forall}t,{\tau}{\in}{\bf R}_{+}\leqno(4.32)$$
i.e.
$${\bf E}[{\theta}_{.}{\vert}({\xi}_{t^{(1)}+{\tau}}
(f^{(1)}).. \
.{\xi}_{t^{(m)}+{\tau}}(f^{(m)})]{\equiv}\leqno(4.33)$$
$${\bf E}[{\theta}_{.}^{({\tau})}{\vert}({\xi}_{t^{(1)}}
(f^{(1)}).. \ .{\xi}_{t^{(m)}}(f^{(m)})]$$
In the particular case where $t^{(1)}=.. \ =t^{(m)}=0,$ the
r.h.s. of this equation depends on ${\theta}_{.}^{({\tau})}$ only
through the value of
${\theta}_{0}^{({\tau})}{\equiv}{\theta}_{\tau},$ by (4.32).
Thus, by (4.30), the equal time correlation functions for the
process ${\xi}_{.}$ are of the form
$${\bf E}[{\theta}_{.}{\vert}{\xi}_{\tau}(f^{(1)})..
 \ .{\xi}_{\tau}(f^{(m)})]{\equiv}
{\bf E}[{\theta}_{\tau}{\vert}{\xi}_{0}(f^{(1)})..
 \ .{\xi}_{0}(f^{(m)}]\leqno(4.34)$$
\vskip 0.2cm
Our next assumption is that the space-time clustering properties
of ${\cal Q}$ render the process ${\xi}_{.}$ Gaussian (cf.
[GVV]), and that the infinite separation of the relevant
relaxation time-scales of the models $M$ and ${\cal Q}$ ensure
that it is Markovian.
\vskip 0.3cm
(II) {\it The process ${\xi}_{.}$ is Gaussian and temporally
Markovian.}
\vskip 0.3cm
It follows immediately from this assumption that the process
${\xi}$ is completely determined by its two-point function. Our
next assumption is the following generalisation of Onsager's
regression hypothesis [On]. 
\vskip 0.3cm
(III) {\it The fluctuation process ${\xi}$ is governed by
precisely the same dynamics as the perturbation, $y_{.},$ to the
deterministic process $q_{.},$ i.e., by (4.11),}
$${\bf E}[{\theta}_{.}{\vert}({\xi}_{t+{\tau}}(f){\xi}_{t}(g)]=
{\bf E}[{\theta}_{.}{\vert}({\xi}_{t}
(T({\theta}_{.}{\vert}t+{\tau},t)^{\star}f){\xi}_{t}(g)]$$
{\it Hence, by (4.34)}
$${\bf E}[{\theta}_{.}{\vert}({\xi}_{t+{\tau}}(f){\xi}_{t}(g)]=
{\bf E}[{\theta}_{t}{\vert}({\xi}_{0}
(T({\theta}_{t}{\vert}t+{\tau},t)^{\star}f){\xi}_{0}(g)]
\leqno(4.35)$$
$$ \ {\forall}f{\in}{\cal S}^{(n)}(X),t{\in}
{\bf R}, \ {\tau}{\in}{\bf R}_{+}$$
\vskip 0.3cm
Thus, the process is determined by the form of $T$ and of the
expectation functional ${\bf E}[{\theta}_{t}{\vert}.]$ on the
algebra generated by ${\xi}_{0}.$ In order to formulate the
action of space translations and scale transformations on the
process, we define
$$f^{(a,{\lambda})}(x):={\lambda}^{-d/2}f({\lambda}^{-1}(x-a))
 \ {\forall}a{\in}X,{\lambda}{\in}{\bf R}_{+}\leqno(4.36)$$
and
$${\xi}_{0}^{(a,{\lambda})}(f)
:={\xi}_{0}(f^{(a,{\lambda})}) \ {\forall}a{\in}X,
{\lambda}{\in}{\bf R}_{+}\leqno(4.37)$$
\vskip 0.3cm
{\bf Proposition 4.2} {\it Under the above assumptions and
definitions,}
$${\bf E}[{\theta}_{b}{\vert}({\xi}_{0}^{(a,{\lambda})}
(T({\theta}_{.}{\vert}b+{\tau},b)^{\star}f)
{\xi}_{0}^{(a,{\lambda})}(g)]=\leqno(4.38)$$
$${\bf E}[{\theta}_{0}^{(a,b,{\lambda})}
{\vert}{\xi}_{0}(T({\theta}_{.}^{(a,b,{\lambda})}
{\vert}{\tau},0)^{\star}f){\xi}_{0}(g)] \ {\forall}a{\in}X, 
 \ b,{\tau}{\in}{\bf R}_{+}, \ {\lambda}{\in}
{\bf R}_{+}$$
{\it with ${\theta}_{.}^{(a,b,{\lambda})}$ as defined by (4.13).}
\vskip 0.3cm
{\bf Proof.} The result is obtained by replacing 
each $f$ by $f^{a{\lambda}},$ in (4.30), and using
equns. (4.13), (4.14), (4.19), (4.27), (4.28), and (4.35)-(4.37).
\vskip 0.3cm
We note now that the local properties of the process ${\xi},$ in
the neighbourhood of a space-time point $(a,b),$ is given by the
form of the l.h.s. of (4.38), in the limit
${\lambda}{\rightarrow}0.$ Moreover, by (4.13), the function
${\theta}_{.}^{(a,b,{\lambda})},$ which occurs there, tends
pointwise to a constant, ${\theta}(a,b),$ in this limit. These
observations leads us to the following {\it local equilibrium}
assumption.
\vskip 0.3cm
(V) 
$${\lim}_{{\lambda}{\rightarrow}0}
{\bf E}[{\theta}_{0}^{(a,b,{\lambda})}
{\vert}{\xi}_{0}(T({\theta}_{.}^{(a,b,{\lambda})}
{\vert}({\tau},0)^{\star}f){\xi}_{0}(g)]=\leqno(4.39)$$
$${\bf E}[{\theta}(a,b){\vert}({\xi}_{0}
(T({\theta}(a,b){\vert}{\tau},0)^{\star}f){\xi}_{0}(g)]
 \ {\forall}f,g{\in}{\cal S}^{(n)}(X), \ {\tau}{\ge}0$$
{\it and further, the r.h.s. of this formula is precisely the
same as for the fluctuations of the field ${\hat q}_{.}$ about
an equilibrium state ${\omega}_{{\theta}(a,b)},$ as defined in
item (6) of ${\S}4.2,$ i.e.} 
$${\bf E}[{\theta}(a,b){\vert}{\xi}_{0}
(T({\theta}(a,b){\vert}{\tau},0)^{\star}f){\xi}_{0}(g)]
{\equiv}\leqno(4.40)$$
$${\lim}_{L\to\infty}{\omega}_{{\theta}(a,b)}
([{\alpha}(L^{r}{\tau}){\hat {\xi}}_{0}(f)]
{\hat{\xi}}_{0}(g))$$
\vskip 0.3cm
Hence, by (4.12) and (4.40), 
$${\bf E}[{\theta}(a,b){\vert}{\xi}_{0}
({\exp}({\cal L}({\theta}(a,b)){\tau}))^{\star}f){\xi}_{0}(g)]
=\leqno(4.41)$$
$${\lim}_{L\to\infty}{\omega}_{{\theta}(a,b)}
([{\alpha}(L^{r}{\tau}){\hat {\xi}}_{0}(f)]
{\hat{\xi}}_{0}(g))$$
\vskip 0.3cm
{\bf 4.4. Consequences of (I)-(V): Generalised Onsager
Relations.} Let ${\cal R}({\theta})$ be the range of the function
${\theta}{\equiv}{\lbrace}{\theta}(a,b){\vert}a{\in}X,b{\in}{\bf
R}_{+}{\rbrace}.$ We shall employ the above theory to obtain
properties of ${\bf E}[{\overline {\theta}}{\vert}.]$ and ${\cal
L}({\overline {\theta}})$ for arbitrary ${\overline
{\theta}}{\in}{\cal R}({\theta}).$ 
\vskip 0.3cm
(a) {\bf Symmetry Property of Time Correlations Functions.} In
view of the microscopic reversibility conditions (4.18), (4.20)
and (4.23), together with the stationarity of
${\omega}_{\overline {\theta}},$ it follows from (4.30), with
${\omega}^{(L)}={\omega}_{\overline {\theta}},$ that
$${\omega}_{\overline {\theta}}({\xi}_{t}(f){\xi}_{0}(g)){\equiv}
{\omega}_{\overline {\theta}}({\xi}_{0}(f){\xi}_{-t}(g)){\equiv}
{\omega}_{\overline {\theta}}({\xi}_{t}(g){\xi}_{0}(f)) \
{\forall}{\overline {\theta}}{\in}{\cal R}({\theta})$$
Hence, by (4.41), we have the symmetry property
$${\bf E}[{\overline {\theta}}{\vert}
{\xi}_{0}({\exp}({\cal L}({\overline {\theta}})^{\star}{\tau})f)
{\xi}_{0}(g)]{\equiv}{\bf E}[{\overline {\theta}}{\vert}
{\xi}_{0}({\exp}({\cal L}
({\overline {\theta}})^{\star}{\tau})g){\xi}_{0}(f)]
 \ {\forall}{\overline {\theta}}{\in}{\cal R}({\theta})
\leqno(4.42)$$
\vskip 0.3cm
(b) {\bf The Static Two-point Function.} It follows immediately
from (4.41) that ${\bf E}[{\overline {\theta}}{\vert}.]$ inherits
the translational invariance of ${\omega}_{\overline {\theta}}.$
Hence, in view of the tempered distribution property of
${\xi}_{0},$ the generalised function
$(x,y)({\in}X^{2}){\rightarrow}{\bf
E}[{\overline {\theta}}{\vert}({\xi}_{0}(x){\xi}_{0}(y)]$ is an 
${\cal S}^{(n){\prime}}(X)-$ class distribution $F(x-y);$ and,
by Prop. 4.2, $F({\lambda}x){\equiv}{\lambda}^{-d}F(x),$
and is therefore of the form $C{\delta}(x),$ where $C$ is an
n-by-n matrix. Thus,
$${\bf E}[{\overline {\theta}}{\vert}{\xi}_{0}(f){\xi}_{0}(g)]=
{\langle}Cf,g{\rangle} \ {\forall}{\overline {\theta}}
{\in}{\cal R}({\theta})\leqno(4.43)$$
where the angular brackets denote the inner product for the
Hilbert space ${\cal H}^{(n)}$ of square integrable functions
from $X$ into ${\bf R}^{n},$ as defined by the formula
$${\langle}f,g{\rangle}={\int}f(x).g(x)dx\leqno(4.44)$$
the dot denoting the ${\bf R}^{n}$ scalar product. Moreover, it
follows [Se5] from a treatment of the linear response
of ${\omega}_{\overline {\theta}}$ to local Hamiltonian
perturbations ${\hat q}(f)$ that, under mild technical
assumptions, $C=B({\theta})^{-1},$ where $B$ is specified in
(4.10). Hence, by (4.43), 
$${\bf E}[{\overline {\theta}}{\vert}{\xi}_{0}(f){\xi}_{0}(g)]=
{\langle}B({\overline {\theta}})^{-1}f,g{\rangle}\leqno(4.45)$$
\vskip 0.3cm
(c) {\bf Generalised Onsager Relations.} It follows immediately
from (4.42) and (4.45) that
$${\langle}B({\overline {\theta}})^{-1}{\exp}
({\cal L}({\overline {\theta}})^{\star}{\tau})f,
g{\rangle}{\equiv}{\langle}B({\overline {\theta}})^{-1}{\exp}
({\cal L}({\overline {\theta}})^{\star}{\tau})g,
f{\rangle}\leqno(4.46)$$
Hence, by (4.9), we have the following result.
\vskip 0.3cm
{\bf Proposition 4.3.} {\it Under the above assumptions,
${\Lambda}$ satisfies the generalised Onsager relation}
$${\langle}{\Lambda}({\overline
{\theta}})^{\star}f,g{\rangle}{\equiv}
{\langle}{\Lambda}({\overline {\theta}})^{\star}g,f{\rangle} \
{\forall}{\overline {\theta}}{\in}{\cal R}({\theta}), \ f,g{\in}
{\cal S}^{(n){\prime}}(X)\leqno(4.47)$$
{\it i.e. ${\Lambda}({\overline {\theta}}),$ considered as an
operator in ${\cal H}^{(n)},$ with domain ${\cal S}^{(n)},$ is
symmetric.}
\vskip 0.3cm
{\bf Comment.} In the case of the non-linear diffusion given by
(4.6a), it follows from (4.12a) that (4.47) reduces to the form
$$L_{kl}({\theta})(x,t))=L_{lk}({\theta})(x,t)) \
{\forall}x{\in}X, \ t{\in}{\bf R}_{+}$$
\vskip 0.5cm
{\bf 5. Concluding Remarks.} I have endeavoured to show here how,
at least in certain domains, a macroscopically-based approach to
statistical mechanics can serve to determine the form imposed by
quantum mechanics on the structure of phenomenological laws. By
contrast with the standard many-body theory, the  microscopic
imput here is limited to very general principles; and this serves
to pare down the conceptual structure of the theory to its
essentials. 
\vskip 0.2cm
Of course, the relative simplicity gained by this approach is
dependent on a number of assumptions, specified in the previous
Sections, that are very difficult to verify constructively.
Furthermore, the dynamical systems treated in ${\S}'s$ 3 and 4
have the simplifying, and rather particular, property of scale
covariance. In the case of the plasma model, this stems from the
fact that the Coulomb potential is given by a power law: in the
case of the non-equilibrium thermodynamics of ${\S}4,$ it is an
assumed property of the macroscopic dynamics.
\vskip 0.2cm
Thus, it is clear that the formulation of a coherent, general
formulation of the statistical mechanics of macroscopic variables
poses deep problems, concerning both its underlying assumptions
and its potential scope. I would hope that inroads into these 
problems may be achieved through the study both of suitable
models and of the relevant general structures. 
\vskip 0.5cm
\centerline {\bf References}
\vskip 0.3cm
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[BCS] J. Bardeen, L. N. Cooper and J. R. Schrieffer: Phys. Rev.
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[DF] B. Deaver and W. M. Fairbank: Phys. Rev. Lett. {\bf 7}, 43
(1961)
\vskip 0.2cm
[GVV] D. Goderis, A. Verbeure and P. Vets: Pp. 178-193 of "Quantum 
Theory and Applications V", Ed. L. Accardi and W. von Waldenfels, 
Springer Lecture Notes in Mathematics 1442, 1990
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[He] K. Hepp: Helv. Phys. Acta {\bf 45}, 237 (1972)
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[HHW] R. Haag, N. M. Hugenholtz and M. Winnink: Commun. Math.
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[Jo] B. D. Josephson: Rev. Mod. Phys. {\bf 36}, 216 (1964)
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[Lo] F. London: Superfluids, Vol. 1, Wiley, London, 1950
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[LL] L. D. Landau and I. M. Lifschitz: "Fluid Mechanics",
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\vfill\eject
[LR] O. E. Lanford and D. Ruelle: Commun. Math. Phys. {\bf 13},
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[Ne] Fluid Dyn. Trans. {\bf 9},229 (1978)
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[NS] H. Narnhofer and G. L. Sewell: Commun. Math. Phys. {\bf 79},
9 (1981)
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[On] L. Onsager: Phys. Rev. {\bf 37}, 405 (1931) and {\bf 38},
2265 (1931)
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[Pe] O. Penrose: Phil. Mag. {\bf 42}, 1373 (1951)
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[PO] O. Penrose and L. Onsager: Phys. Rev. {\bf 104}, 576 (1956)
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[Ri] G. Rickayzen: Phys. Rev. {\bf 115}, 795 (1959)
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[Ru] D. Ruelle: "Statistical Mechanics", W. A. Benjamin Inc., New
York, 1969
[Sc] M. R. Schafroth: Phys. Rev. {\bf 100}, 463 (1955)
\vskip 0.2cm
[Se1] G. L. Sewell: "Quantum Theory of Collective Phenomena",
Oxford University Press, Oxford, 1989 
\vskip 0.2cm
[Se2] G. L. Sewell: J. Stat. Phys. {\bf 61}, 415 (1990)
\vskip 0.2cm
[Se3] G. L. Sewell: "Macroscopic Quantum Theory of
Superconductivity and the Higgs Mechanism", to be published in
the Proceedings of the 1991 Locarno Conference on "Stochastics,
Physics and Geometry".
\vskip 0.2cm
[Se4] G. L. Sewell: "Quantum Plasma Model with Hydrodynamical
Phase Transition", Preprint
\vskip 0.2cm
[Se5] G. L. Sewell:Pp.77-122 of "Large-Scale Molecular Systems: 
Quantum and Stochastic Aspects", Nato ASI Series B, Ed. W. Gans, 
A. Blumen and A. Amann, Plenum, New York and London, 1991
\vskip 0.2cm
[Se6] G. L. Sewell: Lett. Math. Phys. {\bf 6}, 209 (1982)
\vskip 0.2cm
[Se7] G. L. Sewell: J. Math. Phys. {\bf 26}, 2324 (1985)
\vskip 0.2cm
[Sp] H. Spohn: Math. Meth. Appl. Sci. {\bf 3},445 (1981)
\vskip 0.2cm
[Ya] C. N. Yang: Rev. Mod. Phys. {\bf 34}, 694 {1962}
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[ZA] Z. Zou and P. W. Anderson: Phys. Rev. B {\bf 37}, 627 (1987)
\magnification=\magstep1
\hoffset=1.5cm
\vsize=22.5truecm
\hsize=13.7truecm
\centerline {{\bf TOWARDS A MACROSTATISTICAL MECANICS}\footnote
*{Based on a talk given at the Workshop on "Mathematical Physics
Towards the 21st Century", held at Beersheva, Israel, March
14-19, 1993}}
\vskip 1cm
\centerline {{\bf by Geoffrey L. Sewell}\footnote{**}{Partially
supported 
by European Capital and Mobility Contract No. CHRX-CT92-0007}}
\vskip 0.5cm
\centerline {\bf Department of Physics, Queen Mary and Westfield
College}
\vskip 0.5cm
\centerline {\bf Mile End Road, London E1 4NS}
\vskip 1cm
\centerline {\bf ABSTRACT}
\vskip 0.5cm
I discuss the general question of the derivation of the
statistical mechanics of macroscopic variables from the quantum
structures of many-particle systems, as represented in the
thermodynamic limit. I then present a number of strands of such
a macrostatistical mechanics, including (a) a derivation of the
electrodynamics of superconductors from their order structure and
gauge covariance; (b) a hydrodynamics, with transition from
deterministic to stochastic flow, of a quantum plasma model; and
(c) a non-linear generalisation of Onsager's irreversible
thermodynamics.
\vskip 1cm
{\bf 1. Introduction.} The 'miracle' of statistical physics is
that the microscopically chaotic dynamics of many-particle
systems conspires to generate macroscopic laws of relatively
simple structure, such as those of thermodynamics and
hydrodynamics. In view of the complexity of the microscopic
picture and the simplicity of the macroscopic one, it is natural
to seek an approach to the subject, that is centred on macro-
observables, with microscopic imput limited to very general
principles, e.g. conservation laws, ergodicity, etc. Such an
approach would evidently be at the opposite pole from the
conventional microscopically based 'many-body theory'. 
\vskip 0.2cm
In fact, there are already important macroscopically-based 
theories in statistical physics, most notably Onsager's [On]
linear irreversible thermodynamics and Landau's [LL, \ Ch.17]] 
fluctuating hydrodynamics. However, these theories are essentially 
heuristic because, apart from questions of rigour, they lack the 
structures needed for a precise characterisation of their purportedly 
key ingredient of macroscopicality. Moreover, the same thing can be 
said about all works devised within the traditional framework of the 
statistical mechanics of strictly finite systems. 
\vskip 0.2cm
On the other hand, the 'revolution' in statistical mechanics,
based on
the formulation of many-particle systems in the thermodynamic
limit [Ru, HHW, He], have provided us with just the framework
required for a sharp characterisation of macroscopicality, and
even of different levels thereof [Se1, GVV]. My objective here
is to discuss the project of extracting a 'macrostatistical
mechanics' (MSM) from the quantum structures of many-particle
systems, within this framework. In fact, we already have some
strands of such a discipline in the works of Hepp and Lieb [HL] 
on the derivation of the macroscopic dynamics, with non-equibrium
phase transition, of a laser mode; of myself [Se1,Ch.4]
on the formulation of an extended classical thermodynamics with
phase structure; and of the Leuven school [GVV] on macroscopic
fluctuation theory.
\vskip 0.2cm
In this article, I shall bring together further contributions, 
from three different areas, towards an MSM. The first of these, 
$({\S}2),$ consists of a derivation of the electromagnetic 
properties of superconductors from their order structure and 
gauge covariance [Se2,3]. The second, $({\S}3),$ is an extraction
of the hydrodynamics, with non-equilibrium phase transition, of
a 
quantum plasma model [Se4];  and the third, $({\S}4),$ consists
of 
a quantum-statistical derivation of a non-linear generalisation
of 
Onsager's irreversible thermodynamics [Se5]. I shall conclude,
in 
${\S}5,$ with some further brief comments about the project of
a  
macroscopically-centred statistical mechanics.
\vskip 0.5cm
{\bf 2. Macroscopic Quantum Theory of Superconductivity.} At the
{\it phenomenological} level, the principal electrodynamic
properties of superconductors are their capacity to support
persistent electric currents (superconductivity) and their
perfect diagmagnetism (Meissner effect). These two properties are
intimately related, since the Meissner effect is the mechanism
whereby the supercurrents screen the magnetic field they generate
from the interior of the body [Lo]. Thus, superconductivity
arises from the combination of the Meissner effect with the
thermodynamic metastability of the supercurrents and their
magnetic fields. 
\vskip 0.2cm
Although it appears to be widely accepted that the microscopic
theory of Bardeen-Cooper-Schrieffer [BCS] leads to the
electrodynamics of metallic superconductors, the arguments
employed both there and in related works [An, Ri] are radically
flawed in that (a) they are based on totally uncontrolled
approximations, and (b) they violate (exact) gauge covariance of
the second kind {\it at the Hamiltonian level}, and thus do not
even admit precise definition of a local current density. As
regards ceramic, i.e.
high $T_{c},$ superconductivity, the microscopic theory is 
less developed than in the metallic case, and has not yet led to
an electrodynamics.
\vskip 0.2cm
On the other hand, the BCS characterisation of the structure of
the superconductive phase by electron pairing, first proposed by
Schafroth [Sc], has been amply substantiated by experiments on
the Josephson effect [Jo] and the quantisation of trapped
magnetic flux in multiply-connected superconductors [DF].
Moreover, Yang [Ya], generalising ideas of O. Penrose [Pe, PO],
pointed out that this characterisation is captured by the
hypothesis of {\it off-diagonal long range order} (ODLRO). This
is a macroscopic quantum property, representing a well-defined
order structure in a gauge covariant way. Furthermore, it is a
property also possessed by certain ans\"atze, e.g. [ZA], for the
high $T_{c}$ superconductive phase of ceramics.
\vskip 0.2cm
I shall now sketch an approach [Se2,3] I have made to the
electrodynamics of superconductors, based on the assumption of
ODLRO. This is designed to relate the electromagnetic properties
to the order structure of these systems in purely macroscopic 
quantum terms (cf. eqno. (2.8) below). For brevity, I shall confine 
myself here to the derivation of the Meissner effect from ODLRO.
\vskip 0.2cm
{\bf The Model.} We take the quantum model, ${\Sigma},$ to be an
infinitely extended system of electrons, and possibly also of
ions or phonons, in a Euclidean space $X:$ lattice systems may
be formulated analogously. Points in $X$ will generally be
denoted by $x,$ (sometimes by $y,a$ or $b$) and the Lebesgue
measure by $dx.$  It will be assumed that the model enjoys the
properties of gauge covariance of the second kind, and that its
interactions are translationally invariant.
\vskip 0.2cm
The electronic part of ${\Sigma}$ is formulated in terms of a 
quantised field ${\psi}=({\psi}_{\uparrow},{\psi}_{\downarrow}),$
satisfying the canonical anticommutation relations. Thus, in a 
standard way, the $C^{\star}-$algebra ${\cal F}_{el}$ 
of the CAR over the Hilbert space ${\cal H}:=L^{2}(X,dx)$ is
defined by the specifictions that
\vskip 0.2cm
(1) there are linear maps
${\psi}_{\uparrow},{\psi}_{\downarrow},$ 
from ${\cal H}$ into ${\cal F}_{el}$ satisfying the CAR
$${\lbrack}{\psi}_{\alpha}(f),{\psi}_{\beta}(g)^{\star}
{\rbrack}_{+}= {\delta}_{{\alpha},{\beta}}
{\langle}g,f{\rangle}_{\cal H}; \
{\lbrack}{\psi}_{\alpha}(f),{\psi}_{\beta}(g){\rbrack}_{+}
=0\leqno(2.1)$$
\vskip 0.2cm
(2) ${\cal F}_{el}$ is generated by ${\lbrace}{\psi}_{\alpha}(f),
{\psi}_{\alpha}(f)^{\star}{\vert}f{\in}{\cal H}; \ {\alpha}=
{\uparrow},{\downarrow}{\rbrace}.$
\vskip 0.2cm
The algebra ${\cal F}_{el}$ is then the {\it  field algebra} of
the electrons. Space translations and gauge transformations are 
represented by the homomorphisms ${\sigma}_{0}, \ {\gamma}$ of
the additive  groups $X, \ C_{R}^{\infty}(X),$ respectively, into
$Aut({\cal F}_{el}),$ defined by the formulae
$${\sigma}_{0}(a){\psi}(f)={\psi}(f_{a}) \ {\forall}a{\in}X, \ 
with \ f_{a}(x):=f(x-a)\leqno(2.2)$$
and
$${\gamma}({\chi}){\psi}({\phi})={\psi}({\phi}{\exp}(i{\chi}))
\ {\forall}{\chi}{\in}C_{R}^{\infty}(X)\leqno(2.3)$$
The {\it global} gauge automorphisms are those for which ${\chi}$
is constant. 
\vskip 0.2cm
Let ${\cal K}:={\lbrace}f_{1}{\otimes}f_{2}
{\vert}f_{1},f_{2}{\in}{\cal H}{\rbrace}.$ We define the {\it
pair field}
${\Psi}$ to be the mapping of ${\cal K}$ into ${\cal F}_{el}$
given by
$${\Psi}(F)={\psi}_{\uparrow}(f_{1}){\psi}_{\downarrow}(f_{2})
 \ for \ F=f_{1}{\otimes}f_{2}\leqno(2.4)$$
Hence, by (2.2)-(2.4),
$${\sigma}_{0}(a){\Psi}(F)={\Psi}(F_{a}), \ with \
F_{a}(x_{1},x_{2})
=F(x_{1}-a,x_{2}-a)\leqno(2.5)$$
and
$${\gamma}({\chi}){\Psi}(F)={\Psi}(g({\chi})F);\leqno(2.6)$$
$$ \ with \  (g({\chi})F)(x_{1},x_{2}):=F(x_{1},x_{2})
{\exp}i({\chi}(x_{1})+{\chi}(x_{2}))$$
In a standard way, we define the {\it observable algebra} of the
electrons to be the subalgebra ${\cal A}_{el}$ of ${\cal F}_{el}$
that is elementwise invariant under the global gauge
automorphisms
${\psi}{\rightarrow}{\psi}{\exp}(i{\alpha}),$ with ${\alpha}$
constant. Space translations and gauge transformations of the
electronic observables are then given by the 
restrictions to ${\cal A}_{el}$ of the automorphism groups
${\sigma}_{0}(X)$ and ${\gamma}(C_{R}^{\infty}(X)),$
respectively. 
\vskip 0.2cm
The construction of the $C^{\star}-$algebra ${\cal A}_{0}$ of the
observables of the other species of particles of ${\Sigma}$ is 
effected in a similar way. We take the $C^{\star}-$algebra of 
observables of the system to be ${\cal A}:={\cal A}_{el}
{\otimes}{\cal A}_{0}$ and canonically identify ${\cal A}_{el}$
with its subalgebra ${\cal A}_{el}{\otimes}I.$ It is assumed that
the automorphism groups ${\sigma}_{0}(X)$ and
${\gamma}(C_{R}^{\infty}(X))$ extend from ${\cal A}_{el}$ to
${\cal A}.$ 
\vskip 0.2cm
We assume, for simplicity,\footnote *{The more generally valid
assumption of a $W^{\star}-$dynamical system [Se6], employed in
${\S4}$ 
of this article, leads to
precisely the same results as the present one.} that the dynamics
of ${\Sigma},$ as given by a canonical limiting form of that of
finite versions of the system, corresponds to a one-parameter
group ${\lbrace}{\alpha}(t){\vert}t{\in}{\bf R}{\rbrace}$ of
automorphisms of ${\cal A}.$
\vskip 0.2cm
An equilibrium state ${\omega}$ at inverse
temperature ${\beta}$ may then be characterised by the
Kubo-Martin-Schwinger (KMS) conditions [HHW], namely that, for
arbitrary $A,B{\in}{\cal A},$ there is a function $F$ on {\bf C}
that is analytic in the interior of the strip
$Im(z){\in}[0,{\hbar}{\beta}]$ and continuous on its boundaries,
and satisfies the relations
$$F(t)={\omega}(A{\alpha}(t)B); \ F(t+i{\hbar}{\beta})=
{\omega}(({\alpha}(t)A)B)\leqno(2.7)$$
\vskip 0.3cm
{\bf Off-Diagonal Long Range Order.} A state ${\omega}$ is said
to possess the property of off-diagonal long-range order (ODLRO)
if there is a mapping ${\Phi}:{\cal K}{\rightarrow}{\bf C},$ such
that
$${\lim}_{{\vert}y{\vert}\to\infty}[{\omega}({\Psi}(F)^{\star}
{\Psi}(G_{y}))-{\overline {{\Phi}(F)}}{\Phi}(G_{y})]=0
 \ {\forall}F,G{\in}{\cal K}\leqno(2.8)$$
where the bar denotes complex conjugation; and ${\Phi}(G_{y})$
does not 
tend to zero, for all $G{\in}{\cal K}$ as
$y{\rightarrow}{\infty}.$
\vskip 0.2cm
${\Phi}$ is then termed the {\it macroscopic wave function} for
the state ${\omega}.$
\vskip 0.2cm
${\bf Note}$ that, although ${\Psi}(F)$ does not belong to the
observable algebra ${\cal A},$ the argument of ${\omega}$ in
(2.8) does.
\vskip 0.3cm
${\bf Lemma \ 2.1.}$ {\it The ODLRO conditions 
define the macroscopic wave function up to a constant phase 
factor; i.e., if ${\Phi}_{1},{\Phi}_{2}$ are two such functions
satisfying these conditions, for the same state ${\omega},$ then
${\Phi}_{2}={\Phi}_{1}{\exp}(i{\eta}),$ where ${\eta}$ is a 
real-valued constant.}
\vskip 0.3cm
{\bf Proof.} Assuming that ${\Phi}_{1},{\Phi}_{2}$ both 
satisfy the ODLRO conditions with respect to the same state
${\omega},$ it follows from (2.8) that
$${\lim}_{{\vert}y{\vert}\to\infty}[{\overline {{\Phi}_{1}(F)}}
{\Phi}_{1}(G_{y})-{\overline {{\Phi}_{2}(F)}}
{\Phi}_{2}(G_{y})]=0$$
Since this is valid for all $F,G{\in}{\cal K},$ we may
replace $F$ by $F^{\prime}({\in}{\cal K}),$ thereby obtaining 
$${\lim}_{{\vert}y{\vert}\to\infty}
[{\overline {{\Phi}_{1}(F^{\prime})}}
{\Phi}_{1}(G_{y})-{\overline {{\Phi}_{2}(F^{\prime})}}
{\Phi}_{2}(G_{y})]=0$$ 
On multiplying the complex conjugate of the first equation by
${\Phi}_{2}(F^{\prime})$ and that of the second one by
${\Phi}_{2}(F),$ and then taking the difference, we see that
$${\lim}_{{\vert}y{\vert}\to\infty}
{\overline {{\Phi}_{1}(G_{y})}}
[{\Phi}_{1}(F){\Phi}_{2}(F^{\prime}))-{\Phi}_{1}(F^{\prime})
{\Phi}_{2}(F)]=0\leqno(2.9)$$
Since, by the above definition of ODLRO, there are elements $G$
of ${\cal K}$ for which ${\Phi}(G_{y})$ does not tend to zero,
as ${\vert}y{\vert}{\rightarrow}{\infty},$ it follows that the
quantity in the square brackets of (2.9) vanishes. Consequently,
as ${\Phi}_{1,2}$ are non-zero, by the same stipulation, 
$${\Phi}_{2}(F)=c{\Phi}_{1}(F) \ {\forall}F{\in}{\cal K}$$
where c is a complex-valued constant, and since
${\Phi}_{1},{\Phi}_{2}$ both satisfy (2.8), it follows
immediately that this is just a constant phase factor
${\exp}(i{\eta}).$
\vskip 0.3cm
{\bf Gauge and Space Translational Covariance.} Assume now that 
the system is subjected to a classical, static magnetic field,
of induction 
$$B=curlA\leqno(2.10)$$
The time-translational automorphisms then become $A-$dependent,
and we denote them by ${\alpha}_{A}({\bf R}).$ We assume that
they are covariant w.r.t. space translations and gauge
transformations, i.e. that 
$${\sigma}_{0}(a){\alpha}_{A}(t){\sigma}_{0}(-a)=
{\alpha}_{A_{a}}(t); \ with \ A_{a}(x):=A(x-a)\leqno(2.11)$$
and
$${\gamma}({e{\chi}\over {\hbar}c}){\alpha}_{A}(t){\gamma}
({-e{\chi}\over
{\hbar}c})={\alpha}_{A+{\nabla}{\chi}}(t)\leqno(2.12)$$
We note now that, in the particular case where $B$ is uniform,
it may be represented by the vector potential 
$$A={1\over 2}B{\times}x\leqno(2.13)$$
Thus, choosing 
$${\chi}(x)=-{1\over 2}(B{\times}a).x,\leqno(2.14)$$
we see that                                   
$$A+{\nabla}{\chi}=A_{a}\leqno(2.15)$$ 
Hence, defining ${\sigma}:X{\rightarrow}Aut({\cal A})$ by the
formula 
$${\sigma}(a)={\gamma}({e{\chi}\over {\hbar}c}){\sigma}_{0}(a),
\leqno(2.16)$$
it follows from (2.11),(2.12) and (2.16) and the definition of
${\cal A}$ that ${\sigma}$ is a representation of $X$ in
$Aut{\cal A}$ which commutes with the time translations  
${\alpha}_{A}({\bf R}),$ with the gauge fixed by equn. (2.13). 
In view of these properties, we take
${\sigma}$ to be the space translational automorphisms in the
presence of the uniform magnetic field $B.$
\vskip 0.2cm
{\bf Note.} By (2.5),(2.6) and (2.16), the canonical extension
of ${\sigma}$ to ${\cal F}_{el},$ and in particular to the pair
field ${\Psi},$ yields
$${\sigma}(a){\Psi}(F)={\Psi}(s(a)F)\leqno(2.17)$$
where  
$$(s(a)F)(x_{1},x_{2})=F(x_{1}-a,x_{2}-a){\exp}
({ie(B{\times}a).(x_{1}+x_{2})\over 2{\hbar}c})\leqno(2.18)$$
Hence, 
$${\lbrack}s(a),s(b){\rbrack}_{-}=2isin
({eB.(a{\times}b)\over {\hbar}c})s(a+b) \ and \ s(a)s(-a)=
I \ {\forall}a,b{\in}X \leqno(2.19)$$ 
where ${\lbrack},{\rbrack}_{-}$ denotes commutator. Thus, as
${\sigma}(a)$ and ${\sigma}(b)$ do not, in
general intercommute, it follows from (2.17) that the extension
of ${\sigma}(X)$ to the field algebra ${\cal F}_{el}$ is non-
abelian. This is crucial for our derivation of the Meissner
effect below.
\vskip 0.3cm 
{\bf ODLRO and the Meissner Effect.} The essential distinction 
between normal diamagnetism and the Meissner effect is that the 
former can support a uniform, static, non-zero magnetic induction
and 
the latter can not. Thus, we base our derivation of the Meissner 
effect on considerations of the response of an state with ODLRO 
to the action of a uniform magnetic field.
\vskip 0.3cm
${\bf Proposition \ 2.2.}$ {\it The system cannot support
uniform,
non-zero magnetic induction in translationally invariant states
with ODLRO.}
\vskip 0.3cm      
The following Corollary follows immediately from this Propostion
and elementary thermodynamics.
\vskip 0.3cm
${\bf Corollary \ 2.3.}$ {\it Assuming that translational
symmetry 
is preserved in the equilibrium state ${\omega}_{A},$ then either
\vskip 0.2cm
(1) ${\omega}_{A}$ possesses the property of ODLRO and $B=0$
\vskip 0.2cm
or
\vskip 0.2cm
(2) ${\omega}_{A}$ does not possess ODLRO and is normally
diamagnetic.}
\vskip 0.2cm
{\it Further, assuming that, in the absence of a magnetic field,
the free energy density of the ODLRO phase is lower, by
${\Delta},$ than that of the normal one, the former phase will
prevail, and thus the system will exhibit the Meissner effect,
provided that the applied field, $H,$ satisfies the 
condition that} ${\vert}H{\vert}<H_{c}:=(8{\pi}{\Delta})^{1/2}$ 
\vskip 0.3cm
{\bf Comment.} This result signifies that the Meissner effect
arises 
from a rigidity of the macroscopic wave-function in the face of 
an applied magnetic field. This corresponds precisely to 
London's [Lo] idea of rigidity, transferred to the macroscopic
level.
\vskip 0.3cm
${\bf Proof \ of \ Proposition \ 2.2.}$ By the translational
invariance of ${\omega}_{A}$ and the formula (2.17) for
${\sigma},$ 
$${\omega}_{A}({\Psi}(s(a)F)^{\star}{\Psi}(s(a)G)){\equiv}
{\omega}_{A}({\Psi}(F)^{\star}{\Psi}(G))\leqno(2.20)$$
Further, it follows from the ODLRO condition (2.8) and the
definition (2.18) of $s$ that
$${\lim}_{y\to\infty}[{\omega}_{A}({\Psi}(s(a)F)^{\star}
{\Psi}(s(a)G_{y}))-{\overline {{\Phi}(s(a)F)}}
{\Phi}(s(a)G_{y})]=0$$
and hence, by (2.20), that
$${\lim}_{y\to\infty}[{\omega}_{A}({\Psi}(F)^{\star}{\Psi}
(G_{y}))-{\overline {{\Phi}_{a}(F)}}
{\Phi}_{a}(G_{y})]=0\leqno(2.21)$$
where
$${\Phi}_{a}={\Phi}{\circ}s(a)\leqno(2.22)$$
Equation (2.21) implies that ${\Phi}_{a},$ as well as ${\Phi},$
serves as a macroscopic wave function for the state
${\omega}_{A},$ and therefore, by Lemma 2.1 and equn. (2.22),
that 
$${\Phi}{\circ}s(a)=c(a){\Phi}$$
where $c(a)$ is a complex number of unit modulus. Similarly, if
$b$ is another element of $X,$ then
$${\Phi}{\circ}s(b)=c(b){\Phi}$$
where $c(b)$ is also a complex number of unit modulus. Thus, the
last two equations imply that
$${\Phi}{\circ}{\lbrack}s(a),s(b){\rbrack}_{-}=0$$
On taking the composition product of this equation with 
$s(-a-b)$ and using (2.19), we see that, 
$$sin({eB.(a{\times}b)\over {\hbar}c}){\Phi}=0 \ {\forall}
a,b{\in}X$$
from which it follows that {\it either} $B=0$ {\it or}
${\Phi}=0,$ which is the required result.
\vskip 0.5cm
{\bf 3. Hydrodynamical Phase Transition in a Quantum Plasma.} 
\vskip 0.2cm
The quantum Jellium model is a system of electrons, interacting
via
Coulomb forces both with one another and with a uniform,
positively charged, neutralising background. It is thus a model
of a many-particle system with realistic interactions. Its
thermodynamic properties have been established at the level of
mathematical physics [LN]. I shall now sketch a treatment [Se4,7]
leading from its many-particle Schr\"odinger equation to an
Eulerian hydrodynamics with a non-equilibrium phase transition.
The three principal stages of the derivation of this result are
the following. Firstly, we derive a classical Vlasov equation,
governing a {\it large scale} description of its dynamics, from
the microscopic Schr\"odinger equation, together with certain
initial and regularity conditions. Secondly, we observe that the
Vlasov dynamics corresponds precisely to the Liouville equation
for a {\it Lagrangian} hydrodynamics, governing the evolution of
the time (t)-dependent position, $X_{t}(x),$ of a 'fluid
particle'located initially at the point $x.$  Then, finally, we
show that this Lagrangian hydrodynamics reduces to a
deterministic {\it Eulerian} form, provided that a certain
parameter, governing the non-uniformity in the initial density
profile, lies below a certain critical value, and that otherwise
it corresponds to a {\it stochastic} flow.
\vskip 0.3cm
{\bf The Jellium Model,} ${\Sigma}^{(N,L)},$ consists of $N$
electrons 
in a cube, $K^{(L)},$ of side $L,$ with uniform neutralising
positive charge background. We assume periodic boundary
conditions. Our objective is to obtain a quantum theoretical
derivation of the hydrodynamics of ${\Sigma}^{(N,L)}$ in a limit
where $N$ and $L$ tend to infinity and the mean particle density,
$${\overline n}=N/L^{3}\leqno(3.1)$$
remains fixed and finite. 
\vskip 0.3cm
Denoting the gradient operator in $K^{(L)}$ by ${\nabla}^{(L)},$
we represent the position vectors and momenta of the electrons
by the standard multiplicative and differential operators
${\lbrace}X_{j},P_{j}=-i{\hbar}{\nabla}_{j}^{(L)}{\vert}j=1,..
\,N{\rbrace}$ in the Hilbert space ${\cal H}^{(N,L)}$ of square
integrable antisymmetric functions on $(K^{(L)})^{N}.$ The
Hamiltonian is then the self-adjoint operator in ${\cal
H}^{(N,L)}$ given by
$$H^{(N,L)}={-{\hbar}^{2}\over 2m}{\sum}_{j=1}^{N}
{\Delta}_{j}^{(L)}+e^{2}{\sum}_{j,k(>j)=1}^{N}U^{(L)}
(X_{j}-X_{k})\leqno(3.2)$$
where $-e,m$ are the electronic charge and mass, respectively,
${\Delta}^{(L)}$ is the Laplacian for $K^{(L)},$ and $U^{(L)}(X)$
is the difference between ${\vert}X{\vert}^{-1},$ periodicised
w.r.t. $K^{(L)},$ and its space average over that cube, i.e.
$$U^{(L)}(X)={4{\pi}\over L^{3}}{\sum}^{(L)}
{{\exp}(iQ.X)\over Q^{2}}\leqno(3.3)$$
the superscript $(L)$ over ${\Sigma}$ signifying that summation
is taken over the non-zero vectors $Q=(2{\pi}/L)(n_{1},n_{2},
n_{3}),$ with the $n$'s integers. The pure states of
${\Sigma}^{(N,L)}$ are represented by the normalised vectors,
${\Psi}^{(N)}$ in ${\cal H}^{(N,L)}$ and their evolution is
governed by the time-dependent Schr\"odinger equation
$$i{\hbar}{{\partial}{\Psi}_{T}^{(N)}\over {\partial}T}=
H^{(N,L)}{\Psi}_{T}^{(N)}\leqno(3.4)$$
with $T$ the time variable. We shall assume the following initial
kinetic and potential energy bounds for ${\Sigma}^{(N,L)}$-more
precisely, for the family of systems
${\lbrace}{\Sigma}^{(N,L)}{\rbrace},$ with
$N,L$ satisfying (3.1).
\vskip 0.2cm
$(I.1)^{(L)}$ The expectation value of the total kinetic energy
per particle, for the initial state ${\Psi}_{0}^{(N)},$ is less
than some finite $N-$independent constant $B/2m,$ i.e.
$$({\Psi}_{0}^{(N)},P_{1}^{2}{\Psi}_{0}^{(N)})<B\leqno(3.5)$$
\vskip 0.2cm
$(I.2)^{(L)}$ The expectation value of the total potential
energy, for the the initial state ${\Psi}_{0}^{(N)},$ is less
than some finite $N-$independent constant, $e^{2}C/2,$ times
$N^{5/3}.$ This bound corresponds to the electrostatic energy of
a continuous distribution of charge, whose density is a smooth
function of $X/L,$ and signifies that
$$({\Psi}_{0}^{(N)},U^{(L)}(X_{1}-X_{2}){\Psi}_{0}^{(N)})<
CN^{2/3}\leqno(3.6)$$
\vskip 0.2cm
We base our macroscopic description of the model on scales of
length, time and particle momentum given by $L,  \
{\omega}^{-1}$ and $mL{\omega},$ where ${\omega}$ is the
classical plasma frequency, i.e. 
$${\omega}=(4{\pi}{\overline n}e^{2}/m)^{1/2}\leqno(3.7)$$
For this description, we employ the rescaled space and time
coordinates, $x=X/L, \ t={\omega}T,$ respectively. Under this
rescaling, ${\Sigma}^{(N,L)}$ is mapped onto a system
${\Sigma}^{(N)}$ of particles in a unit cube,
$K,$ with periodic boundaries. This rescaling induces a unitary
mapping of ${\cal H}^{(N,L)}$ onto the Hilbert space ${\cal
H}^{(N)}$ of antisymmetric square integrable functions on
$K^{N},$ under which ${\Psi}_{T}^{(N)}$ is transformed to the
state of ${\Sigma}^{(N)}$ given by
$${\psi}_{t}^{(N)}(x_{1},.. \ .,x_{N})=L^{3N/2}
{\Psi}_{{\omega}^{-1}t}^{(N)}(Lx_{1},.. \ .,Lx_{N})\leqno(3.8)$$
The Schr\"odinger equation (3.4) thus transforms to 
$$i{\hbar}_{N}{{\partial}{\psi}_{t}^{(N)}\over {\partial}t}=
H^{(N)}{\psi}_{t}^{(N)}\leqno(3.9)$$
where
$$H^{(N)}={1\over 2}{\sum}_{j=1}^{N}p_{j}^{2} \ + \
N^{-1}{\sum}_{j,k(>j)=1}^{N}U(x_{j}-x_{k})\leqno(3.10)$$
$$p_{j}=-i{\hbar}_{N}{\nabla}_{j}\leqno(3.11)$$
$${\hbar}_{N}={{\hbar}\over mL^{2}{\omega}}=
{{\hbar}\over m{\omega}}({{\overline n}\over
N})^{2/3}\leqno(3.12)$$
is a dimensionless effective 'Planck constant', ${\nabla}$ is the
gradient operator for $K,$ and 
$$U(x)={\sum}_{q}^{(1)}{\exp}(iq.x)/q^{2}\leqno(3.13)$$
the superscript $(1)$ over ${\Sigma}$ signifying that summation 
is taken over the non-zero vectors $2{\pi}(n_{1},n_{2},n_{3}),$ 
with the $n$'s integers. It follows from (2.13) that
$${\Delta}U(x)=1-{\delta}(x)\leqno(3.14)$$
where ${\Delta}$ is the Laplacian and ${\delta}$ the Dirac 
distribution for $K.$
\vskip 0.2cm
We formulate the dynamics of ${\Sigma}^{(N)}$  in
terms of its characteristic functions 
$${\mu}_{t}^{(N,n)}({\xi}_{1},.. \ .,{\xi}_{n};{\eta}_{1},.. \
.,{\eta}_{n}):=\leqno(3.15)$$
$$({\psi}_{t}^{(N)},{\Pi}_{j=1}^{n}
({\exp}(i{\xi}_{j}.p_{j}/2){\exp}(i{\eta}_{j}.x_{j})
{\exp}(i{\xi}_{j}.p_{j}/2)){\psi}_{t}^{(N)})$$
where the ${\xi}'$s and ${\eta}'$s run over the ranges ${\bf
R}^{3}$ and $(2{\pi}{\bf Z})^{3},$ respectively. 
\vskip 0.2cm
The initial condition for ${\Sigma}^{(N)},$  corresponding to
$(I.1)^{(L)}$ for ${\Sigma}^{(N,L)},$ is
\vskip 0.2cm
$(I.1)$
$$({\psi}_{0}^{(N)},p_{1}^{2}{\psi}_{0}^{(N)})
<N^{-2/3}b \ {\rightarrow}0 \ as \
N{\rightarrow}{\infty}\leqno(3.16)$$
{\it where $b$ is a finite constant.}
\vskip 0.2cm
The initial condition $(II.2)^{(L)}$ transforms to the following
form for ${\Sigma}^{(N)},$ which is unaffected by the above
rephasing of the initial state.
\vskip 0.2cm
$(I.2)$
$$({\psi}_{0}^{(N)},U(x_{1}-x_{2}){\psi}_{0}^{(N)})<c
\leqno(3.17)$$
{\it where $c$ is a finite constant. We further assume that}
\vskip 0.2cm
$(I.3)$
$${\lim}_{N\to\infty}{\rho}_{0}^{(N,1)}(x)={\sigma}_{0}(x)
\ {\forall}x{\in}K\leqno(3.18)$$ 
{\it where the function ${\sigma}_{0}$ is continuous; and that}
\vskip 0.2cm
$(I.4)$
$${\lim}_{N\to\infty}({\mu}_{0}^{(N,n)}({\xi}_{1},.. \
.,{\xi}_{n};{\eta}_{1},.. \
.,{\eta}_{n})-{\Pi}_{1}^{n}{\mu}_{0}^{(N,1)}
({\xi}_{j},{\eta}_{j})){\equiv}0\leqno(3.19)$$
This last condition signifies that, in the limit
$N{\rightarrow}{\infty},$ the initial correlations of
${\Sigma}^{(N)}$ are of zero range: the assumption behind
this is that the initial state of the microscopic model
${\Sigma}^{(N,L)},$ carries only short range correlations, as in
a pure phase [Ru, Se1].
\vskip 0.2cm
We note here that it was shown by explicit construction, in the 
Appendix of Ref. [Se7], that the initial conditions (I.1)-(I.4)
are perfectly viable. 
\vskip 0.2cm
The macroscopic dynamics of the Jellium model, then, is
represented by the time-dependence of the characteristic
functions ${\mu}_{t}^{(N,n)},$ in the limit
$N{\rightarrow}{\infty},$ subject to the initial conditions
(I.1-4). 
\vskip 0.2cm
{\bf The Vlasov Dynamics.} We note now that the dynamics of the
model ${\Sigma}^{(N)},$ as given by (3.9) and (3.10), has the
simplifying features that (a) the effective, dimensionless Planck
constant, ${\hbar}_{N},$ governing its quantum behaviour, tends
to zero as $N{\rightarrow}{\infty};$ and (b) the pair interaction
potential scales as $N^{-1}.$ The properties (a) and (b) are
generally the hallmarks of a classical and of a mean field
theory, respectively, in the limit $N{\rightarrow}{\infty}.$ 
In fact, were it not for the Coulomb singularity in $U,$ on e
could infer immediately from [NS, Sp] that the Schr\"odinger
equation (3.9), subject to the initial conditions (I.1-4), led
to a classical (mean field theoretic) Vlasov dynamics in the
limit $N{\rightarrow}{\infty}.$ Specifically, if the two-body
potential $U$ in (3.10) were a twice continuously differentiable
function, then the following results would prevail.
\vskip 0.2cm
(A) [NS, Sp] Pointwise convergence of ${\mu}_{t}^{(N,1)}$,
as $N{\rightarrow}{\infty},$ to the characteristic function,
${\mu}_{t},$ of a classical probability measure $m_{t}$ on
$(K{\times}{\bf R}^{3}),$ that satisfies the
weak Vlasov equation
$${d\over dt}{\int}f(x,v)dm_{t}(x,v)=
{\int}v.{{\partial}\over
{\partial}x}f(x,v)dm_{t}(x,v)\leqno(3.20)$$
$$ \ \ -{\int}{\nabla}U(x-y).
{{\partial}\over {\partial}v}f(x,v)
dm_{t}(x,v)dm_{t}(y,w) \
{\forall}f{\in}C_{0}^{2}(K{\times}{\bf R}^{3})$$
\vskip 0.2cm
(B) [NS, Sp] Pointwise convergence of ${\mu}_{t}^{(N,n)}$ to the
characteristic function of the classical probability measure
$m_{t}{\otimes}m_{t}.. \ .{\otimes}m_{t} \ (n \ factors)$ on
$(K{\times}{\bf R}^{3})^{n}.$
\vskip 0.2cm
(C) [Se7] The initial condition for $m,$ ensuing from (I.1) and
(I.3), that
$${\int}dm_{0}(x,v)f(x,v)={\int}dx{\sigma}_{0}(x)
f(x,0)\leqno(3.21)$$ 
\vskip 0.2cm
(D)[Ne, Se7] The unique determination of the solution of the
Vlasov equation (3.20), subject to the initial condition (3.21),
by that of the Newtonian mean field theoretic problem
$${dX_{t}(x)\over dt}=V_{t}(x); \ {dV_{t}(x)\over dt}=
-{\int}dy{\sigma}_{0}(y){\nabla}U^{\prime}(X_{t}(x)-X_{t}(y));
\leqno(3.22)$$
$$ \ with \ X_{0}(x)=x; \ V_{0}(x)=0$$
according to the formula
$${\int}dm_{t}(x,v)f(x,v)={\int}dx{\sigma}_{0}(x)
f(X_{t}(x),V_{t}(x))\leqno(3.23)$$
\vskip 0.2cm
In order to extend these results to the Jellium model,
${\Sigma}^{(N)},$ we require certain regularity conditions on the
dynamics of the system, that serve to tame the Coulomb
singularity. Accordingly, we invoke certain supplementary
assumptions (R), specified in [Se4], whose essential import is
that the Coulomb repulsion keeps the electrons apart,
sufficiently to ensure that the magnitude of the internal
electric field remains bounded. Under these assumptions, we have
the following result.
\vskip 0.3cm
{\bf Proposition 3.1.}[Se4] {\it The Schr\"odinger equation
(3.10) for ${\Sigma}^{(N)},$ together with the initial conditions
(I.1-4) and the regularity conditions R, leads to the Vlasov
dynamics specified by (A)-(D).}
\vskip 0.3cm
{\bf Eulerian Versus Stochastic Hydrodynamics.} The Newtonian
mean field theory, given by (3.22), corresponds to a {\it
Lagrangian} hydrodynamics, in which $X_{t}(x)$ and $V_{t}(x)$ are
the position and velocity, respectively, of a 'fluid particle';
and the Vlasov equation (3.20) is just the Liouville equation
representing its probabilistic description. Our aim now is to
show that the reducibility of the Vlasov dynamics to a
deterministic Eulerian hydrodynamics depends on the invertibility
of $X_{t}.$ 
\vskip 0.2cm
{\bf Case (a):} $X_{t}$ {\bf Invertible.} In this case, the
Jacobean 
$$J_{t}(x)={{\partial}(X_{t}^{(1)},X_{t}^{(2)},X_{t}^{(3)})\over 
 {\partial}(x^{(1)},x^{(2)},x^{(3)})}\leqno(3.24)$$ 
with $x^{(j)}$ (resp. $X_{t}^{(j)}$) the j'th component $x$
(resp. $X_{t}$), is strictly positive, and so we can define
$${\sigma}_{t}(x)={\sigma}_{0}(X_{t}^{-1}(x))/J_{t}(x);
 \ and \ u_{t}(x)=V_{t}(X_{t}^{-1}(x))\leqno(3.25)$$ 
Thus, since, by (3.23) and (3.25),
$${\int}dm_{t}(x,v)f(x,v)={\int}dx{\sigma}_{t}(x)
f(x,u_{t}(x)) \ {\forall}f{\in}C(K)\leqno(3.26)$$
i.e., formally,
$$dm_{t}(x,v)={\sigma}_{t}(x){\delta}(v-u_{t}(x))dxdv$$
it follows that $u_{t}(x)$ and ${\sigma}_{t}(x)$ are the drift
velocity and normalised particle density, respectively, at
position $x$ and time $t.$ Moreover, by (3.14), (3.22) and
(3.25), $u_{t},{\sigma}_{t}$ evolve according to the 
Euler-cum-Maxwell hydrodynamical equations
$${{\partial}{\sigma}_{t}\over {\partial}t}+{\nabla}.
({\sigma}_{t}u_{t})=0; \ {{\partial}u_{t}\over
{\partial}t}+(u_{t}.{\nabla})u_{t}=E_{t}; \
{\nabla}.E_{t}=({\sigma}_{t}-1)\leqno(3.27)$$
\vskip 0.3cm
{\bf Case (b): $X_{t}$ Non-Invertible.} In this case, we cannot
employ the formulae (3.25) to define the time-dependent
density and drift velocity. Instead, we have to consider the
situation where the equation
$$X_{t}(y)=x$$
has several solutions, labelled by an index set $S,$ for $y$ as
a function of $x$ and $t,$ i.e.
$$y={\lbrace}Y_{t}^{(s)}(x){\vert}s{\in}S{\rbrace}$$
In this case, equation (3.26) implies that
$${\int}dm_{t}(x,v)f(x,v)={\sum}_{s{\in}S}{\int}dx
{\sigma}_{t}^{(s)}(x)f(x,u_{t}^{(s)}(x))\leqno(3.28)$$
where
$${\sigma}_{t}^{(s)}(x)={\sigma}_{0}(Y_{t}^{(s)}(x)){\vert}
K_{t}^{(s)}(x){\vert}; \ \ u_{t}^{(s)}(x)=
V_{t}(Y_{t}^{(s)}(x))\leqno(3.29)$$
and
$$K_{t}^{(s)}(x)={{\partial}(Y_{t}^{(s,1)},Y_{t}^{(s,2)}),
Y_{t}^{(s,3)})\over
{\partial}(x^{(1)},x^{(2)},x^{(3)})}\leqno(3.30)$$
Thus, (3.28) signifies that the macroscopic state of the system
at time t corresponds to a statistical mixture of different
streams, indexed by $S,$ with dnsities ${\sigma}_{t}^{(s)}$ and
drift velocities $u_{t}^{(s)}.$ This implies that the local
density and drift velocity of the system are now {\it
stochastic} variables of a hydrodynamics still governed by the
Vlasov equation.
\vskip 0.2cm
We may summarise the above observations in the following form.
\vskip 0.3cm
{\bf Proposition 3.2.} {\it If $X_{t}$ is invertible, then the
macroscopic dynamics of the system corresponds to a deterministic
hydrodynamics given by the Euler-cum-Maxwell equations (3.27).
Otherwise, it is a stochastic flow, still governed by the Vlasov
equation.}
\vskip 0.3cm
{\bf Example of Hydrodynamic Phase Transition.} Let us now
specialise to the situation where ${\sigma}_{0}$ is a function
of just a single coordinate, $x^{(1)}.$ In this case, as the
regularity conditions (R) imply uniqueness of the solution of
(3.22) [Se4], the flow becomes effectively one-dimensional, with
$X_{1,t}$ dependent on the coordinate $x^{(1)}$ only, and
$X_{t}^{(2)},X_{t}^{(3)}$ remaining fixed at $x^{(2)},x^{(3)},$
respectively. For notational convenience, we shall henceforth
drop the superscript $(1)$ from $X_{t}^{(1)}$ and $x^{(1)},$ and
denote by $X_{t}(x)$ the first component of the vector hitherto
represented by this symbol. Thus, by equations (3.22),
$$X_{t}(x)=x+{\int}_{0}^{t}ds(t-s){\int}_{0}^{1}dy
{\sigma}_{0}(y)F(X_{s}(x)-X_{s}(y))\leqno(3.31)$$
where
$$F(x)=-{\int}_{0}^{1}dx^{(2)}{\int}_{0}^{1}dx^{(3)}
{{\partial}U\over {\partial}x}(x,x^{(2)},x^{(3)})\leqno(3.32)$$
Let
$$J_{t}(x)={{\partial}X_{t}(x)\over {\partial}x}\leqno(3.33)$$
Then, by Prop. 3.2 and the implicit function theorem, the
neccessary and sufficient condition for deterministic
hydrodynamics is that $J_{t}$ has no zeroes. We note also that
the definition (3.33) permits us to re-express (3.31) in the form
$$X_{t}(x)=x+{\int}_{0}^{t}ds(t-s){\int}_{0}^{1}dy
{\sigma}_{0}(y)F({\int}_{y}^{x}dzJ_{s}(z)))
\leqno(3.31)^{\prime}$$
\vskip 0.3cm
{\bf Proposition 3.3.} {\it Under the specified assumptions, the
hydrodynamics of the model is deterministic and given by the
Euler-cum-Maxwell equations (3.27), provided that}
$${\sigma}_{0}(x)>{1\over 2}{\forall}x{\in}[0,1]\leqno(3.34)$$
{\it Otherwise there is a transition to a stochastic flow at a
certain time ${\tau},$ given by the least positive value of $t$
for which} 
$${\sigma}_{0}(x)+(1-{\sigma}_{0}(x)){\cos}(t)=0\leqno(3.35)$$
{\it for some $x{\in}[0,1].$}
\vskip 0.3cm
{\bf Proof.} By (3.13), (3.14) and (3.31)-(3.33),
$$J_{t}(x)=1+{\int}_{0}^{t}ds(t-s)
J_{s}(x)(-1+{\int}_{0}^{1}dy{\sigma}_{0}(y)
{\delta}(X_{s}(x)-X_{s}(y)))\leqno(3.36)$$
where now ${\delta}$ is the Dirac distribution on $[0,1],$
subject to periodic boundary conditions. 
\vskip 0.2cm
Let us first suppose that $X_{t}$ is invertible, i.e. that
$J_{t}$ is strictly positive, over a time interval
$0{\le}t<{\tau}_{0},$ for some positive ${\tau}_{0}.$ In this
case, 
$$J_{s}(x){\delta}(X_{s}(x)-X_{s}(y)){\equiv}{\delta}(x-y) \
{\forall}s{\in}[0,{\tau}_{0})$$
and therefore (3.36) reduces to
$$J_{t}(x)=1+{\int}_{0}^{t}ds(t-s)
({\sigma}_{0}(x)-J_{s}(x))$$
i.e.
$$({d^{2}\over dt^{2}}+1)J_{t}(x)={\sigma}_{0}(x);
 \ with \ J_{0}(x)=1; \ {\dot J}_{0}(x)=0$$
where ${\dot J}_{t}=dJ_{t}/dt.$ Hence,
$$J_{t}(x)={\sigma}_{0}(x)+(1-{\sigma}_{0}(x)){\cos}(t)$$
In view of the non-negativity of ${\sigma}_{0},$ this equation
implies that $J_{t}$ is strictly positive for all $t{\ge}0$
if and only if the condition (3.34) is fulfilled. Otherwise,
$J_{t}$ changes sign at some point $x{\in}[0,1]$ when $t$ reaches
the value ${\tau}$ specified in the statement of the Proposition.
Hence, our assumption of the invertibility of $X_{t}$ is
untenable if (3.34) is violated; and therefore, by Prop. 3.2, the
flow becomes stochastic in this case.
\vskip 0.2cm
The proof (cf. [Se4]) of the converse, i.e. that (3.34) implies
the invertibility of $X_{t}$ and thus the deterministic Euler-
cum-Maxwell flow, stems from the explicit form of our regularity
condition (R).
\vskip 0.5cm
{\bf 4. Macrostatistics and Non-Equilibrium Thermodynamics.}
Here, I present an approach to the general problem of formulating
the structures imposed on non-equilibrium thermodynamics by the
underlying quantum mechanics of many-particle systems.
This is based on a combination of macroscopic and
microscopic treatments of a generic system, ${\Sigma},$ of
particles occupying a Euclidean space $X={\bf R}^{d}.$ The
argument consists of four parts. The first (${\S}4.1$) is a
formulation of a mathematical framework for non-equilibrium
thermodynamics, based on a macroscopic, continuum mechanical
model, $M,$ of ${\Sigma}.$ The second (${\S}4.2$) is an algebraic
quantum statistical formulation of a microscopic model, ${\cal
Q}$ of the same system. The third (${\S}4.3$) consists of a
treatment of the connection between $M$ and ${\cal Q},$ leading
to a {\it classical macrostatistical mechanics} $(CMSM)$ of the
observables of ${\cal Q}$ that correspond to the hydrodynamical
variables of $M.$ This admits a clear formulation of {\it local
equilibrium} conditions and of a generalised version of Onsager's
regression hypothesis [On], namely that the macroscopic fluctuations
about the flow given by $M$ is governed by the same dynamics as
the externally induced 'weak' perturbations to that flow. In
${\S}4.4,$ I derive a generalised form of the Onsager reciprocity
relations for $M$ from the structure of $CMSM.$ The key elements
of this derivation are the symmetry properties of the
model, stemming from the microscopic reversibility of ${\cal Q},$
and the assumption of local equilibrium.
\vskip 0.3cm
{\bf 4.1. The Macroscopic Model, $M$.}  In the
classical thermodynamical description, the {\it equilibrium
states} of ${\Sigma}$ are represented by a certain set
$q=(q_{1},.. \ .,q_{n})$ of intensive variables, corresponding
to global densities of extensive conserved quantities, which we
shall characterise in ${\S}4.2.$ Its entropy density, $s,$ is
then a function of these variables, and its equilibrium
properties of governed by the form of $s.$ The thermodynamic
variables ${\theta}=({\theta}_{1},.. \ .,{\theta}_{n})$ conjugate
to $q$ are defined by the formula
$${\theta}={{\partial}s(q)\over {\partial}q}, \ i.e. \ 
{\theta}_{k}={{\partial}s(q)\over {\partial}q_{k}}, \ 
k=1,2.. \ ,n\leqno(4.1)$$
and the pressure, $p,$ is the function of ${\theta}$ given by
$$p({\theta})=sup_{q}(s(q)-q.{\theta})\leqno(4.2)$$
the dot representing the ${\bf R}^{n}$ inner product. Thus, the
inverse of the formula (4.1) (in the pure phase region) is
$$q=-{{\partial p}({\theta})\over
{\partial}{\theta}}\leqno(4.3)$$
\vskip 0.2cm
In non-equilibrium thermodynamics, the state of the system is
generally not translationally invariant, and the macroscopic
variables $q$ become functions of position, $x,$ and of time,
$t.$ In other words, the macrostate is an n-component {\it
classical field} 
$$q(x,t){\equiv}q_{t}(x){\equiv}
(q_{1,t}(x),.. \ .,q_{n,t}(x)).$$ 
The laws of continuum mechanics, such as those of
hydrodynamics or heat conduction, are then of the {\it
deterministic} form
$${\dot q}_{t}={\cal F}(q_{t})\leqno(4.4)$$ 
where ${\cal F}$ is some functional of the fields $q_{t}.$
Equivalently, defining the local thermodynamic conjugates
${\theta}_{t}$ of the fields $q_{t}$ by
$${\theta}_{t}(x){\equiv}{\theta}(x,t):={{\partial}s(q(x,t))\over 
{\partial q}},\leqno(4.5)$$ 
it follows from (4.3) that the equation of motion (4.4) may be
re-expressed as
$${\dot q}_{t}{\equiv}{d{\dot q}_{t}\over dt}=
{\Phi}({\theta}_{t}); \ {\Phi}:={\cal F}{\circ}
(-{{\partial}p\over {\partial}{\theta}})\leqno(4.6)$$
\vskip 0.2cm
{\bf Note.} The definition (4.5) of ${\theta}_{t}$ does not
require the microstate of ${\Sigma}$ to simulate one of
equilibrium at a local level. On the other hand, some assumption
of local equilibrium will be needed for the determination of
properties of the functional ${\cal F}.$
\vskip 0.2cm
{\bf Example.} An example of a law of the above form is the non-
linear, n-component diffusion given by
$${\dot q}_{k,t}={\sum}_{l=1}^{n}{\nabla}.
(L_{k,l}({\theta}_{t}){\nabla}{\theta}_{l,t})
\leqno(4.6a)$$
\vskip 0.2cm
Returning to the general structure, we impose the following
assumptions on the dynamical law (4.6). 
\vskip 0.2cm
(M.1) {\it The system is confined to a single phase region, i.e.
the values of $q_{t}(x), \ {\theta}_{t}(x)$ always lie in domains
where the functions $s$ and $p$ are $C^{({\infty})}.$}
\vskip 0.3cm
(M.2) {\it ${\cal F}(q)=0$ if $q(x)$ is constant, i.e. the
system is in equilibrium if its local thermodynamic variables
$q(x)$ (or equivalently ${\theta}(x)$) are spatially uniform.}
\vskip 0.3cm
(M.3) {\it The equation of motion (4.6) is covariant w.r.t.
space-time translations and scale transformations
$x{\rightarrow}{\lambda}x,t{\rightarrow}{\lambda}^{r}t,$ with r
a positive integer (=2 in the case of
(4.6a)).}\footnote *{Note that this assumption is not
always fulfilled: for example, the Navier-Stokes equation is not
scale invariant.}  
\vskip 0.3cm
Let $y_{t}, \ {\eta}_{t}$ be 'small' perturbations of $q_{t}, \
{\theta}_{t},$ respectively, and let 
\vskip 0.3cm
$$[{\Lambda}({\theta}){\eta}_{t}](x):=
{d\over dz}{\Phi}({\theta}_{t}(x)+z{\eta}_{t}(x))
{\vert}_{z=0}\leqno(4.7)$$
in the pointwise sense, for the class of functions ${\eta}_{t}$
for which the r.h.s. of this formula is well-defined. Then
it follows from (4.6) and (4.7) that the {\it linearised}
equation of motion for $y_{t}$ is
$${\dot y}_{t}={\cal L}({\theta}_{t})y_{t}\leqno(4.8)$$
where
$${\cal L}({\theta}_{t})={\Lambda}({\theta}_{t})
B({\theta}_{t})\leqno(4.9)$$
and $B({\theta})$ is the Hessian $s^{{\prime}{\prime}}(q),$ i.e.
$$(B({\theta}))y)_{k}:={\sum}_{l=1}^{n}
{{\partial}^{2}s(q)\over {\partial}q_{k}{\partial}q_{l}}
y_{l}\leqno(4.10)$$
The next assumption permits us to formulate the perturbed
dynamics as the evolution of a tempered distribution. Thus,
defining ${\cal S}^{(n)}(X)$ to be the space of the Schwartz
${\cal S}-$class ${\bf R}^{n}-$ valued functions on $X,$ and
${\cal S}^{(n){\prime}}(X)$ to be its dual space, we assume that
\vskip 0.3cm
(M.4) {\it the linear operator ${\cal L}({\theta}_{t})$ extends,
by continuity, to a transformation of ${\cal S}^{(n){\prime}}(X),$
and the formula (4.8), considered now as an equation of motion
in that space, has a unique solution}
$$y_{t}=T({\theta}_{.}{\vert}t,t^{\prime})y_{t^{\prime}} \ 
{\forall}t{\ge}t^{\prime}{\ge}0\leqno(4.11)$$
with
$$T({\theta}_{.}{\vert}t,t_{0}){\equiv}
T({\theta}_{.}{\vert}t,t_{1})T({\theta}_{.}{\vert}t_{1},t_{0})
 \ and \ T({\theta}_{.}{\vert}t,t)=1 \
{\forall}t{\ge}t_{1}{\ge}t_{0}{\ge}0$$ 
\vskip 0.3cm
{\bf Comments.} In the equilibrium case, where
${\theta}_{t}={\overline {\theta}},$ a constant, the
two-parameter
family $T({\theta}{\vert},.,)$ of transformations of ${\cal
S}^{(n){\prime}(X)}$ reduces to a one-parameter semi-group
${\lbrace}{\tilde T}({\overline {\theta}}{\vert}t){\vert}t{\in}
{\bf R}_{+}{\rbrace},$ where
$${\tilde T}({\overline {\theta}}{\vert}(t-t_{0}){\equiv}
T({\overline {\theta}}{\vert}(t,t_{0})=
{\exp}({\cal L}({\overline {\theta}})(t-t_{0}))\leqno(4.12)$$
Further, in the case of the non-linear diffusion (4.6a), ${\cal
L}({\theta}^{(0)})$ reduces to the form
$${\cal L}({\overline {\theta}})=L({\overline {\theta}})
B({\overline {\theta}}){\Delta}; \ L({\overline {\theta}})
=[L_{kl}({\overline {\theta}})];\ B=[B_{kl}({\overline
{\theta}})]
\leqno(4.12a)$$
where ${\Delta}$ is the Laplacian.
\vskip 0.3cm 
The following Proposition follows immediately from
(M.1) and (M.4). 
\vskip 0.2cm
{\bf Proposition 4.1.} {\it The process $y.$ is covariant w.r.t. 
space translations and space-time scale transformations, i.e.
defining}
$${\phi}_{t}^{(a,b,{\lambda})}(x){\equiv}
{\phi}^{(a,b,{\lambda})}(x,t):=
{\phi}(a+{\lambda}x,b+{\lambda}^{r}t), \ for 
\ {\phi}={\theta},q,y\leqno(4.13)$$
and for arbitrary $(a,b,{\lambda}){\in}X{\times}{\bf R}
{\times}{\bf R}_{+}),$ 
$$(T({\theta}_{.}{\vert}t,t^{\prime})
y_{{t}^{\prime}})^{(a,b,{\lambda})}{\equiv}
T({\theta}_{.}^{(a,b,{\lambda})}{\vert}t,t^{\prime})
y_{{t}^{\prime}}^{(a,b,{\lambda})}\leqno(4.14)$$
\vskip 0.3cm 
{\bf 4.2. The Quantum Model,} ${\cal Q}.$ In order to
accommodate the dynamics of both the microscopic and macroscopic
observables of ${\Sigma},$ we construct its quantum model,
${\cal Q},$ as a $W^{\star}-$dynamical system [Se6] $({\cal
A},{\alpha},{\cal N}({\cal A})),$ where ${\cal A}$ is a
$W^{\star}-$algebra of observables, $t{\rightarrow}{\alpha}_{t}$
is a representation of ${\bf R}$ in $Aut({\cal A}),$
corresponding to the dynamics of the system, and ${\cal N}({\cal
A}))$ is the folium of normal states on ${\cal A}.$  In fact,
this algebra is constructed as the weak closure of the largest
locally normal representation, ${\pi},$ of the standard [HHW]
quasi-local $C^{\star}-$algebra, ${\tilde {\cal A}},$ that can
support the dynamical and thermodynamical structures we require.
Thus, we characterise ${\cal A},{\alpha}$ and ${\pi}$ according
to the following prescription.
\vskip 0.2cm
(1) Let $L$ be the set of bounded open regions of X. Then for
each ${\Lambda}{\in}L,$ there is a type-I factor, ${\cal
A}({\Lambda})({\subset}{\cal A}),$ representing the observables
of that region and satisfying the conditions of isotony and local
commutativity; and ${\cal A}$ is the weak closure of ${\cal
A}_{L}:=U_{{\Lambda}{\in}L}{\cal A}({\Lambda}).$ 
We extend automorphisms ${\gamma}$ of ${\cal A}$ to unbounded
observables, $Q,$ affiliated\footnote *{Recall that, if
${\cal M}$ is a $W^{\star}-$algebra of operators in a Hilbert
space ${\cal H},$ then the unbounded operators, in ${\cal H},$
affiliated to ${\cal M}$ are the densely defined ones that
commute with ${\cal M}^{\prime}.$} to this algebra, according to
the formula
$${\exp}(i{\lambda}{\gamma}(Q))={\gamma}({\exp}(i{\lambda}Q))
 \ {\forall}{\lambda}{\in}{\bf R}\leqno(4.15)$$
We similarly extend anti-automorphisms of ${\cal A}$ to its
unbounded affiliates.
\vskip 0.2cm
(2) The local energy observable (Hamiltonian) $H({\Lambda})$ for
the region ${\Lambda}({\in}L)$ is, in general, an unbounded one,
affiliated to ${\cal A}({\Lambda}).$ We assume that ${\cal A}$
is equipped with a Wigner time-reversal anti-automorphism,
${\rho},$ which leaves this observable invariant, i.e.
$${\rho}H({\Lambda})=H({\Lambda})\leqno(4.16)$$ 
\vskip 0.2cm
(3) Space translations are represented by a homomorphism
${\sigma}$ of $X$ into $Aut({\cal A}),$ such that
${\sigma}(x){\cal
A}_{0}({\Lambda}){\equiv}{\cal A}_{0}({\Lambda}+x)$ and
${\sigma}(x)H({\Lambda})=H({\Lambda}+x)),$ this
last condition representing an assumption of translationally
invariant interactions. 
\vskip 0.2cm
(4) The dynamical automorphisms ${\alpha}$ are given by infinite
volume limits of those of finite versions of ${\Sigma},$
according to the formula
$${\alpha}_{t}A=s-{\lim}_{{\Lambda}{\uparrow}}
({\exp}(iH({\Lambda})t/{\hbar})A{\exp}(-iH({\Lambda})t/{\hbar}))
\ {\forall}t{\in}{\bf R}, \ A{\in}
{\cal A}_{L}\leqno(4.17)$$ 
Thus, by the translational covariance of $H({\Lambda}), \
{\alpha}_{t}$ commutes with the space translational
automorphisms, ${\sigma},$ and, by (4.16), satisfies the {\it
microscopic reversibiliy} condition
$${\rho}{\alpha}_{t}{\rho}={\alpha}_{-t} \ {\forall}t
{\in}{\bf R}\leqno(4.18)$$
\vskip 0.2cm
(5) We formulate the thermodynamics of the system in terms a
Hermitian quantum field ${\hat q}=({\hat q}_{1},.. \ .,{\hat
q}_{n}),$ the ${\hat q}_{k}'s$ being densities of
locally conserved quantities. We assume that ${\hat q}$ is a
tempered distribution, affiliated to ${\cal A},$ that transforms
covariantly w.r.t space translations. Thus, ${\hat q}$ is a
mapping of ${\cal S}^{(n)}(X)$ into the
self-adjoint affiliates of ${\cal A}.$ We assume that the
function ${\exp}(i{\hat q}(.))$ is strongly continuous, and
that ${\hat q}$ transforms covariantly w.r.t. space translations,
i.e.
$${\sigma}(x)[{\hat q}(f)]={\hat q}(f_{x}), \ where \
f_{x}(y)=f(x-y) \ {\forall}x,y{\in}X,f{\in}{\cal S}^{(n)}(X)
\leqno(4.19)$$
We assume, for simplicity, that ${\hat q}$ is invariant under
time reversals, i.e. 
$${\rho}{\hat q}(f){\equiv}{\hat q}(f)\leqno(4.20)$$
and that its components satisfy the following commutation
relations, which signify that their space integrals over finite
volumes intercommute, up to 'surface effects'.
$${\lbrack}{\hat q}_{k}(g),q_{l}(h){\rbrack}_{-}=
i{\hbar}j_{k,l}({\nabla}(gh)) \ {\forall}g,h{\in}
{\cal S}(X),\leqno(4.21)$$
where $(gh)(x):=g(x)h(x)$ and $j_{k,l}$ a tempered distribution.
Further, denoting ${\alpha}_{t}[{\hat q}(f)]$ by ${\hat
q}_{t}(f),$ we assume local
conservation law of the form
$${{\partial}{\hat q}_{t}(f)\over
{\partial}t}=j_{t}({\nabla}f)\leqno(4.22)$$
where $j_{t}$ is a tempered distribution.
\vskip 0.2cm
(6) We take the equilibrium thermodynamic variables of ${\cal Q}$
to be the 'observables at infinity' [LR], given by the global
spatial average ${\hat q}^{\infty}$ of ${\hat q}$ over $X.$
Further, denoting by ${\hat s}$ the standard [Ru] entropy density
functional on the translationally invariant states on ${\cal
A}_{0},$ we assume that these observables form a complete
thermodynamic set, in the sense that [Se1, Ch.4] 
\vskip 0.2cm
(a) for each expectation value, $q,$ of ${\tilde q},$ there is
precisely one translationally invariant state $({\in}{\cal
N}({\cal A}))$ that maximises ${\hat s};$ and 
\vskip 0.2cm
(b) no proper subset of ${\hat q}$ possesses this
property. 
\vskip 0.2cm
The equilibrium thermodynamics of the system is thus given by the
form of the resultant entropy density, $s(q).$ We identify $q,s$
with the objects denoted by these symbols in the macroscopic
model $M;$ and, defining ${\theta}$ according to (4.1), we denote
by ${\omega}_{\theta}$ the maximising state of condition (a). We
assume that this state is stationary,\footnote *{The proof of
this is straightforward for lattice systems, since one can show
within the framework of [Se1,Ch.4], that, for these,
${\omega}_{\theta}$ satisfies the KMS conditions, and is
therefore stationary.} i.e. ${\alpha}-$invariant, in view of the
fact that ${\hat q}_{\infty}$ is a globally conserved quantity;
and we designate it as the equilibrium state corresponding to
${\theta},$ i.e. to $q.$ We note that, by (4.18) and the 
thermodynamic completeness condition (a),
$${\omega}_{\theta}{\circ}{\rho}={\omega}_{\theta}\leqno(4.23)$$ 
\vskip 0.2cm
(7) We define ${\pi}$ to be the largest locally normal
representation of ${\tilde {\cal A}}$ that supports the dynamical
group ${\alpha}$ and the quantum field ${\hat q},$ as defined
above.
\vskip 0.3cm
Thus, the quantum model, ${\cal Q},$ is given
by $({\cal A},{\alpha},{\sigma},{\hat q},{\cal N}({\cal A})),$
as specified by the conditions (1)-(7).
\vskip 0.3cm
{\bf 4.3. Relationship between $M$ and ${\cal Q}.$} Our
formulation of this relationship is based on the idea that the
phenomenological law (4.6) corresponds to the dynamics of the
quantum field ${\hat q}$ in a large-scale limit. Thus,
in view of the scale-invariance assumption (M.1) for
$M,$ we introduce a length parameter $L$ and reformulate ${\cal
Q}$ on a length scale $L$ and a time scale $L^{r},$ defining the
quantum field ${\hat q}^{(L)}$ on these scales by the formula
$${\hat q}_{t}^{(L)}(f){\equiv}{\hat q}^{(L)}(f,t):=
{\hat q}(f^{(L)},L^{r}t)\leqno(4.24)$$
where
$$f^{(L)}(x):=L^{-d}f(x/L)\leqno(4.25)$$
We assume that $L$ is also the length scale of spatial variations
of the initial state, ${\omega}^{(L)},$ of ${\cal Q},$ i.e.
that there is a map $A{\rightarrow}{\overline A}$ of ${\cal A}$
into $C(X)$ and a tempered distribution $q_{0}({\in}{\cal
S}^{(n){\prime}}),$ such that
$${\lim}_{L\to\infty}{\omega}^{(L)}({\sigma}(Lx)[A])=
{\overline A}(x) \ {\forall}x{\in}X\leqno(4.26)$$
and
$${\lim}_{L\to\infty}{\omega}^{(L)}
({\hat q}_{0}^{(L)}(f))=q_{0}(f) \ {\forall}
f{\in}{\cal S}^{(n)}\leqno(4.27)$$
We define the fluctuation field ${\hat {\xi}}^{(L)}$ by the
formula
$${\hat {\xi}}_{t}^{(L)}(f):=L^{d/2}({\hat q}_{t}^{(L)}(f)-
{\omega}^{(L)}({\hat q}_{t}^{(L)}(f))), \ {\forall}f{\in}
{\cal S}^{(n)}(X), \ t{\in}{\bf R}\leqno(4.28)$$
Our basic assumptions for the large-scale dynamics of the model
are the following.
\vskip 0.3cm
(I) {\it For each $M-$ process $q_{.},$ there is an equivalence
class of initial states, ${\lbrace}{\omega}^{(L)}{\rbrace},$
parametrised by $L,$ such that
\vskip 0.2cm
(a) the mapping 
$$f^{(1)},.. \ .,f^{(m)};t^{(1)},. \ .,t^{(m)} \
{\rightarrow}{\omega}^{(L)}({\hat{\xi}}_{t^{(1)}}^{(L)}(f^{(1)})
.. \ .({\hat {\xi}}_{t^{(m)}}^{(L)}(f^{(m)}))$$ 
of $({\cal S}^{(n)})^{m}{\times}{\bf R}^{m})$ into ${\bf C}$ is
continuous, for all positive integers $m.$
\vskip 0.2cm
(b) The expectation value of the quantum field ${\hat
q}_{t}^{(L)}$ reduces to that of the classical one, $q_{t},$
of $M$ in the limit $L{\rightarrow}{\infty},$ i.e.}
$${\lim}_{L\to\infty}{\omega}^{(L)}({\hat q}_{t}^{(L)}(f)))=
q_{t}(f) \ {\forall}f{\in}{\cal S}^{(n)}(X)\leqno(4.29)$$
{\it where $q_{t}$ is the solution of (4.4), with initial value
given by (4.27).}
\vskip 0.2cm
{\it (c) The quantum stochastic process ${\hat {\xi}}^{(L)}$
converges to a classical one, ${\xi},$ as
$L{\rightarrow}{\infty},$ i.e.}
$${\lim}_{L\to\infty}{\omega}^{(L)}
({\hat {\xi}}_{t^{(1)}}^{(L)}(f^{(1)})
.. \ .({\hat {\xi}}_{t^{(m)}}^{(L)}(f^{(m)}))=\leqno(4.30)$$
$${\bf E}[{\theta}_{.}{\vert}({\xi}_{t^{(1)}}
(f^{(1)}).. \ .{\xi}_{t^{(m)}}(f^{(m)})]$$
$$ \ {\forall}t^{(1)},. \
.,t^{(m)}{\in}{\bf R}_{+}, \ f^{(1)},.. \ .,f^{(m)}{\in}
{\cal S}^{(n)}(X), \ r{\in}{\bf N}$$
{\it where the expectation functional ${\bf E}[{\theta}_{.}{\vert}.]$
is governed by the restriction of ${\theta}_{.}$ to the close
interval between the minimum and maximum of} ${\lbrace}t^{(1)},.
\ .,t^{(m)}{\rbrace}.$
\vskip 0.3cm
{\bf Comments.} (1) The classicality of the limits of $q^{(L)}$
and ${\xi}^{(L)}$ here are assumed to arise from the commutation
rules (4.21), together with asymptotic abelian properties of
${\cal Q}$ with respect to time.
\vskip 0.2cm
(2) Since (c) implies that the dispersion in ${\hat
q}_{t}^{(L)},$ for the state ${\omega}^{(L)},$ tends to zero as
$L{\rightarrow}{\infty},$ it follows from (b) that the quantum
process ${\hat q}_{t}^{(L)}$ reduces to the classical one,
$q_{t},$ in this limit.
\vskip 0.2cm
(3) It follows from (b) and (c) that ${\xi}_{t}$ is an ${\cal
S}^{(n){\prime}}-$valued random variable.
\vskip 0.3cm
Let 
$${\omega}_{\tau}^{(L)}:={\omega}^{(L)}{\circ}
{\alpha}(L^{r}{\tau}) \
{\forall}{\tau}{\in}{\bf R}_{+}\leqno(4.31)$$
Then ${\lbrace}{\omega}_{t}^{(L)}{\rbrace}$ satisfies the
conditions of (I), and the replacement of ${\omega}^{(L)}$ by
${\omega}_{\tau}^{(L)}$ corresponds to that of ${\theta}_{.}$ by
${\theta}^{({\tau})}$ in (4.30), where 
$${\theta}_{t}^{({\tau})}={\theta}_{t+{\tau}}, \
{\forall}t,{\tau}{\in}{\bf R}_{+}\leqno(4.32)$$
i.e.
$${\bf E}[{\theta}_{.}{\vert}({\xi}_{t^{(1)}+{\tau}}
(f^{(1)}).. \
.{\xi}_{t^{(m)}+{\tau}}(f^{(m)})]{\equiv}\leqno(4.33)$$
$${\bf E}[{\theta}_{.}^{({\tau})}{\vert}({\xi}_{t^{(1)}}
(f^{(1)}).. \ .{\xi}_{t^{(m)}}(f^{(m)})]$$
In the particular case where $t^{(1)}=.. \ =t^{(m)}=0,$ the
r.h.s. of this equation depends on ${\theta}_{.}^{({\tau})}$ only
through the value of
${\theta}_{0}^{({\tau})}{\equiv}{\theta}_{\tau},$ by (4.32).
Thus, by (4.30), the equal time correlation functions for the
process ${\xi}_{.}$ are of the form
$${\bf E}[{\theta}_{.}{\vert}{\xi}_{\tau}(f^{(1)})..
 \ .{\xi}_{\tau}(f^{(m)})]{\equiv}
{\bf E}[{\theta}_{\tau}{\vert}{\xi}_{0}(f^{(1)})..
 \ .{\xi}_{0}(f^{(m)}]\leqno(4.34)$$
\vskip 0.2cm
Our next assumption is that the space-time clustering properties
of ${\cal Q}$ render the process ${\xi}_{.}$ Gaussian (cf.
[GVV]), and that the infinite separation of the relevant
relaxation time-scales of the models $M$ and ${\cal Q}$ ensure
that it is Markovian.
\vskip 0.3cm
(II) {\it The process ${\xi}_{.}$ is Gaussian and temporally
Markovian.}
\vskip 0.3cm
It follows immediately from this assumption that the process
${\xi}$ is completely determined by its two-point function. Our
next assumption is the following generalisation of Onsager's
regression hypothesis [On]. 
\vskip 0.3cm
(III) {\it The fluctuation process ${\xi}$ is governed by
precisely the same dynamics as the perturbation, $y_{.},$ to the
deterministic process $q_{.},$ i.e., by (4.11),}
$${\bf E}[{\theta}_{.}{\vert}({\xi}_{t+{\tau}}(f){\xi}_{t}(g)]=
{\bf E}[{\theta}_{.}{\vert}({\xi}_{t}
(T({\theta}_{.}{\vert}t+{\tau},t)^{\star}f){\xi}_{t}(g)]$$
{\it Hence, by (4.34)}
$${\bf E}[{\theta}_{.}{\vert}({\xi}_{t+{\tau}}(f){\xi}_{t}(g)]=
{\bf E}[{\theta}_{t}{\vert}({\xi}_{0}
(T({\theta}_{t}{\vert}t+{\tau},t)^{\star}f){\xi}_{0}(g)]
\leqno(4.35)$$
$$ \ {\forall}f{\in}{\cal S}^{(n)}(X),t{\in}
{\bf R}, \ {\tau}{\in}{\bf R}_{+}$$
\vskip 0.3cm
Thus, the process is determined by the form of $T$ and of the
expectation functional ${\bf E}[{\theta}_{t}{\vert}.]$ on the
algebra generated by ${\xi}_{0}.$ In order to formulate the
action of space translations and scale transformations on the
process, we define
$$f^{(a,{\lambda})}(x):={\lambda}^{-d/2}f({\lambda}^{-1}(x-a))
 \ {\forall}a{\in}X,{\lambda}{\in}{\bf R}_{+}\leqno(4.36)$$
and
$${\xi}_{0}^{(a,{\lambda})}(f)
:={\xi}_{0}(f^{(a,{\lambda})}) \ {\forall}a{\in}X,
{\lambda}{\in}{\bf R}_{+}\leqno(4.37)$$
\vskip 0.3cm
{\bf Proposition 4.2} {\it Under the above assumptions and
definitions,}
$${\bf E}[{\theta}_{b}{\vert}({\xi}_{0}^{(a,{\lambda})}
(T({\theta}_{.}{\vert}b+{\tau},b)^{\star}f)
{\xi}_{0}^{(a,{\lambda})}(g)]=\leqno(4.38)$$
$${\bf E}[{\theta}_{0}^{(a,b,{\lambda})}
{\vert}{\xi}_{0}(T({\theta}_{.}^{(a,b,{\lambda})}
{\vert}{\tau},0)^{\star}f){\xi}_{0}(g)] \ {\forall}a{\in}X, 
 \ b,{\tau}{\in}{\bf R}_{+}, \ {\lambda}{\in}
{\bf R}_{+}$$
{\it with ${\theta}_{.}^{(a,b,{\lambda})}$ as defined by (4.13).}
\vskip 0.3cm
{\bf Proof.} The result is obtained by replacing 
each $f$ by $f^{a{\lambda}},$ in (4.30), and using
equns. (4.13), (4.14), (4.19), (4.27), (4.28), and (4.35)-(4.37).
\vskip 0.3cm
We note now that the local properties of the process ${\xi},$ in
the neighbourhood of a space-time point $(a,b),$ is given by the
form of the l.h.s. of (4.38), in the limit
${\lambda}{\rightarrow}0.$ Moreover, by (4.13), the function
${\theta}_{.}^{(a,b,{\lambda})},$ which occurs there, tends
pointwise to a constant, ${\theta}(a,b),$ in this limit. These
observations leads us to the following {\it local equilibrium}
assumption.
\vskip 0.3cm
(V) 
$${\lim}_{{\lambda}{\rightarrow}0}
{\bf E}[{\theta}_{0}^{(a,b,{\lambda})}
{\vert}{\xi}_{0}(T({\theta}_{.}^{(a,b,{\lambda})}
{\vert}({\tau},0)^{\star}f){\xi}_{0}(g)]=\leqno(4.39)$$
$${\bf E}[{\theta}(a,b){\vert}({\xi}_{0}
(T({\theta}(a,b){\vert}{\tau},0)^{\star}f){\xi}_{0}(g)]
 \ {\forall}f,g{\in}{\cal S}^{(n)}(X), \ {\tau}{\ge}0$$
{\it and further, the r.h.s. of this formula is precisely the
same as for the fluctuations of the field ${\hat q}_{.}$ about
an equilibrium state ${\omega}_{{\theta}(a,b)},$ as defined in
item (6) of ${\S}4.2,$ i.e.} 
$${\bf E}[{\theta}(a,b){\vert}{\xi}_{0}
(T({\theta}(a,b){\vert}{\tau},0)^{\star}f){\xi}_{0}(g)]
{\equiv}\leqno(4.40)$$
$${\lim}_{L\to\infty}{\omega}_{{\theta}(a,b)}
([{\alpha}(L^{r}{\tau}){\hat {\xi}}_{0}(f)]
{\hat{\xi}}_{0}(g))$$
\vskip 0.3cm
Hence, by (4.12) and (4.40), 
$${\bf E}[{\theta}(a,b){\vert}{\xi}_{0}
({\exp}({\cal L}({\theta}(a,b)){\tau}))^{\star}f){\xi}_{0}(g)]
=\leqno(4.41)$$
$${\lim}_{L\to\infty}{\omega}_{{\theta}(a,b)}
([{\alpha}(L^{r}{\tau}){\hat {\xi}}_{0}(f)]
{\hat{\xi}}_{0}(g))$$
\vskip 0.3cm
{\bf 4.4. Consequences of (I)-(V): Generalised Onsager
Relations.} Let ${\cal R}({\theta})$ be the range of the function
${\theta}{\equiv}{\lbrace}{\theta}(a,b){\vert}a{\in}X,b{\in}{\bf
R}_{+}{\rbrace}.$ We shall employ the above theory to obtain
properties of ${\bf E}[{\overline {\theta}}{\vert}.]$ and ${\cal
L}({\overline {\theta}})$ for arbitrary ${\overline
{\theta}}{\in}{\cal R}({\theta}).$ 
\vskip 0.3cm
(a) {\bf Symmetry Property of Time Correlations Functions.} In
view of the microscopic reversibility conditions (4.18), (4.20)
and (4.23), together with the stationarity of
${\omega}_{\overline {\theta}},$ it follows from (4.30), with
${\omega}^{(L)}={\omega}_{\overline {\theta}},$ that
$${\omega}_{\overline {\theta}}({\xi}_{t}(f){\xi}_{0}(g)){\equiv}
{\omega}_{\overline {\theta}}({\xi}_{0}(f){\xi}_{-t}(g)){\equiv}
{\omega}_{\overline {\theta}}({\xi}_{t}(g){\xi}_{0}(f)) \
{\forall}{\overline {\theta}}{\in}{\cal R}({\theta})$$
Hence, by (4.41), we have the symmetry property
$${\bf E}[{\overline {\theta}}{\vert}
{\xi}_{0}({\exp}({\cal L}({\overline {\theta}})^{\star}{\tau})f)
{\xi}_{0}(g)]{\equiv}{\bf E}[{\overline {\theta}}{\vert}
{\xi}_{0}({\exp}({\cal L}
({\overline {\theta}})^{\star}{\tau})g){\xi}_{0}(f)]
 \ {\forall}{\overline {\theta}}{\in}{\cal R}({\theta})
\leqno(4.42)$$
\vskip 0.3cm
(b) {\bf The Static Two-point Function.} It follows immediately
from (4.41) that ${\bf E}[{\overline {\theta}}{\vert}.]$ inherits
the translational invariance of ${\omega}_{\overline {\theta}}.$
Hence, in view of the tempered distribution property of
${\xi}_{0},$ the generalised function
$(x,y)({\in}X^{2}){\rightarrow}{\bf
E}[{\overline {\theta}}{\vert}({\xi}_{0}(x){\xi}_{0}(y)]$ is an 
${\cal S}^{(n){\prime}}(X)-$ class distribution $F(x-y);$ and,
by Prop. 4.2, $F({\lambda}x){\equiv}{\lambda}^{-d}F(x),$
and is therefore of the form $C{\delta}(x),$ where $C$ is an
n-by-n matrix. Thus,
$${\bf E}[{\overline {\theta}}{\vert}{\xi}_{0}(f){\xi}_{0}(g)]=
{\langle}Cf,g{\rangle} \ {\forall}{\overline {\theta}}
{\in}{\cal R}({\theta})\leqno(4.43)$$
where the angular brackets denote the inner product for the
Hilbert space ${\cal H}^{(n)}$ of square integrable functions
from $X$ into ${\bf R}^{n},$ as defined by the formula
$${\langle}f,g{\rangle}={\int}f(x).g(x)dx\leqno(4.44)$$
the dot denoting the ${\bf R}^{n}$ scalar product. Moreover, it
follows [Se5] from a treatment of the linear response
of ${\omega}_{\overline {\theta}}$ to local Hamiltonian
perturbations ${\hat q}(f)$ that, under mild technical
assumptions, $C=B({\theta})^{-1},$ where $B$ is specified in
(4.10). Hence, by (4.43), 
$${\bf E}[{\overline {\theta}}{\vert}{\xi}_{0}(f){\xi}_{0}(g)]=
{\langle}B({\overline {\theta}})^{-1}f,g{\rangle}\leqno(4.45)$$
\vskip 0.3cm
(c) {\bf Generalised Onsager Relations.} It follows immediately
from (4.42) and (4.45) that
$${\langle}B({\overline {\theta}})^{-1}{\exp}
({\cal L}({\overline {\theta}})^{\star}{\tau})f,
g{\rangle}{\equiv}{\langle}B({\overline {\theta}})^{-1}{\exp}
({\cal L}({\overline {\theta}})^{\star}{\tau})g,
f{\rangle}\leqno(4.46)$$
Hence, by (4.9), we have the following result.
\vskip 0.3cm
{\bf Proposition 4.3.} {\it Under the above assumptions,
${\Lambda}$ satisfies the generalised Onsager relation}
$${\langle}{\Lambda}({\overline
{\theta}})^{\star}f,g{\rangle}{\equiv}
{\langle}{\Lambda}({\overline {\theta}})^{\star}g,f{\rangle} \
{\forall}{\overline {\theta}}{\in}{\cal R}({\theta}), \ f,g{\in}
{\cal S}^{(n){\prime}}(X)\leqno(4.47)$$
{\it i.e. ${\Lambda}({\overline {\theta}}),$ considered as an
operator in ${\cal H}^{(n)},$ with domain ${\cal S}^{(n)},$ is
symmetric.}
\vskip 0.3cm
{\bf Comment.} In the case of the non-linear diffusion given by
(4.6a), it follows from (4.12a) that (4.47) reduces to the form
$$L_{kl}({\theta})(x,t))=L_{lk}({\theta})(x,t)) \
{\forall}x{\in}X, \ t{\in}{\bf R}_{+}$$
\vskip 0.5cm
{\bf 5. Concluding Remarks.} I have endeavoured to show here how,
at least in certain domains, a macroscopically-based approach to
statistical mechanics can serve to determine the form imposed by
quantum mechanics on the structure of phenomenological laws. By
contrast with the standard many-body theory, the  microscopic
imput here is limited to very general principles; and this serves
to pare down the conceptual structure of the theory to its
essentials. 
\vskip 0.2cm
Of course, the relative simplicity gained by this approach is
dependent on a number of assumptions, specified in the previous
Sections, that are very difficult to verify constructively.
Furthermore, the dynamical systems treated in ${\S}'s$ 3 and 4
have the simplifying, and rather particular, property of scale
covariance. In the case of the plasma model, this stems from the
fact that the Coulomb potential is given by a power law: in the
case of the non-equilibrium thermodynamics of ${\S}4,$ it is an
assumed property of the macroscopic dynamics.
\vskip 0.2cm
Thus, it is clear that the formulation of a coherent, general
formulation of the statistical mechanics of macroscopic variables
poses deep problems, concerning both its underlying assumptions
and its potential scope. I would hope that inroads into these 
problems may be achieved through the study both of suitable
models and of the relevant general structures. 
\vskip 0.5cm
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