Interacting Particle Systems

The (second order) integro-differential equation \[ \begin{array}{rl} \partial_t \rho &= \text{div} \left( D(\rho) \nabla \rho+\sigma(\rho) \nabla V[\rho]\right )\\ V[\rho] & = J * \rho \end{array}\] describes at the macroscopic scale the phase segregation in a gas whose particles are interacting at long ranges, as shown by Giacomin and Lebowitz . More concretely, the above equation is the hydrodynamic limit of an interacting particle system evolving by the so called Kawasaki dynamics.

The interacting particle system

At the microscopic level, the system is described by a function

\[ \eta : \Lambda_\gamma \to \{ 0,1\} \]

where $\gamma>0$ taken very small represents the spatial scale and $\Lambda_\gamma $ denotes the finite $d$-dimensional lattice

\[ \Lambda_\gamma = \{ 1,2,...,[\gamma^{-1}]\}^d\]

in other words, a cube inside $\mathbb{Z}^d$ with sides given by $[\gamma^{-1}]$, as $\gamma \to 0$, this exhausts all of $\mathbb{Z}^d$. The set of all posible configurations $\eta$ will be denoted by $\Omega_\gamma$, this is the state space where the (microscopic scale) dynamics takes place. As $\gamma \to 0$ we expect to recover the above as a limiting dynamical system the integro-differential equation listed above, of course first we have to describe the microscopic dynamics.

Given any initial condition $\eta_0 : \Lambda_\gamma \to \{0,1\}$, we consider a stochastic Poisson jump $\{ \eta_t\}_{t\in\mathbb{R}_+}$ process with values in $\Lambda_\gamma$ and which is generated by the operator

\[ L_\gamma f(\eta) = \sum \limits_{x,y\in \Lambda_\gamma} c_\gamma(x,y;\eta) \left (f(\eta^{x,y})-f(\eta) \right )\]

Here $\eta^{x,y}$ denotes the state $\eta$ where the values at $x$ and $y$ have been interchanged and the kernel $c_\gamma(x,y;\eta)$ is defined as

\[c_\gamma(x,y;\eta) = \left \{ \begin{array}{rl} \Phi \left ( \beta\left [ H(\eta^{x,y})-H(\eta) \right ] \right) & \text{ if }\; |x-y|=1\\ 0 & \text{ otherwise } \end{array}\right.\]

The Hydrodynamic limit

If $\mu_\gamma \in \Omega_\gamma$ is such that $\mu_\gamma \to \rho_0 \in L^1(\mathbb{T}^d)$ and if $\rho(x,t)$ solves the integro-differential equation with initial data $\rho_0$, then

\[ \lim \limits_{\gamma \to 0}\;\;P^{\mu_\gamma}_\gamma \left ( \left | \gamma^d \sum\limits_k \phi(\gamma k)\eta_{t\gamma^{-2}}(k)-\int_{\mathbb{T}^d}\phi(x)\rho(x,t)dx \right |>\delta \right ) =0\]