# 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 [1]. 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 $\gamma \to 0$ the dynamical system recovers the integro-differential equation listed above, as explained below.

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

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

## References

1. Lebowitz, Joel; Giacomin, Giambattista (1997), "Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits", Journal of Statistical Physics 87 (1): 37–61, doi:10.1007/BF02181479, ISSN 0022-4715