Interacting Particle Systems

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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 
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