Isaacs equation and Interacting Particle Systems: Difference between pages
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The | 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 <ref name="GL97"/>. This equation not only arises as the limit of the microscopic system but the approximation is good enough to capture both qualitative and quantitative phenomena of the microscopic system <ref name="GL97"/>. More concretely, the above equation arises as the hydrodynamic limit of an interacting particle system evolving by the so called Kawasaki dynamics. Giacomin and Lebowitz also note that unlike the standard Cahn-Hilliard equation, the above integro-differential equation has been shown rigorously to arise as the macroscopic limit of a microscopic model of interacting particles <ref name="GL97"/>. | |||
The | == 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 process with values in $\Lambda_\gamma$ generated by the operator | |||
\[ L_\gamma f(\eta) = \sum \limits_{x,y\in \Lambda_\gamma} c_\gamma(x,y;\eta) \left (f(\eta^{x,y})-f(\eta) \right )\] | |||
where $\eta^{x,y}$ 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 | |||
\[c_\gamma(x,y;\eta) = \left \{ \begin{array}{rl} | |||
\Phi \left ( \beta\left [ H(\eta^{x,y}-H(\eta) \right ] \right) & \text{ if }\; |x-y|=1\\ | |||
0 & \text{ otherwise } | |||
\end{array}\right.\] | |||
== References == | |||
{{reflist|refs= | |||
<ref name="GL97"> {{Citation | last1=Lebowitz | first1=Joel | last2=Giacomin | first2=Giambattista | title=Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits | doi=10.1007/BF02181479 | year=1997 | journal=Journal of Statistical Physics | issn=0022-4715 | volume=87 | issue=1 | pages=37–61}} </ref> | |||
}} | |||
Revision as of 00:35, 1 February 2012
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]. This equation not only arises as the limit of the microscopic system but the approximation is good enough to capture both qualitative and quantitative phenomena of the microscopic system [1]. More concretely, the above equation arises as the hydrodynamic limit of an interacting particle system evolving by the so called Kawasaki dynamics. Giacomin and Lebowitz also note that unlike the standard Cahn-Hilliard equation, the above integro-differential equation has been shown rigorously to arise as the macroscopic limit of a microscopic model of interacting particles [1].
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 process with values in $\Lambda_\gamma$ generated by the operator
\[ L_\gamma f(\eta) = \sum \limits_{x,y\in \Lambda_\gamma} c_\gamma(x,y;\eta) \left (f(\eta^{x,y})-f(\eta) \right )\]
where $\eta^{x,y}$ 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
\[c_\gamma(x,y;\eta) = \left \{ \begin{array}{rl} \Phi \left ( \beta\left [ H(\eta^{x,y}-H(\eta) \right ] \right) & \text{ if }\; |x-y|=1\\ 0 & \text{ otherwise } \end{array}\right.\]