To Do List and Interacting Particle Systems: Difference between pages

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== Things that need to be done ==
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"/>.


We need to come up with some organization for the articles.


The list below can be a starting point to click on links and edit each page. The following are some of the topics that should appear in this wiki.
== The interacting particle system ==


* We better start thinking hard about writing the [[Introduction to nonlocal equations]]. We gotta start somewhere, so any random idea or small thing you want to write should go in there. This is high priority, since if someone reads one page of this wiki, it will likely be this one.
At the microscopic level, the system is described by a function


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


* Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]].
where  $\gamma>0$ taken very small represents the spatial scale and $\Lambda_\gamma $ denotes the finite $d$-dimensional lattice


* Fractional curvatures in conformal geometry.  
\[ \Lambda_\gamma = \{ 1,2,...,[\gamma^{-1}]\}^d\]


* We need  to explain further the  [[Extension technique]] and its connection with fractional powers of the Laplacian and Conformal geometry. Required background: [[Geometric Scattering Theory]], [[Ambient Metric Construction]], [[GJMS Operators]] and [[Singular Yamabe Problem]]..)
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.


* It would be wise (once the wiki is more mature) to add pages about the [[Boltzmann equation]], since it is one of the more "classical" and better known integro-differential equations.
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


* Pages about [[Homogenization]] (local and nonlocal) should appear here too. (CITATIONS still needed)
\[ L_\gamma f(\eta) = \sum \limits_{x,y\in \Lambda_\gamma} c_\gamma(x,y;\eta) \left (f(\eta^{x,y})-f(\eta) \right )\]


* Given the recent works of Osher/Gilboa and Bertozzi/Flenner on Ginzburg-Landau  on graphs we should have an article on the natural similarities between non-local operators and [[elliptic operators on graphs]].
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


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


* [[Aggregation equation]].


* [[Keller-Segel equation]].
== References ==
{{reflist|refs=


* [[Active scalar equations]].
<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>


== (partially) Completed tasks ==
}}
 
* A sort of [[Starting page]] that serves as a "root" for all pages we add (ideally, any page we create should be reachable from here). (UPDATE: See discussion.)
 
* Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]].
 
* Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Alexadroff-Bakelman-Pucci estimates]], some reference to [[free boundary problems]].
 
* A list of [[regularity results for fully nonlinear integro-differential equations|regularity results]] for [[fully nonlinear integro-differential equations]].
 
* Some discussion on models involving [[Levy processes]] and [[stochastic control]].
 
* The [[surface quasi-geostrophic equation]].
 
* A page about [[semilinear equations]].
 
* [[Nonlocal porous medium equation]].
 
* [[drift-diffusion equations]].
 
* [[Nonlocal minimal surfaces]] and [[Nonlocal mean curvature flow]].
 
* There is also plenty of work on [[Dislocation dynamics]] that we ought to add later on.
 
* Phase transitions involving non-local interactions, in particular, pages about [[Particle Systems]], discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional.
 
* It is convenient to have a [[mini second order elliptic wiki]] inside this wiki.
 
* [[open problems]].
 
* Having a [[list of equations]] may make it easier to navigate the wiki.
 
== Other ideas ==
 
 
* Fill up the list of [[upcoming events]] such as conferences, workshops, summer schools.
 
* [[User:Nestor|Nestor]] has started a [[Literature on Nonlocal Equations]] to dump there all papers we want to reference or are already referencing on the wiki.

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

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


References

  1. 1.0 1.1 1.2 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