Stochastic simulation Fokker-Planck equation

Fokker-Planck Equations. Fully stochastic realizations of the Langevin systems and the Monte Carlo simulations to which they give rise. 

Modeling the dynamics of an IP3R on the basis of its subunits leads to various consequences for a cluster of N IP3RS. As long as every IP3R is treated individually and subunits are assigned to individual channels -as has been done in stochastic simulations [6] - the state of the cluster is uniquely determined by the states of its subunits. However, an approach based on a population of subunits not grouped into individual channels is more suitable for the derivation of master equations and Fokker-Planck equations which we would like to use. That requires to determine the number of open channels from the total number of activatable subunits in the subunit population. We assume that the activatable subunits are randomly scattered across the channels. The distribution of the mo activatable subunits on the 4A subunits of a cluster decides upon the value of rio and hence the Ca " concentration. We show in the appendix that this distribution is sharply peaked around its mean value. Therefore, we set m = (rio) = Ua-ria is defined in equation (11.52). 

In recent years, there has been much interest in the nature of the fluctuations in nonequilibrium systems [ ll. Most of the work in this field has consisted of studying the composition fluctuations for a given system through the Master Equation, the Fokker-Planck Equation or a Stochastic Differential Equation. Recently, these methods have been applied to the study of thermal systems by Nicolis, Baras, and Malek Mansour [ 2l In this paper, we review their analysis of the two reservoir model. We discuss a computer simulation which has been developed to study this system and present a confirmation of their thermal fluctuation predictions. 

Exploiting the relation between this stochastic differential equation and its Fokker-Planck equation, it can be shown that the fluctuation-dissipation theorem holds [46], and that the method therefore simulates a canonical ensemble. DPD can be extended to thermalize the perpendicular component of the interparticle velocity as well, thereby allowing more control over the transport properties of the model [49,57]. 

Salis and Kaznessis proposed a hybrid stochastic algorithm that is based on a dynamical partitioning of the set of reactions into fast and slow subsets. The fast subset is treated as a continuous Markov process governed by a multidimensional Fokker-Planck equation, while the slow subset is considered to be a jump or discrete Markov process governed by a CME. The approximation of fast/continuous reactions as a continuous Markov process significantly reduces the computational intensity and introduces a marginal error when compared to the exact jump Markov simulation. This idea becomes very useful in systems where reactions with multiple reaction scales are constantly present. 

There are two conceptually different theoretical approaches for simulation of motional CW EPR spectra. The first is based on the stochastic Liouville equation (SLE) in the Fokker-Planck (FP) form which was developed by Kubo in the early 1960s and the second is the so-called trajectory based approach (see later). 

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