## Theory of direct simulation Monte Carlo method

##### Date

##### Authors

Karabulut, Hasan

Karabulut, Huriye Ariman

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

##### Abstract

##### Description

A treatment of direct simulation Monte Carlo method (DSMC) as a Markov
process with a master equation is given and the corresponding master equation
is derived. A hierarchy of equations for the reduced probability distributions
is derived from the master equation. An equation similar to the Boltzmann
equation for single particle probability distribution is derived using
assumption of molecular chaos. It is shown that starting from an uncorrelated
state, the system remains uncorrelated always in the limit $N\to \infty ,$
where $N$ is the number of particles. Simple applications of the formalism to
direct simulation money games are given as examples to the formalism. The
formalism is applied to the direct simulation of homogenous gases. It is shown
that appropriately normalized single particle probability distribution
satisfies the Boltzmann equation for simple gases and Wang Chang-Uhlenbeck
equation for a mixture of molecular gases. As a consequence of this development
we derive Birds no time counter algorithm. We extend the analysis to the
inhomogenous gases and define a new direct simulation algorithm for this case.
We show that single particle probability distribution satisfies the Boltzmann
equation in our algorithm in the limit $% N\to \infty ,$ $V_{k}\to 0,$ $\Delta
t\to 0$ where $% V_{k}$ is the volume $k^{th}$ cell. We also show that that our
algorithm and Bird's algorithm approach each other in the limit $N_{k}\to
\infty$ where $N_{k}$ is the number of particles in the volume $V_{k}$.

Comment: Submitted to Physica A

Comment: Submitted to Physica A

##### Keywords

Condensed Matter - Statistical Mechanics