Monte Carlo Method

Abstract

The study of heat transfer on short length and time scales involves understanding the behaviour of microscopic heat carriers in order to quantify the average heat and energy flux. In dilute media, an intermediate statistical analysis is based on a local average of these carriers, namely, the velocity distribution function, and the Boltzmann equation. This same equation also governs the dynamical evolution of the local electron density, the number of phonon modes in conduction, and the spectral intensity in radiation. The approach is invaluable, especially when the carrier collision term is easy to integrate and the geometry of the system is simple.

However, if the interaction physics is not given in terms of these densities by a simple relation, there is no choice but to return to the carrier trajectories. This is the goal of the technique known as molecular dynamics. When the geometry is also complex and as a consequence the system size becomes large, the computation time required by molecular dynamics calculations is prohibitive. The Monte Carlo method takes microscopic constituents into account by sampling them in a relevant way, in order to produce the most accurate statistical averages possible on the basis of a limited number of operations. The behaviour of a small number of carriers is thus described independently under realistic physical constraints. Random sampling involves choosing the carriers and the rules governing their behaviour according to probabilistic laws imposed by:

• the laws of physics,

• a reasonable computation time,

• optimal accuracy for the calculated average value.

Today the Monte Carlo method is used in many areas of research. It is a basic tool for statisticians [1] and chemists use it to study equilibrium configurations of structures and molecules [2,3], or to predict phase change phenomena [4].

Monte Carlo calculations of neutron fluxes [5], radiative fluxes [6–8], and matter fluxes in fluid flows [9] can be found in a correspondingly vast literature. The Monte Carlo technique can even be extremely effective in solving transport problems in complex geometries [10]. It has thus become a standard tool for tackling transport problems.

However, its application to heat fluxes, either on very short length and time scales in gases, or in crystals [11,12] remains rather novel. In these contexts, it provides a way of taking into account the finer details of physical interaction mechanisms which standard statistical methods could only do at great cost in analytical complexity.

Whereas the Monte Carlo technique only partially describes systems in macroscopic configurations, by selecting certain energy carriers, it is interesting to note that, on short scales and in certain cases, it provides a direct simulation of all elementary carriers and sometimes an even greater number of carriers.