CONSERVATION OF EXPECTED MOMENTUM AND ENERGY IN MONTE-CARLO PARTICLE SIMULATION

Rarefied gas dynamics can be simulated numerically by Monte Carlo particle methods in which energy and momentum are conserved in expectation, but not exactly, through each collision. The conservation of the expected values of these moments does not imply the conservation of other expected second moments. For example, in Nanbu's method [J. Phys. Soc. Jpn. 49, 2042 (1980)] (which has been proved to converge), the expected value of the temperature decreases through each collision step, and the relaxation of a gas calculated by this scheme leads to the zero temperature state. The decrease in expected temperature is of order O(1/N), where N is the number of simulated particles.