Direct Simulation Scheme Derived from the Boltzmann Equation. IV. Correlation of Velocity

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The collision process described in the first paper of this series is written in the form of the stochastic difference equation for a molecular velocity. By using this stochastic equation the correlation of velocity between a molecule and its collision partner is examined. It is shown that the correlation grows stronger as τ⁄N increases, where τ is the time and N is the number of simulated molecules. The assumption of molecular chaos requires a negligibly small correlation, so that the condition τ⁄N<<1 is necessary for solutions of the stochastic difference equation to agree with solutions of the Boltzmann equation.<BR>Also, the correlation coefficients of the velocities at two time points are obtained. Suppose that N>>1 and τ⁄N<<1. If these velocities belong to a single molecule, the coefficient is exp (−θη)+O(N−1), and if they belong to different molecules, the coefficient is of O(N−1), where θ is a constant and η is the interval between the time points.

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