In recent years a new kind of data has become available through the use of fast computers. Basically the idea underlying these computer “experiments” is quite simple: one solves Newton’s equations for N particles in a box (N of the order of 103) by a finite difference numerical approximation. Hard-sphere interactions are particularly simple since their dynamics consists of straight line motions interrupted by sudden deflections obeying simple geometrical laws. Various results have been obtained this way, in particular, velocity autocorrelation functions of single particles have been determined [30.1]. These correlation functions for short times (of the order of the relaxation time) showed an exponential decrease as expected from the simple theory of Brownian motion, see (4.10). It turned out [30.1], however, that after a few relaxation times, when the correlations had decreased to a few percent of their initial value 〈υ 2〉, the decrease slowed down to a power law, which in three-dimensional space was ∝ t −3/2, see Fig. 30.1. One sees that after sufficiently many collisions the correlation functions decay away very slowly according to a power law.
KeywordsDiffusion Equation Mass Renormalization Velocity Autocorrelation Function Straight Line Motion Velocity Correlation Function
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- 30.3Landau, LX., Lifshitz, EM.: Fluid Mechanics, Course of Theoretical Physics, Vol. 6 ( Pergamon, Oxford 1959 )Google Scholar