Theorem on the Distribution of Short-Time Particle Displacements with Physical Applications

Details Theorem on the Distribution of Short-Time Particle Displacements with Physical Applications Theorem on the Distribution of Short-Time Particle Displacements with Physical Applications

2005-05-30

arxiv.org/abs/cond-mat/0505734v3

J. Stat. Phys. 123, 1--37 (2006)

Physics

Condensed Matter

Statistical Mechanics

A less ambiguous mathematical notation for cumulants was adopted and several passages were reformulated and clarified. 40 page

Scientific paper

10.1007/s10955-006-9027-9

The distribution of the initial short-time displacements of particles is considered for a class of classical systems under rather general conditions on the dynamics and with Gaussian initial velocity distributions, while the positions could have an arbitrary distribution. This class of systems contains canonical equilibrium of a Hamiltonian system as a special case. We prove that for this class of systems the nth order cumulants of the initial short-time displacements behave as the 2n-th power of time for all n>2, rather than exhibiting an nth power scaling. This has direct applications to the initial short-time behavior of the Van Hove self-correlation function, to its non-equilibrium generalizations the Green's functions for mass transport, and to the non-Gaussian parameters used in supercooled liquids and glasses.

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