Difference Boltzmann Equation

TL;DR

This paper derives a Difference Boltzmann Equation using plane wavelets, incorporating two-particle correlations, and demonstrates its relation to the classical Boltzmann Equation in the continuum limit.

Contribution

It introduces a difference form of the Boltzmann Equation in a plane wavelet basis, accounting for correlations and quantized variables, extending kinetic theory.

Findings

The set of plane wavelet functions is complete.

The system is described by particle numbers at quantized positions and momenta.

The equation reduces to classical Boltzmann Equation in the continuous limit.

Abstract

Difference Boltzmann Equation is derived in a plane wavelets representation with account of two-particle correlations. It is shown that the set of plane wavelet orthonormal functions is complete. The set of ket vectors is used as the second quantization basis allowing introducing the positively definite distribution function. It is obtained that inhomogeneous system is described by numbers of particles at quantized positions with quantized momenta. Difference Boltzmann Equation transforms into the classical Boltzmann Equation in the limit, where distribution function varies little in and