Collision Integral for Non-Equilibrium Distributions of 1D Bosons with Non-Linear Dispersions

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Authors

  • Piotr Chudzinski Institute of Fundamental Technological Research, Polish Academy of Sciences, Poland ORCID ID 0000-0003-2362-9963
  • Wieslaw Larecki Institute of Fundamental Technological Research, Polish Academy of Sciences, Poland

Abstract

In order to understand transport phenomena in the quasi-classical regime, the Boltzmann transport equation (BTE) is one of the most frequently used tools. Therein, the key quantity is the collision integral – the quantity that encapsulates the properties of the medium under consideration. Usually the result of this integral is approximated by a single parameter, the relaxation time. However, this leaves one wondering if such a treatment is sufficient, for instance, if the dispersion of bosons is non-linear, what will be the influence of this non-linearity on the BTE. Here, we give a fully analytic solution of the collision integral for 1D bosonic gases with non-linear dispersion and far out of equilibrium. Our analytic result is given in terms of the Lerch transcendent function and it has been obtained for the case of two subsystems (one dragging the other), by taking a maximum-entropy displaced Bose–Einstein ansatz for their distributions. Currently, there are numerous experiments performed far from equilibrium, where distributions are massively shifted and our result may serve as a main building block for deriving distributions of bosons, and later linear and non-linear transport coefficients, in such regimes.

Keywords:

Boltzmann transport equation, bosons with non-linear dispersion, non-equilibrium distributions

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