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

In order to understand transport phenomena in a quasi-classical regime the Boltzmann transport equation (BTE) is one of the most frequently used tools. Therein, the key quantity is a collision integral – the quantity that encapsulates the properties of the medium under consideration. Usually the result of this integral is approximated by one single parameter, the relaxation time. However it leaves one wondering if such situation is sufficient: for instance, if the dispersion of bosons is non-linear then what will be the influence of this non-linearity on the BTE. Here we give a fully analytic solution of the collisions’ 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 sub-systems (one dragging another), by taking a maximum-entropy displaced Bose–Einstein ansatz for their distributions. Currently, there are numerous experiments performed far away from equilibrium, where distributions are massively shifted and our result will serve as a main building block to derive distributions of bosons, and later linear and nonlinear transport coefficients, in such regimes.