Energy fluctuations and transitions in non-equilibrium inter-particle collisions

The energy fluctuations and transitions, as well as the random forces, of a Brownian particle in the short-time limit of one single collision are studied using microscopic kinetic theory. Fluctuations and random forces play fundamental roles in non-equilibrium dynamics and statistical mechanics. We find that the mean-square of random forces is proportional to the energy fluctuation for Brown particles. By considering collision processes between an incident particle and background molecules within the framework of the hard-sphere collision model, we analytically investigate energy fluctuations and transitions based on microscopic kinetic theories. Results for the example of the thermal equilibrium gas show that the energy fluctuations arise from two distinct contributions: kinetic fluctuations resulting from the random selection of scattering cross-section positions, which are proportional to the particle’s initial energy; and thermal fluctuations associated with the velocity distribution of molecules, which are proportional to gas temperature. We demonstrate that the energy flux density is equal to the temperature difference $\Delta T$ multiplied by the thermal conductivity coefficient $\kappa$ in two mixed gases under thermal equilibrium. Therefore, a microscopic kinetic perspective is provided for the second law of thermodynamics in this specific scenario. Our finding helps understand random dynamics and relaxation processes in non-equilibrium systems.