Non-equilibrium Thermodynamics from a Kinetic Standpoint

The behavior of fluid, especially non-ideal fluids and their mixtures is largely conditioned by physical processes that require to downscale to the molecular level for a correct understanding of the underlying physics. Nevertheless, in numerical simulations of molecular dynamics, the round-off errors prevent the simulations from being performed beyond very limited spatial and time ranges. This difficulty increases for very large systems of particles because there is no means to ascertain the exact position and velocity of each particle at the initial time \(t=0\) . The best it is possible to do is to try to prescribe the probability that this particle will have a certain velocity when it is in a position \(\textbf{x}\) at time t. This probabilistic approach led to the Boltzmann equation and the finding that a system of particles ruled by the deterministic Newton’s laws of motion satisfies the second law of thermodynamics. In this chapter, we derive the Boltzmann equation starting from the Liouville equation for large systems of particles. The Boltzmann equation is restricted to material points considered sources of repulsion fields and collisions are binary involving, solely, a pair of particles. It is thus limited to rarefied gases. Considering their molecular nature and the great difficulty of the treatment of multiple collisions, the building up of kinetic equations considering only the information of the molecular scale is still a great scientific challenge. A kinetic model may be understood as a model that makes use of the information from the macroscopic scale to improve the simplifying assumptions that were required when deriving a model considering solely what is known from the molecular scale. Kinetic models are presented in this chapter for the collision term and non-ideal fluids and their mixtures.