Nonequilibrium Entropy Conservation and the Transport Equations of Mass, Momentum, and Energy

Nonequilibrium statistical mechanics or molecular theory has put the transport equations of mass, momentum and energy on a firm or rigorous theoretical foundation that has played a critical role in their use and applications. Here, it is shown that those methods can be extended to nonequilibrium entropy conservation. As already known, the “closure” of the transport equations leads to the theory underlying the phenomenological laws, including Fick’s Law of Diffusion, Newton’s Law of Viscosity, and Fourier’s Law of Heat. In the case of entropy, closure leads to the relationship of entropy flux to heat as well as the Second Law or the necessity of positive entropy generation. It is further demonstrated how the complete set of transport equations, including entropy, can be simplified under physically restrictive assumptions, such as reversible flows and local equilibrium flows. This analysis, in general, yields a complete, rigorous set of transport equations for use in applications. Finally, it is also shown how this basis set of transport equations can be transformed to a new set of nonequilibrium thermodynamic functions, such as the nonequilibrium Gibbs’ transport equation derived here, which may have additional practical utility.

Relative entropy of freely cooling granular gases. A molecular dynamics study

Whereas the original Boltzmann’s H-theorem applies to elastic collisions, its rigorous generalization to the inelastic case is still lacking. Nonetheless, it has been conjectured in the literature that the relative entropy of the velocity distribution function with respect to the homogeneous cooling state (HCS) represents an adequate nonequilibrium entropy-like functional for an isolated freely cooling granular gas. In this work, we present molecular dynamics results reinforcing this conjecture and rejecting the choice of the Maxwellian over the HCS as a reference distribution. These results are qualitatively predicted by a simplified theoretical toy model. Additionally, a Maxwell-demon-like velocity-inversion simulation experiment highlights the microscopic irreversibility of the granular gas dynamics, monitored by the relative entropy, where a short “anti-kinetic” transient regime appears for nearly elastic collisions only.

Numerical Stability with Help from Entropy: Solving a Set of 13 Moment Equations for Shock Tube Problem

The shock structures of a 13 moment generalized hydrodynamics system of rarefied gases are simulated. These are first order hyperbolic equations derived from the Boltzmann equation. The investigated moment system stands out due to having an entropy evolution. In addition, a particular interest arises from the fact that the equations not only contain nonconservative products, but also provide the key to solving this mathematical and numerical issue by means of a simple substitution utilizing the physical entropy evolution. The apparent success of this method warrants investigation and provides a new perspective and starting point for finding general approaches to nonconservative products and irreversible processes. Furthermore, the system shows physically accurate results for low Mach numbers and is able to reveal the nonequilibrium entropy profile across a shock wave.

Effect of Temperature Jump on Nonequilibrium Entropy Generation in a MOSFET Transistor Using Dual-Phase-Lagging Model

This paper investigates the effect of temperature-jump boundary condition on nonequilibrium entropy production under the effect of the dual-phase-lagging (DPL) heat conduction model in a two-dimensional sub-100 nm metal-oxide-semiconductor field effect transistor (MOSFET). The transient DPL model is solved using finite element method. Also, the influences of the governing parameters on global entropy generation for the following cases—(I) constant applied temperature, (II) temperature-jump boundary condition, and (III) a realistic MOSFET with volumetric heat source and adiabatic boundaries—are discussed in detail and depicted graphically. The analysis of our results indicates that entropy generation minimization within a MOSFET can be achieved by using temperature-jump boundary condition and for low values of Knudsen number. A significant reduction of the order of 85% of total entropy production is observed when a temperature-jump boundary condition is adopted.