Entropy production in diffusion-reaction systems: The reactive random Lorentz gas

Abstract

We report the study of a random Lorentz gas with a reaction of isomerization A<==>B between two colors of moving particles elastically bouncing on hard disks. The reaction occurs when the moving particles collide on catalytic disks, which constitute a fraction of all the disks. Under dilute-gas conditions, the reaction-diffusion process is ruled by two coupled Boltzmann-Lorentz equations for the distribution functions of the colors. The macroscopic reaction-diffusion equations with cross-diffusion terms induced by the chemical reaction are derived from the kinetic equations. We use an H theorem of the kinetic theory in order to derive a macroscopic entropy depending on the gradients of color densities and which has a non-negative entropy production in agreement with the second law of thermodynamics.

General self-similar solutions of diffusion equation and related constructions

“…We will see later on that, α < 0 values mean exploding solutions which have only mathematical interest in most cases. It is also clear from (22) that no real solutions can be defined for t < 0.The spreading and decaying properties are visualized on Fig. 1 below.…”

“…Connections with non-equilibrium thermodynamics were analyzed in [20], and [21]. Beyond diffusion Mátyás and Gaspard [22] discussed diffusion with a simple reaction -isomerization -where not only the diffusion coefficient, but the reaction rate is also evaluated. One may also find diffusive processes where the diffusion is determined by the surroundings or the boundaries or the shape of the surface [23][24][25][26] Regarding biological applications, the diffusion equation has an important role at mesoscopic -cell size -scale [27], and also in the design of bioreactors [28].…”

Advanced Analytic Self-Similar Solutions of Regular and Irregular Diffusion Equations

“…In the following case n = −1 will be considered with special attention, because corresponds to a characteristic dependence for dilute systems [7,9].…”

Even and Odd Self-Similar Solutions of the Diffusion Equation for Infinite Horizon

“…Regular diffusion is the cornerstone of many scientific disciplines, such as surface growth [6][7][8], reactions diffusion [9] or even flow problems in porous media. In our last two papers, we gave an exhaustive summary of such processes with numerous relevant reviews [10,11].…”