36th M. Smoluchowski Symposium on Statistical Physics: Soft Matter, Information Processing and Nonequilibrium Fluctuations

Sep 24 – 28, 2023
Faculty of Physics, Astronomy and Applied Computer Science; Jagiellonian University
Europe/Warsaw timezone
Lectures will take place in A1-03 hall (see www.tinyurl.com/36MSSmap).

Global in time existence theorem for the full revised Enskog equation

Sep 26, 2023, 10:05 AM
25m
Faculty of Physics, Astronomy and Applied Computer Science; Jagiellonian University

Faculty of Physics, Astronomy and Applied Computer Science; Jagiellonian University

Lojasiewicza 11 30-348 Kraków Poland
Regular Talk

Speaker

Prof. Jacek Polewczak (Department of Mathematics, California State University, Northridge, CA, 91330-8313, USA)

Description

I prove global in time existence of solutions to the full revised Enskog equation. This equation generalizes the Boltzmann theory to dense gases in two ways:

1. by taking into account the fact that the centers of two colliding spheres are at a distance a, equal to the diameter of hard spheres.

2. by increasing the collision frequency by a factor $Y_0$ which nowadays is identiﬁed with the radial pair correlation function $g_2$ for the system of hard spheres at a uniform equilibrium.

In contrast to the dilute gas mode described by the Boltzmann equation, the Enskog equation includes spatial pair correlation function for hard-spheres potential and depends in a highly non linear way on the local density of dense gas. The full revised Enskog equation refers to the case where $g_2$, the pair the correlation function (for non-uniform equilibrium of hard-spheres) is in general form. In terms of the virial expansion (in local density $n$, spatially dependent) at contact value, $g_2$ reads:

$g_2(n)=1+V_1(n)+V_2(n)+....+V_N(n)+...$,

where the term $V_1(n)$ depends on $n$ linearly, $V_2(n)$ depends on $n$ quadratically, $V_N(n)$ depends on $n$ as $n^N$, and so on.

Circa 30 years ago Arkeryd-Cercignani proved the result for the truncated $g_2$, i.e., when $g_2=1$ (no density dependence). The case with $g_2=1$ refers to the so called Boltzmann-Enskog equation. It differs from the Boltzmann equation only by existence of the shifts in the spatial variable in the collisional integral.

Since then many researchers tried/wanted to prove the result for general form of $g_2$. Dependence of $g_2$ on $n$ requires a different approach and new tools, as compared to Arkeryd-Cercignani proof ([1]). Additionally, this result finally completes and fulfills the existence result for the revised Enskog Equation.

The proof of existence of solutions to the revised Enskog equation is based on two constructions:

1. Construction of an H-functional (see [2]), where the full expansion of $g_2$ is used, but convergence of the series was not addressed.

2. Construction of a special sequence of stochastic kinetic equations (studied in [3]) and the proof that their solutions converge to weak solutions of the revised Enskog equation.

References

[1] L. Arkeryd and C. Cercignani, Global existence in L1 for the Enskog equation and convergence of solutions to solutions of the Boltzmann equation, J. Stat. Phys. 59 (1990), 845–867.

[2] M. Mareschal, J. Bławzdziewicz, and J. Piasecki, Local entropy production from the revised Enskog equation:
General formulation for inhomogeneous ﬂuids, Phys. Rev. Lett. 52 (1984), 1169–1172.

[3] J. Polewczak and G. Stell,. Transport coefﬁcients in some stochastic models of the revised Enskog equation, J. Stat. Phys. 109, (2002) 569–590.

Primary author

Prof. Jacek Polewczak (Department of Mathematics, California State University, Northridge, CA, 91330-8313, USA)

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