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2024 | OriginalPaper | Chapter

The Entropic Journey of Kac’s Model

Author : Amit Einav

Published in: From Particle Systems to Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

The goal of this paper is to review the advances that were made during the last few decades in the study of the entropy, and in particular the entropy method, for Kac’s many particle system.

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previous chapter Some Remarks About the Link Between the Fisher Information and Landau or Landau-Fermi-Dirac Entropy Dissipation

next chapter Entropy Production Bounds for the Kac Model are Uniform in the Number of Particles

As opposed to the rigorous derivation of it from Newtonian mechanics, which is known as Hilbert’s 6th problem.

We will be remiss if we won’t mention that there are many other studies on the convergence to equilibrium, such as the impressive [17].

Equation (3) represents the Navier-Stokes equation without external force.

One can show that this relation is the only viable one if one assumes conservation of energy and momentum.

Bogoliubov-Born-Green-Kirkwood-Yvon.

Kac’s model can be (and have been) extended to a higher dimensional model where this has been rectified (see [16]).

This condition can be reformulated using empirical measures. $\left\{ \mu _N\right\} _{N\in \mathbb {N}}$ will be $\mu _0$-chaotic if and only if for any random vector $\left( X_1,\dots ,X_N \right) $ with law $\mu _N$ we have that the random empirical measure

$$ \mu _N = \frac{1}{N}\sum _{i=1}^N \delta _{X_i} $$

converges in law towards the deterministic measure $\mu _0$.

We call is a relative entropy as it measures a function relative to another, in our case the equilibrium. There are many possibilities for entropies and we refer the reader to [1] where they can find some common examples used in other kinetic equations.

$$ \frac{d}{dt}\left\langle u(t),u_\infty \right\rangle =\left\langle Lu(t),u_\infty \right\rangle =\left\langle u(t),Lu_\infty \right\rangle =0. $$

This is the conservation of mass property for the equation.

We require this condition since only then do we have that

$$ u(t)-u_\infty = u(t)-\left\langle u(t),u_\infty \right\rangle u_\infty \in \left\{ u_\infty \right\} ^{\perp }. $$

I.e. a functional such that $\frac{d}{dt}\mathscr {E}\left( f(t)|f_\infty \right) \le 0$ for any solution, f(t), of the equation.

In a sense we hope that our entropy and its production have truly captured the geometry of the problem and the relevant properties of the evolution.

It is worth to note that $\mathscr {H}_N$ is indeed a relative functional. It measures the entropic distance of $F_Nd\sigma _N$ from $1d\sigma _N$.

The formula presented here was achieved by some simplification of the argument of [20] in the case of the Kac model we present here (see [6] for more details).

We do need to be quite careful here. The independence that chaos guarantees is always “capped” at a given finite marginal. We have no idea what happens with the correlations between a number of elements that is of order of N.

I.e. a central limit theorem for the probability density of the independent sum.

Note that $\int _{\mathbb {R}} v^2 \mathscr {M}_{a}(v)dv = a$ which shows that each part of $f_{\delta }$ carries the same kinetic energy.

In terms of a jump processes, the dependence on $v_i$ and $v_j$ is slightly problematic but we continue as if no additional justification for the proposed model is needed.

We just need to add $\left( 1+v_1^2+v_2^2 \right) ^\gamma $ inside the integral.

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Bobylev, A., Cercignani, C.: On the rate of entropy production for the Boltzmann equation. J. Stat. Phys. 94(3–4), 603–618 (1999)MathSciNetCrossRef

Carlen, E.A., Carvalho, M.C.: Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Stat. Phys. 67(3–4), 575–608 (1992)MathSciNetCrossRef

Carlen, E.A., Carvalho, M.C.: Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Stat. Phys. 74(3–4), 743–782 (1994)MathSciNetCrossRef

Carlen, E.A., Carvalho, M.C., Einav, A.: Entropy production inequalities for the Kac walk. Kinet. Relat. Models 11(2), 219–238 (2018)MathSciNetCrossRef

Carlen, E.A., Carvalho, M.C., Le Roux, J., Loss, M., Villani, C.: Entropy and chaos in the Kac model. Kinet. Relat. Models 3(1), 85–122 (2010)MathSciNetCrossRef

Carlen, E.A., Carvalho, M.C., Loss, M.: Determination of the spectral gap for Kac’s master equation and related stochastic evolution. Acta Math. 191(1), 1–54 (2003)MathSciNetCrossRef

Carlen, E.A., Carvalho, M.C., Loss, M.: Spectral gaps for reversible Markov processes with chaotic invariant measures: the Kac process with hard sphere collisions in three dimensions. Ann. Prob. 48(6), 2807–2844 (2020)MathSciNetCrossRef

Carlen, E.A., Carvalho, M.C., Loss M.: Kinetic theory and the Kac master equation. In Entropy and the quantum II, Contemporary Mathematics, vol. 552, pp. 1–20. American Mathematical Society, Providence, RI (2011)

10.

Carrapatoso, K., Einav, A.: Chaos and entropic chaos in Kac’s model without high moments. Electron. J. Probab. 18(78), 38 (2013)

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Einav, A.: On Villani’s conjecture concerning entropy production for the Kac master equation. Kinet. Relat. Models 4(2), 479–497 (2011)MathSciNetCrossRef

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Gallagher, I., Saint-Raymond, L., Texier, B.: From Newton to Boltzmann: hard spheres and short-range potentials. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2013)

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Hauray, M., Mischler, S.: On Kac’s chaos and related problems. J. Funct. Anal. 266(10), 6055–6157 (2014)MathSciNetCrossRef

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Janvresse, E.: Spectral gap for Kac’s model of Boltzmann equation. Ann. Prob. 29(1), 288–304 (2001)MathSciNetCrossRef

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Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pp. 171–197. University of California Press, Berkeley-Los Angeles, Calif. (1956)

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Jr, McKean, H.P.: An exponential formula for solving Boltmann’s equation for a Maxwellian gas. J. Comb. Theory 2, 358–382 (1967)

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Mischler, S., Mouhot, C.: Kac’s program in kinetic theory. Invent. Math. 193(1), 1–147 (2013)MathSciNetCrossRef

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Toscani, G., Villani, C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203(3), 667–706 (1999)MathSciNetCrossRef

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Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of Mathematical Fluid Dynamics, vol. 1, pp. 71–74. North-Holland, 2002

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Villani, C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234(3), 455–490 (2003)MathSciNetCrossRef

Title: The Entropic Journey of Kac’s Model
Author: Amit Einav
Publisher: Springer International Publishing
Book: From Particle Systems to Partial Differential Equations
Print ISBN: 978-3-031-65194-6

Electronic ISBN: 978-3-031-65195-3

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-65195-3_4

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