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The Entropic journey of Kac's Model (2024)
Presentation / Conference Contribution
Einav, A. (2022, June). The Entropic journey of Kac's Model. Presented at PSPDE X: Particle Systems and Partial Differential Equations X, Braga, Portugal

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.

The Emergence of Order in Many Element Systems (2024)
Journal Article
Einav, A. (2024). The Emergence of Order in Many Element Systems. Journal of Statistical Physics, 191(7), Article 86. https://doi.org/10.1007/s10955-024-03307-7

Our work is dedicated to the introduction and investigation of a new asymptotic correlation relation in the field of mean field models and limits. This new notion, order (as opposed to chaos), revolves around a tendency for self organisation in a giv... Read More about The Emergence of Order in Many Element Systems.

Quantitative Dynamics of Irreversible Enzyme Reaction-Diffusion Systems (2022)
Journal Article
Braukhoff, M., Einav, A., & Quoc Tang, B. (2022). Quantitative Dynamics of Irreversible Enzyme Reaction-Diffusion Systems. Nonlinearity, 35(4), Article 1876. https://doi.org/10.1088/1361-6544/ac4d84

In this work we investigate the convergence to equilibriumfor mass action reactiondiffusion systemswhich model irreversible enzyme reactions. Using the standard entropy method in this situation is not feasible as the irreversibility of the system imp... Read More about Quantitative Dynamics of Irreversible Enzyme Reaction-Diffusion Systems.

Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation (2021)
Journal Article
Arnold, A., Einav, A., Signorello, B., & Wöhrer, T. (2021). Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation. Journal of Statistical Physics, 182(2), Article 41. https://doi.org/10.1007/s10955-021-02702-8

The Goldstein-Taylor equations can be thought of as a simplified version of a BGK system, where the velocity variable is constricted to a discrete set of values. It is intimately related to turbulent fluid motion and the telegrapher’s equation. A det... Read More about Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation.

Indirect Diffusion Effect in Degenerate Reaction-Diffusion Systems (2020)
Journal Article
Einav, A., Morgan, J. J., & Tang, B. Q. (2020). Indirect Diffusion Effect in Degenerate Reaction-Diffusion Systems. SIAM Journal on Mathematical Analysis, 52(5), 4314–4361. https://doi.org/10.1137/20m1319930

In this work we study global well-posedness and large time behavior for a typical reaction-diffusion system, which include degenerate diffusion, and whose nonlinearities arise from chemical reactions. We show that there is an indirect diffusion effec... Read More about Indirect Diffusion Effect in Degenerate Reaction-Diffusion Systems.

Weak Poincaré Inequalities in the Absence of Spectral Gaps (2019)
Journal Article
Ben-Artzi, J., & Einav, A. (2020). Weak Poincaré Inequalities in the Absence of Spectral Gaps. Annales Henri Poincaré, 21(2), 359–375. https://doi.org/10.1007/s00023-019-00858-4

For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called weak Poincaré inequality (WPI), originally introduced by Liggett (Ann Probab 19(3):935–959, 1991). Applicatio... Read More about Weak Poincaré Inequalities in the Absence of Spectral Gaps.

Uniform moment propagation for the Becker--Döring equations (2018)
Journal Article
Cãnizo, J. A., Einav, A., & Lods, B. (2018). Uniform moment propagation for the Becker--Döring equations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 149(4), 995-1015. https://doi.org/10.1017/prm.2018.99

We show uniform-in-time propagation of algebraic and stretched exponential moments for the Becker--Döring equations. Our proof is based upon a suitable use of the maximum principle together with known rates of convergence to equilibrium.

Interpolation of weighted Sobolev spaces (2018)
Journal Article
Cwikel, M., & Einav, A. (2019). Interpolation of weighted Sobolev spaces. Journal of Functional Analysis, 277(7), 2381-2441. https://doi.org/10.1016/j.jfa.2018.11.008

In this work we present a newly developed study of the interpolation of weighted Sobolev spaces by the complex method. We show that in some cases, one can obtain an analogue of the famous Stein–Weiss theorem for weighted Lp spaces. We consider an exa... Read More about Interpolation of weighted Sobolev spaces.

Entropy production inequalities for the Kac Walk (2018)
Journal Article
A. Carlen, E., C. Carvalho, M., & Einav, A. (2018). Entropy production inequalities for the Kac Walk. Kinetic and Related Models, 11(2), 219-238. https://doi.org/10.3934/krm.2018012

Mark Kac introduced what is now called 'the Kac Walk' with the aim of investigating the spatially homogeneous Boltzmann equation by probabilistic means. Much recent work, discussed below, on Kac's program has run in the other direction: using recent... Read More about Entropy production inequalities for the Kac Walk.

On the rates of decay to equilibrium in degenerate and defective Fokker–Planck equations (2018)
Journal Article
Arnold, A., Einav, A., & Wöhrer, T. (2018). On the rates of decay to equilibrium in degenerate and defective Fokker–Planck equations. Journal of Differential Equations, 264(11), 6843-6872. https://doi.org/10.1016/j.jde.2018.01.052

We establish sharp long time asymptotic behaviour for a family of entropies to defective Fokker–Planck equations and show that, much like defective finite dimensional ODEs, their decay rate is an exponential multiplied by a polynomial in time. The no... Read More about On the rates of decay to equilibrium in degenerate and defective Fokker–Planck equations.

On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials (2017)
Journal Article
Cañizo, J. A., Einav, A., & Lods, B. (2018). On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials. Journal of Mathematical Analysis and Applications, 462(1), 801-839. https://doi.org/10.1016/j.jmaa.2017.12.052

In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad's angular cut-off assumption. This is done by an adaptation of the famous entropy method and its vari... Read More about On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials.

Trend to equilibrium for the Becker–Döring equations : an analogue of Cercignani’s conjecture (2017)
Journal Article
equations : an analogue of Cercignani’s conjecture. Analysis & PDE, 10(7), 1663–1708. https://doi.org/10.2140/apde.2017.10.1663

We investigate the rate of convergence to equilibrium for subcritical solutions to the Becker–Döring equations with physically relevant coagulation and fragmentation coefficients and mild assumptions on the given initial data. Using a discrete versio... Read More about Trend to equilibrium for the Becker–Döring equations : an analogue of Cercignani’s conjecture.

On the Cauchy Problem for the Homogeneous Boltzmann–Nordheim Equation for Bosons: Local Existence, Uniqueness and Creation of Moments (2016)
Journal Article
Briant, M., & Einav, A. (2016). On the Cauchy Problem for the Homogeneous Boltzmann–Nordheim Equation for Bosons: Local Existence, Uniqueness and Creation of Moments. Journal of Statistical Physics, 163(5), 1108–1156. https://doi.org/10.1007/s10955-016-1517-9

The Boltzmann–Nordheim equation is a modification of the Boltzmann equation, based on physical considerations, that describes the dynamics of the distribution of particles in a quantum gas composed of bosons or fermions. In this work we investigate t... Read More about On the Cauchy Problem for the Homogeneous Boltzmann–Nordheim Equation for Bosons: Local Existence, Uniqueness and Creation of Moments.

On the Subadditivity of the Entropy on the Sphere (2015)
Journal Article
Einav, A. (2016). On the Subadditivity of the Entropy on the Sphere. Journal of Geometric Analysis, 26(4), 3098–3128. https://doi.org/10.1007/s12220-015-9664-9

We present a refinement of a known entropic inequality on the sphere, finding suitable conditions under which the uniform probability measure on the sphere behaves asymptomatically like the Gaussian measure on RN with respect to the entropy. Addition... Read More about On the Subadditivity of the Entropy on the Sphere.

Chaos and entropic chaos in Kac's model without high moments (2013)
Journal Article
Carrapatoso, K., & Einav, A. (2013). Chaos and entropic chaos in Kac's model without high moments. Electronic Journal of Probability, 18, 1-38. https://doi.org/10.1214/ejp.v18-2683

In this paper we present a new local Lévy Central Limit Theorem, showing convergence to stable states that are not necessarily the Gaussian, and use it to find new and intuitive entropically chaotic families with underlying one-particle function that... Read More about Chaos and entropic chaos in Kac's model without high moments.

A Counter Example to Cercignani’s Conjecture for the d Dimensional Kac Model (2012)
Journal Article
Einav, A. (2012). A Counter Example to Cercignani’s Conjecture for the d Dimensional Kac Model. Journal of Statistical Physics, 148(6), 1076–1103. https://doi.org/10.1007/s10955-012-0565-z

Kac’s d dimensional model gives a linear, many particle, binary collision model from which, under suitable conditions, the celebrated Boltzmann equation, in its spatially homogeneous form, arise as a mean field limit. The ergodicity of the evolution... Read More about A Counter Example to Cercignani’s Conjecture for the d Dimensional Kac Model.

Sharp trace inequalities for fractional Laplacians (2012)
Journal Article
Einav, A., & Loss, M. (2012). Sharp trace inequalities for fractional Laplacians. Proceedings of the American Mathematical Society, 140(12), https://doi.org/10.1090/s0002-9939-2012-11380-2

The sharp trace inequality of José Escobar is extended to traces for the fractional Laplacian on Rn, and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Lieb’s sharp form of the Hardy-L... Read More about Sharp trace inequalities for fractional Laplacians.

On Villani's conjecture concerning entropy production for the Kac Master equation (2011)
Journal Article
Master equation. Kinetic and Related Models, 4(2), 479-497. https://doi.org/10.3934/krm.2011.4.479

In this paper we take an idea presented in recent paper by Carlen, Carvalho, Le Roux, Loss, and Villani ([3]) and push it one step forward to find an exact estimation on the entropy production. The new estimation essentially proves that Villani's con... Read More about On Villani's conjecture concerning entropy production for the Kac Master equation.