arXiv subjects

Amit Einav

Publications and source records attributed to Amit Einav.

At least 19 recordsLinked to original sources

arXiv2025

A notion of partial order in the Choose the Leader model

In this work we continue the study of non-chaotic asymptotic correlations in many element systems and discuss the emergence of a new notion of asymptotic correlation -- partial order -- in the Choose the Leader (CL) system. Similarly to the newly defined notion of order, partial order refers to alignment of the elements in the system -- though it allows for deviation from total adherence. Our presented work revolves around the definition of partial order and shows its emergence in the CL model in its original critical scaling. Furthermore, we discuss the propagation of partial order in the CL model and give a quantitative estimate to the convergence to this state. This new notion (as well as that of order) opens the door to exploring old and new (probabilistic) models of biological and societal nature in a more realistic way.

math-ph↗

arXiv2025

On the equivalence between static and dynamic optimal transport governed by linear control systems

In this paper we revisit a class of optimal transport problems associated to non-autonomous linear control systems. Building on properties of the cost functions on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ derived from suitable variational problems, we show the equivalence between the static and dynamic versions of the corresponding transport problems. Our analysis is constructive in nature and relies on functional analytic properties of the end-point map and the fine properties of the optimal control functions. These lead to some new quantitative estimates which play a crucial role in our investigation.

math.OC↗

arXiv2023

Sharp Decay of the Fisher Information for Degenerate Fokker-Planck Equations

The goal of this work is to find the sharp rate of convergence to equilibrium under the quadratic Fisher information functional for solutions to Fokker-Planck equations governed by a constant drift term and a constant, yet possibly degenerate, diffusion matrix. A key ingredient in our investigation is a recent work of Arnold, Signorello, and Schmeiser, where the $L^2$-propagator norm of such Fokker-Planck equations was shown to be identical to the propagator norm of a finite dimensional ODE which is determined by matrices that are intimately connected to those appearing in the associated Fokker-Planck equations.

math.AP↗

arXiv2023

The emergence of order in many element systems

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 given system and is expected to be observed in biological and societal models. Beyond the definition of this new notion, our work will show its applicability, and propagation, in the so-called Choose the Leader model.

math-ph↗

arXiv2023

Convergence to equilibrium for a degenerate McKean-Vlasov Equation

In this work we study the convergence to equilibrium for a (potentially) degenerate nonlinear and nonlocal McKean-Vlasov equation. We show that the solution to this equation is related to the solution of a linear degenerate and/or defective Fokker-Planck equation and employ recent sharp convergence results to obtain an easily computable (and many times sharp) rates of convergence to equilibrium for the equation in question.

math.AP↗

arXiv2023

The Entropic journey of Kac's Model

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.

math.AP↗

arXiv2022

Generalised Fisher Information in Defective Fokker-Planck Equations

The goal of this work is to introduce and investigate a generalised Fisher Information in the setting of linear Fokker-Planck equations. This functional, which depends on two functions instead of one, exhibits the same decay behaviour as the standard Fisher information, and allows us to investigate different parts of the Fokker-Planck solution via an appropriate decomposition. Focusing almost exclusively on Fokker-Planck equations with constant drift and diffusion matrices, we will use a modification of the well established Bakry-Emery method with this newly defined functional to provide an alternative proof to the sharp long time behaviour of relative entropies of solutions to such equations when the diffusion matrix is positive definite and the drift matrix is defective. This novel approach is different to previous techniques and relies on minimal spectral information on the Fokker-Planck operator, unlike the one presented the authors' previous work, where powerful tools from spectral theory were needed.

math.AP↗

arXiv2021

Quantitative dynamics of irreversible enzyme reaction-diffusion systems

In this work we investigate the convergence to equilibrium for mass action reaction-diffusion systems which model irreversible enzyme reactions. Using the standard entropy method in this situation is not feasible as the irreversibility of the system implies that the concentrations of the substrate and the complex decay to zero. The key idea we utilise in this work to circumvent this issue is to introduce a family of cut-off partial entropy functions which, when combined with the dissipation of a mass like term of the substrate and the complex, yield an explicit exponential convergence to equilibrium. This method is also applicable in the case where the enzyme and complex molecules do not diffuse, corresponding to chemically relevant situation where these molecules are large in size.

math.AP↗

arXiv2020

Large time convergence of the non-homogeneous Goldstein-Taylor Equation

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 detailed understanding of the large time behaviour of the solutions to these equations has been mostly achieved in the case where the relaxation function, measuring the intensity of the relaxation towards equally distributed velocity densities, is constant. The goal of the presented work is to provide a general method to tackle the question of convergence to equilibrium when the relaxation function is not constant, and to do so as quantitatively as possible. In contrast to the usual modal decomposition of the equations, which is natural when the relaxation function is constant, we define a new Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case, that is able to deal with full spatial dependency of the relaxation function. The approach we develop is robust enough that one can apply it to multi-velocity Goldstein-Taylor models, and achieve explicit rates of convergence. The convergence rate we find, however, is not optimal, as we show by comparing our result to the that found in the work of Bernard and Salvarani from 2013.

math.AP↗

arXiv2020

Indirect diffusion effect in degenerate reaction-diffusion systems

In this work we study global well-posedness and large time behaviour for a typical reaction--diffusion system, which include degenerate diffusion, and whose non-linearities arise from chemical reactions. We show that there is an {\it indirect diffusion effect}, i.e. an effective diffusion for the non-diffusive species which is incurred by a combination of diffusion from diffusive species and reversible reactions between the species. Deriving new estimates for such degenerate reaction-diffusion system, we show, by applying the entropy method, that the solution converges exponentially to equilibrium, and provide explicit convergence rates and associated constants.

math.AP↗

arXiv2018

Interpolation of weighted Sobolev spaces

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 $L^{p}$ spaces. We consider an example which gives some indication that this may not be possible in all cases. Our results apply in cases which cannot be treated by methods in earlier papers about interpolation of weighted Sobolev spaces. They include, for example, a proof that $\left[W^{1,p}(\mathbb{R}^{d},\omega_{0}),W^{1,p}(\mathbb{R}^{d},\omega_{1})\right]_{\theta}=W^{1,p}(\mathbb{R}^{d},\omega_{0}^{1-\theta}\omega_{1}^{\theta})$ whenever $\omega_{0}$ and $\omega_{1}$ are continuous and their quotient is the exponential of a Lipschitz function. We also mention some possible applications of such interpolation in the study of convergence in evolution equations.

math.FA↗

arXiv2018

Weak Poincar\'e inequalities in the absence of spectral gaps

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 Poincar\'e inequality" (WPI), originally introduced by Liggett [Ann. Probab., 1991]. Applications to general classes of constant coefficient pseudodifferential operators are studied. Particular examples are the heat semigroup and the semigroup generated by the fractional Laplacian in the whole space, where the optimal decay rates are recovered. Moreover, the classical Nash inequality appears as a special case of the WPI for the heat semigroup.

math.FA↗

arXiv2017

On The Rates of Decay to Equilibrium in Degenerate and Defective Fokker-Planck Equations

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 novelty of our study lies in the amalgamation of spectral theory and a quantitative non-symmetric hypercontractivity result, as opposed to the usual approach of the entropy method.

math.AP↗

arXiv2017

Uniform moment propagation for the Becker-D\"oring equation

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

math-ph↗

arXiv2017

On the rate of convergence to equilibrium for the linear boltzmann equation with soft potentials

In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad's angular cutoff assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cutoff case and conjecture what we believe to be the right rate of convergence in that case.

math.AP↗

arXiv2016

Entropy production inequalities for the Kac Walk

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 results on the Boltzmann equation, or its one-dimensional analog, the non-linear Kac-Boltzmann equation, to prove results for the Kac Walk. Here we investigate new functional inequalities for the Kac Walk pertaining to entropy production, and introduce a new form of `chaoticity'. We then show how these entropy production inequalities imply entropy production inequalities for the Kac-Boltzmann equation. This results validate Kac's program for proving results on the non-linear Boltzmann equation via analysis of the Kac Walk, and they constitute a partial solution to the `Almost' Cercignani Conjecture on the sphere.

math-ph↗

arXiv2015

Trend to Equilibrium for the Becker-D\"oring Equations: An Analogue of Cercignani's Conjecture

In this work we investigate the rate of convergence to equilibrium for subcritical solutions to the Becker-D\"oring equations with physically relevant coagulation and fragmentation coefficients and mild assumptions on the given initial data. Using a discrete version of the log-Sobolev inequality with weights we show that in the case where the coagulation coefficient grows linearly and the detailed balance coefficients are of typical form, one can obtain a linear functional inequality for the dissipation of the relative free energy. This results in showing Cercignani's conjecture for the Becker-D\"oring equations and consequently in an exponential rate of convergence to equilibrium. We also show that for all other typical cases one can obtain an 'almost' Cercignani's conjecture that results in an algebraic rate of convergence to equilibrium. Additionally, we show that if one assumes an exponential moment condition one can recover Jabin and Niethammer's rate of decay to equilibrium, i.e. an exponential to some fractional power of $t$.

math-ph↗

arXiv2014

On the Subadditivity of the Entropy on the Sphere

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 $\mathbb{R}^N$ with respect to the entropy.

math.FA↗