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Eric A. Carlen

Publications and source records attributed to Eric A. Carlen.

At least 37 records · Page 2Linked to original sources
arXiv2021

A simplified approach to the repulsive Bose gas from low to high densities and its numerical accuracy

In 1963, a Simple Approach was developed to study the ground state energy of an interacting Bose gas. It consists in the derivation of an Equation, which is not based on perturbation theory, and which gives the exact expansion of the energy at low densities. This Equation is expressed directly in the thermodynamic limit, and only involves functions of $3$ variables, rather than $3N$. Here, we revisit this approach, and show that the Equation yields accurate predictions for various observables for all densities. Specifically, in addition to the ground state energy, we have shown that the Simple Approach gives predictions for the condensate fraction, two-point correlation function, and momentum distribution. We have carried out a variety of tests by comparing the predictions of the Equation with Quantum Monte Carlo calculations, and have found remarkable agreement. We thus show that the Simple Approach provides a new theoretical tool to understand the behavior of the many-body Bose gas, not only in the small and large density ranges, which have been studied before, but also in the range of intermediate density, for which little is known.

cond-mat.quant-gas↗
arXiv2021

Dissipation for a non-convex gradient flow problem of a Patlack-Keller-Segel type for densities on $\mathbb{R}^n$, $n\geq 3$

We study an evolution equation that is the gradient flow in the $2$-Wasserstien metric of a non-convex functional for densities in $\mathbb{R}^n$ with $n \geq 3$. Like the Patlack-Keller-Segel system on $\mathbb{R}^2$, this evolution equation features a competition between the dispersive effects of diffusion, and the accretive effects of a concentrating drift. We determine a parameter range in which the diffusion dominates, and all mass leaves any fixed compact subset of $\mathbb{R}^n$ at an explicit polynomial rate.

math.AP↗
arXiv2020

Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $C^*$-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein--Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

math.OA↗
arXiv2020

Complete positivity and self-adjointness

We specify the structure of completely positive operators and quantum Markov semigroup generators that are symmetric with respect to a family of inner products, also providing new information on the order strucure an extreme points in some previously studied cases.

math.FA↗
arXiv2019

Recovery map stability for the Data Processing Inequality

The Data Processing Inequality (DPI) says that the Umegaki relative entropy $S(ρ||σ) := {\rm Tr}[ρ(\log ρ- \log σ)]$ is non-increasing under the action of completely positive trace preserving (CPTP) maps. Let ${\mathcal M}$ be a finite dimensional von Neumann algebra and ${\mathcal N}$ a von Neumann subalgebra if it. Let ${\mathcal E}_τ$ be the tracial conditional expectation from ${\mathcal M}$ onto ${\mathcal N}$. For density matrices $ρ$ and $σ$ in ${\mathcal N}$, let $ρ_{\mathcal N} := {\mathcal E}_τρ$ and $σ_{\mathcal N} := {\mathcal E}_τσ$. Since ${\mathcal E}_τ$ is CPTP, the DPI says that $S(ρ||σ) \geq S(ρ_{\mathcal N}||σ_{\mathcal N})$, and the general case is readily deduced from this. A theorem of Petz says that there is equality if and only if $σ= {\mathcal R}_ρ(σ_{\mathcal N} )$, where ${\mathcal R}_ρ$ is the Petz recovery map, which is dual to the Accardi-Cecchini coarse graining operator ${\mathcal A}_ρ$ from ${\mathcal M} $ to ${\mathcal N} $. In it simplest form, our bound is $$S(ρ||σ) - S(ρ_{\mathcal N} ||σ_{\mathcal N} ) \geq \left(\frac{1}{8π}\right)^{4} \|Δ_{σ,ρ}\|^{-2} \| {\mathcal R}_{ρ_{\mathcal N}} -σ\|_1^4 $$ where $Δ_{σ,ρ}$ is the relative modular operator. We also prove related results for various quasi-relative entropies. Explicitly describing the solutions set of the Petz equation $σ= {\mathcal R}_ρ(σ_{\mathcal N} )$ amounts to determining the set of fixed points of the Accardi-Cecchini coarse graining map. Building on previous work, we provide a throughly detailed description of the set of solutions of the Petz equation, and obtain all of our results in a simple self, contained manner.

math.OA↗
arXiv2018

Inequalities for $L^p$-norms that sharpen the triangle inequality and complement Hanner's Inequality

In 2006 Carbery raised a question about an improvement on the naïve norm inequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$ this is an equality, but when the supports of $f$ and $g$ are disjoint the factor $2^{p-1}$ is not needed. Carbery's question concerns a proposed interpolation between the two situations for $p>2$. The interpolation parameter measuring the overlap is $\|fg\|_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all $p$.

math.FA↗
arXiv2018

Chaos, Ergodicity and Equilibria in a Quantum Kac Model

We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the $N$ particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory, and a quantitative study the rate of destruction of entanglement in a class of quantum Markov semigroups describing binary interactions.

math-ph↗
arXiv2018

Inequalities for quantum divergences and the Audenaert-Datta conjecture

Given two density matrices $ρ$ and $σ$, there are a number of different expressions that reduce to the $α$-Rényi relative entropy of $ρ$ with respect to $σ$ in the classical case; i.e., when $ρ$ and $σ$ commute. Only those expressions for which the Data Processing Inequality (DPI) is valid are of potential interest as quantum divergences in quantum information theory. Audenaert and Datta have made a conjecture on the validity of the DPI for an interesting family of quantum generalizations of the $α$ - Rényi relative entropies, the $α-z$ - Rényi relative entropies. They and others have contributed to the partial solution of this conjecture. We review the problem, its context, and the methods that have been used to obtain the results that are known at present, presenting a unified treatment of developments that have unfolded in a number of different papers.

math-ph↗
arXiv2018

Recovery and the Data Processing Inequality for quasi-entropies

We prove number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies defined in terms of an operator monotone decreasing functions. Included in our results is a bound in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras where the results are already non-trivial and of interest in quantum information theory.

math-ph↗
arXiv2018

Some trace inequalities for exponential and logarithmic functions

Consider a function $F(X,Y)$ of pairs of positive matrices with values in the positive matrices such that whenever $X$ and $Y$ commute $F(X,Y)= X^pY^q.$ Our first main result gives conditions on $F$ such that ${\rm Tr}[ X \log (F(Z,Y))] \leq {\rm Tr}[X(p\log X + q \log Y)]$ for all $X,Y,Z$ such that ${\rm Tr} Z = {\rm Tr} X$. (Note that $Z$ is absent from the right side of the inequality.) We give several examples of functions $F$ to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables $X,Y,Z$ instead of just $X,Y$ alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy $D(X||Y) = {\rm Tr} [X(\log X-\log Y])$, and two others, the Donald relative entropy $D_D(X||Y)$, and the Belavkin-Stasewski relative entropy $D_{BS}(X||Y)$. They are known to satisfy $D_D(X||Y) \leq D(X||Y)\leq D_{BS}(X||Y)$. We prove that the Donald relative entropy provides the sharp upper bound, independent of $Z$, on ${\rm Tr}[ X \log (F(Z,Y))]$ in a number of cases in which $(Z,Y)$ is homogeneous of degree $1$ in $Z$ and $-1$ in $Y$. We also investigate the Legendre transforms in $X$ of $D_D(X||Y)$ and $D_{BS}(X||Y)$, and show how our results for these lead to new refinements of the Golden-Thompson inequality.

math-ph↗
arXiv2017

Quantitative Bounds on the Rate of Approach to Equilibrium for some One-Dimensional Stochastic Non-Linear Schrödinger Equations

We establish quantitative bounds on the rate of approach to equilibrium for a system with infinitely many degrees of freedom evolving according to a one-dimensional focusing nonlinear Schrödinger equation with diffusive forcing. Equilibrium is described by a generalized grand canonical ensemble. Our analysis also applies to the easier case of defocusing nonlinearities

math-ph↗
arXiv2017

A dual form of the sharp Nash inequality and its weighted generalization

The well known duality between the Sobolev inequality and the Hardy-Littlewood-Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three norms instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the $L^2$ norm to the infimal convolution of the $L^\infty $ and $H^{-1}$ norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also yields the sharp form of some new, weighted generalizations of the Nash inequality as well as the dual of these weighted variants.

math.FA↗
arXiv2017

On multi-dimensional hypocoercive BGK models

We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically meaningful rates. In fact, we not only estimate the exponential rate, but also the second time scale governing the time one must wait before one begins to see the exponential relaxation in the L1 distance. This waiting time phenomenon, with a long plateau before the exponential decay "kicks in" when starting from initial data that is well-concentrated in phase space, is familiar from work of Aldous and Diaconis on Markov chains, but is new in our continuous phase space setting. Our strategies are based on the entropy and spectral methods, and we introduce a new "index of hypocoercivity" that is relevant to models of our type involving jump processes and not only diffusion. At the heart of our method is a decomposition technique that allows us to adapt Lyapunov's direct method to our continuous phase space setting in order to construct our entropy functionals. These are used to obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.

math.AP↗
arXiv2017

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

We study a class of ergodic quantum Markov semigroups on finite-dimensional unital $C^*$-algebras. These semigroups have a unique stationary state $σ$, and we are concerned with those that satisfy a quantum detailed balance condition with respect to $σ$. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to $σ$ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the $2$-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical $2$-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.

math.OA↗
arXiv2017

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↗
arXiv2017

Approach to the steady state in kinetic models with thermal reservoirs at different temperatures

We continue the investigation of kinetic models of a system in contact via stochastic interactions with several spatially homogeneous thermal reservoirs at different temperatures. Considering models different from those investigated in earlier work, we explicitly compute the unique spatially uniform non-equilibrium steady state (NESS) and prove that it is approached exponentially fast from any uniform initial state. This leaves open the question of whether there exist NESS that are not spatially uniform. Making a further simplification of our models, we then prove non-existence of such NESS and exponential approach to the unique spatially uniform NESS (with a computably boundable rate). The method of proof relies on refined Doeblin estimates and other probabilisitic techniques, and is quite different form the analysis in earlier work that was based on contraction mapping methods.

math-ph↗