The theory of phase transitions is one of the branches of statistical physics in which smoothness and continuity play an important role. In fact, phase transitions are characterized mathematically by the degree of non...The theory of phase transitions is one of the branches of statistical physics in which smoothness and continuity play an important role. In fact, phase transitions are characterized mathematically by the degree of non-analyticity of the thermodynamic potentials associated with the given system. In this paper, we propose a method that is not based on cluster expansions for computing the higher derivatives of the free energy and estimating the error between the finite and infinite volume free energy in certain continuum gas models. Our approach is suitable for a direct proof of the analyticity of the pressure or free energy in certain models of Kac-type. The methods known up to now strongly rely on the validity of the cluster expansion. An extension of the method to classical lattice gas models is also discussed.展开更多
In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserste...In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a(K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds.Our result improves an important result due to Lott and Villani(2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.展开更多
文摘The theory of phase transitions is one of the branches of statistical physics in which smoothness and continuity play an important role. In fact, phase transitions are characterized mathematically by the degree of non-analyticity of the thermodynamic potentials associated with the given system. In this paper, we propose a method that is not based on cluster expansions for computing the higher derivatives of the free energy and estimating the error between the finite and infinite volume free energy in certain continuum gas models. Our approach is suitable for a direct proof of the analyticity of the pressure or free energy in certain models of Kac-type. The methods known up to now strongly rely on the validity of the cluster expansion. An extension of the method to classical lattice gas models is also discussed.
基金supported by a Postdoctoral Fellowship at Beijing Normal University and China Postdoctoral Science Foundation(Grant No.2017M610797)supported by National Natural Science Foundation of China(Grant Nos.11771430 and 11371351)Key Laboratory of Random Complex Structures and Data Science,Chinese Academy of Sciences(Grant No.2008DP173182).
文摘In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a(K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds.Our result improves an important result due to Lott and Villani(2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.
