Pub Date : 2023-06-07DOI: 10.21711/217504322023/em3811
G. Jona-Lasinio
In his ninth memoir Clausius summarizes the two principles of thermodynamics as follows:"The whole mechanical theory of heat rests on two fundamental theorems: that of equivalence of heat and work, and that of equivalence of transformations."This paper contains an introduction to Clausius' approach to entropy as illustrated in his original articles and describes an analogy in the macroscopic fluctuation theory of non-equilibrium diffusive systems.
{"title":"On Clausius’ approach to entropy and analogies in non-equilibrium","authors":"G. Jona-Lasinio","doi":"10.21711/217504322023/em3811","DOIUrl":"https://doi.org/10.21711/217504322023/em3811","url":null,"abstract":"In his ninth memoir Clausius summarizes the two principles of thermodynamics as follows:\"The whole mechanical theory of heat rests on two fundamental theorems: that of equivalence of heat and work, and that of equivalence of transformations.\"This paper contains an introduction to Clausius' approach to entropy as illustrated in his original articles and describes an analogy in the macroscopic fluctuation theory of non-equilibrium diffusive systems.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130361130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-26DOI: 10.21711/217504322023/em382
G. Bellettini, S. Kholmatov
We consider the geometric evolution of a network in the plane, flowing by anisotropic curvature. We discuss local existence of a classical solution in the presence of several smooth anisotropies. Next, we discuss some aspects of the polycrystalline case.
{"title":"Some aspects of anisotropic curvature flow of planar partitions","authors":"G. Bellettini, S. Kholmatov","doi":"10.21711/217504322023/em382","DOIUrl":"https://doi.org/10.21711/217504322023/em382","url":null,"abstract":"We consider the geometric evolution of a network in the plane, flowing by anisotropic curvature. We discuss local existence of a classical solution in the presence of several smooth anisotropies. Next, we discuss some aspects of the polycrystalline case.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126851352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-25DOI: 10.21711/217504322023/em3814
D. Tsagkarogiannis
We review some recent progress on applications of Cluster Expansions. We focus on a system of classical particles living in a continuous medium and interacting via a stable and tempered pair potential. We review the cluster expansion in both the canonical and the grand canonical ensemble and compute thermodynamic quantities such as the pressure, the free energy as well as various correlation functions. We derive the equation of state either by performing inversion of the density-activity series or directly in the canonical ensemble. Further applications to the liquid state expansions and the relevant closures are discussed, in particular their convergence in the gas regime.
{"title":"Cluster expansions, trees, inversions and correlations","authors":"D. Tsagkarogiannis","doi":"10.21711/217504322023/em3814","DOIUrl":"https://doi.org/10.21711/217504322023/em3814","url":null,"abstract":"We review some recent progress on applications of Cluster Expansions. We focus on a system of classical particles living in a continuous medium and interacting via a stable and tempered pair potential. We review the cluster expansion in both the canonical and the grand canonical ensemble and compute thermodynamic quantities such as the pressure, the free energy as well as various correlation functions. We derive the equation of state either by performing inversion of the density-activity series or directly in the canonical ensemble. Further applications to the liquid state expansions and the relevant closures are discussed, in particular their convergence in the gas regime.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"505 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117226582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-02DOI: 10.21711/217504322023/em3810
T. Gobron, E. Saada
Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in the simple exclusion process. We consider here general exclusion processes where jump rates from an occupied site to an empty one depend not only on the location of the jump but also possibly on the whole configuration. These processes include in particular exclusion processes with speed change introduced by F. Spitzer in [18]. For such processes we derive necessary and sufficient conditions for attractiveness, through the construction of a coupled process under which discrepancies do not increase. We emphasize the fact that basic coupling is never attractive for this class of processes, except in the case of simple exclusion, and that the coupled processes presented here necessarily differ from it. We study various examples, for which we determine the set of extremal translation invariant and invariant probability measures.
{"title":"Couplings and attractiveness for general exclusion processes","authors":"T. Gobron, E. Saada","doi":"10.21711/217504322023/em3810","DOIUrl":"https://doi.org/10.21711/217504322023/em3810","url":null,"abstract":"Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in the simple exclusion process. We consider here general exclusion processes where jump rates from an occupied site to an empty one depend not only on the location of the jump but also possibly on the whole configuration. These processes include in particular exclusion processes with speed change introduced by F. Spitzer in [18]. For such processes we derive necessary and sufficient conditions for attractiveness, through the construction of a coupled process under which discrepancies do not increase. We emphasize the fact that basic coupling is never attractive for this class of processes, except in the case of simple exclusion, and that the coupled processes presented here necessarily differ from it. We study various examples, for which we determine the set of extremal translation invariant and invariant probability measures.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133124505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-27DOI: 10.21711/217504322023/em385
P. Buttà, M. Pulvirenti, S. Simonella
In [Phys. Rev. 94 (1954), 511-525], P.L. Bhatnagar, E.P. Gross and M. Krook introduced a kinetic equation (the BGK equation), effective in physical situations where the Knudsen number is small compared to the scales where Boltzmann's equation can be applied, but not enough for using hydrodynamic equations. In this paper, we consider the stochastic particle system (inhomogeneous Kac model) underlying Bird's direct simulation Monte Carlo method (DSMC), with tuning of the scaled variables yielding kinetic and/or hydrodynamic descriptions. Although the BGK equation cannot be obtained from pure scaling, it does follow from a simple modification of the dynamics. This is proposed as a mathematical interpretation of some arguments in [Phys. Rev. 94 (1954), 511-525], complementing previous results in [Arch. Ration. Mech. Anal. 240 (2021), 785-808] and [Kinet. Relat. Models 16 (2023), 269-293].
{"title":"From particle systems to the BGK equation","authors":"P. Buttà, M. Pulvirenti, S. Simonella","doi":"10.21711/217504322023/em385","DOIUrl":"https://doi.org/10.21711/217504322023/em385","url":null,"abstract":"In [Phys. Rev. 94 (1954), 511-525], P.L. Bhatnagar, E.P. Gross and M. Krook introduced a kinetic equation (the BGK equation), effective in physical situations where the Knudsen number is small compared to the scales where Boltzmann's equation can be applied, but not enough for using hydrodynamic equations. In this paper, we consider the stochastic particle system (inhomogeneous Kac model) underlying Bird's direct simulation Monte Carlo method (DSMC), with tuning of the scaled variables yielding kinetic and/or hydrodynamic descriptions. Although the BGK equation cannot be obtained from pure scaling, it does follow from a simple modification of the dynamics. This is proposed as a mathematical interpretation of some arguments in [Phys. Rev. 94 (1954), 511-525], complementing previous results in [Arch. Ration. Mech. Anal. 240 (2021), 785-808] and [Kinet. Relat. Models 16 (2023), 269-293].","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134150799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-20DOI: 10.21711/217504322023/em381
S. Albeverio, B. Rüdiger, P. Sundar
Given the existence of a solution f(t; x; v)_{t in mathbb{R}_+^0} of the Boltzmann equation for hard spheres, we introduce a stochastic differential equation driven by a Poisson random measure that depends on f(t; x; v). The marginal distributions of its solution solves a linearized Boltzmann equation in the weak form. Further, if the distributions admit a probability density, we establish, under suitable conditions, that the density at each t coincides with f(t; x; v). The stochastic process is therefore called the Boltzmann process.
给定解f(t;x;v) {t in mathbb{R}_+^0}，我们引入了一个由泊松随机测度驱动的随机微分方程，它依赖于f(t);x;v).其解的边际分布以弱形式解线性化玻尔兹曼方程。进一步，如果分布允许一个概率密度，我们建立，在适当的条件下，密度在每个t与f(t;x;因此，随机过程称为玻尔兹曼过程。
{"title":"On the construction and identification of Boltzmann processes","authors":"S. Albeverio, B. Rüdiger, P. Sundar","doi":"10.21711/217504322023/em381","DOIUrl":"https://doi.org/10.21711/217504322023/em381","url":null,"abstract":"Given the existence of a solution f(t; x; v)_{t in mathbb{R}_+^0} of the Boltzmann equation for hard spheres, we introduce a stochastic differential equation driven by a Poisson random measure that depends on f(t; x; v). The marginal distributions of its solution solves a linearized Boltzmann equation in the weak form. Further, if the distributions admit a probability density, we establish, under suitable conditions, that the density at each t coincides with f(t; x; v). The stochastic process is therefore called the Boltzmann process.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"215 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116523799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-30DOI: 10.21711/217504322023/em3812
T. Komorowski, J. Lebowitz, S. Olla, Marielle Simon
We summarize and extend some of the results obtained recently for the microscopic and macroscopic behavior of a pinned harmonic chain, with random velocity flips at Poissonian times, acted on by a periodic force {at one end} and in contact with a heat bath at the other end. Here we consider the case where the system is in contact with two heat baths at different temperatures and a periodic force is applied at any position. This leads in the hydrodynamic limit to a heat equation for the temperature profile with a discontinuous slope at the position where the force acts. Higher dimensional systems, unpinned cases and anharmonic interactions are also considered.
{"title":"On the conversion of work into heat: microscopic models and macroscopic equations","authors":"T. Komorowski, J. Lebowitz, S. Olla, Marielle Simon","doi":"10.21711/217504322023/em3812","DOIUrl":"https://doi.org/10.21711/217504322023/em3812","url":null,"abstract":"We summarize and extend some of the results obtained recently for the microscopic and macroscopic behavior of a pinned harmonic chain, with random velocity flips at Poissonian times, acted on by a periodic force {at one end} and in contact with a heat bath at the other end. Here we consider the case where the system is in contact with two heat baths at different temperatures and a periodic force is applied at any position. This leads in the hydrodynamic limit to a heat equation for the temperature profile with a discontinuous slope at the position where the force acts. Higher dimensional systems, unpinned cases and anharmonic interactions are also considered.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126063663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-20DOI: 10.21711/217504322023/em387
P. Ferrari, C. Franceschini, Dante G. E. Grevino, H. Spohn
We study the hydrodynamics of the hard rod model proposed by Boldrighini, Dobrushin and Soukhov by describing the displacement of each quasiparticle with respect to the corresponding ideal gas particle as a height difference in a related field. Starting with a family of nonhomogeneous Poisson processes contained in the position-velocity-length space $mathbb{R}^3$, we show laws of large numbers for the quasiparticle positions and the length fields, and the joint convergence of the quasiparticle fluctuations to a Levy Chentsov field. We allow variable rod lengths, including negative lengths.
{"title":"Hard rod hydrodynamics and the Lévy Chentsov field","authors":"P. Ferrari, C. Franceschini, Dante G. E. Grevino, H. Spohn","doi":"10.21711/217504322023/em387","DOIUrl":"https://doi.org/10.21711/217504322023/em387","url":null,"abstract":"We study the hydrodynamics of the hard rod model proposed by Boldrighini, Dobrushin and Soukhov by describing the displacement of each quasiparticle with respect to the corresponding ideal gas particle as a height difference in a related field. Starting with a family of nonhomogeneous Poisson processes contained in the position-velocity-length space $mathbb{R}^3$, we show laws of large numbers for the quasiparticle positions and the length fields, and the joint convergence of the quasiparticle fluctuations to a Levy Chentsov field. We allow variable rod lengths, including negative lengths.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127195269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-08DOI: 10.21711/217504322023/em388
T. Funaki, P. Meurs, S. Sethuraman, K. Tsunoda
We derive the hydrodynamic limit of Glauber-Kawasaki dynamics. The Kawasaki part is simple and describes independent movement of the particles with hard core exclusive interactions. It is speeded up in a diffusive space-time scaling. The Glauber part describes the birth and death of particles. It is set to favor two levels of particle density with a preference for one of the two. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, the limiting particle density instantly takes either of the two favored density values. The interface which separates these two values evolves with constant speed (Huygens' principle). Similar hydrodynamic limits have been derived in four recent papers. The crucial difference with these papers is that we consider Glauber dynamics which has a preferences for one of the two favored density values. As a result, we observe limiting dynamics on a shorter time scale, and the evolution is different from the mean curvature flow obtained in the four previous papers. While several steps in our proof can be adopted from these papers, the proof for the propagation of the interface is new.
{"title":"Constant-speed interface flow from unbalanced Glauber-Kawasaki dynamics","authors":"T. Funaki, P. Meurs, S. Sethuraman, K. Tsunoda","doi":"10.21711/217504322023/em388","DOIUrl":"https://doi.org/10.21711/217504322023/em388","url":null,"abstract":"We derive the hydrodynamic limit of Glauber-Kawasaki dynamics. The Kawasaki part is simple and describes independent movement of the particles with hard core exclusive interactions. It is speeded up in a diffusive space-time scaling. The Glauber part describes the birth and death of particles. It is set to favor two levels of particle density with a preference for one of the two. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, the limiting particle density instantly takes either of the two favored density values. The interface which separates these two values evolves with constant speed (Huygens' principle). Similar hydrodynamic limits have been derived in four recent papers. The crucial difference with these papers is that we consider Glauber dynamics which has a preferences for one of the two favored density values. As a result, we observe limiting dynamics on a shorter time scale, and the evolution is different from the mean curvature flow obtained in the four previous papers. While several steps in our proof can be adopted from these papers, the proof for the propagation of the interface is new.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"88 7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126308390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-23DOI: 10.21711/217504322023/em386
Gioia Carinci, Simone Floreani, C. Giardinà, F. Redig
Inspired by the recent work of Bertini and Posta, who introduced the boundary driven Brownian gas on $[0,1]$, we study boundary driven systems of independent particles in a general setting, including particles jumping on finite graphs and diffusion processes on bounded domains in $mathbb{R}^d$. We prove duality with a dual process that is absorbed at the boundaries, thereby creating a general framework that unifies dualities for boundary driven systems in the discrete and continuum setting. We use duality first to show that from any initial condition the systems evolve to the unique invariant measure, which is a Poisson point process with intensity the solution of a Dirichlet problem. Second, we show how the boundary driven Brownian gas arises as the diffusive scaling limit of a system of independent random walks coupled to reservoirs with properly rescaled intensity.
{"title":"Boundary driven Markov gas: duality and scaling limits","authors":"Gioia Carinci, Simone Floreani, C. Giardinà, F. Redig","doi":"10.21711/217504322023/em386","DOIUrl":"https://doi.org/10.21711/217504322023/em386","url":null,"abstract":"Inspired by the recent work of Bertini and Posta, who introduced the boundary driven Brownian gas on $[0,1]$, we study boundary driven systems of independent particles in a general setting, including particles jumping on finite graphs and diffusion processes on bounded domains in $mathbb{R}^d$. We prove duality with a dual process that is absorbed at the boundaries, thereby creating a general framework that unifies dualities for boundary driven systems in the discrete and continuum setting. We use duality first to show that from any initial condition the systems evolve to the unique invariant measure, which is a Poisson point process with intensity the solution of a Dirichlet problem. Second, we show how the boundary driven Brownian gas arises as the diffusive scaling limit of a system of independent random walks coupled to reservoirs with properly rescaled intensity.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125098174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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