Pub Date : 2021-03-02DOI: 10.1007/978-3-030-61346-4_3
U. Brauer, Lavi Karp
{"title":"The Non-isentropic Relativistic Euler System Written in a Symmetric Hyperbolic Form","authors":"U. Brauer, Lavi Karp","doi":"10.1007/978-3-030-61346-4_3","DOIUrl":"https://doi.org/10.1007/978-3-030-61346-4_3","url":null,"abstract":"","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90071870","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 : 2020-12-17DOI: 10.11606/T.45.2020.TDE-06012021-103444
T. Raszeja
Countable Markov shifts, denoted by $Sigma_A$ for a 0-1 infinite matrix $A$, are central objects in symbolic dynamics and ergodic theory. R. Exel and M. Laca introduced the corresponding operator algebras, a generalization of the Cuntz-Krieger algebras for infinite countable alphabet, and the set $X_A$, a kind of Generalized Markov Shift (GMS) that coincides with $Sigma_A$ in the locally compact case. The set $Sigma_A$ is dense in $X_A$, and its complement, a set of finite allowed words, is dense in $X_A$ when non-empty. We develop the thermodynamic formalism for $X_A$, introducing the notion of conformal measure in it, and exploring its connections with the usual formalism for $Sigma_A$. New phenomena appear, as different types of phase transitions and new conformal measures undetected by the classical thermodynamic formalism for $A$ not row-finite. Given a potential $F$ and inverse of temperature $beta$, we study the existence of conformal measures $mu_{beta}$ associated to $beta F$. We present examples where there exists a critical $beta_c$ s. t. we have existence of conformal probabilities satisfying $mu_{beta}(Sigma_A)=0$ for every $beta > beta_c$ and, on the weak$^*$ topology, the set of conformal probabilities for $beta >beta_c$ collapses to the standard conformal probability $mu_{beta_c}$, $mu_{beta_c}(Sigma_A)=1$, for the limit $betatobeta_c$. We study in detail the generalized renewal shift and modifications of it. We highlight the bijection between infinite emitters of the alphabet and extremal conformal probabilities for this class of renewal type shifts. We prove the existence and uniqueness of the eigenmeasure probability of the Ruelle's transformation at low enough temperature for a particular potential on the generalized renewal shift; such measures are not detected on the standard renewal shift since for low temperatures, $beta F$ is transient.
{"title":"Thermodynamic formalism for generalized countable Markov shifts","authors":"T. Raszeja","doi":"10.11606/T.45.2020.TDE-06012021-103444","DOIUrl":"https://doi.org/10.11606/T.45.2020.TDE-06012021-103444","url":null,"abstract":"Countable Markov shifts, denoted by $Sigma_A$ for a 0-1 infinite matrix $A$, are central objects in symbolic dynamics and ergodic theory. R. Exel and M. Laca introduced the corresponding operator algebras, a generalization of the Cuntz-Krieger algebras for infinite countable alphabet, and the set $X_A$, a kind of Generalized Markov Shift (GMS) that coincides with $Sigma_A$ in the locally compact case. The set $Sigma_A$ is dense in $X_A$, and its complement, a set of finite allowed words, is dense in $X_A$ when non-empty. We develop the thermodynamic formalism for $X_A$, introducing the notion of conformal measure in it, and exploring its connections with the usual formalism for $Sigma_A$. New phenomena appear, as different types of phase transitions and new conformal measures undetected by the classical thermodynamic formalism for $A$ not row-finite. Given a potential $F$ and inverse of temperature $beta$, we study the existence of conformal measures $mu_{beta}$ associated to $beta F$. We present examples where there exists a critical $beta_c$ s. t. we have existence of conformal probabilities satisfying $mu_{beta}(Sigma_A)=0$ for every $beta > beta_c$ and, on the weak$^*$ topology, the set of conformal probabilities for $beta >beta_c$ collapses to the standard conformal probability $mu_{beta_c}$, $mu_{beta_c}(Sigma_A)=1$, for the limit $betatobeta_c$. We study in detail the generalized renewal shift and modifications of it. We highlight the bijection between infinite emitters of the alphabet and extremal conformal probabilities for this class of renewal type shifts. We prove the existence and uniqueness of the eigenmeasure probability of the Ruelle's transformation at low enough temperature for a particular potential on the generalized renewal shift; such measures are not detected on the standard renewal shift since for low temperatures, $beta F$ is transient.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81925549","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 : 2020-12-15DOI: 10.11606/T.45.2020.TDE-04012021-102503
Gregorio Luis Dalle Vedove Nosaki
In equilibrium statistical mechanics or thermodynamics formalism one of the main objectives is to describe the behavior of families of equilibrium measures for a potential parametrized by the inverse temperature $beta$. Here we consider equilibrium measures as the shift invariant measures that maximizes the pressure. Other constructions already prove the chaotic behavior of these measures when the system freezes, that is, when $betarightarrow+infty$. One of the most important examples was given by Chazottes and Hochman where they prove the non-convergence of the equilibrium measures for a locally constant potential when the dimension is bigger then 3. In this work we present a construction of a bidimensional example described by a finite alphabet and a locally constant potential in which there exists a subsequence $(beta_k)_{kgeq 0}$ where the non-convergence occurs for any sequence of equilibrium measures at inverse of temperature $beta_k$ when $beta_krightarrow+infty$. In order to describe such an example, we use the construction described by Aubrun and Sablik which improves the result of Hochman used in the construction of Chazottes and Hochman.
{"title":"Chaos and Turing machines on bidimensional models at zero temperature","authors":"Gregorio Luis Dalle Vedove Nosaki","doi":"10.11606/T.45.2020.TDE-04012021-102503","DOIUrl":"https://doi.org/10.11606/T.45.2020.TDE-04012021-102503","url":null,"abstract":"In equilibrium statistical mechanics or thermodynamics formalism one of the main objectives is to describe the behavior of families of equilibrium measures for a potential parametrized by the inverse temperature $beta$. Here we consider equilibrium measures as the shift invariant measures that maximizes the pressure. Other constructions already prove the chaotic behavior of these measures when the system freezes, that is, when $betarightarrow+infty$. One of the most important examples was given by Chazottes and Hochman where they prove the non-convergence of the equilibrium measures for a locally constant potential when the dimension is bigger then 3. In this work we present a construction of a bidimensional example described by a finite alphabet and a locally constant potential in which there exists a subsequence $(beta_k)_{kgeq 0}$ where the non-convergence occurs for any sequence of equilibrium measures at inverse of temperature $beta_k$ when $beta_krightarrow+infty$. In order to describe such an example, we use the construction described by Aubrun and Sablik which improves the result of Hochman used in the construction of Chazottes and Hochman.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85815397","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}
In this paper, we investigate the $phi^4$ model with cutoffs. By introducing a spatial cutoff and a momentum cutoff, the total Hamiltonian is a self-adjoint operator on a boson Fock space. Under regularity conditions of the momentum cutoff, we obtain the first order expansion of a non-degenerate ground state energy of the total Hamiltonian.
{"title":"The first order expansion of a ground state energy of the ϕ4 model with cutoffs","authors":"Toshimitsu Takaesu","doi":"10.1063/5.0040022","DOIUrl":"https://doi.org/10.1063/5.0040022","url":null,"abstract":"In this paper, we investigate the $phi^4$ model with cutoffs. By introducing a spatial cutoff and a momentum cutoff, the total Hamiltonian is a self-adjoint operator on a boson Fock space. Under regularity conditions of the momentum cutoff, we obtain the first order expansion of a non-degenerate ground state energy of the total Hamiltonian.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90962640","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}
The theory of strict deformation quantization of the two sphere $S^2subsetmathbb{R}^3$ is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by $H_N$ and where $N$ indicates the number of sites. Indeed, since the fibers $A_{1/N}=M_{N+1}(mathbb{C})$ and $A_0=C(S^2)$ form a continuous bundle of $C^*$-algebras over the base space $I={0}cup 1/mathbb{N}^*subset[0,1]$, one can define a strict deformation quantization of $A_0$ where quantization is specified by certain quantization maps $Q_{1/N}: tilde{A}_0 rightarrow A_{1/N}$, with $tilde{A}_0$ a dense Poisson subalgebra of $A_0$. Given now a sequence of such $H_N$, we show that under some assumptions a sequence of eigenvectors $psi_N$ of $H_N$ has a classical limit in the sense that $omega_0(f):=lim_{Ntoinfty}langlepsi_N,Q_{1/N}(f)psi_Nrangle$ exists as a state on $A_0$ given by $omega_0(f)=frac{1}{n}sum_{i=1}^nf(Omega_i)$, where $n$ is some natural number. We give an application regarding spontaneous symmetry breaking (SSB) and moreover we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere $S^2$.
{"title":"The classical limit of mean-field quantum spin systems","authors":"Christiaan J. F. van de Ven","doi":"10.1063/5.0021120","DOIUrl":"https://doi.org/10.1063/5.0021120","url":null,"abstract":"The theory of strict deformation quantization of the two sphere $S^2subsetmathbb{R}^3$ is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by $H_N$ and where $N$ indicates the number of sites. Indeed, since the fibers $A_{1/N}=M_{N+1}(mathbb{C})$ and $A_0=C(S^2)$ form a continuous bundle of $C^*$-algebras over the base space $I={0}cup 1/mathbb{N}^*subset[0,1]$, one can define a strict deformation quantization of $A_0$ where quantization is specified by certain quantization maps $Q_{1/N}: tilde{A}_0 rightarrow A_{1/N}$, with $tilde{A}_0$ a dense Poisson subalgebra of $A_0$. Given now a sequence of such $H_N$, we show that under some assumptions a sequence of eigenvectors $psi_N$ of $H_N$ has a classical limit in the sense that $omega_0(f):=lim_{Ntoinfty}langlepsi_N,Q_{1/N}(f)psi_Nrangle$ exists as a state on $A_0$ given by $omega_0(f)=frac{1}{n}sum_{i=1}^nf(Omega_i)$, where $n$ is some natural number. We give an application regarding spontaneous symmetry breaking (SSB) and moreover we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere $S^2$.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85906811","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}
We study semigroups generated by two-dimensional relativistic Hamiltonians with magnetic field. In particular, for compactly supported radial magnetic field we show how the long time behaviour of the associated heat kernel depends on the flux of the field. Similar questions are addressed for Aharonov-Bohm type magnetic field.
{"title":"Heat kernel estimates for two-dimensional relativistic Hamiltonians with magnetic field","authors":"H. Kovařík","doi":"10.4171/ecr/18-1/16","DOIUrl":"https://doi.org/10.4171/ecr/18-1/16","url":null,"abstract":"We study semigroups generated by two-dimensional relativistic Hamiltonians with magnetic field. In particular, for compactly supported radial magnetic field we show how the long time behaviour of the associated heat kernel depends on the flux of the field. Similar questions are addressed for Aharonov-Bohm type magnetic field.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85448215","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}
We characterize symmetry transformations of Lagrangian extremals generating `on shell' conservation laws. We relate symmetry transformations of extremals to Jacobi fields and study symmetries of higher variations by proving that a pair given by a symmetry of the $l$-th variation of a Lagrangian and by a Jacobi field of the $s$-th variation of the same Lagrangian (with $s
{"title":"Symmetry transformations of extremals and higher conserved quantities: Invariant Yang–Mills connections","authors":"Luca Accornero, M. Palese","doi":"10.1063/5.0038533","DOIUrl":"https://doi.org/10.1063/5.0038533","url":null,"abstract":"We characterize symmetry transformations of Lagrangian extremals generating `on shell' conservation laws. We relate symmetry transformations of extremals to Jacobi fields and study symmetries of higher variations by proving that a pair given by a symmetry of the $l$-th variation of a Lagrangian and by a Jacobi field of the $s$-th variation of the same Lagrangian (with $s<l$) is associated with an `of shell' conserved current. The conserved current associated with two symmetry transformations is constructed and, as a case of study, its expression for invariant Yang--Mills connections on Minkowski space-times is obtained.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87155080","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}
Pierre-Antoine Bernard, N. Crampé, Dounia Shaaban Kabakibo, L. Vinet
The generic Heun operator of Lie type is identified as a certain $BC$-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We show also that these Bethe roots are intimately associated to the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of $O(3)$ and the computation of the entanglement entropy for free Fermions on the Krawtchouk chain.
{"title":"Heun operator of Lie type and the modified algebraic Bethe ansatz","authors":"Pierre-Antoine Bernard, N. Crampé, Dounia Shaaban Kabakibo, L. Vinet","doi":"10.1063/5.0041097","DOIUrl":"https://doi.org/10.1063/5.0041097","url":null,"abstract":"The generic Heun operator of Lie type is identified as a certain $BC$-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We show also that these Bethe roots are intimately associated to the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of $O(3)$ and the computation of the entanglement entropy for free Fermions on the Krawtchouk chain.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74215965","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}
We present a fresh look at the methods introduced by Boccato, Brennecke, Cenatiempo, and Schlein concerning the trapped Bose gas and give a conceptually very simple and concise proof of BEC in the Gross-Pitaevskii limit for small interaction potentials.
{"title":"Another proof of BEC in the GP-limit","authors":"C. Hainzl","doi":"10.1063/5.0039123","DOIUrl":"https://doi.org/10.1063/5.0039123","url":null,"abstract":"We present a fresh look at the methods introduced by Boccato, Brennecke, Cenatiempo, and Schlein concerning the trapped Bose gas and give a conceptually very simple and concise proof of BEC in the Gross-Pitaevskii limit for small interaction potentials.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88005426","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 : 2020-11-06DOI: 10.1103/physreva.103.013502
P. Brandão, O. Korotkova
A theoretical framework is developed for scattering of scalar radiation from stationary, three-dimensional media with correlation functions of scattering potentials obeying $mathcal{PT}$-symmetry. It is illustrated that unlike in scattering from deterministic $mathcal{PT}$-symmetric media, its stationary generalization involves two mechanisms leading to symmetry breaking in the statistics of scattered radiation, one stemming from the complex-valued medium realizations and the other - from the complex-valued degree of medium's correlation.
{"title":"Scattering theory for stationary materials with \u0000PT\u0000 symmetry","authors":"P. Brandão, O. Korotkova","doi":"10.1103/physreva.103.013502","DOIUrl":"https://doi.org/10.1103/physreva.103.013502","url":null,"abstract":"A theoretical framework is developed for scattering of scalar radiation from stationary, three-dimensional media with correlation functions of scattering potentials obeying $mathcal{PT}$-symmetry. It is illustrated that unlike in scattering from deterministic $mathcal{PT}$-symmetric media, its stationary generalization involves two mechanisms leading to symmetry breaking in the statistics of scattered radiation, one stemming from the complex-valued medium realizations and the other - from the complex-valued degree of medium's correlation.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73800484","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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