We prove that for all but a measure zero set of local Hamiltonians, starting from random product states at sufficiently high but finite temperature, with overwhelming probability expectation values of observables equilibrate such that at sufficiently long times, fluctuations around the stationary value are exponentially small in the system size.
{"title":"Random product states at high temperature equilibrate exponentially well","authors":"Yichen Huang","doi":"arxiv-2409.08436","DOIUrl":"https://doi.org/arxiv-2409.08436","url":null,"abstract":"We prove that for all but a measure zero set of local Hamiltonians, starting\u0000from random product states at sufficiently high but finite temperature, with\u0000overwhelming probability expectation values of observables equilibrate such\u0000that at sufficiently long times, fluctuations around the stationary value are\u0000exponentially small in the system size.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260524","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}
This short book is an elementary course on entropy, leading up to a calculation of the entropy of hydrogen gas at standard temperature and pressure. Topics covered include information, Shannon entropy and Gibbs entropy, the principle of maximum entropy, the Boltzmann distribution, temperature and coolness, the relation between entropy, expected energy and temperature, the equipartition theorem, the partition function, the relation between expected energy, free energy and entropy, the entropy of a classical harmonic oscillator, the entropy of a classical particle in a box, and the entropy of a classical ideal gas.
{"title":"What is Entropy?","authors":"John C. Baez","doi":"arxiv-2409.09232","DOIUrl":"https://doi.org/arxiv-2409.09232","url":null,"abstract":"This short book is an elementary course on entropy, leading up to a\u0000calculation of the entropy of hydrogen gas at standard temperature and\u0000pressure. Topics covered include information, Shannon entropy and Gibbs\u0000entropy, the principle of maximum entropy, the Boltzmann distribution,\u0000temperature and coolness, the relation between entropy, expected energy and\u0000temperature, the equipartition theorem, the partition function, the relation\u0000between expected energy, free energy and entropy, the entropy of a classical\u0000harmonic oscillator, the entropy of a classical particle in a box, and the\u0000entropy of a classical ideal gas.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260520","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}
Baoying Wang, Roger Ayats, Kengo Deguchi, Alvaro Meseguer, Fernando Mellibovsky
We analyse in the Taylor-Couette system, a canonical flow that has been studied extensively for over a century, a parameter regime exhibiting dynamics that can be approximated by a simple discrete map. The map has exceptionally neat mathematical properties, allowing to prove its chaotic nature as well as the existence of infinitely many unstable periodic orbits. Remarkably, the fluid system and the discrete map share a common catalog of unstable periodic solutions with the tent map, a clear indication of topological conjugacy. A sufficient number of these solutions enables the construction of a conjugacy homeomorphism, which can be used to predict the probability density function of direct numerical simulations. These results rekindle Hopf's aspiration of elucidating turbulence through the study of recurrent patterns.
{"title":"Mathematically established chaos in fluid dynamics: recurrent patterns forecast statistics","authors":"Baoying Wang, Roger Ayats, Kengo Deguchi, Alvaro Meseguer, Fernando Mellibovsky","doi":"arxiv-2409.09234","DOIUrl":"https://doi.org/arxiv-2409.09234","url":null,"abstract":"We analyse in the Taylor-Couette system, a canonical flow that has been\u0000studied extensively for over a century, a parameter regime exhibiting dynamics\u0000that can be approximated by a simple discrete map. The map has exceptionally\u0000neat mathematical properties, allowing to prove its chaotic nature as well as\u0000the existence of infinitely many unstable periodic orbits. Remarkably, the\u0000fluid system and the discrete map share a common catalog of unstable periodic\u0000solutions with the tent map, a clear indication of topological conjugacy. A\u0000sufficient number of these solutions enables the construction of a conjugacy\u0000homeomorphism, which can be used to predict the probability density function of\u0000direct numerical simulations. These results rekindle Hopf's aspiration of\u0000elucidating turbulence through the study of recurrent patterns.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260518","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 first part of the paper explains how to encode a one-cocycle and a two-cocycle on a group $G$ with values in its representation by networks of planar trivalent graphs with edges labelled by elements of $G$, elements of the representation floating in the regions, and suitable rules for manipulation of these diagrams. When the group is a semidirect product, there is a similar presentation via overlapping networks for the two subgroups involved. M. Kontsevich and J.-L. Cathelineau have shown how to interpret the entropy of a finite random variable and infinitesimal dilogarithms, including their four-term functional relations, via 2-cocycles on the group of affine symmetries of a line. We convert their construction into a diagrammatical calculus evaluating planar networks that describe morphisms in suitable monoidal categories. In particular, the four-term relations become equalities of networks analogous to associativity equations. The resulting monoidal categories complement existing categorical and operadic approaches to entropy.
{"title":"Entropy, cocycles, and their diagrammatics","authors":"Mee Seong Im, Mikhail Khovanov","doi":"arxiv-2409.08462","DOIUrl":"https://doi.org/arxiv-2409.08462","url":null,"abstract":"The first part of the paper explains how to encode a one-cocycle and a\u0000two-cocycle on a group $G$ with values in its representation by networks of\u0000planar trivalent graphs with edges labelled by elements of $G$, elements of the\u0000representation floating in the regions, and suitable rules for manipulation of\u0000these diagrams. When the group is a semidirect product, there is a similar\u0000presentation via overlapping networks for the two subgroups involved. M. Kontsevich and J.-L. Cathelineau have shown how to interpret the entropy\u0000of a finite random variable and infinitesimal dilogarithms, including their\u0000four-term functional relations, via 2-cocycles on the group of affine\u0000symmetries of a line. We convert their construction into a diagrammatical calculus evaluating\u0000planar networks that describe morphisms in suitable monoidal categories. In\u0000particular, the four-term relations become equalities of networks analogous to\u0000associativity equations. The resulting monoidal categories complement existing\u0000categorical and operadic approaches to entropy.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260523","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 the Fr"ohlich polaron in the regime of strong coupling and prove the asymptotically sharp lower bound on the effective mass $m_{mathrm{eff}}(alpha)geq alpha^4 m_{mathrm{LP}}-Calpha^{4-epsilon}$, where $m_{mathrm{LP}}$ is an explicit constant. Together with the corresponding upper bound, which has been verified recently in [5], we confirm the validity of the celebrated Landau-Pekar formula [12] from 1948 for the effective mass $underset{alpharightarrow infty}{lim}alpha^{-4}m_{mathrm{eff}}(alpha)=m_{mathrm{LP}}$ as conjectured by Spohn [25] in 1987.
{"title":"Proof of the Landau-Pekar Formula for the effective Mass of the Polaron at strong coupling","authors":"Morris Brooks","doi":"arxiv-2409.08835","DOIUrl":"https://doi.org/arxiv-2409.08835","url":null,"abstract":"We study the Fr\"ohlich polaron in the regime of strong coupling and prove\u0000the asymptotically sharp lower bound on the effective mass\u0000$m_{mathrm{eff}}(alpha)geq alpha^4 m_{mathrm{LP}}-Calpha^{4-epsilon}$,\u0000where $m_{mathrm{LP}}$ is an explicit constant. Together with the\u0000corresponding upper bound, which has been verified recently in [5], we confirm\u0000the validity of the celebrated Landau-Pekar formula [12] from 1948 for the\u0000effective mass $underset{alpharightarrow\u0000infty}{lim}alpha^{-4}m_{mathrm{eff}}(alpha)=m_{mathrm{LP}}$ as\u0000conjectured by Spohn [25] in 1987.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260521","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 a Schr"odinger-like equation for the anharmonic potential $x^{2 alpha}+ell(ell+1) x^{-2}-E$ when the anharmonicity $alpha$ goes to $+infty$. When $E$ and $ell$ vary in bounded domains, we show that the spectral determinant for the central connection problem converges to a special function written in terms of a Bessel function of order $ell+frac{1}{2}$ and its zeros converge to the zeros of that Bessel function. We then study the regime in which $E$ and $ell$ grow large as well, scaling as $Esim alpha^2 varepsilon^2$ and $ellsim alpha p$. When $varepsilon$ is greater than $1$ we show that the spectral determinant for the central connection problem is a rapidly oscillating function whose zeros tend to be distributed according to the continuous density law $frac{2p}{pi}frac{sqrt{varepsilon^2-1}}{varepsilon}$. When $varepsilon$ is close to $1$ we show that the spectral determinant converges to a function expressed in terms of the Airy function $operatorname{Ai}(-)$ and its zeros converge to the zeros of that function. This work is motivated by and has applications to the ODE/IM correspondence for the quantum KdV model.
{"title":"ODE/IM correspondence in the semiclassical limit: Large degree asymptotics of the spectral determinants for the ground state potential","authors":"Gabriele Degano","doi":"arxiv-2409.07866","DOIUrl":"https://doi.org/arxiv-2409.07866","url":null,"abstract":"We study a Schr\"odinger-like equation for the anharmonic potential $x^{2\u0000alpha}+ell(ell+1) x^{-2}-E$ when the anharmonicity $alpha$ goes to\u0000$+infty$. When $E$ and $ell$ vary in bounded domains, we show that the\u0000spectral determinant for the central connection problem converges to a special\u0000function written in terms of a Bessel function of order $ell+frac{1}{2}$ and\u0000its zeros converge to the zeros of that Bessel function. We then study the\u0000regime in which $E$ and $ell$ grow large as well, scaling as $Esim alpha^2\u0000varepsilon^2$ and $ellsim alpha p$. When $varepsilon$ is greater than $1$\u0000we show that the spectral determinant for the central connection problem is a\u0000rapidly oscillating function whose zeros tend to be distributed according to\u0000the continuous density law\u0000$frac{2p}{pi}frac{sqrt{varepsilon^2-1}}{varepsilon}$. When $varepsilon$\u0000is close to $1$ we show that the spectral determinant converges to a function\u0000expressed in terms of the Airy function $operatorname{Ai}(-)$ and its zeros\u0000converge to the zeros of that function. This work is motivated by and has\u0000applications to the ODE/IM correspondence for the quantum KdV model.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216323","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 introduce a class of examples which provide an affine generalization of the nonholonomic problem of a convex body rolling without slipping on the plane. We investigate dynamical aspects of the system such as existence of first integrals, smooth invariant measure and integrability, giving special attention to the cases in which the convex body is a dynamically balanced sphere or a body of revolution.
{"title":"Affine generalizations of the nonholonomic problem of a convex body rolling without slipping on the plane","authors":"M. Costa Villegas, L. C. García-Naranjo","doi":"arxiv-2409.08072","DOIUrl":"https://doi.org/arxiv-2409.08072","url":null,"abstract":"We introduce a class of examples which provide an affine generalization of\u0000the nonholonomic problem of a convex body rolling without slipping on the\u0000plane. We investigate dynamical aspects of the system such as existence of\u0000first integrals, smooth invariant measure and integrability, giving special\u0000attention to the cases in which the convex body is a dynamically balanced\u0000sphere or a body of revolution.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216320","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 a construction of scattering for wave-type equations with singular potentials on the whole space $mathbb{R}^n$ in a framework of weak-$L^p$ spaces. First, we use an Yamazaki-type estimate for wave groups on Lorentz spaces and fixed point arguments to prove the global well-posedness for wave-type equations on weak-$L^p$ spaces. Then, we provide a corresponding scattering results in such singular framework. Finally, we use also the dispersive estimates to establish the polynomial stability and improve the decay of scattering.
{"title":"Interpolation scattering for wave equations with singular potentials and singular data","authors":"Tran Thi Ngoc, Pham Truong Xuan","doi":"arxiv-2409.07867","DOIUrl":"https://doi.org/arxiv-2409.07867","url":null,"abstract":"In this paper we investigate a construction of scattering for wave-type\u0000equations with singular potentials on the whole space $mathbb{R}^n$ in a\u0000framework of weak-$L^p$ spaces. First, we use an Yamazaki-type estimate for\u0000wave groups on Lorentz spaces and fixed point arguments to prove the global\u0000well-posedness for wave-type equations on weak-$L^p$ spaces. Then, we provide a\u0000corresponding scattering results in such singular framework. Finally, we use\u0000also the dispersive estimates to establish the polynomial stability and improve\u0000the decay of scattering.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216326","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 a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. These Scattering Quantum Walks model the discrete dynamics of a system on the edges of the graph, with a scattering process at each vertex governed by the scattering matrix assigned to it. We show that Scattering Quantum Walks encompass several known Quantum Walks. Additionally, we introduce two classes of Open Scattering Quantum Walks on arbitrary graphs, also parameterized by scattering matrices: one class defined on the edges and the other on the vertices of the graph. We show that these walks give rise to proper Quantum Channels and describe their main spectral and dynamical properties, relating them to naturally associated classical Markov chains.
{"title":"Unitary and Open Scattering Quantum Walks on Graphs","authors":"Alain Joye","doi":"arxiv-2409.08428","DOIUrl":"https://doi.org/arxiv-2409.08428","url":null,"abstract":"We study a class of Unitary Quantum Walks on arbitrary graphs, parameterized\u0000by a family of scattering matrices. These Scattering Quantum Walks model the\u0000discrete dynamics of a system on the edges of the graph, with a scattering\u0000process at each vertex governed by the scattering matrix assigned to it. We\u0000show that Scattering Quantum Walks encompass several known Quantum Walks.\u0000Additionally, we introduce two classes of Open Scattering Quantum Walks on\u0000arbitrary graphs, also parameterized by scattering matrices: one class defined\u0000on the edges and the other on the vertices of the graph. We show that these\u0000walks give rise to proper Quantum Channels and describe their main spectral and\u0000dynamical properties, relating them to naturally associated classical Markov\u0000chains.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260522","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}
It is known that there are two inequivalent $mathbb{Z}_2^2$-graded $osp(1|2)$ Lie superalgebras. Their affine extensions are investigated and it is shown that one of them admits two central elements, one is non-graded and the other is $(1,1)$-graded. The affine $mathbb{Z}_2^2$-$osp(1|2)$ algebras are used by the Sugawara construction to study possible $mathbb{Z}_2^2$-graded extensions of the Virasoro algebra. We obtain a $mathbb{Z}_2^2$-graded Virasoro algebra with a non-trivially graded central element. Throughout the investigation, invariant bilinear forms on $mathbb{Z}_2^2$-graded superalgebras play a crucial role, so a theory of invariant bilinear forms is also developed.
{"title":"Affine extensions of $mathbb{Z}_2^2$-graded $osp(1|2)$ and Virasoro algebra","authors":"N. Aizawa, J. Segar","doi":"arxiv-2409.07938","DOIUrl":"https://doi.org/arxiv-2409.07938","url":null,"abstract":"It is known that there are two inequivalent $mathbb{Z}_2^2$-graded\u0000$osp(1|2)$ Lie superalgebras. Their affine extensions are investigated and it\u0000is shown that one of them admits two central elements, one is non-graded and\u0000the other is $(1,1)$-graded. The affine $mathbb{Z}_2^2$-$osp(1|2)$ algebras\u0000are used by the Sugawara construction to study possible $mathbb{Z}_2^2$-graded\u0000extensions of the Virasoro algebra. We obtain a $mathbb{Z}_2^2$-graded\u0000Virasoro algebra with a non-trivially graded central element. Throughout the\u0000investigation, invariant bilinear forms on $mathbb{Z}_2^2$-graded\u0000superalgebras play a crucial role, so a theory of invariant bilinear forms is\u0000also developed.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216322","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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