We contribute an extension of large-deviation results obtained in [N.J.B. Aza, J.-B. Bru, W. de Siqueira Pedra, A. Ratsimanetrimanana, J. Math. Pures Appl. 125 (2019) 209] on conductivity theory at atomic scale of free lattice fermions in disordered media. Disorder is modeled by (i) a random external potential, like in the celebrated Anderson model, and (ii) a nearest-neighbor hopping term with random complex-valued amplitudes. In accordance with experimental observations, via the large deviation formalism, our previous paper showed in this case that quantum uncertainty of microscopic electric current densities around their (classical) macroscopic value is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. Here, the quantum fluctuations of linear response currents are shown to exist in the thermodynamic limit and we mathematically prove that they are related to the rate function of the large deviation principle associated with current densities. We also demonstrate that, in general, they do not vanish (in the thermodynamic limit) and the quantum uncertainty around the macroscopic current density disappears exponentially fast with an exponential rate proportional to the squared deviation of the current from its macroscopic value and the inverse current fluctuation, with respect to growing space (volume) scales.
我们提供了在[N.J.B.]中得到的大偏差结果的扩展阿扎,J.-B。Bru, W. de Siqueira Pedra, A. Ratsimanetrimanana, J. Math。[2]张建军，张建军。无序介质中自由晶格费米子的原子尺度电导率理论。光子学报，125(2019):209]。无序由(i)一个随机的外部电位，如著名的安德森模型，和(ii)一个具有随机复值振幅的最近邻跳跃项来建模。根据实验观察，通过大偏差形式，我们之前的论文表明，在这种情况下，微观电流密度在其(经典)宏观值周围的量子不确定性被抑制，相对于施加外电场的晶格区域的体积而言，其速度呈指数级增长。本文证明了线性响应电流的量子涨落存在于热力学极限，并从数学上证明了它们与电流密度相关的大偏差原理的速率函数有关。我们还证明，在一般情况下，它们不会消失(在热力学极限下)，宏观电流密度周围的量子不确定性以指数速度消失，其指数速率与电流与其宏观值的平方偏差和反向电流波动成正比，相对于增长的空间(体积)尺度。
{"title":"Quantum Fluctuations and Large Deviation Principle for Microscopic Currents of Free Fermions in Disordered Media","authors":"J. Bru, W. Pedra, A. Ratsimanetrimanana","doi":"10.2140/PAA.2020.2.205","DOIUrl":"https://doi.org/10.2140/PAA.2020.2.205","url":null,"abstract":"We contribute an extension of large-deviation results obtained in [N.J.B. Aza, J.-B. Bru, W. de Siqueira Pedra, A. Ratsimanetrimanana, J. Math. Pures Appl. 125 (2019) 209] on conductivity theory at atomic scale of free lattice fermions in disordered media. Disorder is modeled by (i) a random external potential, like in the celebrated Anderson model, and (ii) a nearest-neighbor hopping term with random complex-valued amplitudes. In accordance with experimental observations, via the large deviation formalism, our previous paper showed in this case that quantum uncertainty of microscopic electric current densities around their (classical) macroscopic value is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. Here, the quantum fluctuations of linear response currents are shown to exist in the thermodynamic limit and we mathematically prove that they are related to the rate function of the large deviation principle associated with current densities. We also demonstrate that, in general, they do not vanish (in the thermodynamic limit) and the quantum uncertainty around the macroscopic current density disappears exponentially fast with an exponential rate proportional to the squared deviation of the current from its macroscopic value and the inverse current fluctuation, with respect to growing space (volume) scales.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75087250","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 study generalized time-fractional diffusion equations on the Poincar`e half plane $mathbb{H}_2^+$. The time-fractional operators here considered are fractional derivatives of a function with respect to another function, that can be obtained by starting from the classical Caputo-derivatives essentially by means of a deterministic change of variable. We obtain an explicit representation of the fundamental solution of the generalized-diffusion equation on $mathbb{H}_2^+$ and provide a probabilistic interpretation related to the time-changed hyperbolic Brownian motion. We finally include an explicit result regarding the non-linear case admitting a separating variable solution.
{"title":"A note on generalized fractional diffusion equations on Poincaré half plane","authors":"R. Garra, F. Maltese, E. Orsingher","doi":"10.7153/fdc-2021-11-07","DOIUrl":"https://doi.org/10.7153/fdc-2021-11-07","url":null,"abstract":"In this paper we study generalized time-fractional diffusion equations on the Poincar`e half plane $mathbb{H}_2^+$. The time-fractional operators here considered are fractional derivatives of a function with respect to another function, that can be obtained by starting from the classical Caputo-derivatives essentially by means of a deterministic change of variable. We obtain an explicit representation of the fundamental solution of the generalized-diffusion equation on $mathbb{H}_2^+$ and provide a probabilistic interpretation related to the time-changed hyperbolic Brownian motion. We finally include an explicit result regarding the non-linear case admitting a separating variable solution.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81379456","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 apply symmetric function theory to study random processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes are degenerations of Macdonald processes introduced by Borodin and Corwin. Through this connection, we obtain explicit formulae for the distribution of singular values of a deterministic matrix multiplied by a truncated Haar orthogonal or symplectic matrix under conditions where the latter factor acts as a rank $1$ perturbation. Consequently, we generalize the recent Kieburg-Kuijlaars-Stivigny formula for the joint singular value density of a product of truncated unitary matrices to symplectic and orthogonal symmetry classes. Specializing to products of two symplectic matrices with a rank $1$ perturbative factor, we show that the squared singular values form a Pfaffian point process.
{"title":"Product Matrix Processes With Symplectic and Orthogonal Invariance via Symmetric Functions","authors":"Andrew Ahn, E. Strahov","doi":"10.1093/IMRN/RNAB045","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB045","url":null,"abstract":"We apply symmetric function theory to study random processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes are degenerations of Macdonald processes introduced by Borodin and Corwin. Through this connection, we obtain explicit formulae for the distribution of singular values of a deterministic matrix multiplied by a truncated Haar orthogonal or symplectic matrix under conditions where the latter factor acts as a rank $1$ perturbation. Consequently, we generalize the recent Kieburg-Kuijlaars-Stivigny formula for the joint singular value density of a product of truncated unitary matrices to symplectic and orthogonal symmetry classes. Specializing to products of two symplectic matrices with a rank $1$ perturbative factor, we show that the squared singular values form a Pfaffian point process.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74020197","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}
An autonomous dynamical system is described by a system of second order differential equations whose solution gives the trajectories of the system. The solution is facilitated by the use of first integrals (FIs) that are used to reduce the order of the system of differential equations and, if there are enough of them, to determine the solution. Therefore, it is important that there exists a systematic method to determine the FIs. On the other hand, a system of second order differential equations defines a kinetic energy, which provides a symmetric second order tensor called kinetic metric of the system. This metric via its symmetries brings into the scene the numerous methods of differential geometry and hence it is apparent that one should manage to relate the determination of the FIs to the symmetries of the kinetic metric. The subject of this work is to provide a theorem that realizes this scenario. The method we follow considers the generic quadratic FI of the form $I=K_{ab}(t,q^{c})dot{q}^{a}dot{q}^{b}+K_{a}(t,q^{c})dot{q}^{a} +K(t,q^{c})$ where $K_{ab}(t,q^{c}), K_{a}(t,q^{c}), K(t,q^{c})$ are unknown tensor quantities and requires $dI/dt = 0$. This condition leads to a system of differential equations involving the coefficients of $I$ whose solution provides all possible quadratic FIs of this form. We demonstrate the application of the theorem in the classical cases of the geodesic equations and the generalized Kepler potential. We also obtain and discuss the time-dependent FIs.
{"title":"Quadratic first integrals of autonomous conservative dynamical systems","authors":"M. Tsamparlis, Antonios Mitsopoulos","doi":"10.1063/1.5141392","DOIUrl":"https://doi.org/10.1063/1.5141392","url":null,"abstract":"An autonomous dynamical system is described by a system of second order differential equations whose solution gives the trajectories of the system. The solution is facilitated by the use of first integrals (FIs) that are used to reduce the order of the system of differential equations and, if there are enough of them, to determine the solution. Therefore, it is important that there exists a systematic method to determine the FIs. On the other hand, a system of second order differential equations defines a kinetic energy, which provides a symmetric second order tensor called kinetic metric of the system. This metric via its symmetries brings into the scene the numerous methods of differential geometry and hence it is apparent that one should manage to relate the determination of the FIs to the symmetries of the kinetic metric. The subject of this work is to provide a theorem that realizes this scenario. The method we follow considers the generic quadratic FI of the form $I=K_{ab}(t,q^{c})dot{q}^{a}dot{q}^{b}+K_{a}(t,q^{c})dot{q}^{a} +K(t,q^{c})$ where $K_{ab}(t,q^{c}), K_{a}(t,q^{c}), K(t,q^{c})$ are unknown tensor quantities and requires $dI/dt = 0$. This condition leads to a system of differential equations involving the coefficients of $I$ whose solution provides all possible quadratic FIs of this form. We demonstrate the application of the theorem in the classical cases of the geodesic equations and the generalized Kepler potential. We also obtain and discuss the time-dependent FIs.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82993655","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 : 2020-07-19DOI: 10.1007/978-3-030-53305-2_5
L. Fehér, I. Marshall
{"title":"On the bi-Hamiltonian Structure of the Trigonometric Spin Ruijsenaars–Sutherland Hierarchy","authors":"L. Fehér, I. Marshall","doi":"10.1007/978-3-030-53305-2_5","DOIUrl":"https://doi.org/10.1007/978-3-030-53305-2_5","url":null,"abstract":"","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78203091","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 complex Monge-Ampere equation $(CMA)$ in a two-component form is treated as a bi-Hamiltonian system. We present explicitly the first nonlocal symmetry flow in the hierarchy of this system. An invariant solution of $CMA$ with respect to this nonlocal symmetry is constructed which, being a noninvariant solution in the usual sense, does not undergo symmetry reduction in the number of independent variables. We also construct the corresponding 4-dimensional anti-self-dual (ASD) gravitational metric with either Euclidean or neutral signature. It admits no Killing vectors which is one of characteristic features of the famous gravitational instanton $K3$.
{"title":"Nonlocal symmetry of CMA generates ASD Ricci-flat metric with no Killing vectors","authors":"M. Sheftel","doi":"10.1063/5.0022021","DOIUrl":"https://doi.org/10.1063/5.0022021","url":null,"abstract":"The complex Monge-Ampere equation $(CMA)$ in a two-component form is treated as a bi-Hamiltonian system. We present explicitly the first nonlocal symmetry flow in the hierarchy of this system. An invariant solution of $CMA$ with respect to this nonlocal symmetry is constructed which, being a noninvariant solution in the usual sense, does not undergo symmetry reduction in the number of independent variables. We also construct the corresponding 4-dimensional anti-self-dual (ASD) gravitational metric with either Euclidean or neutral signature. It admits no Killing vectors which is one of characteristic features of the famous gravitational instanton $K3$.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79687137","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 : 2020-07-15DOI: 10.1007/978-3-030-31531-3_14
K. Cherednichenko, A. Kiselev, Luis O. Silva
{"title":"Scattering theory for a class of non-selfadjoint extensions of symmetric operators","authors":"K. Cherednichenko, A. Kiselev, Luis O. Silva","doi":"10.1007/978-3-030-31531-3_14","DOIUrl":"https://doi.org/10.1007/978-3-030-31531-3_14","url":null,"abstract":"","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72572965","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}
Very loosely, $mathbb{Z}_2^n$-manifolds are `manifolds' with $mathbb{Z}_2^n$-graded coordinates and their sign rule is determined by the scalar product of their $mathbb{Z}_2^n$-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian $mathbb{Z}_2^n$-manifold, i.e., a $mathbb{Z}_2^n$-manifold equipped with a Riemannian metric that may carry non-zero $mathbb{Z}_2^n$-degree. We show that the basic notions and tenets of Riemannian geometry directly generalise to the setting of $mathbb{Z}_2^n$-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.
{"title":"Riemannian Structures on Z 2 n -Manifolds","authors":"A. Bruce, J. Grabowski","doi":"10.3390/math8091469","DOIUrl":"https://doi.org/10.3390/math8091469","url":null,"abstract":"Very loosely, $mathbb{Z}_2^n$-manifolds are `manifolds' with $mathbb{Z}_2^n$-graded coordinates and their sign rule is determined by the scalar product of their $mathbb{Z}_2^n$-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian $mathbb{Z}_2^n$-manifold, i.e., a $mathbb{Z}_2^n$-manifold equipped with a Riemannian metric that may carry non-zero $mathbb{Z}_2^n$-degree. We show that the basic notions and tenets of Riemannian geometry directly generalise to the setting of $mathbb{Z}_2^n$-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81631994","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 effect of repulsive pair interactions on Bose-Einstein condensation in a well-established random one-dimensional system known as the Luttinger-Sy model at positive temperature. We study separately hard core interactions as well as a class of more general repulsive interactions, also allowing for a scaling of certain interaction parameters in the thermodynamic limit. As a main result, we prove in both cases that for sufficiently strong interactions all eigenstates of the non-interacting one-particle Luttinger-Sy Hamiltonian as well as any sufficiently localized one-particle state are almost surely not macroscopically occupied.
{"title":"On the effect of repulsive pair interactions on Bose–Einstein condensation in the Luttinger–Sy model","authors":"J. Kerner, M. Pechmann","doi":"10.1090/proc/15424","DOIUrl":"https://doi.org/10.1090/proc/15424","url":null,"abstract":"In this paper we investigate the effect of repulsive pair interactions on Bose-Einstein condensation in a well-established random one-dimensional system known as the Luttinger-Sy model at positive temperature. We study separately hard core interactions as well as a class of more general repulsive interactions, also allowing for a scaling of certain interaction parameters in the thermodynamic limit. As a main result, we prove in both cases that for sufficiently strong interactions all eigenstates of the non-interacting one-particle Luttinger-Sy Hamiltonian as well as any sufficiently localized one-particle state are almost surely not macroscopically occupied.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76210570","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 consider Fredholm determinants of matrix convolution operators associated to matrix versions of the $n - $th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painleve II hierarchy, defined through a matrix valued version of the Lenard operators. In particular, the Riemann-Hilbert technique used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitely written in terms of these matrix valued Lenard operators and some solution of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy convolution operators.
{"title":"A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants","authors":"Sofia Tarricone","doi":"10.3842/sigma.2021.002","DOIUrl":"https://doi.org/10.3842/sigma.2021.002","url":null,"abstract":"We consider Fredholm determinants of matrix convolution operators associated to matrix versions of the $n - $th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painleve II hierarchy, defined through a matrix valued version of the Lenard operators. In particular, the Riemann-Hilbert technique used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitely written in terms of these matrix valued Lenard operators and some solution of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy convolution operators.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73988619","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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