{"title":"Markov generators as non-Hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and singular value decompositions","authors":"Cécile Monthus","doi":"10.1088/1742-5468/ad613a","DOIUrl":null,"url":null,"abstract":"Continuity equations associated with continuous-time Markov processes can be considered as Euclidean Schrödinger equations, where the non-Hermitian quantum Hamiltonian <inline-formula>\n<tex-math><?CDATA $\\boldsymbol{H} = {\\mathbf{div}}{\\boldsymbol{J}}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">H</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">div</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">J</mml:mi></mml:mrow></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad613aieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> is naturally factorized into the product of the divergence operator <inline-formula>\n<tex-math><?CDATA ${\\mathbf{div}}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">div</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad613aieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> and the current operator <bold>\n<italic toggle=\"yes\">J</italic>\n</bold>. For non-equilibrium Markov jump processes in a space of <italic toggle=\"yes\">N</italic> configurations with <italic toggle=\"yes\">M</italic> links and <inline-formula>\n<tex-math><?CDATA $C = M-(N-1)\\unicode{x2A7E} 1$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mi>M</mml:mi><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mtext>⩾</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad613aieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> independent cycles, this factorization of the <italic toggle=\"yes\">N</italic> × <italic toggle=\"yes\">N</italic> Hamiltonian <inline-formula>\n<tex-math><?CDATA ${\\boldsymbol{H}} = {\\boldsymbol{I}}^{\\dagger}{\\boldsymbol{J}}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">H</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">J</mml:mi></mml:mrow></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad613aieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> involves the incidence matrix <bold>\n<italic toggle=\"yes\">I</italic>\n</bold> and the current matrix <bold>\n<italic toggle=\"yes\">J</italic>\n</bold> of size <italic toggle=\"yes\">M</italic> × <italic toggle=\"yes\">N</italic>, so that the supersymmetric partner <inline-formula>\n<tex-math><?CDATA ${\\hat{\\boldsymbol{H}}} = {\\boldsymbol{J}}{\\boldsymbol{I}}^{\\dagger}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">H</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">J</mml:mi></mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad613aieqn5.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> governing the dynamics of the currents existing on the <italic toggle=\"yes\">M</italic> links is of size <italic toggle=\"yes\">M</italic> × <italic toggle=\"yes\">M</italic>. To better understand the relations between the spectral decompositions of these two Hamiltonians <inline-formula>\n<tex-math><?CDATA $\\boldsymbol{H} = {\\boldsymbol{I}}^{\\dagger}{\\boldsymbol{J}}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">H</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">J</mml:mi></mml:mrow></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad613aieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> and <inline-formula>\n<tex-math><?CDATA ${\\hat {\\boldsymbol{H}}} = {\\boldsymbol{J}}{\\boldsymbol{I}}^{\\dagger}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">H</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">J</mml:mi></mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad613aieqn7.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> with respect to their bi-orthogonal basis of right and left eigenvectors that characterize the relaxation dynamics towards the steady state and the steady currents, it is useful to analyze the properties of the singular value decomposition of the two rectangular matrices <bold>\n<italic toggle=\"yes\">I</italic>\n</bold> and <bold>\n<italic toggle=\"yes\">J</italic>\n</bold> of size <italic toggle=\"yes\">M</italic> × <italic toggle=\"yes\">N</italic> and the interpretations in terms of discrete Helmholtz decompositions. This general framework concerning Markov jump processes can be adapted to non-equilibrium diffusion processes governed by Fokker–Planck equations in dimension <italic toggle=\"yes\">d</italic>, where the number <italic toggle=\"yes\">N</italic> of configurations, the number <italic toggle=\"yes\">M</italic> of links and the number <inline-formula>\n<tex-math><?CDATA $C = M-(N-1)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mi>M</mml:mi><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad613aieqn8.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> of independent cycles become infinite, while the two matrices <bold>\n<italic toggle=\"yes\">I</italic>\n</bold> and <bold>\n<italic toggle=\"yes\">J</italic>\n</bold> become first-order differential operators acting on scalar functions to produce vector fields.","PeriodicalId":17207,"journal":{"name":"Journal of Statistical Mechanics: Theory and Experiment","volume":"7 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Mechanics: Theory and Experiment","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1742-5468/ad613a","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
Abstract
Continuity equations associated with continuous-time Markov processes can be considered as Euclidean Schrödinger equations, where the non-Hermitian quantum Hamiltonian H=divJ is naturally factorized into the product of the divergence operator div and the current operator J. For non-equilibrium Markov jump processes in a space of N configurations with M links and C=M−(N−1)⩾1 independent cycles, this factorization of the N × N Hamiltonian H=I†J involves the incidence matrix I and the current matrix J of size M × N, so that the supersymmetric partner H^=JI† governing the dynamics of the currents existing on the M links is of size M × M. To better understand the relations between the spectral decompositions of these two Hamiltonians H=I†J and H^=JI† with respect to their bi-orthogonal basis of right and left eigenvectors that characterize the relaxation dynamics towards the steady state and the steady currents, it is useful to analyze the properties of the singular value decomposition of the two rectangular matrices I and J of size M × N and the interpretations in terms of discrete Helmholtz decompositions. This general framework concerning Markov jump processes can be adapted to non-equilibrium diffusion processes governed by Fokker–Planck equations in dimension d, where the number N of configurations, the number M of links and the number C=M−(N−1) of independent cycles become infinite, while the two matrices I and J become first-order differential operators acting on scalar functions to produce vector fields.
与连续时间马尔可夫过程相关的连续性方程可视为欧几里得薛定谔方程,其中非赫米态量子哈密顿方程 H=divJ 自然被因子化为发散算子 div 与电流算子 J 的乘积。对于具有 M 个链接和 C=M-(N-1)⩾1 个独立循环的 N 个构型空间中的非平衡马尔可夫跃迁过程,N × N 哈密顿H=I†J 的这种因式分解涉及大小为 M × N 的入射矩阵 I 和电流矩阵 J,因此支配存在于 M 个链接上的电流动力学的超对称伙伴 H^=JI† 的大小为 M × M。为了更好地理解这两个哈密顿的频谱分解 H=I†J 和 H^=JI† 与它们的左右特征向量双正交基础之间的关系,这两个特征向量描述了向稳态的弛豫动力学和稳态电流,分析大小为 M × N 的两个矩形矩阵 I 和 J 的奇异值分解的性质以及离散亥姆霍兹分解的解释是有用的。这个关于马尔科夫跃迁过程的一般框架可以适用于由 d 维福克-普朗克方程控制的非平衡扩散过程,在此过程中,配置数 N、链接数 M 和独立循环数 C=M-(N-1) 变得无限大,而两个矩阵 I 和 J 成为作用于标量函数的一阶微分算子,从而产生矢量场。
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