{"title":"弱扰动二维伊辛模型中单位时间演化的宏观热化--罗思-特乌费尔-图穆尔卡-沃格尔定理的应用","authors":"Hal Tasaki","doi":"arxiv-2409.09395","DOIUrl":null,"url":null,"abstract":"To demonstrate the implication of the recent important theorem by Roos,\nTeufel, Tumulka, and Vogel [1] in a simple but nontrivial example, we study\nthermalization in the two-dimensional Ising model in the low-temperature phase.\nWe consider the Hamiltonian $\\hat{H}_L$ of the standard ferromagnetic Ising\nmodel with the plus boundary conditions and perturb it with a small\nself-adjoint operator $\\lambda\\hat{V}$ drawn randomly from the space of\nself-adjoint operators on the whole Hilbert space. Suppose that the system is\ninitially in a classical spin configuration with a specified energy that may be\nvery far from thermal equilibrium. It is proved that, for most choices of the\nrandom perturbation, the unitary time evolution\n$e^{-i(\\hat{H}_L+\\lambda\\hat{V})t}$ brings the initial state into thermal\nequilibrium after a sufficiently long and typical time $t$, in the sense that\nthe measurement result of the magnetization density at time $t$ almost\ncertainly coincides with the spontaneous magnetization expected in the\ncorresponding equilibrium.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"5 2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Macroscopic thermalization by unitary time-evolution in the weakly perturbed two-dimensional Ising model --- An application of the Roos-Teufel-Tumulka-Vogel theorem\",\"authors\":\"Hal Tasaki\",\"doi\":\"arxiv-2409.09395\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"To demonstrate the implication of the recent important theorem by Roos,\\nTeufel, Tumulka, and Vogel [1] in a simple but nontrivial example, we study\\nthermalization in the two-dimensional Ising model in the low-temperature phase.\\nWe consider the Hamiltonian $\\\\hat{H}_L$ of the standard ferromagnetic Ising\\nmodel with the plus boundary conditions and perturb it with a small\\nself-adjoint operator $\\\\lambda\\\\hat{V}$ drawn randomly from the space of\\nself-adjoint operators on the whole Hilbert space. Suppose that the system is\\ninitially in a classical spin configuration with a specified energy that may be\\nvery far from thermal equilibrium. It is proved that, for most choices of the\\nrandom perturbation, the unitary time evolution\\n$e^{-i(\\\\hat{H}_L+\\\\lambda\\\\hat{V})t}$ brings the initial state into thermal\\nequilibrium after a sufficiently long and typical time $t$, in the sense that\\nthe measurement result of the magnetization density at time $t$ almost\\ncertainly coincides with the spontaneous magnetization expected in the\\ncorresponding equilibrium.\",\"PeriodicalId\":501312,\"journal\":{\"name\":\"arXiv - MATH - Mathematical Physics\",\"volume\":\"5 2 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Mathematical Physics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.09395\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Mathematical Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09395","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Macroscopic thermalization by unitary time-evolution in the weakly perturbed two-dimensional Ising model --- An application of the Roos-Teufel-Tumulka-Vogel theorem
To demonstrate the implication of the recent important theorem by Roos,
Teufel, Tumulka, and Vogel [1] in a simple but nontrivial example, we study
thermalization in the two-dimensional Ising model in the low-temperature phase.
We consider the Hamiltonian $\hat{H}_L$ of the standard ferromagnetic Ising
model with the plus boundary conditions and perturb it with a small
self-adjoint operator $\lambda\hat{V}$ drawn randomly from the space of
self-adjoint operators on the whole Hilbert space. Suppose that the system is
initially in a classical spin configuration with a specified energy that may be
very far from thermal equilibrium. It is proved that, for most choices of the
random perturbation, the unitary time evolution
$e^{-i(\hat{H}_L+\lambda\hat{V})t}$ brings the initial state into thermal
equilibrium after a sufficiently long and typical time $t$, in the sense that
the measurement result of the magnetization density at time $t$ almost
certainly coincides with the spontaneous magnetization expected in the
corresponding equilibrium.