Jitendra Kethepalli, Manas Kulkarni, Anupam Kundu, Satya N Majumdar, David Mukamel, Grégory Schehr
{"title":"密闭 1d 短程里兹气体的全计数统计","authors":"Jitendra Kethepalli, Manas Kulkarni, Anupam Kundu, Satya N Majumdar, David Mukamel, Grégory Schehr","doi":"10.1088/1742-5468/ad66c5","DOIUrl":null,"url":null,"abstract":"We investigate the full counting statistics of a harmonically confined 1d short range Riesz gas consisting of <italic toggle=\"yes\">N</italic> particles in equilibrium at finite temperature. The particles interact with each other through a repulsive power-law interaction with an exponent <italic toggle=\"yes\">k</italic> > 1 which includes the Calogero–Moser model for <italic toggle=\"yes\">k</italic> = 2. We examine the probability distribution of the number of particles in a finite domain <inline-formula>\n<tex-math><?CDATA $[-W, W]$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad66c5ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> called number distribution, denoted by <inline-formula>\n<tex-math><?CDATA $\\mathcal{N}(W, N)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad66c5ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>. We analyze the probability distribution of <inline-formula>\n<tex-math><?CDATA $\\mathcal{N}(W, N)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad66c5ieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> and show that it exhibits a large deviation form for large <italic toggle=\"yes\">N</italic> characterized by a speed <inline-formula>\n<tex-math><?CDATA $N^{\\frac{3k+2}{k+2}}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad66c5ieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> and by a large deviation function (LDF) of the fraction <inline-formula>\n<tex-math><?CDATA $c = \\mathcal{N}(W, N)/N$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad66c5ieqn5.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> of the particles inside the domain and <italic toggle=\"yes\">W</italic>. We show that the density profiles that create the large deviations display interesting shape transitions as one varies <italic toggle=\"yes\">c</italic> and <italic toggle=\"yes\">W</italic>. This is manifested by a third-order phase transition exhibited by the LDF that has discontinuous third derivatives. Monte–Carlo simulations based on Metropolis–Hashtings (MH) algorithm show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of <inline-formula>\n<tex-math><?CDATA $\\mathcal{N}(W, N)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad66c5ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>, obtained from our field theoretic calculations are Gaussian distributed with a variance that scales as <inline-formula>\n<tex-math><?CDATA $N^{\\nu_k}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:msub><mml:mi>ν</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad66c5ieqn7.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>, with <inline-formula>\n<tex-math><?CDATA $\\nu_k = (2-k)/(2+k)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mi>ν</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad66c5ieqn8.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>. We also present some numerical findings on the mean and the variance. Furthermore, we adapt our formalism to study the index distribution (where the domain is semi-infinite <inline-formula>\n<tex-math><?CDATA $(-\\infty, W])$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo>,</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad66c5ieqn9.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>, linear statistics (the variance), thermodynamic pressure and bulk modulus.","PeriodicalId":17207,"journal":{"name":"Journal of Statistical Mechanics: Theory and Experiment","volume":"45 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Full counting statistics of 1d short range Riesz gases in confinement\",\"authors\":\"Jitendra Kethepalli, Manas Kulkarni, Anupam Kundu, Satya N Majumdar, David Mukamel, Grégory Schehr\",\"doi\":\"10.1088/1742-5468/ad66c5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We investigate the full counting statistics of a harmonically confined 1d short range Riesz gas consisting of <italic toggle=\\\"yes\\\">N</italic> particles in equilibrium at finite temperature. The particles interact with each other through a repulsive power-law interaction with an exponent <italic toggle=\\\"yes\\\">k</italic> > 1 which includes the Calogero–Moser model for <italic toggle=\\\"yes\\\">k</italic> = 2. We examine the probability distribution of the number of particles in a finite domain <inline-formula>\\n<tex-math><?CDATA $[-W, W]$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mo stretchy=\\\"false\\\">[</mml:mo><mml:mo>−</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy=\\\"false\\\">]</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad66c5ieqn1.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> called number distribution, denoted by <inline-formula>\\n<tex-math><?CDATA $\\\\mathcal{N}(W, N)$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad66c5ieqn2.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>. We analyze the probability distribution of <inline-formula>\\n<tex-math><?CDATA $\\\\mathcal{N}(W, N)$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad66c5ieqn3.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> and show that it exhibits a large deviation form for large <italic toggle=\\\"yes\\\">N</italic> characterized by a speed <inline-formula>\\n<tex-math><?CDATA $N^{\\\\frac{3k+2}{k+2}}$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad66c5ieqn4.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> and by a large deviation function (LDF) of the fraction <inline-formula>\\n<tex-math><?CDATA $c = \\\\mathcal{N}(W, N)/N$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad66c5ieqn5.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> of the particles inside the domain and <italic toggle=\\\"yes\\\">W</italic>. We show that the density profiles that create the large deviations display interesting shape transitions as one varies <italic toggle=\\\"yes\\\">c</italic> and <italic toggle=\\\"yes\\\">W</italic>. This is manifested by a third-order phase transition exhibited by the LDF that has discontinuous third derivatives. Monte–Carlo simulations based on Metropolis–Hashtings (MH) algorithm show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of <inline-formula>\\n<tex-math><?CDATA $\\\\mathcal{N}(W, N)$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mi>W</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad66c5ieqn6.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>, obtained from our field theoretic calculations are Gaussian distributed with a variance that scales as <inline-formula>\\n<tex-math><?CDATA $N^{\\\\nu_k}$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:msub><mml:mi>ν</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad66c5ieqn7.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>, with <inline-formula>\\n<tex-math><?CDATA $\\\\nu_k = (2-k)/(2+k)$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mi>ν</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad66c5ieqn8.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>. We also present some numerical findings on the mean and the variance. Furthermore, we adapt our formalism to study the index distribution (where the domain is semi-infinite <inline-formula>\\n<tex-math><?CDATA $(-\\\\infty, W])$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\\\"normal\\\">∞</mml:mi><mml:mo>,</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy=\\\"false\\\">]</mml:mo><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad66c5ieqn9.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>, linear statistics (the variance), thermodynamic pressure and bulk modulus.\",\"PeriodicalId\":17207,\"journal\":{\"name\":\"Journal of Statistical Mechanics: Theory and Experiment\",\"volume\":\"45 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/ad66c5\",\"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}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Mechanics: Theory and Experiment","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1742-5468/ad66c5","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
摘要
我们研究了由 N 个在有限温度下处于平衡状态的粒子组成的谐约束 1d 短程里兹气体的全计数统计。粒子通过指数为 k > 1 的幂律斥力相互作用相互影响,其中包括 k = 2 的卡洛吉罗-莫泽模型。我们研究的是有限域 [-W,W] 中粒子数量的概率分布,称为数量分布,用 N(W,N) 表示。我们分析了 N(W,N)的概率分布,并证明它在大 N 的情况下表现出大偏差形式,其特征是速度 N3k+2k+2 和域内粒子分数 c=N(W,N)/N 的大偏差函数 (LDF)。基于 Metropolis-Hashtings(MH)算法的蒙特卡洛模拟结果表明,我们对相应密度曲线的分析表达式与之非常吻合。我们发现,通过场论计算得到的 N(W,N)的典型波动是高斯分布的,其方差与 Nνk 成比例,νk=(2-k)/(2+k)。我们还给出了一些关于均值和方差的数值结果。此外,我们还调整了形式主义,以研究指数分布(域为半无限(-∞,W])、线性统计(方差)、热力学压力和体积模量。
Full counting statistics of 1d short range Riesz gases in confinement
We investigate the full counting statistics of a harmonically confined 1d short range Riesz gas consisting of N particles in equilibrium at finite temperature. The particles interact with each other through a repulsive power-law interaction with an exponent k > 1 which includes the Calogero–Moser model for k = 2. We examine the probability distribution of the number of particles in a finite domain [−W,W] called number distribution, denoted by N(W,N). We analyze the probability distribution of N(W,N) and show that it exhibits a large deviation form for large N characterized by a speed N3k+2k+2 and by a large deviation function (LDF) of the fraction c=N(W,N)/N of the particles inside the domain and W. We show that the density profiles that create the large deviations display interesting shape transitions as one varies c and W. This is manifested by a third-order phase transition exhibited by the LDF that has discontinuous third derivatives. Monte–Carlo simulations based on Metropolis–Hashtings (MH) algorithm show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of N(W,N), obtained from our field theoretic calculations are Gaussian distributed with a variance that scales as Nνk, with νk=(2−k)/(2+k). We also present some numerical findings on the mean and the variance. Furthermore, we adapt our formalism to study the index distribution (where the domain is semi-infinite (−∞,W]), linear statistics (the variance), thermodynamic pressure and bulk modulus.
期刊介绍:
JSTAT is targeted to a broad community interested in different aspects of statistical physics, which are roughly defined by the fields represented in the conferences called ''Statistical Physics''. Submissions from experimentalists working on all the topics which have some ''connection to statistical physics are also strongly encouraged.
The journal covers different topics which correspond to the following keyword sections.
1. Quantum statistical physics, condensed matter, integrable systems
Scientific Directors: Eduardo Fradkin and Giuseppe Mussardo
2. Classical statistical mechanics, equilibrium and non-equilibrium
Scientific Directors: David Mukamel, Matteo Marsili and Giuseppe Mussardo
3. Disordered systems, classical and quantum
Scientific Directors: Eduardo Fradkin and Riccardo Zecchina
4. Interdisciplinary statistical mechanics
Scientific Directors: Matteo Marsili and Riccardo Zecchina
5. Biological modelling and information
Scientific Directors: Matteo Marsili, William Bialek and Riccardo Zecchina