A system of interacting walkers on a two-dimensional space where the dynamics of each walker are governed by the opinions of agents of a three-state opinion dynamics model are considered. Such walks, inspired by Ising-like models and opinions dynamics models, are usually considered in one-dimensional virtual spaces. Here, the mapping is done in such a way that the walk is directed along the $y$ axis while it can move either way along the $x$ axis. We explore the properties of such walks as the parameter representing the noise in the opinion dynamics model, responsible for a continuous phase transition, is varied. The walk features show marked differences as the system crosses the critical point. The bivariate distribution of the displacements below the critical point is a modified biased Gaussian function of x and y which is symmetric about the X axis. The marginal probability distributions can be extracted and the scaling forms of different quantities, showing power law behaviour, are obtained. The directed nature of the walk is reflected in the marginal distributions as well as in the exponents.
本研究考虑了二维空间中相互作用的步行者系统，其中每个步行者的动态都受三态舆论动力学模型中代理人意见的支配。这种行走受类似 Ising 模型和舆论动力学模型的启发，通常在一维虚拟空间中考虑。在这里，映射的方式是让行走沿 y$ 轴定向，同时可以沿 x$ 轴任一方向移动。我们探讨了随着舆论动力学模型中负责连续相变的噪声参数的变化，这种行走的特性。临界点以下位移的双变量分布是 x 和 y 的修正偏置高斯函数，它与 X 轴对称。可以提取边际概率分布，并得到不同量的缩放形式，显示出幂律行为。边际分布和指数都反映了行走的定向性质。
{"title":"On interactive anisotropic walks in two dimensions generated from a three state opinion dynamics model","authors":"Surajit Saha, Parongama Sen","doi":"arxiv-2409.10413","DOIUrl":"https://doi.org/arxiv-2409.10413","url":null,"abstract":"A system of interacting walkers on a two-dimensional space where the dynamics\u0000of each walker are governed by the opinions of agents of a three-state opinion\u0000dynamics model are considered. Such walks, inspired by Ising-like models and\u0000opinions dynamics models, are usually considered in one-dimensional virtual\u0000spaces. Here, the mapping is done in such a way that the walk is directed along\u0000the $y$ axis while it can move either way along the $x$ axis. We explore the\u0000properties of such walks as the parameter representing the noise in the opinion\u0000dynamics model, responsible for a continuous phase transition, is varied. The\u0000walk features show marked differences as the system crosses the critical point.\u0000The bivariate distribution of the displacements below the critical point is a\u0000modified biased Gaussian function of x and y which is symmetric about the X\u0000axis. The marginal probability distributions can be extracted and the scaling forms\u0000of different quantities, showing power law behaviour, are obtained. The\u0000directed nature of the walk is reflected in the marginal distributions as well\u0000as in the exponents.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268519","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}
Applying a time-periodic magnetic field to the standard ferromagnetic Curie-Weiss model brings the spin system in a steady out-of-equilibrium condition. We recall how the hysteresis gets influenced by the amplitude and the frequency of that field, and how an amplitude- and frequency-dependent (dynamical) critical temperature can be discerned. The dissipated power measures the area of the hysteresis loop and changes with temperature. The excess heat determines a nonequilibrium specific heat giving the quasistatic response. We compute that specific heat, which appears to diverge at the critical temperature, quite different from the equilibrium case.
{"title":"Specific heat of the driven Curie-Weiss model","authors":"Elena Rufeil Fiori, Christian Maes, Robbe Vidts","doi":"arxiv-2409.10198","DOIUrl":"https://doi.org/arxiv-2409.10198","url":null,"abstract":"Applying a time-periodic magnetic field to the standard ferromagnetic\u0000Curie-Weiss model brings the spin system in a steady out-of-equilibrium\u0000condition. We recall how the hysteresis gets influenced by the amplitude and\u0000the frequency of that field, and how an amplitude- and frequency-dependent\u0000(dynamical) critical temperature can be discerned. The dissipated power\u0000measures the area of the hysteresis loop and changes with temperature. The\u0000excess heat determines a nonequilibrium specific heat giving the quasistatic\u0000response. We compute that specific heat, which appears to diverge at the\u0000critical temperature, quite different from the equilibrium case.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"198 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268520","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}
Flocking is a prime example of how robust collective behavior can emerge from simple interaction rules. The flocking transition has been studied extensively since the inception of the original Vicsek model. Here, we introduce a novel self-propelled particle model with quenched disorder in the pairwise alignment interaction couplings akin to a spin glass model. We show that the presence of quenched disorder can promote (rather than destroy) the emergence of global polar order. In particular, we show that our model can display a flocking phase even when the majority of the interaction couplings are anti-aligning. Activity is the key ingredient to reduce frustration in the system as it allows local particle clustering combined with self-organization of the particles to favor neighborhoods with strong cooperative interactions.
{"title":"Disordered Yet Directed: The Emergence of Polar Flocks with Disordered Interactions","authors":"Eloise Lardet, Raphaël Voituriez, Silvia Grigolon, Thibault Bertrand","doi":"arxiv-2409.10768","DOIUrl":"https://doi.org/arxiv-2409.10768","url":null,"abstract":"Flocking is a prime example of how robust collective behavior can emerge from\u0000simple interaction rules. The flocking transition has been studied extensively\u0000since the inception of the original Vicsek model. Here, we introduce a novel\u0000self-propelled particle model with quenched disorder in the pairwise alignment\u0000interaction couplings akin to a spin glass model. We show that the presence of\u0000quenched disorder can promote (rather than destroy) the emergence of global\u0000polar order. In particular, we show that our model can display a flocking phase\u0000even when the majority of the interaction couplings are anti-aligning. Activity\u0000is the key ingredient to reduce frustration in the system as it allows local\u0000particle clustering combined with self-organization of the particles to favor\u0000neighborhoods with strong cooperative interactions.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252067","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}
Yating Wang, Enmai Lei, Yu-Han Ma, Z. C. Tu, Geng Li
Active matter represents a class of non-equilibrium systems that constantly dissipate energy to produce directed motion. The thermodynamic control of active matter holds great potential for advancements in synthetic molecular motors, targeted drug delivery, and adaptive smart materials. However, the inherently non-equilibrium nature of active matter poses a significant challenge in achieving optimal control with minimal energy cost. In this work, we extend the concept of thermodynamic geometry, traditionally applied to passive systems, to active matter, proposing a systematic geometric framework for minimizing energy cost in non-equilibrium driving processes. We derive a cost metric that defines a Riemannian manifold for control parameters, enabling the use of powerful geometric tools to determine optimal control protocols. The geometric perspective reveals that, unlike in passive systems, minimizing energy cost in active systems involves a trade-off between intrinsic and external dissipation, leading to an optimal transportation speed that coincides with the self-propulsion speed of active matter. This insight enriches the broader concept of thermodynamic geometry. We demonstrate the application of this approach by optimizing the performance of an active monothermal engine within this geometric framework.
{"title":"Thermodynamic Geometric Control of Active Matter","authors":"Yating Wang, Enmai Lei, Yu-Han Ma, Z. C. Tu, Geng Li","doi":"arxiv-2409.09994","DOIUrl":"https://doi.org/arxiv-2409.09994","url":null,"abstract":"Active matter represents a class of non-equilibrium systems that constantly\u0000dissipate energy to produce directed motion. The thermodynamic control of\u0000active matter holds great potential for advancements in synthetic molecular\u0000motors, targeted drug delivery, and adaptive smart materials. However, the\u0000inherently non-equilibrium nature of active matter poses a significant\u0000challenge in achieving optimal control with minimal energy cost. In this work,\u0000we extend the concept of thermodynamic geometry, traditionally applied to\u0000passive systems, to active matter, proposing a systematic geometric framework\u0000for minimizing energy cost in non-equilibrium driving processes. We derive a\u0000cost metric that defines a Riemannian manifold for control parameters, enabling\u0000the use of powerful geometric tools to determine optimal control protocols. The\u0000geometric perspective reveals that, unlike in passive systems, minimizing\u0000energy cost in active systems involves a trade-off between intrinsic and\u0000external dissipation, leading to an optimal transportation speed that coincides\u0000with the self-propulsion speed of active matter. This insight enriches the\u0000broader concept of thermodynamic geometry. We demonstrate the application of\u0000this approach by optimizing the performance of an active monothermal engine\u0000within this geometric framework.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252068","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}
Soumen Das, Shankar Ghosh, Tridib Sadhu, Juliane U Klamser
Active-matter systems are inherently out-of-equilibrium and perform mechanical work by utilizing their internal energy sources. Breakdown of time-reversal symmetry (BTRS) is a hallmark of such dissipative non-equilibrium dynamics. We introduce a robust, experimentally accessible, noninvasive, quantitative measure of BTRS in terms of the Kullback-Leibler divergence in collision events, demonstrated in our novel artificial active matter, comprised of battery-powered spherical rolling robots whose energetics in different modes of motion can be measured with high precision. Our dimensionless measure characterizes how dissipation and internal energetics are influenced by kinematic constraints from interactions with the environment. We propose this measure of BTRS as an empirical estimate of the distance from equilibrium. An energetic insight into this departure of active matter from equilibrium comes from our demonstration of a non-trivial fluctuation symmetry, which reveals a potentially universal thermodynamic characteristic of active energetics. As a many-body consequence of BTRS in our experimental active system, we demonstrate the emergence of activity-induced herding, which has no equilibrium analogue.
{"title":"Role of kinematic constraints in the time reversal symmetry breaking of a model active matter","authors":"Soumen Das, Shankar Ghosh, Tridib Sadhu, Juliane U Klamser","doi":"arxiv-2409.10425","DOIUrl":"https://doi.org/arxiv-2409.10425","url":null,"abstract":"Active-matter systems are inherently out-of-equilibrium and perform\u0000mechanical work by utilizing their internal energy sources. Breakdown of\u0000time-reversal symmetry (BTRS) is a hallmark of such dissipative non-equilibrium\u0000dynamics. We introduce a robust, experimentally accessible, noninvasive,\u0000quantitative measure of BTRS in terms of the Kullback-Leibler divergence in\u0000collision events, demonstrated in our novel artificial active matter, comprised\u0000of battery-powered spherical rolling robots whose energetics in different modes\u0000of motion can be measured with high precision. Our dimensionless measure\u0000characterizes how dissipation and internal energetics are influenced by\u0000kinematic constraints from interactions with the environment. We propose this\u0000measure of BTRS as an empirical estimate of the distance from equilibrium. An\u0000energetic insight into this departure of active matter from equilibrium comes\u0000from our demonstration of a non-trivial fluctuation symmetry, which reveals a\u0000potentially universal thermodynamic characteristic of active energetics. As a\u0000many-body consequence of BTRS in our experimental active system, we demonstrate\u0000the emergence of activity-induced herding, which has no equilibrium analogue.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268521","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}
O. A. Dobush, M. P. Kozlovskii, R. V. Romanik, I. V. Pylyuk
Thermodynamic response functions, namely the isothermal compressibility, the thermal pressure coefficient, and the thermal expansion coefficient, are calculated for a many-particle system interacting through a modified Morse potential. These calculations are based on an equation of state previously derived for a cell fluid model in the grand canonical ensemble. The calculated quantities are presented graphically as functions of density and the effective chemical potential.
{"title":"Thermodynamic response functions in a cell fluid model","authors":"O. A. Dobush, M. P. Kozlovskii, R. V. Romanik, I. V. Pylyuk","doi":"arxiv-2409.09786","DOIUrl":"https://doi.org/arxiv-2409.09786","url":null,"abstract":"Thermodynamic response functions, namely the isothermal compressibility, the\u0000thermal pressure coefficient, and the thermal expansion coefficient, are\u0000calculated for a many-particle system interacting through a modified Morse\u0000potential. These calculations are based on an equation of state previously\u0000derived for a cell fluid model in the grand canonical ensemble. The calculated\u0000quantities are presented graphically as functions of density and the effective\u0000chemical potential.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252069","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 independent searchers competing for a target under restarts and find that introduction of restarts tends to enhance the search efficiency of an already efficient searcher. As a result, the difference between the search probabilities of the individual searchers increases when the system is subject to restarts. This result holds true independent of the identity of individual searchers or the specific details of the distribution of restart times. However, when only one of a pair of searchers is subject to restarts while the other evolves in an unperturbed manner, a concept termed as subsystem restarts, we find that the search probability exhibits a nonmonotonic dependence on the restart rate. We also study the mean search time for a pair of run and tumble and Brownian searchers when only the run and tumble particle is subject to restarts. We find that, analogous to restarting the whole system, the mean search time exhibits a nonmonotonic dependence on restart rates.
{"title":"Competing random searchers under restarts","authors":"R. K. Singh, R. Metzler, T. Sandev","doi":"arxiv-2409.09390","DOIUrl":"https://doi.org/arxiv-2409.09390","url":null,"abstract":"We study independent searchers competing for a target under restarts and find\u0000that introduction of restarts tends to enhance the search efficiency of an\u0000already efficient searcher. As a result, the difference between the search\u0000probabilities of the individual searchers increases when the system is subject\u0000to restarts. This result holds true independent of the identity of individual\u0000searchers or the specific details of the distribution of restart times.\u0000However, when only one of a pair of searchers is subject to restarts while the\u0000other evolves in an unperturbed manner, a concept termed as subsystem restarts,\u0000we find that the search probability exhibits a nonmonotonic dependence on the\u0000restart rate. We also study the mean search time for a pair of run and tumble\u0000and Brownian searchers when only the run and tumble particle is subject to\u0000restarts. We find that, analogous to restarting the whole system, the mean\u0000search time exhibits a nonmonotonic dependence on restart rates.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"193 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252071","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 nonequilibrium steady states (NESSs) in the weakly-coupled XXZ model in contact with two heat baths at different temperatures. We show that the density matrix can be represented using only projection operators specified by the quantum numbers of the quantum group $U_q(sl_2)$ in a very good approximation. By using this property, we numerically calculate physical quantities such as temperature profile, energy current, and correlation functions, for the spin chain consisting of several hundred spins. We analytically derive the exact density matrix in the limit $q rightarrow 0$.
{"title":"Nonequilibrium Steady States in the Weakly-Coupled XXZ Model","authors":"Shimpei Senda, Tatsuro Sunami, Yuto Matsumoto, Ayumu Sugita","doi":"arxiv-2409.09411","DOIUrl":"https://doi.org/arxiv-2409.09411","url":null,"abstract":"We study nonequilibrium steady states (NESSs) in the weakly-coupled XXZ model\u0000in contact with two heat baths at different temperatures. We show that the\u0000density matrix can be represented using only projection operators specified by\u0000the quantum numbers of the quantum group $U_q(sl_2)$ in a very good\u0000approximation. By using this property, we numerically calculate physical\u0000quantities such as temperature profile, energy current, and correlation\u0000functions, for the spin chain consisting of several hundred spins. We\u0000analytically derive the exact density matrix in the limit $q rightarrow 0$.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252070","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 concept of dynamical freezing is a phenomenon where a suitable set of local observables freezes under a strong periodic drive in a quantum many-body system. This happens because of the emergence of approximate but perpetual conservation laws when the drive is strong enough. In this work, we probe the resilience of dynamical freezing to random perturbations added to the relative phases between the interfering states (elements of a natural basis) in the time-evolving wave function after each drive cycle. We study this in an integrable Ising chain in a time-periodic transverse field. Our key finding is, that the imprinted phase noise melts the dynamically frozen state, but the decay is "slow": a stretched-exponential decay rather than an exponential one. Stretched-exponential decays (also known as Kohlrausch relaxation) are usually expected in complex systems with time-scale hierarchies due to strong disorders or other inhomogeneities resulting in jamming, glassiness, or localization. Here we observe this in a simple translationally invariant system dynamically frozen under a periodic drive. Moreover, the melting here does not obliterate the entire memory of the initial state but leaves behind a steady remnant that depends on the initial conditions. This underscores the stability of dynamically frozen states.
{"title":"Stretched-Exponential Melting of a Dynamically Frozen State Under Imprinted Phase Noise in the Ising Chain in a Transverse Field","authors":"Krishanu Roychowdhury, Arnab Das","doi":"arxiv-2409.09128","DOIUrl":"https://doi.org/arxiv-2409.09128","url":null,"abstract":"The concept of dynamical freezing is a phenomenon where a suitable set of\u0000local observables freezes under a strong periodic drive in a quantum many-body\u0000system. This happens because of the emergence of approximate but perpetual\u0000conservation laws when the drive is strong enough. In this work, we probe the\u0000resilience of dynamical freezing to random perturbations added to the relative\u0000phases between the interfering states (elements of a natural basis) in the\u0000time-evolving wave function after each drive cycle. We study this in an\u0000integrable Ising chain in a time-periodic transverse field. Our key finding is,\u0000that the imprinted phase noise melts the dynamically frozen state, but the\u0000decay is \"slow\": a stretched-exponential decay rather than an exponential one.\u0000Stretched-exponential decays (also known as Kohlrausch relaxation) are usually\u0000expected in complex systems with time-scale hierarchies due to strong disorders\u0000or other inhomogeneities resulting in jamming, glassiness, or localization.\u0000Here we observe this in a simple translationally invariant system dynamically\u0000frozen under a periodic drive. Moreover, the melting here does not obliterate\u0000the entire memory of the initial state but leaves behind a steady remnant that\u0000depends on the initial conditions. This underscores the stability of\u0000dynamically frozen states.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252073","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 angular diffusion in a classical $d-$dimensional inertial XY model with interactions decaying with the distance between spins as $r^{-alpha}$, wiht $alphageqslant 0$. After a very short-time ballistic regime, with $sigma_theta^2sim t^2$, a super-diffusive regime, for which $sigma_theta^2sim t^{alpha_D}$, with $alpha_D simeq 1text{.}45$ is observed, whose duration covers an initial quasistationary state and its transition to a second plateau characterized by the Boltzmann-Gibbs temperature $T_text{BG}$. Long after $T_text{BG}$ is reached, a crossover to normal diffusion, $sigma_theta^2sim t$, is observed. We relate, for the first time, via the expression $alpha_D = 2/(3 - q)$, the anomalous diffusion exponent $alpha_D$ with the entropic index $q$ characterizing the time-averaged angles and momenta probability distribution functions (pdfs), which are given by the so called $q-$Gaussian distributions, $f_q(x)propto e_q(-beta x^2)$, where $e_q (u) equiv [1 + (1 - q)u]^{frac{1}{1 - q}}$ ($e_1(u) = exp(u)$). For fixed size $N$ and large enough times, the index $q_theta$ characterizing the angles pdf approaches unity, thus indicating a final relaxation to Boltzmann-Gibbs equilibrium. For fixed time and large enough $N$, the crossover occurs in the opposite sense.
{"title":"Diffusion crossover from/to $q$-statistics to/from Boltzmann-Gibbs statistics in the classical inertial $α$-XY ferromagnet","authors":"Antonio Rodríguez, Constantino Tsallis","doi":"arxiv-2409.08992","DOIUrl":"https://doi.org/arxiv-2409.08992","url":null,"abstract":"We study the angular diffusion in a classical $d-$dimensional inertial XY\u0000model with interactions decaying with the distance between spins as\u0000$r^{-alpha}$, wiht $alphageqslant 0$. After a very short-time ballistic\u0000regime, with $sigma_theta^2sim t^2$, a super-diffusive regime, for which\u0000$sigma_theta^2sim t^{alpha_D}$, with $alpha_D simeq 1text{.}45$ is\u0000observed, whose duration covers an initial quasistationary state and its\u0000transition to a second plateau characterized by the Boltzmann-Gibbs temperature\u0000$T_text{BG}$. Long after $T_text{BG}$ is reached, a crossover to normal\u0000diffusion, $sigma_theta^2sim t$, is observed. We relate, for the first time,\u0000via the expression $alpha_D = 2/(3 - q)$, the anomalous diffusion exponent\u0000$alpha_D$ with the entropic index $q$ characterizing the time-averaged angles\u0000and momenta probability distribution functions (pdfs), which are given by the\u0000so called $q-$Gaussian distributions, $f_q(x)propto e_q(-beta x^2)$, where\u0000$e_q (u) equiv [1 + (1 - q)u]^{frac{1}{1 - q}}$ ($e_1(u) = exp(u)$). For\u0000fixed size $N$ and large enough times, the index $q_theta$ characterizing the\u0000angles pdf approaches unity, thus indicating a final relaxation to\u0000Boltzmann-Gibbs equilibrium. For fixed time and large enough $N$, the crossover\u0000occurs in the opposite sense.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","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":"142252106","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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