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Chen W, Greiner C, Xu Z. Persistent nonequilibrium effects in generalized Langevin dynamics of nonrelativistic and relativistic particles. Phys Rev E 2023; 107:064131. [PMID: 37464650 DOI: 10.1103/physreve.107.064131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2023] [Accepted: 05/30/2023] [Indexed: 07/20/2023]
Abstract
Persistent nonequilibrium effects such as the memory of the initial state, the ballistic diffusion, and the break of the equipartition theorem and the ergodicity in Brownian motions are investigated by analytically solving the generalized Langevin equation of nonrelativistic Brownian particles with colored noise. These effects can also be observed in the Brownian motion of relativistic particles by numerically solving the generalized Langevin equation for specially chosen memory kernels. Our analyses give rise to think about the possible anomalous motion of heavy quarks in relativistic heavy-ion collisions.
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Affiliation(s)
- Weiguo Chen
- Department of Physics, Tsinghua University and Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
| | - Carsten Greiner
- Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität Frankfurt, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
| | - Zhe Xu
- Department of Physics, Tsinghua University and Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
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Albers T, Radons G. Nonergodicity of d-dimensional generalized Lévy walks and their relation to other space-time coupled models. Phys Rev E 2022; 105:014113. [PMID: 35193310 DOI: 10.1103/physreve.105.014113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2021] [Accepted: 12/23/2021] [Indexed: 06/14/2023]
Abstract
We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)PRLTAO0031-900710.1103/PhysRevLett.58.1100] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent letter [Phys. Rev. Lett. 120, 104501 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.104501], give detailed interpretations, and in particular emphasize three surprising results. First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-averaged squared displacements show superdiffusion. Third, we recognize that the fluctuations of the time-averaged squared displacements can become so large that the ergodicity breaking parameter diverges, what we call infinitely strong ergodicity breaking. This phenomenon can also occur for paramter values where the lag-time dependence of the mean-squared displacements is linear indicating normal diffusion. In order to numerically determine the full distribution of time-averaged squared displacements, we use importance sampling. For an embedding of our findings into existing results in the literature, we define a more general model which we call variable speed generalized Lévy walk and which includes well-known models from the literature as special cases such as the space-time coupled Lévy flight or the anomalous Drude model. We discuss and interpret our findings regarding the generalized Lévy walk in detail and compare them with the nonergodicity of the other space-time coupled models following from the more general model.
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Affiliation(s)
- Tony Albers
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany and Institute of Mechatronics, 09126 Chemnitz, Germany
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Cheng CH, Gemao B, Lai PY. Phase transitions in Ehrenfest urn model with interactions: Coexistence of uniform and nonuniform states. Phys Rev E 2020; 101:012123. [PMID: 32069609 DOI: 10.1103/physreve.101.012123] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2019] [Indexed: 11/07/2022]
Abstract
A model based on the classic noninteracting Ehrenfest urn model with two urns is generalized to M urns with the introduction of interactions for particles within the same urn. As the inter-particle interaction strength is varied, phases of different levels of nonuniformity emerge and their stabilities are calculated analytically. In particular, coexistence of locally stable uniform and nonuniform phases connected by first-order transition occurs. The phase transition threshold and energy barrier can be derived exactly together with the phase diagram obtained analytically. These analytic results are further confirmed by Monte Carlo simulations.
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Affiliation(s)
- Chi-Ho Cheng
- Department of Physics, National Changhua University of Education, Changhua 500, Taiwan, Republic of China
| | - Beverly Gemao
- Department of Physics and Center for Complex Systems, National Central University, Chung-Li District, Taoyuan City 320, Taiwan, Republic of China.,Physics Department, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines
| | - Pik-Yin Lai
- Department of Physics and Center for Complex Systems, National Central University, Chung-Li District, Taoyuan City 320, Taiwan, Republic of China
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Albers T, Radons G. Exact Results for the Nonergodicity of d-Dimensional Generalized Lévy Walks. PHYSICAL REVIEW LETTERS 2018; 120:104501. [PMID: 29570320 DOI: 10.1103/physrevlett.120.104501] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2017] [Indexed: 06/08/2023]
Abstract
We provide analytical results for the ensemble-averaged and time-averaged squared displacement, and the randomness of the latter, in the full two-dimensional parameter space of the d-dimensional generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)PRLTAO0031-900710.1103/PhysRevLett.58.1100]. In certain regions of the parameter plane, we obtain surprising results such as the divergence of the mean-squared displacements, the divergence of the ergodicity breaking parameter despite a finite mean-squared displacement, and subdiffusion which appears superdiffusive when one only considers time averages.
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Affiliation(s)
- Tony Albers
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
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Agarwal K, Gopalakrishnan S, Knap M, Müller M, Demler E. Anomalous diffusion and griffiths effects near the many-body localization transition. PHYSICAL REVIEW LETTERS 2015; 114:160401. [PMID: 25955037 DOI: 10.1103/physrevlett.114.160401] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/20/2014] [Indexed: 06/04/2023]
Abstract
We explore the high-temperature dynamics of the disordered, one-dimensional XXZ model near the many-body localization (MBL) transition, focusing on the delocalized (i.e., "metallic") phase. In the vicinity of the transition, we find that this phase has the following properties: (i) local magnetization fluctuations relax subdiffusively; (ii) the ac conductivity vanishes near zero frequency as a power law; and (iii) the distribution of resistivities becomes increasingly broad at low frequencies, approaching a power law in the zero-frequency limit. We argue that these effects can be understood in a unified way if the metallic phase near the MBL transition is a quantum Griffiths phase. We establish scaling relations between the associated exponents, assuming a scaling form of the spin-diffusion propagator. A phenomenological classical resistor-capacitor model captures all the essential features.
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Affiliation(s)
- Kartiek Agarwal
- Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
| | | | - Michael Knap
- Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
- ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
| | - Markus Müller
- The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Eugene Demler
- Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
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Lapas LC, Ferreira RMS, Rubí JM, Oliveira FA. Anomalous law of cooling. J Chem Phys 2015; 142:104106. [DOI: 10.1063/1.4914872] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Ferreira RMS, Santos MVS, Donato CC, Andrade JS, Oliveira FA. Analytical results for long-time behavior in anomalous diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:021121. [PMID: 23005736 DOI: 10.1103/physreve.86.021121] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2011] [Revised: 03/25/2012] [Indexed: 05/26/2023]
Abstract
We investigate through a generalized Langevin formalism the phenomenon of anomalous diffusion for asymptotic times, and we generalized the concept of the diffusion exponent. A method is proposed to obtain the diffusion coefficient analytically through the introduction of a time scaling factor λ. We obtain as well an exact expression for λ for all kinds of diffusion. Moreover, we show that λ is a universal parameter determined by the diffusion exponent. The results are then compared with numerical calculations and very good agreement is observed. The method is general and may be applied to many types of stochastic problem.
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Affiliation(s)
- R M S Ferreira
- Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, 60455-900 Fortaleza, Ceará, Brazil
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Magdziarz M, Weron A. Anomalous diffusion: testing ergodicity breaking in experimental data. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:051138. [PMID: 22181399 DOI: 10.1103/physreve.84.051138] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2011] [Revised: 10/28/2011] [Indexed: 05/31/2023]
Abstract
Recent advances in single-molecule experiments show that various complex systems display nonergodic behavior. In this paper, we show how to test ergodicity and ergodicity breaking in experimental data. Exploiting the so-called dynamical functional, we introduce a simple test which allows us to verify ergodic properties of a real-life process. The test can be applied to a large family of stationary infinitely divisible processes. We check the performance of the test for various simulated processes and apply it to experimental data describing the motion of mRNA molecules inside live Escherichia coli cells. We show that the data satisfy necessary conditions for mixing and ergodicity. The detailed analysis is presented in the supplementary material.
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Affiliation(s)
- Marcin Magdziarz
- Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wroclaw, Poland.
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Weron A, Magdziarz M. Generalization of the Khinchin theorem to Lévy flights. PHYSICAL REVIEW LETTERS 2010; 105:260603. [PMID: 21231638 DOI: 10.1103/physrevlett.105.260603] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2010] [Revised: 11/17/2010] [Indexed: 05/30/2023]
Abstract
One of the most fundamental theorems in statistical mechanics is the Khinchin ergodic theorem, which links the ergodicity of a physical system with the irreversibility of the corresponding autocorrelation function. However, the Khinchin theorem cannot be successfully applied to processes with infinite second moment, in particular, to the relevant class of Lévy flights. Here, we solve this challenging problem. Namely, using the recently developed measure of dependence called Lévy correlation cascade, we derive a version of the Khinchin theorem for Lévy flights. This result allows us to verify the Boltzmann hypothesis for systems displaying Lévy-flight-type dynamics.
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Affiliation(s)
- Aleksander Weron
- Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wroclaw, Poland.
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Wierling A, Sawada I. Wave-number dependent current correlation for a harmonic oscillator. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:051107. [PMID: 21230437 DOI: 10.1103/physreve.82.051107] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2010] [Indexed: 05/30/2023]
Abstract
The wave-number k dependent current-correlation function is considered for a harmonic oscillator model. An explicit analytic expression for the Laplace transformed correlation function is derived. It is compared with numerical solutions and results obtained by the recurrence relation method. Several limiting cases such as the long-wavelength limit k→0 and the deep inelastic limit k→∞ are discussed in detail. In particular, we show that the deep inelastic limit allows for an explicit summation of the continued fraction. An approximation scheme for the recurrants at intermediate values of k is also considered.
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Affiliation(s)
- August Wierling
- Institut für Physik, Universität Rostock, 18051 Rostock, Germany.
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Abstract
The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by nonstationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and suggest a possible generalization of Khinchin's theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation, we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.
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Allegrini P, Bologna M, Fronzoni L, Grigolini P, Silvestri L. Experimental quenching of harmonic stimuli: universality of linear response theory. PHYSICAL REVIEW LETTERS 2009; 103:030602. [PMID: 19659260 DOI: 10.1103/physrevlett.103.030602] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2009] [Revised: 06/16/2009] [Indexed: 05/28/2023]
Abstract
We show that liquid crystals in the weak turbulence electroconvective regime respond to harmonic perturbations with oscillations whose intensity decay with an inverse power law of time. We use the results of this experiment to prove that this effect is the manifestation of a form of linear response theory (LRT) valid in the out-of-equilibrium case, as well as at thermodynamic equilibrium where it reduces to the ordinary LRT. We argue that this theory is a universal property, which is not confined to physical processes such as turbulent or excitable media, and that it holds true in all possible conditions, and for all possible systems, including complex networks, thereby establishing a bridge between statistical physics and all the fields of research in complexity.
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Affiliation(s)
- Paolo Allegrini
- Dipartimento di Fisica E. Fermi, Università di Pisa and INFM CRS-SOFT, Largo Pontecorvo 3, 56127 Pisa, Italy
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Lapas LC, Morgado R, Vainstein MH, Rubí JM, Oliveira FA. Khinchin theorem and anomalous diffusion. PHYSICAL REVIEW LETTERS 2008; 101:230602. [PMID: 19113535 DOI: 10.1103/physrevlett.101.230602] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2008] [Revised: 10/30/2008] [Indexed: 05/26/2023]
Abstract
A recent Letter [M. H. Lee, Phys. Rev. Lett. 98, 190601 (2007)] has called attention to the fact that irreversibility is a broader concept than ergodicity, and that therefore the Khinchin theorem [A. I. Khinchin, (Dover, New York, 1949)] may fail in some systems. In this Letter we show that for all ranges of normal and anomalous diffusion described by a generalized Langevin equation the Khinchin theorem holds.
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Affiliation(s)
- Luciano C Lapas
- Instituto de Física, Universidade de Brasília, Caixa Postal 04513, 70919-970 Brasília, Distrito Federal, Brazil
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