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Wen B, Li MG, Liu J, Bao JD. Ergodic Measure and Potential Control of Anomalous Diffusion. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1012. [PMID: 37509959 PMCID: PMC10377995 DOI: 10.3390/e25071012] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2023] [Revised: 06/15/2023] [Accepted: 06/27/2023] [Indexed: 07/30/2023]
Abstract
In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given process, the time average of an observable quantity might be a random variable, whose distribution width narrows with time, and one wonders how long it takes for the convergence rate to become a constant. This is also the premise of ergodic establishment, because the ensemble average is always equal to the constant. We focus on the time-dependent fluctuation width for the time average of both the velocity and kinetic energy of a force-free particle described by the generalized Langevin equation, where the stationary velocity autocorrelation function is considered. Subsequently, the shortest time scale can be estimated for a system transferring from a stationary state to an effective ergodic state. Moreover, a logarithmic spatial potential is used to modulate the processes associated with free ballistic diffusion and the control of diffusion, as well as the minimal realization of the whole power-law regime. The results presented suggest that non-ergodicity mimics the sparseness of the medium and reveals the unique role of logarithmic potential in modulating diffusion behavior.
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Affiliation(s)
- Bao Wen
- Institutes of Science and Development, Chinese Academy of Sciences, Beijing 100190, China
- School of Public Policy and Management, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Ming-Gen Li
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Jian Liu
- Department of Physics, Beijing Technology and Business University, Beijing 100048, China
| | - Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
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2
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Ferreira RMS. From generalized Langevin stochastic dynamics to anomalous diffusion. Phys Rev E 2022; 106:054157. [PMID: 36559457 DOI: 10.1103/physreve.106.054157] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2022] [Accepted: 11/08/2022] [Indexed: 06/17/2023]
Abstract
Scaling methods are fundamental in all branches of physics. In stochastic process, we usually try to describe the long time behavior of a given time correlation function. In this work we investigate a scaling method for anomalous diffusion in systems with memory that produces good results for long and intermediate times. We will initially present a generalization of the diffusion exponent. Then, we present an asymptotic method to obtain an analytical expression for the diffusion coefficient by introducing a time scale factor λ(t). We found an exact expression for the function λ(t), which allows us to describe the diffusive process. For large times, λ(t) becomes a universal parameter determined by the diffusion exponent. In turn, the analytical results are then compared to the numerical results, with a good matching. Then, we'll show the practical effects of scaling. An important first result is that λ(t) quickly converges to a constant. Another very important point was the classification of new forms of diffusion due to the generalized exponent. In previous works, we verified the existence of ergodic ballistic diffusion with diffusion exponent α=2^{-}. Here, we verify the existence of the nonergodic ballistic diffusion type with the obtainment of the diffusion coefficient α=2. Finally, we show that the scaling works. This method is general and can be applied to various types of stochastic problems.
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Affiliation(s)
- Rogelma M S Ferreira
- Centro de Ciências Exatas e Tecnológicas, Universidade Federal do Recôncavo da Bahia, 44.380-000, Cruz das Almas, Bahia, Brasil
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3
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Bao JD, Li Y, Marchesoni F. Consistent Hamiltonian models for space-momentum diffusion. Phys Rev E 2022; 105:L052105. [PMID: 35706220 DOI: 10.1103/physreve.105.l052105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2021] [Accepted: 05/15/2022] [Indexed: 06/15/2023]
Abstract
We develop a unified Hamiltonian approach to the diffusion of a particle coupled to a dissipative environment, an archetypal model widely invoked to interpret condensed phase phenomena, such as polymerization and cold-atom diffusion in optical lattices. By appropriate choices of the coupling functions, we reformulate phenomenological diffusion models by adding otherwise ignored space-momentum terms. We thus numerically predict a variety of diffusion regimes, from diffusion saturation to superballistic diffusion. With reference to ultracold atoms in optical lattices, we also show that time correlated external noises prevent superdiffusion from exceeding Richardson's law. Some of these results are unexpected and call for experimental validation.
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Affiliation(s)
- Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Yunyun Li
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
| | - Fabio Marchesoni
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
- Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy
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Kanazawa K, Sornette D. Nonuniversal Power Law Distribution of Intensities of the Self-Excited Hawkes Process: A Field-Theoretical Approach. PHYSICAL REVIEW LETTERS 2020; 125:138301. [PMID: 33034505 DOI: 10.1103/physrevlett.125.138301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/06/2020] [Revised: 06/29/2020] [Accepted: 08/25/2020] [Indexed: 06/11/2023]
Abstract
The Hawkes self-excited point process provides an efficient representation of the bursty intermittent dynamics of many physical, biological, geological, and economic systems. By expressing the probability for the next event per unit time (called "intensity"), say of an earthquake, as a sum over all past events of (possibly) long-memory kernels, the Hawkes model is non-Markovian. By mapping the Hawkes model onto stochastic partial differential equations that are Markovian, we develop a field theoretical approach in terms of probability density functionals. Solving the steady-state equations, we predict a power law scaling of the probability density function of the intensities close to the critical point n=1 of the Hawkes process, with a nonuniversal exponent, function of the background intensity ν_{0} of the Hawkes intensity, the average timescale of the memory kernel and the branching ratio n. Our theoretical predictions are confirmed by numerical simulations.
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Affiliation(s)
- Kiyoshi Kanazawa
- Faculty of Engineering, Information and Systems, The University of Tsukuba, Tennodai, Tsukuba, Ibaraki 305-8573, Japan
| | - Didier Sornette
- ETH Zurich, Department of Management, Technology and Economics, Zurich 8092, Switzerland
- Tokyo Tech World Research Hub Initiative, Institute of Innovative Research, Tokyo Institute of Technology, Tokyo 152-8550, Japan
- Institute of Risk Analysis, Prediction and Management, Academy for Advanced Interdisciplinary Studies, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China
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Effect of Self-Oscillation on Escape Dynamics of Classical and Quantum Open Systems. ENTROPY 2020; 22:e22080839. [PMID: 33286610 PMCID: PMC7517439 DOI: 10.3390/e22080839] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/29/2020] [Revised: 07/15/2020] [Accepted: 07/15/2020] [Indexed: 01/07/2023]
Abstract
We study the effect of self-oscillation on the escape dynamics of classical and quantum open systems by employing the system-plus-environment-plus-interaction model. For a damped free particle (system) with memory kernel function expressed by Zwanzig (J. Stat. Phys. 9, 215 (1973)), which is originated from a harmonic oscillator bath (environment) of Debye type with cut-off frequency wd, ergodicity breakdown is found because the velocity autocorrelation function oscillates in cosine function for asymptotic time. The steady escape rate of such a self-oscillated system from a metastable potential exhibits nonmonotonic dependence on wd, which denotes that there is an optimal cut-off frequency makes it maximal. Comparing results in classical and quantum regimes, the steady escape rate of a quantum open system reduces to a classical one with wd decreasing gradually, and quantum fluctuation indeed enhances the steady escape rate. The effect of a finite number of uncoupled harmonic oscillators N on the escape dynamics of a classical open system is also discussed.
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Bao JD. Generalized Einstein relations and conditions for anomalous relaxation. Phys Rev E 2019; 100:052149. [PMID: 31869967 DOI: 10.1103/physreve.100.052149] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2019] [Indexed: 11/07/2022]
Abstract
The generalized Einstein relation (GER) for nonergodic processes is investigated within the framework of the generalized Langevin equation. The conditions for anomalous relaxation such as long-tail decay and non-vanishing velocity autocorrelation function (VAF) are proposed and distinguished. For the stationary nonergodic process, if the initial preparation of the particle velocity is non-thermal, an asymptotic GER occurs in a departure from the usual result. It is shown that the GER holding is a necessary condition rather than a full condition for the system being close to equilibrium. For the nonergodic process of the second type due to cutoff of high frequencies, the VAF oscillates with time, the GER holds but the equilibrium fails in the long-time limit. Applications to some practical examples confirm the present theoretical findings.
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Affiliation(s)
- Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China
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7
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Dybiec B, Gudowska-Nowak E, Barkai E, Dubkov AA. Lévy flights versus Lévy walks in bounded domains. Phys Rev E 2017; 95:052102. [PMID: 28618620 DOI: 10.1103/physreve.95.052102] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2016] [Indexed: 05/26/2023]
Abstract
Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities is the discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. As a consequence, a well-developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flight and Lévy walk models on bounded domains, examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time, and stationary probability density functions. It is demonstrated that the similarity of the models is affected by the type of boundary conditions and the value of the stability index defining the asymptotics of the jump length distribution.
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Affiliation(s)
- Bartłomiej Dybiec
- Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland
| | - Ewa Gudowska-Nowak
- Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland
| | - Eli Barkai
- Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel
| | - Alexander A Dubkov
- Radiophysical Department, Lobachevsky State University, Gagarin ave. 23, 603950 Nizhni Novgorod, Russia
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8
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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: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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9
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Bouzat S. Influence of molecular motors on the motion of particles in viscoelastic media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062707. [PMID: 25019814 DOI: 10.1103/physreve.89.062707] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2013] [Indexed: 06/03/2023]
Abstract
We study theoretically and by numerical simulations the motion of particles driven by molecular motors in a viscoelastic medium representing the cell cytoplasm. For this, we consider a generalized Langevin equation coupled to a stochastic stepping dynamics for the motors that takes into account the action of each motor separately. In the absence of motors, the model produces subdiffusive motion of particles characterized by a power-law scaling of the mean square displacement versus the lag time as t^{α}, with 0<α<1, similar to that observed in cells. Our results show how the action of the motors can induce a transition to a superdiffusive regime at large lag times with the characteristics of those found in experiments reported in the literature. We also show that at small lag times, the motors can act as static crosslinkers that slow down the natural subdiffusive transport. An analysis of previously reported experimental data in the relevant time scales provides evidence of this phenomenon. Finally, we study the effect of a harmonic potential representing an optical trap, and we show a way to approach to a macroscopic description of the active transport in cells. This last point stresses the relevance of the molecular motors for generating not only directed motion to specific targets, but also fast diffusivelike random motion.
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Affiliation(s)
- Sebastián Bouzat
- Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche (CNEA), (8400) Bariloche, Río Negro, Argentina
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10
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Mullen RG, Shea JE, Peters B. Communication: An existence test for dividing surfaces without recrossing. J Chem Phys 2014; 140:041104. [DOI: 10.1063/1.4862504] [Citation(s) in RCA: 42] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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11
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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: 0.7] [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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12
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Gupta S, Bandyopadhyay M. Quantum Langevin equation of a charged oscillator in a magnetic field and coupled to a heat bath through momentum variables. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:041133. [PMID: 22181113 DOI: 10.1103/physreve.84.041133] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2011] [Indexed: 05/31/2023]
Abstract
We obtain the quantum Langevin equation (QLE) of a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a quantum heat bath through momentum variables. The bath is modeled as a collection of independent quantum harmonic oscillators. The QLE involves a random force which does not depend on the magnetic field, and a quantum-generalized classical Lorentz force. These features are also present in the QLE for the case of particle-bath coupling through coordinate variables. However, significant differences are also observed. For example, the mean force in the QLE is characterized by a memory function that depends explicitly on the magnetic field. The random force has a modified form with correlation and commutator different from those in the case of coordinate-coordinate coupling. Moreover, the coupling constants, in addition to appearing in the random force and in the mean force, also renormalize the inertial term and the harmonic potential term in the QLE.
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Affiliation(s)
- Shamik Gupta
- Laboratoire de Physique de l'École Normale Supérieure de Lyon, Université de Lyon, CNRS, 46 Allée d'Italie, 69364 Lyon cédex 07, France
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13
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Bai ZW. Quantum heat-fluctuation theorem of a reduced system: an exactly solvable case. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:021101. [PMID: 21405812 DOI: 10.1103/physreve.83.021101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2010] [Revised: 12/15/2010] [Indexed: 05/30/2023]
Abstract
For a system in which free quantum Brownian particles interact with a non-Markovian heat bath with product initial states, the quantum heat-fluctuation theorems and the Jarzynski relation of the reduced system are derived using the path integral technique. They are not a direct generalization of the classical case. The validity of the second law of quantum thermodynamics is shown. The forms of these theorems and relations have to do with the specified nonequilibrium initial states, and are dependent on the structure of the bath (ergodic or nonergodic) in the quantum case as well as the classical case under some initial states.
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Affiliation(s)
- Zhan-Wu Bai
- Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China.
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14
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Siegle P, Goychuk I, Talkner P, Hänggi P. Markovian embedding of non-Markovian superdiffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:011136. [PMID: 20365352 DOI: 10.1103/physreve.81.011136] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2009] [Indexed: 05/07/2023]
Abstract
We consider different Markovian embedding schemes of non-Markovian stochastic processes that are described by generalized Langevin equations and obey thermal detailed balance under equilibrium conditions. At thermal equilibrium, superdiffusive behavior can emerge if the total integral of the memory kernel vanishes. Such a situation of vanishing static friction is caused by a super-Ohmic thermal bath. One of the simplest models of ballistic superdiffusion is determined by a biexponential memory kernel that was proposed by [Bao J. Stat. Phys. 114, 503 (2004)]. We show that this non-Markovian model has infinitely many different four-dimensional Markovian embeddings. Implementing numerically the simplest one, we demonstrate that (i) the presence of a periodic potential with arbitrarily low barriers changes the asymptotic large-time behavior from free ballistic superdiffusion into normal diffusion; (ii) an additional biasing force renders the asymptotic dynamics superdiffusive again. The development of transients that display a qualitatively different behavior compared to the true large-time asymptotics presents a general feature of this non-Markovian dynamics. These transients though may be extremely long. As a consequence, they can be even mistaken as the true asymptotics. We find that such intermediate asymptotics exhibit a giant enhancement of superdiffusion in tilted washboard potentials and it is accompanied by a giant transient superballistic current growing proportional to t(alpha(eff)) with an exponent alpha(eff) that can exceed the ballistic value of 2.
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Affiliation(s)
- Peter Siegle
- Institute of Physics, University of Augsburg, Augsburg, Germany
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15
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Muehsam DJ, Pilla AA. A Lorentz model for weak magnetic field bioeffects: part I--thermal noise is an essential component of AC/DC effects on bound ion trajectory. Bioelectromagnetics 2009; 30:462-75. [PMID: 19437460 DOI: 10.1002/bem.20494] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
We have previously employed the Lorentz-Langevin model to describe the effects of weak exogenous magnetic fields via the classical Lorentz force on a charged ion bound in a harmonic oscillator potential, in the presence of thermal noise forces. Previous analyses predicted that microT-range fields give rise to a rotation of the oscillator orientation at the Larmor frequency and bioeffects were based upon the assumption that the classical trajectory of the bound charge itself could modulate a biochemical process. Here, it is shown that the thermal component of the motion follows the Larmor trajectory. The results show that the Larmor frequency is independent of the thermal noise strength, and the motion retains the form of a coherent oscillator throughout the binding lifetime, rather than devolving into a random walk. Thermal equilibration results in a continual increase in the vibrational amplitude of the rotating oscillator towards the steady-state amplitude, but does not affect the Larmor orbit. Thus, thermal noise contributes to, rather than inhibits, the effect of the magnetic field upon reactivity. Expressions are derived for the ensemble average of position and the velocity of the thermal component of the oscillator motion. The projection of position and velocity onto a Cartesian axis measures the nonuniformity of the Larmor trajectory and is illustrated for AC and combined AC/DC magnetic fields, suggesting a means of interpreting resonance phenomena. It is noted that the specific location and height of resonances are dependent upon binding lifetime and initial AC phase.
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Affiliation(s)
- David J Muehsam
- Department of Biomedical Engineering, Columbia University, New York, New York, USA
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16
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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: 0.9] [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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17
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Goychuk I, Hänggi P. Anomalous escape governed by thermal 1/f noise. PHYSICAL REVIEW LETTERS 2007; 99:200601. [PMID: 18233128 DOI: 10.1103/physrevlett.99.200601] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2007] [Indexed: 05/07/2023]
Abstract
We present an analytic study for subdiffusive escape of overdamped particles out of a cusp-shaped parabolic potential well which are driven by thermal, fractional Gaussian noise with a 1/omega 1-alpha power spectrum. This long-standing challenge becomes mathematically tractable by use of a generalized Langevin dynamics via its corresponding non-Markovian, time-convolutionless master equation: We find that the escape is governed asymptotically by a power-law whose exponent depends exponentially on the ratio of barrier height and temperature. This result is in distinct contrast to a description with a corresponding subdiffusive fractional Fokker-Planck approach, thus providing experimentalists an amenable testbed to differentiate between the two escape scenarios.
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Affiliation(s)
- I Goychuk
- University of Augsburg, Institute of Physics, Universitätsstr. 1, D-86135 Augsburg, Germany
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18
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Moix JM, Hernandez R, Pollak E. Momentum and Velocity Autocorrelation Functions of a Diatomic Molecule Are Not Necessarily Proportional to Each Other. J Phys Chem B 2007; 112:213-8. [PMID: 17824637 DOI: 10.1021/jp0730951] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023]
Abstract
We present a computation of the classical momentum and velocity correlation functions of Br2 considered as an idealized molecular wire connecting dissipated lead atoms at each end of the dimer. It is demonstrated that coupling of the diatomic relative momentum to the leads may result in momenta that are not equal to the mass-weighted velocity. These differences show up in numerical simulations of both the average value and time correlations of the bond momentum and velocity. These observations are supported by analytical predictions for the average temperature of the diatomic. They imply that the "standard recipes" for modeling the system with a generalized Langevin equation are insufficient.
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Affiliation(s)
- Jeremy M Moix
- Center for Computational Molecular Sciences & Technology, School of Chemistry & Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA
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19
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Lee MH. Why irreversibility is not a sufficient condition for ergodicity. PHYSICAL REVIEW LETTERS 2007; 98:190601. [PMID: 17677614 DOI: 10.1103/physrevlett.98.190601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2006] [Revised: 01/16/2007] [Indexed: 05/16/2023]
Abstract
Khinchin's theorem of ergodicity is examined by means of linear response theory. The resulting ergodic condition shows that, contrary to the theorem, irreversibility is not a sufficient condition for ergodicity. By the recurrence relations method, we prove that irreversibility is broader in scope than ergodicity, showing why it can only be a necessary condition for ergodicity.
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Affiliation(s)
- M Howard Lee
- Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA.
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Lee MH. Birkhoff's theorem, many-body response functions, and the ergodic condition. PHYSICAL REVIEW LETTERS 2007; 98:110403. [PMID: 17501029 DOI: 10.1103/physrevlett.98.110403] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2006] [Revised: 12/18/2006] [Indexed: 05/15/2023]
Abstract
The ergodic hypothesis is viewed as one that is physically measurable through scattering by an external probe. Linear response theory is used to derive a general ergodic condition on the response functions. It is shown that the same condition is also implied by Birkhoff's theorem. This coincidence allows us to shed light on the abstract terms of the theorem via classical many-body models.
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Affiliation(s)
- M Howard Lee
- Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA.
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