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Kasuga K, Yoshimori A. Nonintegral form of the reciprocal relation associated with violation of the fluctuation response relation. Phys Rev E 2023; 108:034109. [PMID: 37849163 DOI: 10.1103/physreve.108.034109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2022] [Accepted: 08/15/2023] [Indexed: 10/19/2023]
Abstract
We extend Onsager's reciprocal relation to systems in a nonequilibrium steady state. While Onsager's reciprocal relation concerns the kinetic (Onsager) coefficient, the extended reciprocal relation concerns violation of the fluctuation response relation (FRR) for mechanical and thermal perturbations. This extended relation holds at each frequency when the extent of the FRR violation is expressed in a frequency domain. This nonintegral form distinguishes the extended relation from previous relations expressed by integration over a frequency. To obtain this relation, we consider one-particle one-dimensional systems described by an overdamped Langevin equation with a force driving the system away from equilibrium. We assume a special property of the potential in the system. From this Langevin equation, we obtain the Fokker-Planck (FP) equation describing the time evolution of the distribution function of the particle. Using the FP equation, we calculate the responses of the particle velocity and heat current by applying time-dependent perturbations of the driving force and temperature. We express the extent of the FRR violation in terms of these responses with time correlation functions and expand them in powers of the FP operator. This reciprocal relation is valid far from equilibrium. One can also confirm this reciprocal relation through experiments with systems such as colloidal suspensions because the FRR violation can be experimentally observed.
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Affiliation(s)
- Kotaro Kasuga
- Department of Physics, Niigata University, Niigata 950-2181, Japan
| | - Akira Yoshimori
- Department of Physics, Niigata University, Niigata 950-2181, Japan
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Diezemann G. Nonlinear response theory for Markov processes. IV. The asymmetric double-well potential model revisited. Phys Rev E 2022; 106:064122. [PMID: 36671146 DOI: 10.1103/physreve.106.064122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2022] [Accepted: 12/02/2022] [Indexed: 06/17/2023]
Abstract
The dielectric response of noninteracting dipoles is discussed in the framework of the classical model of stochastic reorientations in an asymmetric double-well potential (ADWP). In the nonlinear regime, this model exhibits some pecularities in the static response. We find that the saturation behavior of the symmetric double-well potential model does not follow the Langevin function and only in the linear regime are the standard results recovered. If a finite asymmetry is assumed, then the nonlinear susceptibilities are found to change the sign at a number of characteristic temperatures that depend on the magnitude of the asymmetry, as has been observed earlier for the third-order and fifth-order responses. If the kinetics of the barrier crossing in the ADWP model is described as a two-state model, then we can give analytical expressions for the values of the characteristic temperatures. The results for the response obtained from a (numerical) solution of the Fokker-Planck equation for the Brownian motion in a model ADWP behaves very similarly to the two-state model for high barriers. For small barriers no clear-cut timescale separation between the barrier crossing process and the intrawell relaxation exists and the model exhibits a number of timescales. In this case, the frequency-dependent linear susceptibility at low temperatures is dominated by the fast intrawell transitions and at higher temperatures by the barrier crossing kinetics. We find that for nonlinear susceptibilities the latter process appears to be more important and the intrawell transitions play only a role at the lowest temperatures.
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Affiliation(s)
- Gregor Diezemann
- Department Chemie, Johannes Gutenberg Universität Mainz, 55128 Mainz, Germany
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Wang SW. Inferring energy dissipation from violation of the fluctuation-dissipation theorem. Phys Rev E 2018; 97:052125. [PMID: 29906903 DOI: 10.1103/physreve.97.052125] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2017] [Indexed: 06/08/2023]
Abstract
The Harada-Sasa equality elegantly connects the energy dissipation rate of a moving object with its measurable violation of the Fluctuation-Dissipation Theorem (FDT). Although proven for Langevin processes, its validity remains unclear for discrete Markov systems whose forward and backward transition rates respond asymmetrically to external perturbation. A typical example is a motor protein called kinesin. Here we show generally that the FDT violation persists surprisingly in the high-frequency limit due to the asymmetry, resulting in a divergent FDT violation integral and thus a complete breakdown of the Harada-Sasa equality. A renormalized FDT violation integral still well predicts the dissipation rate when each discrete transition produces a small entropy in the environment. Our study also suggests a way to infer this perturbation asymmetry based on the measurable high-frequency-limit FDT violation.
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Affiliation(s)
- Shou-Wen Wang
- Beijing Computational Science Research Center, Beijing, 100094, China and Department of Engineering Physics, Tsinghua University, Beijing, 100086, China
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Diezemann G. Nonlinear response theory for Markov processes. II. Fifth-order response functions. Phys Rev E 2017; 96:022150. [PMID: 28950644 DOI: 10.1103/physreve.96.022150] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2017] [Indexed: 06/07/2023]
Abstract
The nonlinear response of stochastic models obeying a master equation is calculated up to fifth order in the external field, thus extending the third-order results obtained earlier [G. Diezemann, Phys. Rev. E 85, 051502 (2012)PLEEE81539-375510.1103/PhysRevE.85.051502]. For sinusoidal fields the 5ω component of the susceptibility is computed for the model of dipole reorientations in an asymmetric double well potential and for a trap model with a Gaussian density of states. For most realizations of the models a hump is found in the higher-order susceptibilities. In particular, for the asymmetric double well potential model there are two characteristic temperature regimes showing the occurrence of such a hump as compared to a single characteristic regime in the case of the third-order response. In the case of the trap model the results strongly depend on the variable coupled to the field. As for the third-order response, the low-frequency limit of the susceptibility plays a crucial role with respect to the occurrence of a hump. The findings are discussed in light of recent experimental results obtained for supercooled liquids. The differences found for the third-order and the fifth-order response indicate that nonlinear response functions might serve as a powerful tool to discriminate among the large number of existing models for glassy relaxation.
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Affiliation(s)
- Gregor Diezemann
- Institut für Physikalische Chemie, Universität Mainz, Duesbergweg 10-14, 55128 Mainz, Germany
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Speck T. Thermodynamic formalism and linear response theory for nonequilibrium steady states. Phys Rev E 2016; 94:022131. [PMID: 27627270 DOI: 10.1103/physreve.94.022131] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/07/2016] [Indexed: 11/07/2022]
Abstract
We study the linear response in systems driven away from thermal equilibrium into a nonequilibrium steady state with nonvanishing entropy production rate. A simple derivation of a general response formula is presented under the condition that the generating function describes a transformation that (to lowest order) preserves normalization and thus describes a physical stochastic process. For Markov processes we explicitly construct the conjugate quantities and discuss their relation with known response formulas. Emphasis is put on the formal analogy with thermodynamic potentials and some consequences are discussed.
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Affiliation(s)
- Thomas Speck
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
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Wang SW, Kawaguchi K, Sasa SI, Tang LH. Entropy Production of Nanosystems with Time Scale Separation. PHYSICAL REVIEW LETTERS 2016; 117:070601. [PMID: 27563943 DOI: 10.1103/physrevlett.117.070601] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2016] [Indexed: 06/06/2023]
Abstract
Energy flows in biomolecular motors and machines are vital to their function. Yet experimental observations are often limited to a small subset of variables that participate in energy transport and dissipation. Here we show, through a solvable Langevin model, that the seemingly hidden entropy production is measurable through the violation spectrum of the fluctuation-response relation of a slow observable. For general Markov systems with time scale separation, we prove that the violation spectrum exhibits a characteristic plateau in the intermediate frequency region. Despite its vanishing height, the plateau can account for energy dissipation over a broad time scale. Our findings suggest a general possibility to probe hidden entropy production in nanosystems without direct observation of fast variables.
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Affiliation(s)
- Shou-Wen Wang
- Beijing Computational Science Research Center, Beijing 100094, China
- Department of Engineering Physics, Tsinghua University, Beijing 100086, China
| | - Kyogo Kawaguchi
- Department of Systems Biology, Harvard Medical School, Boston, Massachusetts 02115, USA
| | - Shin-Ichi Sasa
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan
| | - Lei-Han Tang
- Beijing Computational Science Research Center, Beijing 100094, China
- Department of Physics and Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Hong Kong, China
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Seifert U. Stochastic thermodynamics, fluctuation theorems and molecular machines. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2012; 75:126001. [PMID: 23168354 DOI: 10.1088/0034-4885/75/12/126001] [Citation(s) in RCA: 1175] [Impact Index Per Article: 97.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/18/2023]
Abstract
Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.
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Affiliation(s)
- Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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Diezemann G. Nonlinear response theory for Markov processes: simple models for glassy relaxation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:051502. [PMID: 23004762 DOI: 10.1103/physreve.85.051502] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2012] [Indexed: 06/01/2023]
Abstract
The theory of nonlinear response for Markov processes obeying a master equation is formulated in terms of time-dependent perturbation theory for the Green's functions and general expressions for the response functions up to third order in the external field are given. The nonlinear response is calculated for a model of dipole reorientations in an asymmetric double well potential, a standard model in the field of dielectric spectroscopy. The static nonlinear response is finite with the exception of a certain temperature T_{0} determined by the value of the asymmetry. In a narrow temperature range around T_{0}, the modulus of the frequency-dependent cubic response shows a peak at a frequency on the order of the relaxation rate and it vanishes for both low frequencies and high frequencies. At temperatures at which the static response is finite (lower and higher than T_{0}), the modulus is found to decay monotonously from the static limit to zero at high frequencies. In addition, results of calculations for a trap model with a Gaussian density of states are presented. In this case, the cubic response depends on the specific dynamical variable considered and also on the way the external field is coupled to the kinetics of the model. In particular, a set of different dynamical variables that gives rise to identical shapes of the linear susceptibility and only to different temperature dependencies of the relaxation times is considered. It is found that the frequency dependence of the nonlinear response functions, however, strongly depends on the particular choice of the variables. The results are discussed in the context of recent theoretical and experimental findings regarding the nonlinear response of supercooled liquids and glasses.
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Affiliation(s)
- Gregor Diezemann
- Institut für Physikalische Chemie, Universität Mainz, Jakob-Welder-Weg 11, 55128 Mainz, Germany
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Diezemann G, Heuer A. Memory effects in the relaxation of the Gaussian trap model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:031505. [PMID: 21517505 DOI: 10.1103/physreve.83.031505] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2010] [Indexed: 05/30/2023]
Abstract
We investigate the memory effect in a simple model for glassy relaxation, a trap model with a Gaussian density of states. In this model, thermal equilibrium is reached at all finite temperatures and we therefore can consider jumps from low to high temperatures in addition to the quenches usually considered in aging studies. We show that the evolution of the energy following the Kovacs protocol can approximately be expressed as a difference of two monotonously decaying functions and thus show the existence of a so-called Kovacs hump whenever these functions are not single exponentials. It is well established that the Kovacs effect also occurs in the linear response regime, and we show that most of the gross features do not change dramatically when large temperature jumps are considered. However, there is one distinguishing feature that only exists beyond the linear regime, which we discuss in detail. For the memory experiment with inverted temperatures, i.e., jumping up and then down again, we find a very similar behavior apart from an opposite sign of the hump.
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Affiliation(s)
- Gregor Diezemann
- Institut für Physikalische Chemie, Universität Mainz, Welderweg 11, DE-55099 Mainz, Federal Republic of Germany
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Corberi F, Lippiello E, Sarracino A, Zannetti M. Fluctuation-dissipation relations and field-free algorithms for the computation of response functions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:011124. [PMID: 20365340 DOI: 10.1103/physreve.81.011124] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/21/2009] [Indexed: 05/29/2023]
Abstract
We discuss the relation between the fluctuation-dissipation relation derived by Chatelain and Ricci-Tersenghi [C. Chatelain, J. Phys. A 36, 10739 (2003); F. Ricci-Tersenghi, Phys. Rev. E 68, 065104(R) (2003)] and that by Lippiello-Corberi-Zannetti [E. Lippiello, F. Corberi, and M. Zannetti, Phys. Rev. E 71, 036104 (2005)]. In order to do that, we rederive the fluctuation-dissipation relation for systems of discrete variables evolving in discrete time via a stochastic nonequilibrium Markov process. The calculation is carried out in a general formalism comprising the Chatelain, Ricci-Tersenghi, result and that by Lippiello-Corberi-Zannetti as special cases. The applicability, generality, and experimental feasibility of the two approaches are thoroughly discussed. Extending the analytical calculation to the variance of the response function, we show the advantage of field-free numerical methods with respect to the standard method, where the perturbation is applied. We also show that the signal-to-noise ratio is better (by a factor square root of 2) in the algorithm of Lippiello-Corberi-Zannetti with respect to that of Chatelain-Ricci Tersenghi.
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Affiliation(s)
- Federico Corberi
- Dipartimento di Matematica ed Informatica and INFN, Gruppo Collegato di Salerno and CNISM, Unitá di Salerno, Università di Salerno, Via Ponte don Melillo, Fisciano (SA), Italy
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Baiesi M, Maes C, Wynants B. Fluctuations and response of nonequilibrium states. PHYSICAL REVIEW LETTERS 2009; 103:010602. [PMID: 19659132 DOI: 10.1103/physrevlett.103.010602] [Citation(s) in RCA: 128] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2009] [Revised: 06/03/2009] [Indexed: 05/28/2023]
Abstract
A generalized fluctuation-response relation is found for thermal systems driven out of equilibrium. Its derivation is independent of many details of the dynamics, which is only required to be first order. The result gives a correction to the equilibrium fluctuation-dissipation theorem, in terms of the correlation between observable and excess in dynamical activity caused by the perturbation. Previous approaches to this problem are recovered and extended in a unifying scheme.
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Affiliation(s)
- Marco Baiesi
- Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, 3001, Belgium
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de Oliveira MJ. Fluctuation-dissipation relation for stochastic dynamics without detailed balance. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:011114. [PMID: 17677417 DOI: 10.1103/physreve.76.011114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2007] [Indexed: 05/16/2023]
Abstract
We show that the fluctuation-dissipation relation can be established for a class of stochastic dynamics that lack detailed balance. This class comprises lattice spin models whose time evolution is governed by a master equation with a one-spin-flip transition rate having the up-down symmetry. The relation is obtained by the introduction of a multiplicative perturbation of the transition rate, which reduces to the usual perturbation when detailed balance is fulfilled. As a part of the derivation we set up an equivalent two-spin-flip stochastic dynamics that conserves the magnetization.
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Affiliation(s)
- Mário J de Oliveira
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo, São Paulo, Brazil
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Diezemann G, Böhmer R. Aging in a free-energy landscape model for glassy relaxation. II. Fluctuation-dissipation relations. J Chem Phys 2006; 124:214507. [PMID: 16774423 DOI: 10.1063/1.2202351] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Several fluctuation-dissipation relations are investigated for a simple free-energy landscape model designed to describe the primary relaxation in supercooled liquids. The calculations of the response and of the correlation functions are performed for a quench from a high temperature to a low temperature. In the model, all dynamical quantities reach equilibrium after long times, but for times shorter than the re-equilibration time they do not exhibit time-translational invariance and the fluctuation-dissipation theorem is violated. Two measures for these violations are considered. One such measure is given by the slope in a plot of the integrated response versus the correlation function and another one by the so-called fluctuation-dissipation ratio. It is found that these measures do not coincide and furthermore are not independent of the dynamical variable considered in the calculation. We propose to determine the fluctuation-dissipation ratio experimentally via measurements of the deuteron spin-lattice relaxation rate and the dielectric loss.
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Affiliation(s)
- Gregor Diezemann
- Institut für Physikalische Chemie, Universität Mainz, 55099 Mainz, Federal Republic of Germany.
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