Fluctuation Theorem of Information Exchange Within an Ensemble of Paths Conditioned on Correlated-Microstates
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Fluctuation Theorem for Nonequilibrium Reactions
Journal of Chemical Physics 120 (2004) 8898-8905 Fluctuation theorem for nonequilibrium reactions Pierre Gaspard Center for Nonlinear Phenomena and Complex Systems, Universit´eLibre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium A fluctuation theorem is derived for stochastic nonequilibrium reactions ruled by the chemical master equation. The theorem is expressed in terms of the generating and large-deviation functions characterizing the fluctuations of a quantity which measures the loss of detailed balance out of thermodynamic equilibrium. The relationship to entropy production is established and discussed. The fluctuation theorem is verified in the Schl¨oglmodel of far-from-equilibrium bistability. PACS numbers: 82.20.Uv; 05.70.Ln; 02.50.Ey I. INTRODUCTION Reacting systems can be driven out of equilibrium when in contact with several particle reservoirs or chemiostats generating fluxes of matter across the system. The fluxes are caused by the differences of chemical potentials between the chemiostats. Such an open system may be thought of as a reactor with inlets for reactants and an outlet for the products. In this case, the open system is driven out of equilibrium at the boundaries with the chemiostats. Even if detailed balance is satisfied for all the reactions in the bulk of the reactor, the nonequilibrium boundary conditions will break detailed balance for the reactions establishing the contact with the chemiostats. According to the second law of thermodynamics, the resulting nonequilibrium states are characterized by the production of entropy inside the open system. Above the nanoscale, the reactions taking place in the system can be described in terms of the numbers of molecules of the different species. -
Fluctuation Theorems for Quantum Master Equations
PHYSICAL REVIEW E 73, 046129 ͑2006͒ Fluctuation theorems for quantum master equations Massimiliano Esposito* and Shaul Mukamel Department of Chemistry, University of California, Irvine, California 92697, USA ͑Received 17 November 2005; published 24 April 2006͒ A quantum fluctuation theorem for a driven quantum subsystem interacting with its environment is derived based solely on the assumption that its reduced density matrix obeys a closed evolution equation—i.e., a quantum master equation ͑QME͒. Quantum trajectories and their associated entropy, heat, and work appear naturally by transforming the QME to a time-dependent Liouville space basis that diagonalizes the instanta- neous reduced density matrix of the subsystem. A quantum integral fluctuation theorem, a steady-state fluc- tuation theorem, and the Jarzynski relation are derived in a similar way as for classical stochastic dynamics. DOI: 10.1103/PhysRevE.73.046129 PACS number͑s͒: 05.30.Ch, 05.70.Ln, 03.65.Yz I. INTRODUCTION restricted situations ͓24–27͔. A quantum exchange fluctua- tion theorem has also been considered in ͓28͔. Some inter- The fluctuation theorems and the Jarzynski relation are esting considerations of the quantum definition of work in some of a handful of powerful results of nonequilibrium sta- the previous studies have been made in ͓29͔. tistical mechanics that hold far from thermodynamic equilib- It should be noted that the dynamics of an isolated rium. Originally derived in the context of classical mechan- ͑whether driven or not͒ quantum system is unitary and its ics ͓1͔, the Jarzynski relation has been subsequently von Neumann entropy is time independent. Therefore, fluc- extended to stochastic dynamics ͓2͔. -
Thermodynamic Derivation of the Fluctuation Theorem and Jarzynski
Thermodynamic derivation of the fluctuation theorem and Jarzynski equality MAARTEN H. P. AMBAUM ∗ Department of Meteorology, University of Reading, United Kingdom Abstract A thermodynamic expression for the analog of the canonical ensemble for nonequilibrium systems is described based on a purely information theoretical interpretation of entropy. It is shown that this nonequilibrium canonical distribu- tion implies some important results from nonequilibrium thermodynamics, specifically, the fluctuation theorem and the Jarzynski equality. Those results are therefore expected to be more widely applicable, for example, to macroscopic systems. 1 Introduction Earth system may be governed by thermodynamic con- straints on entropy production [11, 12, 13, 14, 15]. The The derivations of the fluctuation theorem [1, 2] and the theoretical underpinning of those thermodynamic con- Jarzynski equality [3] appear to depend on the underlying straints is still lacking. The presence of a fluctuation the- microscopic Hamiltonian dynamics. From this it would orem for such systems would be of great importance. follow that these theorems are only relevant to micro- Here we demonstrate that the information-theoretical scopic systems, with their associated definitions of en- definition of entropy implies the fluctuation theorem and tropy and temperature. In contrast, a statistical mechani- the Jarzynski equality. It is shown that these results are cal description of macroscopic systems often depends on due to the counting properties of entropy rather than the more general forms of entropy, primarily information en- dynamics of the underlying system. As such, both these tropy [4, 5, 6]. Two notable examples from fluid dynamics results are applicable to a much wider class of problems, are the statistical mechanics of point vortices [7] and the specifically, macroscopic systems for which we can define statistical mechanics of two-dimensional incompressible an entropy and which are thermostatted in some general flows[8]. -
Fluctuation Theorem
Fluctuation Theorem Giovanni Gallavotti Fluctuation Theorem: a simple consequence of a time reversal symmetry; it deals with motions which are chaotic in the strong mathematical sense of being hyperbolic and transitive (ie are generated by smooth hyperbolic evolutions on a smooth compact surface (the “phase space”) and with a dense trajectory, also called Anosov systems) and “furthermore are time reversible”. In such systems any initial data, with the exception of a set of zero volume in phase space, have the same statistical properties in the sense that all smooth observables admit a time average independent of the initial data and expressed as an integral with respect to a probability distribution on phase space, called the ”natural stationary state”, or simply the “stationary state”. The theorem provides, asymptotically in the observation time, a quantitative and param- eter free relation between the stationary state probability of observing a value of the average entropy production rate and its opposite. Although there are quite a few examples of mechanical systems which are hyperbolic and transitive in the above mathematical sense, the fluctuation theorem ac- quires physical interest only in connection with the chaotic hypothesis. Under the latter general assumption, combined with time reversal, it pre- dicts a ”universal relation” between an entropy creation rate value and its opposite, accessible to simulations and possibly to laboratory experiments. The basis for the physical interpretation of the theorem as a property of stationary states in nonequilibrium statistical mechanics is developed here. Statistical Mechanics of Nonequilibrium Stationary States. Thermostats In nonequilibrium statistical mechanics the molecules of a system are sub- ject to nonconservative forces whose work is dissipated in the form of heat supplied to other systems kept at constant temperature: the ”thermostats” with which the system is in contact. -
Equalities and Inequalities : Irreversibility and the Second Law of Thermodynamics at the Nanoscale
S´eminairePoincar´eXV Le Temps (2010) 77 { 102 S´eminairePoincar´e Equalities and Inequalities : Irreversibility and the Second Law of Thermodynamics at the Nanoscale Christopher Jarzynski Department of Chemistry and Biochemistry and Institute for Physical Science and Technology University of Maryland College Park MD 20742, USA 1 Introduction On anyone's list of the supreme achievements of the nineteenth-century science, both Maxwell's equations and the second law of thermodynamics surely rank high. Yet while Maxwell's equations are widely viewed as done, dusted, and uncontro- versial, the second law still provokes lively arguments, long after Carnot published his Reflections on the Motive Power of Fire (1824) and Clausius articulated the increase of entropy (1865). The puzzle at the core of the second law is this : how can microscopic equations of motion that are symmetric with respect to time-reversal give rise to macroscopic behavior that clearly does not share this symmetry ? Of course, quite apart from questions related to the origin of \time's arrow", there is a nuts-and-bolts aspect to the second law. Together with the first law, it provides a set of tools that are indispensable in practical applications ranging from the design of power plants and refrigeration systems to the analysis of chemical reactions. The past few decades have seen growing interest in applying these laws and tools to individual microscopic systems, down to nanometer length scales. Much of this interest arises at the intersection of biology, chemistry and physics, where there has been tremendous progress in uncovering the mechanochemical details of biomolecular processes. -
Is Turbulence a State of Maximal Dissipation? Martin Mihelich, Davide Faranda, Didier Paillard, Bérengère Dubrulle
Is Turbulence a State of Maximal Dissipation? Martin Mihelich, Davide Faranda, Didier Paillard, Bérengère Dubrulle To cite this version: Martin Mihelich, Davide Faranda, Didier Paillard, Bérengère Dubrulle. Is Turbulence a State of Maximal Dissipation?. Entropy, MDPI, 2017, 10.3390/e19040154. hal-01460706v2 HAL Id: hal-01460706 https://hal.archives-ouvertes.fr/hal-01460706v2 Submitted on 5 Apr 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Article Is Turbulence a State of Maximum Energy Dissipation? Martin Mihelich 1,†, Davide Faranda 2, Didier Paillard2 and Bérengère Dubrulle 1,* 1 SPEC, CEA, CNRS, Université Paris-Saclay, CEA Saclay 91191 Gif sur Yvette cedex, France 2 LSCE-IPSL, CEA Saclay l’Orme des Merisiers, CNRS UMR 8212 CEA-CNRS-UVSQ, Université Paris-Saclay, 91191 Gif-sur-Yvette, France * Correspondence: [email protected]; Tel.: +33-169 087 247 † Current address: [email protected] Academic Editor: name Version March 22, 2017 submitted to Entropy 1 Abstract: Turbulent flows are known to enhance turbulent transport. It has then even been 2 suggested that turbulence is a state of maximum energy dissipation. -
A Statistical Inference of the Principle of Maximum Entropy Production
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 2 March 2021 doi:10.20944/preprints202103.0110.v1 Article A statistical inference of the Principle of Maximum Entropy Production Martijn Veening 1,† 1 EntropoMetrics; [email protected] † Current address: Electrastraat 21, 9801 WD Zuidhorn, The Netherlands 1 Abstract: The maximization of entropy S within a closed system is accepted as an inevitability (as 2 the second law of thermodynamics) by statistical inference alone. The Maximum Entropy Production 3 Principle (MEPP) states that not only S maximizes, but S˙ as well: a system will dissipate as fast as 4 possible. There is still no consensus on the general validity of this MEPP, even though it shows 5 remarkable explanatory power (both qualitatively and quantitatively), and has been empirically 6 demonstrated for many domains. In this theoretical paper I provide a generalization of entropy 7 gradients, to show that the MEPP actually follows from the same statistical inference, as that of 8 the 2nd law of thermodynamics. For this generalization I only use the concepts of super-statespaces 9 and microstate-density. These concepts also allow for the abstraction of ’Self Organizing Criticality’ 10 to a bifurcating local difference in this density, and allow for a generalization of the fundamentally 11 unresolved concepts of ’chaos’ and ’order’. 12 Keywords: entropy production maximization, complexity, thermodynamics, nonlinear dynamics, 13 statistical mechanics, autopoiesis, entropy, entropy production, life 14 1. Introduction 15 In the last few decades, several attempts have been made to explain the occurrence 16 and stability of what we call ’living systems’, or ’order out of chaos’. -
Arxiv:1904.01064V3 [Cond-Mat.Stat-Mech] 20 Jan 2020 Tion Theorem (TFT); in [15], a Detailed fluctuation The- Understanding of Eps, E.G
Unified formalism for entropy productions and fluctuation relations Ying-Jen Yang∗ and Hong Qiany Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, USA Stochastic entropy production, which quantifies the difference between the probabilities of trajec- tories of a stochastic dynamics and its time reversals, has a central role in nonequilibrium thermo- dynamics. In the theory of probability, the change in the statistical properties of observables due to reversals can be represented by a change in the probability measure. We consider operators on the space of probability measure that induce changes in the statistical properties of a process, and for- mulate entropy productions in terms of these change-of-probability-measure (CPM) operators. This mathematical underpinning of the origin of entropy productions allows us to achieve an organization of various forms of fluctuation relations: All entropy productions have a non-negative mean value, admit the integral fluctuation theorem, and satisfy a rather general fluctuation relation. Other results such as the transient fluctuation theorem and detailed fluctuation theorems then are derived from the general fluctuation relation with more constraints on the operator of a entropy production. We use a discrete-time, discrete-state-space Markov process to draw the contradistinction among three reversals of a process: time reversal, protocol reversal and the dual process. The properties of their corresponding CPM operators are examined, and the domains of validity of various fluctuation relations for entropy productions in physics and chemistry are revealed. We also show that our CPM operator formalism can help us rather easily extend other fluctuations relations for excess work and heat, discuss the martingale properties of entropy productions, and derive the stochastic integral formulas for entropy productions in constant-noise diffusion process with Girsanov theorem. -
The Fluctuation Theorem and Green-Kubo Relations
The Fluctuation Theorem and Green-Kubo Relations Debra J. Searles* and Denis J. Evans# *Department of Chemistry, University of Queensland, Brisbane, QLD 4072, Australia #Research School of Chemistry, Australian National University, GPO Box 414, Canberra, ACT 2601, Australia Abstract Green-Kubo and Einstein expressions for the transport coefficients of a fluid in a nonequilibrium steady state can be derived using the Fluctuation Theorem and by assuming the probability distribution of the time-averaged dissipative flux is Gaussian. These expressions are consistent with those obtained using linear response theory and are valid in the linear regime. It is shown that these expressions are however, not valid in the nonlinear regime where the fluid is driven far from equilibrium. We advance an argument for why these expressions are only valid in the linear response, zero field limit. 2 I. INTRODUCTION In 1993 Evans, Cohen and Morriss [1], ECM2, gave a quite general formula for the logarithm of the probability ratio that in a nonequilibrium steady state, the time averaged dissipative flux takes on a value, J+ () t , to minus that value, namely, J−()() t= − J + t . That is they gave a formula for ln[p ( J+ ( t )) / p ( J − ( t ))] from a natural invariant measure [1, 2]. This formula gives an analytic expression for the probability that, for a finite system and for a finite time, the dissipative flux flows in the reverse direction to that required by the Second Law of Thermodynamics. The formula has come to be known as the Fluctuation Theorem, FT. Surprisingly perhaps, it is valid far from equilibrium in the nonlinear response regime [1]. -
Nonequilibrim Fluctuation Theorems and Thermodynamic Second Laws
Nonequilibrim Fluctuation Theorems and Thermodynamic Second Laws Hyunggyu Park School of Physics, KIAS Abstract Most processes in nature are nonequilibrium (NEQ) processes, which include key dynamic processes in biological cells and social networks, as well as the usual physical phenomena. Nevertheless, our understanding of NEQ dynamic processes, except for those near equilibrium (EQ) has been quite primitive. Recently, various interesting results have been reported regarding phenomena far from EQ. These have opened a new understanding of NEQ processes. Among many others, the most dramatic advance was achieved on the issue of the thermodynamic second law, which is known as the law of entropy increase or irreversibility. First, the inequality relation of the thermodynamic second law, ⌬S Ն0, was replaced by the equality relation, e-⌬S =1, through which the inequality relation could be automatically derived. Detailed information on the probability distribution function of ⌬S was given by using the so-called Gallavotti-Cohen symmetry. More surprisingly, the total entropy production could be divided into two distinct terms, each of which satisfies its own thermodynamic 2nd law. This implies a more intricate structure in the irreversibility. In addition, many other new thermodynamic second laws and corresponding equality relations have been established. These equality relations are called the fluctuation theorems. Prelude Most processes in nature are non-equilibrium processes. The ‘equally likely postulate’ formed by Boltzmann and Gibbs a century ago can be applied only to equilibrium processes, and there is currently a lack of a general ensemble theorem that can describe irreversible dynamic processes from one equilibrium state to another, or a steady state that in itself is in non-equilibrium. -
Jarzynski Equality: Connections to Thermodynamics and the Second Law
PHYSICAL REVIEW E 75, 011133 ͑2007͒ Jarzynski equality: Connections to thermodynamics and the second law Benoit Palmieri and David Ronis* Department of Chemistry, McGill University, 801 Sherbrooke Ouest, Montréal, Québec, Canada H3A 2K6 ͑Received 6 September 2006; published 31 January 2007͒ The one-dimensional expanding ideal gas model is used to compute the exact nonequilibrium distribution function. The state of the system during the expansion is defined in terms of local thermodynamics quantities. The final equilibrium free energy, obtained a long time after the expansion, is compared against the free energy that appears in the Jarzynski equality. Within this model, where the Jarzynski equality holds rigorously, the free energy change that appears in the equality does not equal the actual free energy change of the system at any time of the process. More generally, the work bound that is obtained from the Jarzynski equality is an upper bound to the upper bound that is obtained from the first and second laws of thermodynamics. The cancellation of the dissipative ͑nonequilibrium͒ terms that result in the Jarzynski equality is shown in the framework of response theory. This is used to show that the intuitive assumption that the Jarzynski work bound becomes equal to the average work done when the system evolves quasistatically is incorrect under some conditions. DOI: 10.1103/PhysRevE.75.011133 PACS number͑s͒: 05.20.Ϫy, 05.70.Ln, 05.40.Ϫa I. INTRODUCTION tained, as above, by the ratio of two canonical partition func- tions defined at the same temperature, but for the two differ- Only a few relations are exactly satisfied for processes in ent work parameters a and b. -
Jarzynski Equality and Crooks Relation for Local Models of Air-Sea Interaction Achim Wirth1 and Florian Lemarié:2 1Univ
https://doi.org/10.5194/esd-2020-82 Preprint. Discussion started: 15 December 2020 c Author(s) 2020. CC BY 4.0 License. Jarzynski equality and Crooks relation for local models of air-sea interaction Achim Wirth1 and Florian Lemarié:2 1Univ. Grenoble Alpes, CNRS, Grenoble INP, LEGI, 38000 Grenoble, France 2Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble,France Correspondence: [email protected] Abstract. We show using a hierarchy of local models of air-sea interaction that the most prominent of the work theorems, the Jarzynski equality and the Crooks relation can be applied to air-sea interaction. In the more idealized models, with and without a Coriolis force, the variability is provided from a Gaussian white-noise which modifies the shear between the atmosphere and the ocean. The dynamics is Gaussian and the Jarzynski equality and Crooks relation can be obtained analytically solving 5 stochastic differential equations. The more involved model consists of interacting atmospheric and oceanic boundary-layers, where only the dependence on the vertical direction is resolved, the turbulence is modeled through standard turbulent models and the stochasticity comes from a randomized drag coefficient. It is integrated numerically and can give rise to a non-Gaussian dynamics. Also in this case the Jarzynski equality allows for calculating a dynamic-beta βD of the turbulent fluctuations (the 1 equivalent of the thermodynamic-beta β = (kBT )− in thermal fluctuations). The Crooks relation gives the βD as a function of 10 the magnitude of the work fluctuations. It is well defined (constant) in the Gaussian models and can show a slight variation in the more involved models.