On asymptotic behavior of work distributions for driven Brownian motion

Holubec, Viktor; Lips, Dominik; Ryabov, Artem; Chvosta, Petr; Maass, Philipp

doi:10.1140/epjb/e2015-60635-x

Cited by 9 publications

(14 citation statements)

References 19 publications

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“…The generating function ψ s (t) provides an alternative way to get access to the properties of the work distribution function [43], so the central problem is to solve the FKE (6).Frequent collision approximation and the reduced Feynman-Kac equation-Unfortunately, a general exact solution of the FKE (6) is difficult to obtain, due to the complexities arising from both the number of variables and the non-analycity of the expressions (h(ξ)). In fact, besides the driven overdamped Brownian HO [52,53], the V-potential [54], and the logarithmic-harmonic potential [54,55], the only analytically solvable model in ST so far seems to be the breathing overdamped Brownian HO [43,44]. Even for such a model, an exact solution is usually unavailable unless the initial distribution is Gaussian.Accordingly, we need to make further approximations to obtain analytic results in certain interesting regimes.…”

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confidence: 99%

Stochastic Thermodynamics of a Particle in a Box

2016

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The piston system (particles in a box) is the simplest and paradigmatic model in traditional thermodynamics. However, the recently established framework of stochastic thermodynamics (ST) fails to apply to this model system due to the embedded singularity in the potential. In this Letter we study the stochastic thermodynamics of a particle in a box by adopting a novel coordinate transformation technique. Through comparing with the exact solution of a breathing harmonic oscillator, we obtain analytical results of work distribution for an arbitrary protocol in the linear response regime, and verify various predictions of the Fluctuation-Dissipation Relation. When applying to the Brownian Szilard's engine model, we obtain the optimal protocol λt = λ02 t/τ for a given sufficiently long total time τ . Our study not only establishes a paradigm for studying ST of a particle in a box, but also bridges the long-standing gap in the development of ST.PACS numbers: 05.70. Ln, 05.10.Gg Introduction.-When opening any textbook of thermodynamics [1], the piston system [2], or the classical ideal gas inside a rigid-wall potential is the simplest and an archetypal model used to illustrate various thermodynamic processes and cycles. In the context of traditional thermodynamics, due to the macroscopic size of the system, fluctuations are usually vanishingly small. There work and heat are phenomenological variables and the microscopic equation of motion (EOM) is not directly relevant.When considering a small system, however, fluctuations become important and the EOM becomes essential [3]. In recent years, substantial developments in the field of nonequilibrium thermodynamics in small systems [4] have been made. One of them is the formulation of the so-called stochastic thermodynamics (ST) [5][6][7], where stochastic dynamics is incorporated into thermodynamics. For small systems, e.g., a Brownian particle in a controllable potential, a coherent framework of thermodynamics at the trajectory level is constructed. Fluctuating thermodynamic variables, such as work, heat and entropy production, are identified as functionals of individual trajectories [8][9][10][11], based on which one can in principle calculate their distributions in arbitrary driven processes [12,13], and thus go beyond the traditional thermodynamics. In the linear response regime, the work distribution is Gaussian and satisfies the FluctuationDissipation relations (FDRs) [12,14]. What is more, even in arbitrarily far from equilibrium processes, some exact fluctuation relations concerning work, heat and entropy production are discovered [10,[15][16][17][18][19][20]. Experimentally, these fluctuation relations have been verified in various systems including a Brownian particle in a soft-wall potential [21][22][23][24], exemplified by a charged colloidal particle trapped by an optical tweezer. The essential point of these developments in thermodynamics is the microscopic definition of work, heat and entropy at the trajectory level.However, the usual microscopic definiti...

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confidence: 99%

Stochastic Thermodynamics of a Particle in a Box

2016

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“…Our solutions are exact in the low noise ( β → ∞ ) limit.Potential U(x(t), λ(t))A e −B |W | Table 1: Asymptotic behaviour of work distributions. List of asymptotic forms of P (W ) known so far adapted from [22]. A, B, C are constants that depend upon the explicit form of the driving protocol and the duration of the protocol T .It is however hard to find an exact expression for the full work distribution P (W ) except in the few cases mentioned above.…”

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“…In addition, they also obtain the analytic solution to the asymptotic form of P (W ) in an absolute value potential (V-potential). Table 1 is adapted from [22], where the so far known results for the asymptotic forms of work distributions for various driving protocols are listed. In all the above cases the driving protocol λ(t) considered, is a deterministic function of time.In this paper, we will first revisit a model with a deterministic driving protocol, the breathing parabola [1], to familiarize the reader with the methods of EN theory.…”

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confidence: 99%

Asymptotics of work distributions in a stochastically driven system

Manikandan

Krishnamurthy

2017

Eur. Phys. J. B

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Abstract. We determine the asymptotic forms of work distributions at arbitrary times T , in a class of driven stochastic systems using a theory developed by Nickelsen and Engel (EN theory) [D. Nickelsen and A. Engel, Eur. Phys. J. B 82, 207 (2011)], which is based on the contraction principle of large deviation theory. In this paper, we extend the theory, previously applied in the context of deterministically driven systems, to a model in which the driving is stochastic. The models we study are described by overdamped Langevin equations and the work distributions in path integral form, are characterised by having quadratic augmented actions. We first illustrate EN theory, for a deterministically driven system -the breathing parabola model, and show that within its framework, the Crooks fluctuation theorem manifests itself as a reflection symmetry property of a certain characteristic polynomial, which also determines the exact moment-generating-function at arbitrary times. We then extend our analysis to a stochastically driven system, studied in references [S. Sabhapandit, EPL 89, 60003 (2010) (2014)], for both equilibrium and non-equilibrium steady state initial distributions. In both cases we obtain new analytic solutions for the asymptotic forms of (dissipated) work distributions at arbitrary T . For dissipated work in the steady state, we compare the large T asymptotic behaviour of our solution to the functional form obtained in reference [New J. Phys. 16, 095001 (2014)]. In all cases, special emphasis is placed on the computation of the pre-exponential factor and the results show excellent agreement with numerical simulations. Our solutions are exact in the low noise (β → ∞) limit.

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“…Namely, for a piece-wise constant protocol, where k(t) and T (t) involve a single jump [136] or two jumps [7,137]; in the slow-driving limit ( k(t) small compared to the relaxation time of the system) where the work PDF is Gaussian for any k(t) [37,138]; and if k(t) is a rational function of time [40,41]. In addition to exact approaches, theories were developed to predict asymptotics of work PDFs for small and large values of work [40,133,[139][140][141][142]. Also an Onsager-Machlup-type theory has been applied to obtain approximate solutions of the problem in Ref.…”

Section: Overdamped Harmonic Oscillatormentioning

confidence: 99%

Fluctuations in heat engines

Holubec,

Ryabov

2021

Preprint

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Investigation of fundamental limitations of heat to work conversion at the macroscale gave birth to the second law of classical thermodynamics. Recent years have witnessed an enormous effort to develop and describe microscopic heat engines based on systems as small as individual colloidal particles or quantum dots. At such scales, fluctuations are an inherent part of dynamics and thermodynamics. The fundamentally stochastic character of the second law then yields interesting implications on statistics of output power and efficiency of the small heat engines. The general properties, symmetries, and limitations of these statistics have become revealed, understood, and tested only during the last few years. Will their complete understanding ignite a similar revolution as the discovery of the second law? Here, we review the known general results concerning fluctuations in the performance of small heat engines. To make the discussion more transparent, we illustrate the main abstract findings on exactly solvable models and provide a thorough theoretical introduction for newcomers to the field.

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On asymptotic behavior of work distributions for driven Brownian motion

Cited by 9 publications

References 19 publications

Stochastic Thermodynamics of a Particle in a Box

Stochastic Thermodynamics of a Particle in a Box

Asymptotics of work distributions in a stochastically driven system

Fluctuations in heat engines

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