Work distribution in a time-dependent logarithmic–harmonic potential: exact results and asymptotic analysis

Ryabov, Artem; Dierl, Marcel; Chvosta, Petr; Einax, Mario; Maass, Philipp

doi:10.1088/1751-8113/46/7/075002

Cited by 37 publications

(54 citation statements)

References 25 publications

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“…In addition, a bra (ket) vector is normalized by calculating the scalar product with the corresponding ket (bra) vector as is shown in Eq. (20), but the normalization of the probability density is determined by dxρ(x, t) = 1.…”

Section: Bi-orthogonal Systemmentioning

confidence: 99%

“…On the other hand, when the work is expressed as an analytic function of the control parameter, we can apply the variational scheme to find the optimized protocol [4,5]. In fact, the optimization has been exclusively studied for the harmonic potential which can be solved exactly [15][16][17][18][19][20][21][22][23][24]. [45] In an exact calculation, the macroscopic quantities of the Brownian particle is expressed as the functions of the moments of the position of the Brownian particle [15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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Perturbative expansion of irreversible work in Fokker–Planck equationà laquantum mechanics

Koide¹

2017

J. Phys. A: Math. Theor.

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We discuss the systematic expansion of the solution of the Fokker-Planck equation with the help of the eigenfunctions of the time-dependent Fokker-Planck operator.The expansion parameter is the time derivative of the external parameter which controls the form of an external potential. Our expansion corresponds to the perturbative calculation of the adiabatic motion in quantum mechanics. With this method, we derive a new formula to calculate the irreversible work order by order, which is expressed as the expectation value with a pseudo density matrix. Applying this method to the case of the harmonic potential, we show that the first order term of the expansion gives the exact result. Because we do not need to solve the coupled differential equations of moments, our method simplifies the calculations of various functions such as the fluctuation of the irreversible work per unit time. We further investigate the exact optimized protocol to minimize the irreversible work by calculating its variation with respect to the control parameter itself.

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Section: Bi-orthogonal Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Perturbative expansion of irreversible work in Fokker–Planck equationà laquantum mechanics

Koide¹

2017

J. Phys. A: Math. Theor.

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“…The method of functional determinants simplifies this problem to instead determining the solutions of ELEs, which are linear ordinary differential equations. In [17], for the special case of a Brownian particle in a logarithmic harmonic potential, the authors obtained the asymptotic form of work distribution in terms of the solution of a Riccati differential equation. We are not aware of any other analytic methods for performing the exact finite-time computation of the asymptotic form including the pre-factor, in Langevin systems.…”

Section: Comparison With Results In [4]mentioning

confidence: 99%

“…The operator A therefore has a zero-mode which is nothing but the optimal trajectory itself. Writing integration variable y as a series expansion in terms of the normalized eigenfunctions φ n (t) of A as, y(·) = Σ n c n φ n (t) (17) and then performing the Gaussian integrals over the expansion parameters c n and r, it can be shown that the zero-mode of A gets omitted naturally and does not cause any problems to the integral in Eq. (16).…”

Section: The Pre-exponential Factormentioning

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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“…Next we apply the rotating frame transformation (2). The key property making this problem analytically tractable and free of the aforementioned positivity issues is that the 7 superoperators {H i , D i } form a closed algebra [55] (see also [56]). In particular, the sets {H i } and {D i }, when taken separately, satisfy independent algebras:…”

Section: Application To a Harmonic Oscillatormentioning

confidence: 99%

Lindblad-Floquet description of finite-time quantum heat engines

2018

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The operation of autonomous finite-time quantum heat engines rely on the existence of a stable limit cycle in which the dynamics becomes periodic. The two main questions that naturally arise are therefore whether such a limit cycle will eventually be reached and, once it has, what is the state of the system within the limit cycle. In this paper we show that the application of Floquet's theory to Lindblad dynamics offers clear answers to both questions. By moving to a generalized rotating frame, we show that it is possible to identify a single object, the Floquet Liouvillian, which encompasses all operating properties of the engine. First, its spectrum dictates the convergence to a limit cycle. And second, the state within the limit cycle is precisely its zero eigenstate, therefore reducing the problem to that of determining the steady-state of a time-independent master equation.To illustrate the usefulness of this theory, we apply it to a harmonic oscillator subject to a time-periodic work protocol and time-periodic dissipation, an open-system generalization of the Ermakov-Lewis theory. The use of this theory to implement a finite-time Carnot engine subject to continuous frequency modulations is also discussed.

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Work distribution in a time-dependent logarithmic–harmonic potential: exact results and asymptotic analysis

Cited by 37 publications

References 25 publications

Perturbative expansion of irreversible work in Fokker–Planck equationà laquantum mechanics

Perturbative expansion of irreversible work in Fokker–Planck equationà laquantum mechanics

Asymptotics of work distributions in a stochastically driven system

Lindblad-Floquet description of finite-time quantum heat engines

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