Validity of nonequilibrium work relations for the rapidly expanding quantum piston

Abstract

Recent work by Teifel and Mahler [Eur. Phys. J. B 75, 275 (2010)] raises legitimate concerns regarding the validity of quantum nonequilibrium work relations in processes involving moving hard walls. We study this issue in the context of the rapidly expanding one-dimensional quantum piston. Utilizing exact solutions of the time-dependent Schrödinger equation, we find that the evolution of the wave function can be decomposed into static and dynamic components, which have simple semiclassical interpretations in terms of particle-piston collisions. We show that nonequilibrium work relations remains valid at any finite piston speed, provided both components are included, and we study explicitly the work distribution for this model system. In the past two decades, much attention has been devoted to theoretical predictions and experimental investigations regarding the fluctuations of small systems away from thermal equilibrium. These predictions include the nonequilbrium work relation [1,2] and the corresponding fluctuation theorem derived by Crooks [3][4][5] ρ F (+W ) ρ R (−W ) = e β(W −∆F ) ,arXiv:1112.5798v1 [cond-mat.stat-mech]

Interference of identical particles and the quantum work distribution

“…In this article, we extend our previous studies [40,42,44,53,54] to multiparticle systems. We will show that for noninteracting particles, the transition amplitudes between many-particle eigenstates can be constructed from the transition amplitudes between single-particle eigenstates.…”

“…For the sake of simplicity we will continue our discussion for two analytically solvable examples. For single-particle systems analogous studies include the 1D piston system with a moving wall [44,69] and the 1D harmonic oscillator with a time-dependent angular frequency [40,42,70].…”

“…Previous studies of quantum work relations have been mostly focused on single-particle quantum systems, such as dragged harmonic oscillators [38,39], parametric harmonic oscillators [40][41][42][43], a single particle in a time-dependent piston [44], two-level systems [45], and the parametric Morse oscillator [46]. However, the interplay of quantum work and quantum statistical properties, e.g., the Fermi-Dirac statistics or the Bose-Einstein statistics, have not been fully explored yet (but see Refs.…”

Quantum-Classical Correspondence Principle for Work Distributions

“…In recent years, interest in the nonequilibrium thermodynamics of small systems [1][2][3][4] has motivated careful examinations of how to define quantum work . In this context one often considers a process in which a quantum system evolves under the Schrödinger equation as its Hamiltonian is varied in timefor instance, a quantum particle in a piston undergoing compression or expansion [51]. It is typically assumed that the system is initialized in thermal equilibrium, and the question becomes, how do we appropriately define the work performed on the system during a single realization of this process?…”

Bernoulli's formula and Poisson's equations for a confined quantum gas: Effects due to a moving piston

“…Since the kinetic theory of Boltzmann gas tells that a moving piston does not play a role in the equation of states, we shall investigate the nonadiabatic dynamics in the quantum heat engine. While in recent years there appeared papers which treated the quantum engine, they were either concerned with a quantum analog of Carnot's engine [4][5][6][7] or with a quantum analog of nonequilibrium work relation (i.e., fluctuation theorem) [8,9]. And no work so far was engaged in nonadiabatic force and pressure due to a moving piston and in the statistical treatment of a noninteracting Fermi gas.…”