Fluctuation theorems and atypical trajectories

In this work, we have studied simple models that can be solved analytically to illustrate various fluctuation theorems. These fluctuation theorems provide symmetries individually to the distributions of physical quantities like the classical work ($W_c$), thermodynamic work ($W$), total entropy ($\Delta s_{tot}$) and dissipated heat ($Q$), when the system is driven arbitrarily out of equilibrium. All these quantities can be defined for individual trajectories. We have studied the number of trajectories which exhibit behaviour unexpected at the macroscopic level. As the time of observation increases, the fraction of such atypical trajectories decreases, as expected at macroscale. Nature of distributions for the thermodynamic work and the entropy production in nonlinear models may exhibit peak (most probable value) in the atypical regime without violating the expected average behaviour. However, dissipated heat and classical work exhibit peak in the regime of typical behaviour only.Comment: 14 pages, 7 figure

The Steady State Distribution of the Master Equation

The steady states of the master equation are investigated. We give two expressions for the steady state distribution of the master equation a la the Zubarev-McLennan steady state distribution, i.e., the exact expression and an expression near equilibrium. The latter expression obtained is consistent with recent attempt of constructing steady state theormodynamics.Comment: 6 pages, No figures. A mistake was correcte

Noncyclic and nonadiabatic geometric phase for counting statistics

We propose a general framework of the geometric-phase interpretation for counting statistics. Counting statistics is a scheme to count the number of specific transitions in a stochastic process. The cumulant generating function for the counting statistics can be interpreted as a `phase', and it is generally divided into two parts: the dynamical phase and a remaining one. It has already been shown that for cyclic evolution the remaining phase corresponds to a geometric phase, such as the Berry phase or Aharonov-Anandan phase. We here show that the remaining phase also has an interpretation as a geometric phase even in noncyclic and nonadiabatic evolution.Comment: 12 pages, 1 figur

Non-equilibrium work relations

This is a brief review of recently derived relations describing the behaviour of systems far from equilibrium. They include the Fluctuation Theorem, Jarzynski's and Crooks' equalities, and an extended form of the Second Principle for general steady states. They are very general and their proofs are, in most cases, disconcertingly simple.Comment: Brief Summer School Lecture Note

Fluctuation theorems for harmonic oscillators

We study experimentally the thermal fluctuations of energy input and dissipation in a harmonic oscillator driven out of equilibrium, and search for Fluctuation Relations. We study transient evolution from the equilibrium state, together with non equilibrium steady states. Fluctuations Relations are obtained experimentally for both the work and the heat, for the stationary and transient evolutions. A Stationary State Fluctuation Theorem is verified for the two time prescriptions of the torque. But a Transient Fluctuation Theorem is satisfied for the work given to the system but not for the heat dissipated by the system in the case of linear forcing. Experimental observations on the statistical and dynamical properties of the fluctuation of the angle, we derive analytical expressions for the probability density function of the work and the heat. We obtain for the first time an analytic expression of the probability density function of the heat. Agreement between experiments and our modeling is excellent

Path-integral analysis of fluctuation theorems for general Langevin processes

We examine classical, transient fluctuation theorems within the unifying framework of Langevin dynamics. We explicitly distinguish between the effects of non-conservative forces that violate detailed balance, and non-autonomous dynamics arising from the variation of an external parameter. When both these sources of nonequilibrium behavior are present, there naturally arise two distinct fluctuation theorems.Comment: 24 pages, one figur

Fluctuation theorems for stochastic dynamics

Fluctuation theorems make use of time reversal to make predictions about entropy production in many-body systems far from thermal equilibrium. Here we review the wide variety of distinct, but interconnected, relations that have been derived and investigated theoretically and experimentally. Significantly, we demonstrate, in the context of Markovian stochastic dynamics, how these different fluctuation theorems arise from a simple fundamental time-reversal symmetry of a certain class of observables. Appealing to the notion of Gibbs entropy allows for a microscopic definition of entropy production in terms of these observables. We work with the master equation approach, which leads to a mathematically straightforward proof and provides direct insight into the probabilistic meaning of the quantities involved. Finally, we point to some experiments that elucidate the practical significance of fluctuation relations.Comment: 48 pages, 2 figures. v2: minor changes for consistency with published versio

Energy gaps in quantum first-order mean-field-like transitions: The problems that quantum annealing cannot solve

We study first-order quantum phase transitions in models where the mean-field traitment is exact, and the exponentially fast closure of the energy gap with the system size at the transition. We consider exactly solvable ferromagnetic models, and show that they reduce to the Grover problem in a particular limit. We compute the coefficient in the exponential closure of the gap using an instantonic approach, and discuss the (dire) consequences for quantum annealing.Comment: 6 pages, 3 figure

Fluctuation relations and coarse-graining

We consider the application of fluctuation relations to the dynamics of coarse-grained systems, as might arise in a hypothetical experiment in which a system is monitored with a low-resolution measuring apparatus. We analyze a stochastic, Markovian jump process with a specific structure that lends itself naturally to coarse-graining. A perturbative analysis yields a reduced stochastic jump process that approximates the coarse-grained dynamics of the original system. This leads to a non-trivial fluctuation relation that is approximately satisfied by the coarse-grained dynamics. We illustrate our results by computing the large deviations of a particular stochastic jump process. Our results highlight the possibility that observed deviations from fluctuation relations might be due to the presence of unobserved degrees of freedom.Comment: 19 pages, 6 figures, very minor change

Stick-slip motion of solids with dry friction subject to random vibrations and an external field

We investigate a model for the dynamics of a solid object, which moves over a randomly vibrating solid surface and is subject to a constant external force. The dry friction between the two solids is modeled phenomenologically as being proportional to the sign of the object's velocity relative to the surface, and therefore shows a discontinuity at zero velocity. Using a path integral approach, we derive analytical expressions for the transition probability of the object's velocity and the stationary distribution of the work done on the object due to the external force. From the latter distribution, we also derive a fluctuation relation for the mechanical work fluctuations, which incorporates the effect of the dry friction.Comment: v1: 23 pages, 9 figures; v2: Reference list corrected; v3: Published version, typos corrected, references adde