Generalized fluctuation theorems for classical systems

Abstract

Fluctuation theorems have a very special place in the study of non equilibrium dynamics of physical systems. The form in which it is used most extensively is the Gallavoti-Cohen Fluctuation Theorem[1] which is in terms of the distribution of the work p(W )/p(−W ) = exp(αW ). We derive the general form of the fluctuation theorems for an arbitrary Gaussian Markov process and find conditions when the parameter α becomes a universal parameter 1/kT . As an application we consider fluctuation theorems for classical cyclotron motion of an electron in a parabolic potential. The motion of the electron is described by four coupled Langevin equations and thus is non-trivial. The generalized theorems are equally valid for non-equilibrium steady states.

Independent-oscillator model and the quantum Langevin equation for an oscillator: a review

“…Following this, we will discuss a situation where the system (the oscillator) and the bath are connected via momentum-dependent couplings [48][49][50][51]; the specific case of a gauge-invariant model of a three-dimensional oscillator coupled to a heat bath via momentum variables in the presence of a vector potential is carefully analyzed [52,53]. Finally, we discuss the timely topic of fluctuation theorems [54][55][56][57][58][59][60][61][62][63][64][65], focusing on work fluctuations in the context of dissipative quantum oscillators, for such relations are intimately related to the second law of thermodynamics. As it turns out, for simple situations such as the Brownian motion as described by a linear Langevin equation which has a built-in noise term, the fluctuation theorem takes the simple form of the Gallavotti-Cohen kind [54,56,60].…”

“…Finally, we discuss the timely topic of fluctuation theorems [54][55][56][57][58][59][60][61][62][63][64][65], focusing on work fluctuations in the context of dissipative quantum oscillators, for such relations are intimately related to the second law of thermodynamics. As it turns out, for simple situations such as the Brownian motion as described by a linear Langevin equation which has a built-in noise term, the fluctuation theorem takes the simple form of the Gallavotti-Cohen kind [54,56,60]. We analyze this for the case of the dissipative quantum cyclotron motion of a free particle and also its classical counterpart [60].…”

“…As it turns out, for simple situations such as the Brownian motion as described by a linear Langevin equation which has a built-in noise term, the fluctuation theorem takes the simple form of the Gallavotti-Cohen kind [54,56,60]. We analyze this for the case of the dissipative quantum cyclotron motion of a free particle and also its classical counterpart [60].…”

Quantum fluctuation theorem for dissipative cyclotron motion

“…The statement reads: p{W }/p{−W } ∼ exp(αW ), where α is a positive temperature-dependent factor whose form is by no means universal. In fact, in [21] we showed that only for a one-dimensional process wherein the force acts on a single-component displacement vector does α reduce to 1/(K B T ) where K B is the Boltzmann constant. Now the question is: can one develop a similar formulation in the quantum case?…”

“…We have earlier shown the validity of the conventional fluctuation-dissipation theorems for the classical dissipative cyclotron motion described by Eq. ( 2) [21]. As stated before, here we examine if the fluctuation theorems are applicable to the quantized motion in a dissipative environment.…”