Fluctuation-dissipation theorem

About: Fluctuation-dissipation theorem is a research topic. Over the lifetime, 605 publications have been published within this topic receiving 25604 citations. The topic is also known as: FDT.

Statistical-Mechanical Theory of Irreversible Processes : I. General Theory and Simple Applications to Magnetic and Conduction Problems

Abstract: A general type of fluctuation-dissipation theorem is discussed to show that the physical quantities such as complex susceptibility of magnetic or electric polarization and complex conductivity for electric conduction are rigorously expressed in terms of time-fluctuation of dynamical variables associated with such irreversible processes. This is a generalization of statistical mechanics which affords exact formulation as the basis of calculation of such irreversible quantities from atomistic theory. The general formalism of this statistical-mechanical theory is examined in detail. The response, relaxation, and correlation functions are defined in quantummechanical way and their relations are investigated. The formalism is illustrated by simple examples of magnetic and conduction problems. Certain sum rules are discussed for these examples. Finally it is pointed out that this theory may be looked as a generalization of the Einstein relation.

The fluctuation-dissipation theorem

Abstract: The linear response theory has given a general proof of the fluctuation-dissipation theorem which states that the linear response of a given system to an external perturbation is expressed in terms of fluctuation properties of the system in thermal equilibrium. This theorem may be represented by a stochastic equation describing the fluctuation, which is a generalization of the familiar Langevin equation in the classical theory of Brownian motion. In this generalized equation the friction force becomes retarded or frequency-dependent and the random force is no more white. They are related to each other by a generalized Nyquist theorem which is in fact another expression of the fluctuation-dissipation theorem. This point of view can be applied to a wide class of irreversible process including collective modes in many-particle systems as has already been shown by Mori. As an illustrative example, the density response problem is briefly discussed.

Irreversibility and Generalized Noise

Abstract: A relation is obtained between the generalized resistance and the fluctuations of the generalized forces in linear dissipative systems. This relation forms the extension of the Nyquist relation for the voltage fluctuations in electrical impedances. The general formalism is illustrated by applications to several particular types of systems, including Brownian motion, electric field fluctuations in the vacuum, and pressure fluctuations in a gas.

The Fluctuation Theorem

Abstract: The question of how reversible microscopic equations of motion can lead to irreversible macroscopic behaviour has been one of the central issues in statistical mechanics for more than a century. The basic issues were known to Gibbs. Boltzmann conducted a very public debate with Loschmidt and others without a satisfactory resolution. In recent decades there has been no real change in the situation. In1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamicaluctuations in thermostatted dissipa- tive non-equilibrium systems. The relation was derived heuristically and applied to the special case of dissipative non-equilibrium systems subject to constant energy `thermostatting'. These restrictions meant that the full importance of the Theorem was not immediately apparent. Within a few years, derivations of the Theorem were improved but it has only been inthe last few of years that the generality ofthe Theorem has been appreciated. We now know that the Second Law of Thermo- dynamics can be derived assuming ergodicity at equilibrium, and causality. We take the assumption of causality to be axiomatic. It is causality which ultimately is responsible for breaking time reversal symmetry and which leads to the possibility of irreversible macroscopic behaviour. The Fluctuation Theorem does much more than merely prove that in large systems observed for long periods of time, the Second Law is overwhelmingly likely to be valid. The Fluctuation Theorem quanti®es the probability of observing Second Law violations in small systems observed for a short time. Unlike the Boltzmann equation, the FT is completely consistent with Loschmidt's observa- tion that for time reversible dynamics, every dynamical phase space trajectory and its conjugate time reversedanti-trajectory', are both solutions of the underlying equations of motion. Indeed the standard proofs of the FT explicitly consider conjugate pairs of phase space trajectories. Quantitative predictions made by the Fluctuation Theorem regarding the probability of Second Law violations have been con®rmed experimentally, both using molecular dynamics computer simula- tion and very recently in laboratory experiments.

Fluctuating hydrodynamics and Brownian motion

Abstract: The Langevin equation describing Brownian motion is considered as a contraction from the more fundamental, but still phenomenological, description of an incompressible fluid governed by fluctuating hydrodynamics in which a Brownian particle with stick boundary condition is immersed. First, the derivation of fluctuating hydrodynamics is reconsidered to clarify certain ambiguities as to the treatment of boundaries. Subsequently the contraction is carried out. Since Brownian particles of arbitrary shape are considered, rotations and translations are in general coupled. The symmetry of the 6×6 friction tensorγ ij (t) is proved for arbitrary shape without appeal to microscopic arguments. This symmetry is then used to prove that the fluctuation-dissipation theorem on the contracted level (nonwhite noise in general) follows from the corresponding statement on the level of fluctuating hydrodynamics (white noise). The condition under which the contracted description reduces to the classical Langevin equation is given, and the connection between our theory and related work is discussed.