The fluctuation–dissipation theorem

A system in equilibrium will in general exhibit microscopic fluctuations about the equilibrium state. The fluctuation–dissipation theorem relates the spectrum of these fluctuations to a solution of the macroscopic equation describing the approach to equilibrium from a non-equilibrium state. The aim here is to show exactly what the theorem is and how it is to be used. An account of the quantum version of the theorem is given in three parts, depending on the solution of the macroscopic equation used to express the fluctuations: the relaxation function, the response function or the Green function for continuous systems. Each part is illustrated with an example: charge fluctuations in an RLC circuit for the first two and electric field fluctuations in vacuum for the last.     

Computer simulation of the non-local dielectric function of polar liquids: boundary conditions revisited

For the calculation of ε(k) via the correlation function of the bound charge density fluctuations or the ratio of the bound charge density to an external charge density, a generalized form of the fluctuation dissipation theorem is derived which accounts for different types of boundary conditions through a modified Green's function. It is demonstrated for the case of an interaction site model of water that it is possible to obtain consistent results for the non-local dielectric function with different types of boundary conditions. This is shown by calculating ε(k) both from the correlation function 〈ρ_b(k)ρ_b(-k)〉 and from the dielectric response to an external electric field.     

A generalization of the fluctuation-dissipation theorem for the case of excess noise

Electrical fluctuations proportional to the square of the external field are investigated in the case of quasiequilibrium conditions. The binary correlation fluctuation proportional to the square of the external field is expressed in terms of the zero-field quaternary correlation function for both quantum and classical cases. This result can be considered as a generalization of the fluctuation-dissipation theorem. As applied to the electrical $\text{1}\text{f}$ noise the novel method investigation both in theory and in experiment is presented. The frequency dependence of zero-field quaternary potential correlation function is suggested to be investigated instead of binary correlation function proportional to the square of external field.     

Theoretical Treatment of Nonstationary Scattering by Phonons and Polaritons Part I. Derivation of the Basic Equations

The equations of motion of a two-level system and a harmonic oscillator coupled to a dissipative system are discussed; the dissipative system is assumed to consist of a large number of radiation oscillators. Special equations for the determination of the correlation functions of the fluctuation forces are derived under the condition of large time values, for which the atomic system has “forgotten” its initial state. The expectation values of the correlation functions are connected with the damping constant and the population operator of the excited state of the atomic system is in thermal equilibrium. Taking into account the influence of the coherent radiation field on the atomic system, the basic equations for the treatment of the nonstationary Raman scattering by polaritons are derived; the temporal range of validity is discussed. Using a time-dependent “variable” Fourier transformation, the nonstationary time- and spacedependent spectral densities are related to the correlation functions of the fields; here the Wiener-Khintchine theorem is applied in a nonstationary form. The limiting cases of the stationary scattering process as well as the usually introduced correlators of the slowly varying amplitudes are discussed.     

Extension of the fluctuation theorem

Heat fluctuations are studied in a dissipative system with both deterministic and stochastic components for a simple model: a Brownian particle dragged through water by a moving potential. An extension of the stationary state fluctuation theorem is derived. For infinite time, this reduces to the conventional fluctuation theorem only for small fluctuations; for large fluctuations, it gives a much larger ratio of the probabilities of the particle to absorb rather than supply heat. This persists for finite times and should be observable in experiments similar to a recent one carried out by Wang et al.     

Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains

Asymptotic fluctuation theorems are statements of a Gallavotti-Cohen symmetry in the rate function of either the time-averaged entropy production or heat dissipation of a process. Such theorems have been proved for various general classes of continuous-time deterministic and stochastic processes, but always under the assumption that the forces driving the system are time independent, and often relying on the existence of a limiting ergodic distribution. In this paper we extend the asymptotic fluctuation theorem for the first time to inhomogeneous continuous-time processes without a stationary distribution, considering specifically a finite state Markov chain driven by periodic transition rates. We find that for both entropy production and heat dissipation, the usual Gallavotti-Cohen symmetry of the rate function is generalized to an analogous relation between the rate functions of the original process and its corresponding backward process, in which the trajectory and the driving protocol have been time-reversed. The effect is that spontaneous positive fluctuations in the long time average of each quantity in the forward process are exponentially more likely than spontaneous negative fluctuations in the backward process, and vice-versa, revealing that the distributions of fluctuations in universes in which time moves forward and backward are related. As an additional result, the asymptotic time-averaged entropy production is obtained as the integral of a periodic entropy production rate that generalizes the constant rate pertaining to homogeneous dynamics.     

Fluctuation theorem for the mutation process in in vitro evolution

A proposition based on the fluctuation theorem in thermodynamics is formulated to quantitatively describe molecular evolution processes in biology.Although we cannot give full proof of its generality,we demonstrate via computer simulation its applicability in an example of DNA in vitro evolution.According to this theorem,the evolution process is a series of exponentially rare fluctuations fixed by the force of natural selection.     

Green-Kubo Formulas with Symmetrized Correlation Functions for Quantum Systems in Steady States: The Shear Viscosity of a Fluid in a Steady Shear Flow

For a quantum system in a steady state with a constant current of heat or particles driven by a temperature or chemical potential difference between two reservoirs attached to the system, the fluctuation theorem for the current was previously shown to lead to the Green-Kubo formula for the linear response coefficient for the current expressed in terms of the symmetrized correlation function of the current density operator. In this article, we show that for a quantum system in a steady state with a constant rate of work done on the system, the fluctuation theorem for a quantity induced in the system also leads to the Green-Kubo formula expressed in terms of the symmetrized correlation function of the induced quantity. As an example, we consider a fluid in a steady shear flow driven by a constant velocity of a solid plate moving above the fluid.     

Structural fluctuations in nanoparticles

Standard methods of thermodynamics are used to investigate fluctuation conditions for individual nanoparticles. This analysis leads to a theorem on fluctuations of nanoparticles in the vicinity of the phase transformation temperature stating that fluctuations are possible only between an equilibrium and a non-equilibrium phase. The basic conditions used to describe fluctuations are well suited to calculate the limits for fluctuations as function of temperature and particle size.     

Hydrodynamics and the fluctuation theorem

The fluctuation theorem is a pivotal result of statistical physics. It quantifies the probability of observing fluctuations which are in violation of the second law of thermodynamics. More specifically, it quantifies the ratio of the probabilities of observing entropy-producing and entropy-consuming fluctuations measured over a finite volume and time span in terms of the rate of entropy production in the system, the measurement volume, and time. We study the fluctuation theorem in computer simulations of planar shear flow. The simulations are performed by employing the method of multiparticle collision dynamics, which captures both thermal fluctuations and hydrodynamic interactions. The main outcome of our analysis is that the fluctuation theorem is verified at any averaging time provided that the measurement volume exhibits a specific dependence on a hydrodynamic time scale.     

The ballast resistor; An electro-thermal instability in a conducting wire II; Fluctuations around homogeneous stationary states

The correlation function of temperature fluctuations around homogeneous stationary states of the ballast resistor is evaluated by extending the theory of thermal fluctuations around equilibrium states to non-equilibrium situations. It is found that equal-time temperature fluctuations become correlated over large distances if one approaches the critical point. At the “critical point” of the ballast resistor the correlation length and the amplitude of the temperature fluctuations diverge. Furthermore the theory predicts a critical slowing down for the time-dependent temperature fluctuation correlation function. The electric noise of the system is also analyzed.     

Theoretical Analysis of the Fluctuation Theorem Applied to Electric Circuits

We carry out a theoretical analysis of the fluctuations in the Johnson-Nyquist noise in an RC electric circuit being driven by a non-ideal constant current source. Using the appropriate Langevin equation, we derive the fluctuation theorem for the system. This analysis exhibits some interesting features. Firstly, the average of the dissipation function is not necessarily just an instantaneous entropy production, i.e. the work being done on the circuit by the external power source divided by the ambient temperature. Secondly, by appropriate selection of the values for the circuit components, the transient response of the system can be made identical to the nonequilibrium steady state response. In the limit of an ideal current source, the average of the dissipation function reduces to the entropy production. Our analysis accurately reproduces published experimental results.     

Experimental test of the fluctuation theorem for a driven two-level system with time-dependent rates

A single defect center in diamond periodically excited by a laser is shown to provide a simple realization for a system obeying a fluctuation theorem for nonthermal noise. The distribution of these fluctuations is distinctly non-Gaussian, which has also been verified by numerical calculation. For driving protocols symmetric under time reversal a more restricted form of the theorem holds, which is also known from entropy fluctuations caused by thermal noise.     

Nuclear dissipation with residual interactions studied by means of the Mori formalism

We apply correlation function techniques to the calculation of nuclear friction within the framework of a linear response theory. We make use of the fluctuation dissipation theorem to relate the response function to the correlation function which is evaluated by exploiting projection operator techniques. We go beyond the one-body dissipation approximation in the sense that we have taken into account the decay of particle-hole excitations into more complicated configurations. A rather simple formula for the frequency and temperature dependence of the friction coefficient is derived which we have applied to the high energy fission of²³⁸U. The friction coefficients for deformations around the first and second minima of the fission barrier have been calculated using this approach.     

The onset of superconductivity in the time dependent Ginzburg-Landau theory

The fluctuations of the local order parameter above the superconducting transition temperature give rise to singularities in the electrical conductivity and the diamagnetic susceptibility. Using the time dependent Ginzburg-Landau equation the fluctuation of the current density is calculated. By means of the fluctuation-dissipation theorem and a dispersion relation the electromagnetic response function is then determined for small frequencies and wave-numbers. The dynamical conductivity for bulk material, thin films, and thin wires shows an increasing peak at zero frequency the width of which decreases as the transition temperature is approached. This structure should be observable in microwave experiments.     

Deviations from uniform power law scaling in nonstationary time series

A classic problem in physics is the analysis of highly nonstationary time series that typically exhibit long-range correlations. Here we test the hypothesis that the scaling properties of the dynamics of healthy physiological systems are more stable than those of pathological systems by studying beat-to-beat fluctuations in the human heart rate. We develop techniques based on the Fano factor and Allan factor functions, as well as on detrended fluctuation analysis, for quantifying deviations from uniform power-law scaling in nonstationary time series. By analyzing extremely long data sets of up to N = 10(5) beats for 11 healthy subjects, we find that the fluctuations in the heart rate scale approximately uniformly over several temporal orders of magnitude. By contrast, we find that in data sets of comparable length for 14 subjects with heart disease, the fluctuations grow erratically, indicating a loss of scaling stability.     

A stochastic approach to the kinetic theory of gases

A theory of fluctuations in non-equilibrium diluted gases is presented. The velocity distribution function is treated as a stochastic variable and a master equation for its probability is derived. This evolution equation is based on two processes: binary hard sphere collisions and free flow. A mean-field approximation leads to a non-linear master equation containing explicitly a parameter which represents the spatial correlation length of the fluctuations. An infinite hierarchy of equations for the successive moments is found. If the correlation length is sufficiently short a truncation after the first equation is possible and this leads to the Boltzmann kinetic equation. The associated probability distribution is Poissonian. As to the fluctuation of the macroscopic quantities, an approximation scheme permits to recover the Langevin approach of fluctuating hydrodynamics near equilibrium and its fluctuation-dissipation relations.     

Quantum work fluctuation theorem: Nonergodic Brownian motion case

The work fluctuations of a quantum Brownian particle driven by an external force in a general nonergodic heat bath are studied under a general initial state. The exact analytical expression of the work probability distribution function is derived. Results show the existence of a quantum asymptotic fluctuation theorem, which is in general not a direct generalization of its classical counterpart. The form of this theorem is dependent on the structure of the heat bath and the specified initial condition.     

超音速湍流脉动流场气动光学效应分析

对高超音速飞行器在大气中飞行时所产生的湍流脉动气动光学效应进行了理论分析与计算.根据CFD计算流场时所使用的湍流模型及其控制方程,推导出流场的折射率脉动方差控制方程.用统计方法,求出该脉动流场的系综平均光学传递函数及相位均方差,从不同角度描述了湍流脉动的气动光学效应.计算结果表明,气动光学传输函数的幅度响应函数具有低通特征,使所获得的图像发生像模糊,而相位响应函数则导致红外成像相位非线性偏移.此外,在相同飞行高度下,马赫数越高,图像模糊越严重.电弧风洞实验结果验证了本文理论分析的正确性.