Fluctuation-dissipation theorem temperatures

Thus, the requirement that the Brownian particle becomes equilibrated with the surrounding fluid fixes the unknown value of, and provides an expression for it in tenns of the friction coefficient, the thennodynamic temperature of the fluid, and the mass of the Brownian particle. Equation (A3.1.63) is the simplest and best known example of a fluctuation-dissipation theorem, obtained by using an equilibrium condition to relate the strengtii of the fluctuations to the frictional forces acting on the particle [22]. 

For an individual molecule, fluctuations of the instantaneous electronic charge density away from its quantum mechanical average are characterized by the fluctuation-dissipation theorem (3, 4). The molecule is assumed to be in equilibrium with a radiation bath at temperature T then in the final step of the derivation, the limit is taken as T — 0. The fluctuation correlations, which are defined by... 

The concept of a nonequilibrium temperature has stimulated a lot of research in the area of glasses. This line of research has been promoted by Cugliandolo and Kurchan in the study of mean-held models of spin glasses [161, 162] that show violations of the fluctuation-dissipation theorem (FDT) in the NEAS. The main result in the theory is that two-time correlations C t,t ) and responses R t, f ) satisfy a modihed version of the FDT. It is customary to introduce the effective temperature through the fluctuation-dissipation ratio (FDR) [163] dehned as... 

X. Because there is damping of nuclear motion into e-h pairs, excited e-h pairs at electron temperature Te must also be able to excite nuclear coordinates by fluctuating forces Fx that satisfy the second fluctuation dissipation theorem given as... 

Remark. It is easily seen that the second term of (5.2) by itself causes the norm of if/ to change. In order that this is compensated by the fluctuating term the two terms must be linked, as is done by the relation U = V V. This resembles the classical fluctuation-dissipation theorem, which links both terms by the requirement that the fluctuations compensate the energy loss so as to establish the equilibrium. The difference is that the latter requirement involves the temperature T of the environment that makes it possible to suppress the fluctuations by taking T = 0 without losing the damping. This is the reason why in classical theory deterministic equations with damping exist, see XI.5. 

Here M is a mobility coefficient, which is assumed to be constant and r/(r.t) is the random thermal noise term, which for a system in equilibrium at temperature T satisfies the fluctuation-dissipation theorem. The free energy functional is taken to be of a Ginzburg-Landau form. In the notation of Qi and Wang (1996,1997) it is given by... 

For any reservoir in equilibrium the fluctuation-dissipation theorem provides the relation between the symmetrized and antisymmetrized correlators of the noise Sx(x) = Ax(x) coth(w/2T). Yet, the temperature dependence of Sx and Ax may vary depending on the type of the environment. For an oscillator bath, Ax (also called the spectral density Jx(x)) is temperature-independent, so that Sx(x) = Jx(x)coth(x/2T). On the other hand, for a spin bath Sx is temperature-independent and is related to the spins density of states, while Ax([Pg.14]

This is a specific example of the fundamental fluctuation-dissipation theorem that relates the random force / (fluctuation) to the friction constant 7 (dissipation) to ensure that any initial state eventually evolves into a state in thermal equilibrium with the fluid at temperature T. With this requirement, we obtain from Eq. (11.12) the final result... 

The important fluctuation-dissipation theorem that expresses a balance between the dissipation and fluctuating forces to ensure the approach to equilibrium at a specified temperature is generalized to the expression that in the mass-weighted coordinates has the form... 

The aim of this chapter is to show how the concepts of FDT violation and effective temperature can be illustrated in the framework of the above quoted system, as done experimentally in Ref. 12 and theoretically in Refs. 15-19. We do not discuss here the vast general domain of aging effects in glassy systems, which are reviewed in Refs. 2-4. Since the present contribution should be understood by beginners in the field, some relevant fundamental topics of equilibrium statistical physics—namely, on the one hand, the statistical description of a system coupled to an environment and, on the other hand, the fluctuation-dissipation theorem (in a time domain formulation)—are first recalled. Then, questions specifically related to out-of-equilibrium dynamics, such as the description of aging effects by means of an effective temperature, are taken up in the framework of the above-quoted model system. 

In Section III, we recall the fluctuation-dissipation theorem (FDT), valid for dynamical variables evolving in equilibrium. We provide its formulation in the time domain (in the whole range of temperatures). The choice of the time-domain formulation is motivated by the fact that, when out-of-equilibrium variables are concerned, the necessary modification of the equilibrium FDT can conveniently be carried out by introducing a fluctuation-dissipation ratio defined in terms of time-dependent quantities. 

We compute below the velocity and displacement correlation functions, first, of a classical, then, of a quantal, Brownian particle. In contrast to its velocity, which thermalizes, the displacement x(t) — x(l0) of the particle with respect to its position at a given time never attains equilibrium (whatever the temperature, and even at T = 0). The model allows for a discussion of the corresponding modifications of the fluctuation-dissipation theorem. 

One usually studies diffusion in a thermal bath by writing two fluctuation-dissipation theorems, generally referred to as the first and second FDTs (using the Kubo terminology [30,31]). As recalled for instance in Ref. 57, the first FDT expresses a necessary condition for a thermometer in contact solely with the system to register the temperature of the bath. As for the second FDT, it expresses the fact that the bath itself is in equilibrium. 

Summing up, when the particle environment is a thermal bath, the two fluctuation-dissipation theorems are valid. In both theorems the bath temperature T plays an essential role. In its form (157) or (159) (Einstein relation), the first FDT involves the spectral density of a dynamical variable linked to the particle (namely its velocity), while, in its form (161) (Nyquist formula), the second FDT involves the spectral density of the random force, which is a dynamical variable of the bath. 

The fully general situation of a particle diffusing in an out-of-equilibrium environment is much more difficult to describe. Except for the particular case of a stationary environment, the motion of the diffusing particle cannot be described by the generalized Langevin equation (22). A more general equation of motion has to be used. The fluctuation-dissipation theorems are a fortiori not valid. However, one can try to extend these relations with the help of an age- and frequency-dependent effective temperature, such as proposed and discussed, for instance, in Refs. 5 and 6. 

The starting point for the proposed new approach is an exact formula [238], [239], based on the adiabatic connection formula and the zero-temperature fluctuation-dissipation theorem, relating the groundstate xc energy to the inter-... 

Molecular dynamics simulations entail integrating Newton s second law of motion for an ensemble of atoms in order to derive the thermodynamic and transport properties of the ensemble. The two most common approaches to predict thermal conductivities by means of molecular dynamics include the direct and the Green-Kubo methods. The direct method is a non-equilibrium molecular dynamics approach that simulates the experimental setup by imposing a temperature gradient across the simulation cell. The Green-Kubo method is an equilibrium molecular dynamics approach, in which the thermal conductivity is obtained from the heat current fluctuations by means of the fluctuation-dissipation theorem. Comparisons of both methods show that results obtained by either method are consistent with each other [55]. Studies have shown that molecular dynamics can predict the thermal conductivity of crystalline materials [24, 55-60], superlattices [10-12], silicon nanowires [7] and amorphous materials [61, 62]. Recently, non-equilibrium molecular dynamics was used to study the thermal conductivity of argon thin films, using a pair-wise Lennard-Jones interatomic potential [56]. 

The Kramers model consists of a classical particle of mass m moving on a one-dimensional potential surface V(x) (Fig. 1) under the influence of Markovian random force R(t) and damping y, which are related to each other and to the temperature T by the fluctuation dissipation theorem. 

In Section II we will review thermodynamics and the fluctuation-dissipation theorem for excess heat production based on the Boltzmann equilibrium distribution. We will also mention the nonequilibrium work relation by Jarzynski. In Section III, we will extend the fluctuation-dissipation theorem for the superstatisitcal equilibrium distribution. The fluctuation-dissipation theorem can be written as a superposition of correlation functions with different temperatures. When the decay constant of a correlation function depends on temperature, we can expect various behaviors in the excess heat. In Section IV, we will consider the case of the microcanonical equilibrium distribution. We will numerically show the breaking of nonergodic adiabatic invariant in the mixed phase space. In the last section, we will conclude and comment. 

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