Fluctuation-dissipation theorem effects

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

The Langevin dynamics method simulates the effect of individual solvent molecules through the noise W, which is assumed to be Gaussian. The friction coefficient r is related to the autocorrelation function of W through the fluctuation-dissipation theorem,... 

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

The averaged potential energy (V) includes contributions from fluctuations in the charge density at all real frequencies. The fluctuation-dissipation theorem restricts the contributing frequencies to co = -o), but allows for all real co. The effects on the energy are contained in the term 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... 

In equilibrium, or in a stable state, the magnitude of the fluctuations is the outcome of the competition between the jumps and the macroscopic return to equilibrium. Both effects are represented by the second and first term, respectively, on the right of (4.2b). This is the basis of the Einstein relation (VIII.3.9) and of the fluctuation-dissipation theorem. 

Let us first analyze the effect of the bias field on the magnetic SR in the framework of the linear response theory formulated for SR [96,97,99]. The main idea of the linear response treatment is a direct use of the fluctuation-dissipation theorem, which expresses the thermal (fluctuational) power spectrum Qn(a>) of the magnetic moment of the system, namely, magnetic noise, through the imaginary component of its linear dynamic susceptibility ImX = to a weak probing ac field of an arbitrary frequency co as... 

We can run the cause-effect connection the other way. The natural motions of the charges within a material will necessarily create electric fields whose time-varying spectral properties are those known from how the materials absorb the energy of applied fields (the "fluctuation-dissipation theorem"). It is the correlations between these spontaneously occurring electric fields and their source charges that create van der Waals forces. At a deeper level, we can even think of all these charge or field fluctuations as results or distortions of the electromagnetic fields that would occur spontaneously in vacuum devoid of matter. 

This chapter relates to some recent developments concerning the physics of out-of-equilibrium, slowly relaxing systems. In many complex systems such as glasses, polymers, proteins, and so on, temporal evolutions differ from standard laws and are often much slower. Very slowly relaxing systems display aging effects [1]. This means in particular that the time scale of the response to an external perturbation, and/or of the associated correlation function, increases with the age of the system (i.e., the waiting time, which is the time elapsed since the preparation). In such situations, time-invariance properties are lost, and the fluctuation-dissipation theorem (FDT) does not hold. 

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. 

When out-of-equilibrium dynamic variables are concerned, as will be the case in the following sections of this chapter, the equilibrium fluctuation-dissipation theorem is not applicable. In order to discuss properties such as the aging effects which manifest themselves by the loss of time translational invariance in... 

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. 

If the random force obeys the fluctuation dissipation theorem, then the effective Hamiltonian must take the form... 

At any temperature, x(T)/Xc gives the effective number of free spins and is related by the fluctuation-dissipation theorem to the sum of all static spin correlations. The complicated X(T)/Xc behavior in Fig. 17 thus points to disordered exchanges in the power-law regime below 40 °K the flat region 40 < T < 140 K suggests that some 8% are essentially free and follow the Curie law, Eq. (7) the T > 140 K regime... 

We have thus seen that the requirement that the friction y and the random force 7 (Z) together act to bring the system to thennal equilibrium at long time, naturally leads to a relation between them, expressed by Eq. (8.20). This is a relation between fluctuations and dissipation in the system, which constitutes an example of the fluctuation-dissipation theorem (see also Chapter 11). In effect, the requirement that Eq. (8.20) holds is equivalent to the condition of detailed balance, imposed on transition rates in models described by master equations, in order to satisfy the requirement that thermal equilibrium is reached at long time (see Section 8.3). 

In the following sections we will apply the theory to a resonance decaying into a continuum (Section 2.2) and to several resonances decaying into one or several continua (Section 2.3). The physics is described and understood by means of energy-independent effective Hamiltonians (16) and from the method of moments. In Section 2.2 the use of a unique two-dimensional matrix representation (n = 2) of the effective Hamiltonian will allow us to produce the most basic Breit-Wigner and Eano profiles as well as an elementary formulation of the fluctuation-dissipation theorem. In Section 2.3 more elaborate matrix representations (n = 3) will be used to investigate... 

The derivation of the fluctuation-dissipation theorem (38) was based on the simple model Hamiltonian (30). This effective Hamiltonian is employed again to investigate asymmetric profiles in spectroscopy implying quantum interferences with the continuum. 

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