Fluctuation-dissipation theorems theory

R. Kubo, Statistical-mechanical theory of irreversible processes. 1. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan 12, 570 (1957) R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966). 

As mentioned, this equivalence is a consequence of the fluctuation-dissipation theorem (the general basis of linear response theory [51]). In (12.68), we have dropped nonlinear terms and we have not indicated for which state Variance (rj) is computed (because the reactant and product state results only differ by nonlinear terms). We see that A A, AAstat, and AAr x are all linked and are all sensitive to the model parameters, with different computational routes giving a different sensitivity for AArtx. 

For systems close to equilibrium the non-equilibrium behaviour of macroscopic systems is described by linear response theory, which is based on the fluctuation-dissipation theorem. This theorem defines a relationship between rates of relaxation and absorption and the correlation of fluctuations that occur spontaneously at different times in equilibrium systems. 

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 emphasizing the need for satisfying the equipartition theorem, the linear response theory provides a connection for stationary processes through the fluctuation-dissipation theorem. 

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 q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

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... 

Although the Zwanzig and Mori techniques are closely related and, from a purely formal point of view, completely equivalent, the elegant properties of the Mori theory such as the generalized fluctuation-dissipation theorem imply the physical system under study to be linear, whereas this is not necessary in the Zwanzig approach. This is the main reason we shall be able to face nonlinear problems within the context of a Fokker-Planck approach (see also the discussion of the next section). An illuminating approach of this kind can be found in a paper by Zwanzig and Bixon, which has also to be considered an earlier example of the continued fraction technique iq>plied to a non-Hermitian case. This method has also been fruitfully applied to the field of polymer dynamics. 

Here the first two terms just give ma = Force as mass times the second time derivative of the friction equal to the F as the negative derivative of potential, y is the memory friction, and F(t) is the random force. Thus the complex dynamics of all degrees of freedom other than the reaction coordinate are included in a statistical treatment, and the reaction coordinate plus environment are modeled as a modified one-dimensional system. What allows realistic simulation of complex systems is that the statistics of the environment can in fact be calculated from a formal prescription. This prescription is given by the Fluctuation-Dissipation theorem, which yields the relation between the friction and the random force. In particular, this theory shows how to calculate the memory friction from a relatively short-time classical simulation of the reaction coordinate. The Quantum Kramers approach. 

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... 

Equation (38) provides a simple form of the fluctuation-dissipation theorem that relates the fluctuations to dissipation. Here the fluctuation in energy AE is related to the dissipation coefficient E appearing in the exponential decay of P t) exp ( - jt). The same expression (38) was derived by Cohen-Tannoudji et al. from a "coarse-grained" expression of the rate of variation of the density matrix (operator) describing a statistical mixture of states (see D in Chapter IV of Ref. [7]). Because the expression (38) was derived by perturbation, the conditions of validity of perturbation theory must be satisfied. The eigenfunctions associated with Si and Si are... 

A simple expression of the fluctuation-dissipation theorem was established and its range of validity was determined within the standard perturbation theory. 

The first-principle method is being developed for systems with long-range dispersion forces. There are two ways to include dispersion forces in first-principle calculations. A semiempirical van der Waals interaction can be taken into account in ab initio calculations. It is realized by using the Lenard-Jones potential of the form (11.26). The second approach is based on the adiabatic connection fluctuation-dissipation theorem. This theory includes seamless long-range dispersion forces... 

By using linear-response theories, dispersion interactions can be calculated directly in the framework of the Kohn-Sham method. The adiabatic connection/fluctuation-dissipation theorem (AC/FDT) method is a linear-response theory for exactly calculating dispersion interactions within the framework of the Kohn-Sham method (Langreth and Perdew 1975). In this AC/FDT method, electron correlation is calculated as the energy response quantity for the spontaneous fluctuations of electronic motions coming from the perturbation of the interelectronic interactions, as follows ... 

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