Fluctuation-dissipation model

The positive-definite matrix Bfp will be determined by the structure of the fluid flow around the particle, and will more than likely be significantly anisotropic (Tenneti et al, 2012). Note that we have not included the fluid-velocity-fluctuation-dissipation model in Eq. (4.104) when writing the GPBE. Here, for clarity, we will focus exclusively on the interdependence of the fluid and particle velocities, for which it suffices to consider a ID velocity phase space for Vp and Vf. The GPBE for this case is given by... 

When the fluid-velocity-fluctuation-dissipation model in Eq. (4.104) is included, an additional variance reduction mechanism will be present. 

Arimoto, Y., Wada, T., Ohta, M., Abe, Y. Fluctuation-dissipation model for synthesis of superheavy elements. Phys. Rev. C59, 796-809 (1999)... 

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. 

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... 

G. N. Bochkov and Y. E. Kuzovlev, Non-linear fluctuation relations and stochastic models in nonequilibrium thermodynamics. 1. Generalized fluctuation-dissipation theorem. Physica A 106, 443-J79 (1981). 

E. Sciortino and P. Tartaglia, Extension of the fluctuation-dissipation theorem to the physical aging of a model glass-forming liquid. Phys. Rev. Lett. 86, 107-110 (2001). 

Schieber JD (1992) Do internal viscosity models satisfy the fluctuation-dissipation theorem J Non-Newton Fluid Mech 45 47-61... 

Behavior of the Fluctuation-Dissipation Ratio and of the Effective Temperature in the Langevin Model... 

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. 

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. 

Chapter VI illustrates what a mesoscopic level of description is meant to contain. It involves the variables of interest (usually assumed to be slow) plus a suitable set of auxiliary variables, whose role is to mimic the influence of the thermal bath on the variables of interest themselves. This level of description (reduced model theory) is less detailed than the truly microscopic one, because an overwhelming number of microscopic degrees of freedom are simulated with fluctuation-dissipation processes of standard type. The mesoscopic level, however, is still detailed enough to preserve the essential information without which the theoretical investigation becomes difficult and obscure. A new class of non-Gaussian equilibrium properties is proven to be responsible for the acceleration of the fall transient described in Chapter V. To obtain these results, use is made both of the theoretical tools already mentioned and of computer simulation (one-dimensional for translation and two-dimensional for rotation). 

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. 

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). 

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