Fluctuation-dissipation theorems models

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... 

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. 

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). 

According to the fluctuation-dissipation theorem [1], the electrical polarizability of polyelectrolytes is related to the fluctuations of the dipole moment generated in the counterion atmosphere around the polyions in the absence of an applied electric field [2-4], Here we calculate the fluctuations by computer simulation to determine anisotropy of the electrical polarizability Aa of model DNA fragments in salt-free aqueous solutions [5-7]. The Metropolis Monte Carlo (MC) Brownian dynamics method [8-12] is applied to calculate counterion distributions, electric potentials, and fluctuations of counterion polarization. 

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. 

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. 

Based on the fluctuation-dissipation theorem, the equilibrium-simulated Gs t) is predicted to be equivalent to the step strain-simulated Gs t) in the linear region. In Fig. 16.2, the equilibrium-simulated Gs(t) curves for two-bead, five-bead and ten-bead Rouse chains are also shown. These equilibrium-simulated Gs(t) results are in perfect agreement with the step strain-simulated results and the Rouse theoretical curves, illustrating the fluctuation-dissipation theorem as applied to the Rouse model and confirming the validity of the Monte Carlo simulations. 

Although there are several techniques for estimating the reaction rate constants based upon the deterministic model, these methods are usually rather complicated, and the results cannot be statistically characterised. That is why from time to time estimates based upon one or another stochastic model are suggested. Such a suggestion has earlier been described under the name fluctuation-dissipation theorem , and similar methods have been presented by Mulloolly (1971, 1972, 1973), Hilden (1974), and Matis and Hartley (1971). 

This model for describing transmitter-receptor interaction can be identified with the stochastic model of closed compartment systems. Adopting the fluctuation-dissipation theorem the conductance spectrum of the postsynap-tic cell is determined by three qualitatively different factors ... 

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and exotic begins to become common . Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. 

Equilibrium is a state of matter that results from spatial uniformity. In contrast, when there are concentration differences or gradients, particles will flow. In these cases, the rate of flow is proportional to the gradient. The proportionality constant between the flow rate and the gradient is a transport property for particle flow, this property is the diffusion constant. Diffusion can be modelled at the microscopic level as a random flight of the particle. The diffusion constant describes the mean square displacement of a particle per unit time. The fluctuation-dissipation theorem describes how transport properties are related to the ensemble-averaged fluctuations of the system in equilibrium. 

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