Autocorrelation, fluctuation-dissipation

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,... 

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. ... 

We may also use the results to check that the expressions satisfy the fluctuation-dissipation theorem. The Laplace transform of the autocorrelation function for the fluctuating forces is... 

We see that a calculation of Ar involves a Laplace transform of the time-dependent friction kernel. This may typically be determined in a molecular dynamics (MD) simulation, where the autocorrelation function of the random force (R(O)R(t)) may be determined, which then allows us to determine (f) using the fluctuation-dissipation theorem in Eq. (11.58). Note that Eq. (11.85) is an implicit equation for Ar that in general must be solved by iteration. In the absence of friction we see from Eq. (11.85)... 

Another advantage of the simulation is its abihty to make direct tests on the range of validity of basic thermodynamical theorems such as the fluctuation-dissipation theorem. In the second paper of the series by Evans, he considers these points for the simplest type of torque mentioned above, —X F. Consider the return to equilibrium of a dynamical variable A after taking off at r = 0 the constant torque appUed prior to this instant in time. If the torque is removed instantaneously, the first fluctuation-dissipation theorem implies that the normalized fall transient will decay with the same dependence as the autocorrelation function (A(t)A(O))- Al /(A 0)) — Therefore,... 

From Eq. (11.3.19) we note that the memory function is proportional to the autocorrelation function of the random force. This is called the second fluctuation-dissipation theorem (Kubo, 1966). 

The potential of mean force will typically have two wells, corresponding to reactants and products, separated by a barrier. To set the notation, we denote the location of the reactant well, the barrier, and the product well by qa, q, and qb, respectively. One usually expects that the dynamics will be governed by the behavior of the system around the barrier top. Thus the standard procedure (48,49) for generating a GLE is to restrict the system to the barrier top q = q and determine the force autocorrelation function of all other degrees of freedom. The force is just VV, so by using molecular dynamics constrained to the barrier top one can compute the force autocorrelation function (VV(f) W(0)). One then models the true dynamics in terms of a GLE in which the time-dependent friction function is determined through the fluctuation dissipation relation, Eq. (5). 

Ae in terms of the low-density coefficients (equations (5.3) and (5.4)) accordingly contains additional terms. The first, kinetic contribution is the only important term at low densities and scales in time as for diffusion. The final term is the contribution from the potential part alone and the middle term is the cross contribution of the kinetic and potential part. The presence of these terms, and their functional dependence, can be demonstrated simply from a derivation of these expressions by the fluctuation-dissipation theorem, which gives the transport properties in terms of an autocorrelation function of the appropriate flux (see 5.4.1). For thermal conductivity, for example, the flux involves the sum of kinetic and potential eneigies. The autocorrelation of this flux involves the product of the flux at two different times, producing three different terms which can be shown to have the same dependence on density and g(a) as above. 

The methods for calculating A/Zexc( ). an equilibrium property of the system, have already been discussed. The dynamic aspects of the permeation process are captured in D (2 ). The fluctuation-dissipation theorem provides the connection between the local diffusion constant and the time autocorrelation function of the random force acting on the solute. 

The thermal conductivity of a material can be calculated directly from equilibrium molecular dynamics (EMD) simulation based on the linear response theory Green-Kubo relationship. " The fluctuation-dissipation theorem provides a connection between the energy dissipation in irreversible processes and the thermal fluctuations in equilibrium. The thermal conductivity tensor. A, can be expressed in terms of heat current autocorrelation correlation functions (HCACFs),/, ... 

This equilibrium MD approach uses current fluctuations to calculate the thermal conductivity k via the fluctuation dissipation theorem. The MD approach is used to compute the autocorrelation function of the heat flux, which is related to the thermal conductivity by the Green-Kubo formula given by ... 

Another form of the fluctuation-dissipation theorem involves time correlation functions. If the external perturbation is applied at t = —oo and turned off at t = 0, the system will move from a steady state back to equilibrium. The dynamic relaxation of property A from its original steady state, (A(0)), to equilibrium, A)o, can be shown to be congruent to the normalized time autocorrelation function of equilibrium fluctuations. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

As A x was supposed stationary the integral is independent of time. The effect of the fluctuations is therefore to renormalize A0 by adding a constant term of order a2 to it. The added term is the integrated autocorrelation function of At. In particular, if one has a non-dissipative system described by A0, this additional term due to the fluctuations is usually dissipative. This relation between dissipation and the autocorrelation function of fluctuations is analogous to the Green-Kubo relation in many-body systems 510 but not identical to it, because there the fluctuations are internal, rather than added as a separate term as in (2.1). 

E>vib and Our also show up in the theory of spontaneous Raman spectroscopy describing fluctuations of the molecular system. The functions enter the CARS interaction involving vibrational excitation with subsequent dissipation as a consequence of the dissipation-fluctuation theorem and further approximations (21). Equations (2)-(5) refer to a simplified picture a collective, delocalized character of the vibrational mode is not included in the theoretical treatment. It is also assumed that vibrational and reori-entational relaxation are statistically independent. On the other hand, any specific assumption as to the time evolution of vib (or or), e.g., if exponential or nonexponential, is made unnecessary by the present approach. Homogeneous or inhomogeneous dephasing are included as special cases. It is the primary goal of time-domain CARS to determine the autocorrelation functions directly from experimental data. 

This surprising result prompted Mazenko, Ramaswamy and Toner to examine the anharmonic fluctuation effects in the hydrodynamics of smectics. We have already shown that the undulation modes are purely dissipative with a relaxation rate given by (5.3.39). To calculate the effect of these slow, thermally excited modes on the viscosities, we recall that a distortion u results in a force normal to the layers given by (5.3.32). This is the divergence of a stress, which, from (5.3.53), contains the non-linear term 0,(Vj uf. Thus, there is a non-linear contribution (Vj uf to the stress. Now the viscosity at frequency co is the Fourier transform of a stress autocorrelation function, so that At (co), the contribution of the undulations to the viscosity, can be evaluated. It was shown by Mazenko et that Atj(co) 1 /co. In other words, the damping of first and second sounds in smectics, which should go as >/(oo)oo , will now vary linearly as co at low frequencies. 

With proper selection of the scattering geometry, the two dissipative modes can be observed separately and the decay time of the autocorrelation function equals the relaxation time of the chosen fluctuation mode. However, the... 

This equation is used to obtain reasonable models for the fluctuating force if the dissipation term L(r) is known. In order to obtain L(r), one uses experimental information on the phonon density of states g a>) which is related to the cosine-transform of the velocity autocorrelation function... 

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