Diffusion fluctuation-dissipation theorems

The frequency correlation time xm corresponds to the time it takes for a single vibrator to sample all different cavity sizes. The fluctuation-dissipation theorem (144) shows that this time can be found by calculating the time for a vertically excited v = 0 vibrator to reach the minimum in v = 1. This calculation is carried out by assuming that the solvent responds as a viscoelastic continuum to the outward push of the vibrator. At early times, the solvent behaves elastically with a modulus Goo. The push of the vibrator launches sound waves (acoustic phonons) into the solvent, allowing partial expansion of the cavity. This process corresponds to a rapid, inertial solvent motion. At later times, viscous flow of the solvent allows the remaining expansion to occur. The time for this diffusive motion is related to the viscosity rj by Geo and the net force constant at the cavity... 

One usually studies diffusion in a thermal bath by writing two fluctuation-dissipation theorems, generally referred to as the first and second FDTs (using the Kubo terminology [30,31]). As recalled for instance in Ref. 57, the first FDT expresses a necessary condition for a thermometer in contact solely with the system to register the temperature of the bath. As for the second FDT, it expresses the fact that the bath itself is in equilibrium. 

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. 

The physics behind this relation is the fluctuation-dissipation theorem the same random kicks of the surrounding molecules cause both Brownian diffusion and the viscous dissipation leading to the frictional force. It is -instructive to calculate the time scale t required for the particle to move a... 

At the same time diffusion coefficient satisfies Einstein relation Dl = kaT/m connecting in that way two dissipative effects due to conformational changes and fractal geometry of filler surface producing that change. Our system obeys the generalized fluctuation-dissipation theorem. As a comment on the obtained distribution function we can say that from NMR experiments it is indeed possible to obtain Boltzmann distribution of energy sites (within the accuracy of the data) in the form I — Iq exp(395 + 50/F). 

The first terra on the right side comes from a kind of surface tension and tends to smooth the surface, while the second term is a Gaussian fluctuating white noise satisfying the fluctuation dissipation theorem. Equation (31.6) leads for it = 2 to Ds = 1-5. Also numerically, this result is well verified for random deposition with surface diffusion [34—38]. For d = 3, we find from Eq. (31.6) = 3. In... 

Aepd is a kinetic Onsager coefficient, and rj denotes noise that satisfies the fluctuation-dissipation theorem. The Fourier transform of this new diffusion equation is simply ... 

Not surprisingly, the temporal behavior in Eqs. 68 and 69 is the same (see also Eqs. 67 and 50), a consequence of the fluctuation-dissipation theorem, which relates the diffusion in the absence of an external field to the drift in such a field. 

This equation is the simplest form of the so-called fluctuation-dissipation theorem. The diffusion, which is a typical equilibrium phenomenon, is related to the friction, a typical energy dissipation phenomenon. 

Within the linear response approximation, the rate of transport (mass, momentum, or energy) through a system is proportional to the gradient (of concentration, velocity, and temperature), with the transport coefficient being the proportionality constant. This proportionality constant can be computed using equilibrium description of the system through the so-called fluctuation dissipation theorems. One such equation, relating equilibrium fluctuations to the diffusion constant, is Einstein s well-known equation ... 

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. 

The friction coefficient is the inverse particle s relaxation time, jS = 9py/(2pp ), where py is the fluid s dynamic viscosity. Since the Langevin equations are linear, particle velocity and position may be formally solved as functionals of the random force, and in the diffusive limit f >> i. e., for times much larger than the particle relaxation time, they allow for the analytical evaluation of ensemble averaged products of particle position and velocity and two-point correlation functions, in terms of the random-force strength q. The authors carefully justify why they use the classical (equilibrium) form of the fluctuation-dissipation theorem (FDT) in a Langevin description the time scale of the white noise is considered to be much shorter than the time scale of the imjxjsed flow. Thus, the non-equilibrium corrections would be of the order of the ratio of the fluid molecular relaxation time to the time scale of the imposed shear and may be neglected. In this case both the time scales are clearly separated and q may be determined solely from the classical form of the FDT,... 

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. 

Of course, viscoelasticity (response to an external mechanical excitation) and diffusion (response to thermal fluctuations) must have their origin in the same dynamical process at the molecular level, and any model proposed to explain one aspect must also consistently account for the other (as a consequence of the fluctuation dissipation theorem). The elaboration of such models has remained a challenge to polymer scientists for many years, and is indeed a complicated many-body problem. 

The superscript identifies the conformation at the beginnin of the time-step. For small timesteps At this should be reasonable to do. Fj in the above equation is the force exerted on particle j. The so-called spurious drift , i.e., the third term in the r.h.s. of eq. (3.20) usually vanishes, since most diffusion tensors which have been used in the literature have zero divergence (this is directly related to the assumption of incompressible flow). p] At) is the random displacement by the coupling to the heat bath. The crucial difficulty comes from the connection of the displacement by the heat bath and the hydrodynamic interaction tensor Dy via the fluctuation dissipation theorem. This fixes the first two moments to be... 

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