Diffusing particles, fluctuation-dissipation

In the classical case, a modified FDT associated with the particle displacement can be written in the form (68). For a diffusing particle, the fluctuation-dissipation ratio X(t, tw) is given by Eq. (90), rewritten here for convenience as... 

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. 

For a particle evolving in a thermal bath, we focused our interest on the particle displacement, a dynamic variable which does not equilibrate with the bath, even at large times. As far as this variable is concerned, the equilibrium FDT does not hold. We showed how one can instead write a modified FDT relating the displacement response and correlation functions, provided that one introduces an effective temperature, associated with this dynamical variable. Except in the classical limit, the effective temperature is not simply proportional to the bath temperature, so that the FDT violation cannot be reduced to a simple rescaling of the latter. In the classical limit and at large times, the fluctuation-dissipation ratio T/Teff, which is equal to 1 /2 for standard Brownian motion, is a self-similar function of the ratio of the observation time to the waiting time when the diffusion is anomalous. 

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... 

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 key point in these studies is that the presence of flow introduces non-thermal diffusion effects that can be related to the hydrod5mamic viscosity and to hydrodynamic interactions between particles in the case of concentrated suspensions. These effects may modify the fluctuation-dissipation relations and, consequently, the corresponding expressions for the thermal energy and other state variables like pressure or chemical potential. This may lead to what is known as nonequUibrium state equations (Onuki, 2004). 

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,... 

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... 

A second catch is the noise. If one observes the movements of a colloidal particle, the Brownian motion will be evident. There may be a constant drift in the dynamics, but the movement will be irregular. Likewise, if one observes a phase-separating liquid mixture on the mesoscale, the concentration levels would not be steady, but fluctuating. The thermodynamic mean-field model neglects all fluctuations, but they can be restored in the dynamical equations, similar to added noise in particle Brownian dynamics models. The result is a set of stochastic diffusion equations, with an additional random noise source tj [20]. In principle, the value and spectrum of the noise is dictated by a fluctuation dissipation theorem, but usually one simply takes a white noise source. 

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... 

Brownian diffusion is neglected compared with turbulent transport. The left-hand side represents Dpc/Dt, the Stokes or substantive derivative of p--. The first term on the right-hand side is the turbulent diffusion of The second term —2v p j Vp7 is generally positive and represents the generation of p by transfer from the mean How. The third term. 2p//fl, is the contribution of variations in the mte of gas-to-particle conversion by chemical reaction to the rate of production of p . The la.st term is the decrease of mean square fluctuations pj due to the action of small scale diffusion (dissipation). Thus three types of terms appear on the right-hand side of (13,16), the balance equation for Pi (i) turbulent diffusion of p, and tnmsfer from the mean (low to p.. which alTeci... 

In the mesoscale model, setting Tf = 0 forces the fluid velocity seen by the particles to be equal to the mass-average fluid velocity. This would be appropriate, for example, for one-way coupling wherein the particles do not disturb the fluid. In general, fluctuations in the fluid generated by the presence of other particles or microscale turbulence could be modeled by adding a phase-space diffusion term for Vf, similar to those used for macroscale turbulence (Minier Peirano, 2001). The time scale Tf would then correspond to the dissipation time scale of the microscale turbulence. 

Fluctuations are caused by the motion of molecules. Molecular motion is responsible for dissipative processes such as diffusion, heat transport and chemical reactions. Fluctuations in the number of particles in a definite volume play a key role in maintaining equilibrium. Fluctuation in the number of particles A in a volume A V is given by 8N /N For a gas under normal conditions, this is of the order 4 x 10 when A V = (1 p,m). For liquids and solids, the same value of fluctuation will correspond to... 

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