Fluctuation hydrodynamics

In this section we discuss the frequency spectrum of excitations on a liquid surface. Wliile we used linearized equations of hydrodynamics in tire last section to obtain the density fluctuation spectrum in the bulk of a homogeneous fluid, here we use linear fluctuating hydrodynamics to derive an equation of motion for the instantaneous position of the interface. We tlien use this equation to analyse the fluctuations in such an inliomogeneous system, around equilibrium and around a NESS characterized by a small temperature gradient. More details can be found in [9, 10]. 

A great deal of research remains to be done in this area. We are currently extending in the study of spatial correlations in the non-equilibrium fluids to time correlations with the hope of establishing a correspondence between MD and fluctuating hydrodynamic theory. We are also using these systems to study the roles of viscosity and conductivity in fluid behavior under different external constraints. Finally, we plan to continue our research into the formation of spatial structures in fluids. 

The lack of correlation between the fluctuating stress tensor and the fluctuating heat flux in the third expression is an example of the Curie principle for the fluctuations. These equations for fluctuating hydrodynamics are arrived at by a procedure very similar to that exhibited in the preceding section for diffusion. A crucial ingredient is the equation for entropy production in a fluid... 

Expressions for the nonequilibrium fluctuations in liquids can be obtained on the basis of fluctuating hydrodynamics [ 11,121. Fluctuating hydrodynamics assumes that the fluctuations in... 

The nonequilibrium concentration fluctuations can be measured experimentally by dynamic light scattering. Fluctuating hydrodynamics predicts that the time-dependent correlations function C(k,t) ofthe scattered light is given by... 

Yamakawa, ff., and Yoshizaki, T., Effects of fluctuating hydrodynamic interaction on the hydrodynamic radius expansion factor of polymer chains, Macromolecides, 28, 3604—3608 (1995). 

The formalism of thermodynamically consistent dissipative particle dynamics represents a consistent discrete model for the Lagrangian fluctuating hydrodynamics. Equation (26.45)-Equation (26.46) conserve the mass, momentum, energy and volume. The irreversible (produced) entropy S is a strictly inerting function of time in the absence of fluctuations. Thermal fluctuations represented by F, S are consistently included, which lead to an increase of the entropy and to correct for the Einstein distribution function [31]. 

In any case, rotation means fluctuating hydrodynamic forces on the surface elements of the particle. The maximum stress for spherical particles can be calculated as (Raasch 1962) ... 

Fixman studied in some detail the properties of Gaussian chains of up to 56 monomers, comparing fluctuating hydrodynamics to pre-averaged hydrodynamics (this latter case can be done analytically). It was observed that the effect of fluctuating hydrodynamic interactions seems to decrease the diffusion constant in the long time, long chain Umit only weakly ( 5%). 

The first results concern the comparison between the Kirkwood formula for the diffusion constant and the diffusion constant obtained from pre-averaged BD simulations. Within the error bars, the numbers are identical. However, for fluctuating hydrodynamics a diffusion constant systematically above the Kirkwood value was found, at variance with the variational bound. Since the data seem to be rather accurate, this might be an indication that something is wrong with the rigorous bound. For a discussion, see Ref. 38. These questions must be regarded as completely unresolved today. 

In this section we will study the connection between the LB equation, (34), and the equations of fluctuating hydrodynamics [42],... 

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