Fluctuations and stochastic noise in systems with hyperbolic mass transport
Introduction
Memory effects in generalized transport equations play a relevant role at high frequency or high speed of perturbations. The influence of the non-vanishing relaxation time of the diffusion flux on the propagation of fast crystallization fronts has been studied [1], [2] in consistency with extended thermodynamics [3]. The memory effects play an important role in the propagation of phase interfaces during fast phase transitions [4].
Fluctuations for both slow variables (i.e., temperature, solute density, etc.) and fast variables (e.g., diffusion flux and heat flux) have been considered frequently. The fluctuations of the heat flux and the viscous pressure were stressed for the first time by Landau and Lifshitz [5], who derived the expressions for their correlation. In Refs. [6], [7], the role of rapid fluctuations of the heat flux as a stochastic noise has been considered within the extended thermodynamic formalism. In these works, a unified description of slow and fast heat fluctuations has been made [8] for equilibrium and nonequilibrium steady states. The same idea about separation of slow and fast variables to study stochastic noise in a system of particles with inertia has been realized within a supersymmetric path-integral representation [9].
In the present paper, we consider the system of equations for the solute density (concentration) and the flux applied to the phenomenon of density fluctuations in an equilibrium state. For frequencies lower compared to , the usual diffusional description (with ) is satisfactory, but for frequencies comparable or higher than , the finite value of the relaxation time cannot be ignored. These frequencies are experimentally accessible, for instance, by means of neutron scattering experiments.
Besides density fluctuations, we explore the fluctuations of the diffusion flux and investigate their role in two different kinds of descriptions. In the first of them, the shortest measurable time interval is shorter than , and the diffusion flux behaves as an independent fluctuating variable. In the second description, is larger than , and the fluctuating part of the flux behaves as a stochastic noise in the evolution equation for the density.
The paper is organized as follows. In Section 2, the features of the hyperbolic transport are described. In Section 3, an explicit form of the fluctuation spectra for solute density and solute diffusion flux are derived. The fluctuations of the density and of diffusion flux in the two time domains and are specially emphasized in Section 4. Finally, in Section 5 we present a summary of our conclusions.
Section snippets
Hyperbolic transport
The system under study is described by the particle balance equation:where n is the particle density of a solute in a binary system, the diffusion flux, and t the time. The diffusion flux is assumed to be described by a relaxational Maxwell–Cattaneo equation [1], [2], [3], [4]where and D are the relaxation time and diffusion constant, respectively. The relaxation term is negligible for steady states or low-frequency perturbations. It becomes dominant at high
Power spectra of density and flux fluctuations
Let us define the correlation functions for the fluctuations of n and in the following usual formwhere is the position vector of a point in the system. Since we consider equilibrium (homogeneous, time-invariant state), one hasi.e., the correlation functions depend only on relative distances and on the difference in time . We are interested in the Fourier
Stochastic noise in hyperbolic transport
In the former section we have analyzed the fluctuations of density and of diffusion flux, with special emphasis to the former one. Here, in contrast, we focus our attention on a more conceptual question related to the fluctuations of the flux or, more concretely, to their conceptual interpretation. We shall see that, depending on the value of the relaxation time and of the observational time scale, such fluctuations can be interpreted in two different ways. To discuss these ideas we must
Conclusions
The power spectra of a solute number density and a solute diffusion flux have been reviewed. The latter has a nonvanishing relaxation time leading to a hyperbolic transport equation for the evolution of the density.
Several interpretations of the stochastic noise related to fast variables eliminated from the description have been examined. Our aim was not to review the well-known results of adiabatic elimination of slow variables, but, rather, to show up the role of variables which are
Acknowledgements
D. J. acknowledges financial support from the Dirección General de Investigación of the Spanish Ministry of Science and Technology BFM 2003-06033 and the Direcció General de Recerca of the Generalitat of Catalonia under Grant 2001 SGR-00186. P. G. acknowledges financial support from the German Research Foundation (DFG—Deutsche Forschungsgemeinschaft) under the Project no. HE 1601/13.
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