Nonequilibrium Fluctuations of Intensive Variables in a Brownian Gas
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
The dynamics of a Brownian gas is usually analyzed by assuming that it is embedded in a system which is in thermodynamic equilibrium at a fixed temperature. In this work we consider the more general case where the system is itself in contact with a heat bath in a non-equilibrium state. As a result, fluctuations in one of the intensive parameters of the system (temperature) may occur. It is assumed that the intensity, wavelength and time scale of these external fluctuations are such that the Brownian gas is in local thermodynamic equilibrium with the system, but not with the entire heat bath. As a result, the Brownian motion takes place in two time scales 1.
The dynamics of these fluctuations cannot be described in terms of the Onsager-Machlup theory, since this formalism was formulated to account only for the fluctuations in the extensive variables of the system. Therefore, we construct an approach based on Mesoscopic Non-Equilibrium Thermodynamics (MNET) 2,3 which is able to describe the dynamics of these intensive fluctuations. With this point of view, we first derive a probability distribution for the velocities of the Brownian particles that incorporates the effects of the non-equilibrium temperature fluctuations. We then show that this effect is expressed in terms of an effective Maxwell-Boltzmann factor (EMBF) which has its origin in the contraction of the space of mesoscopic variables. We also show that it coincides with an EMBF which has been recently postulated in the literature 1. Then we derive a Fokker-Planck equation describing the relaxation mesoscopic dynamics of the Brownian gas in the space of the local intensive temperature fluctuations, positions and the velocities of the Brownian particles. Considering that the Brownian particles also experiences a harmonic force, we average over the velocities and obtain a stochastic multiplicative equation for the local reduced distribution function, which is then solved by using an approximate but analytical method. This yields an equation for an effective local probability distribution, which incorporates all the effects of the induced intensive fluctuations. From it we derive and solve equations for the mass density field, the density-density correlation, the van Hove function and the structure factor of the system. Finally, we compare our results with those of other approaches, specially with those arising from generalized statistics.\\
1) C. Beck, Phys. Rev. Lett. 87 (2001) 180601\\
2) P. Mazur, Physica A 274 (1999) 491-504\\
3) I. Santamaria Holek, R. F. Rodriguez, Physica A 366 (2006) 141-148