Topics in nonequilibrium statistical mechanics : I. Collective behavior of coupled parametric oscillators, II. Hydrodynamic fluctuations in Kolmogorov flow

Abstract

This thesis is divided into two parts. The first part is devoted to the study of an analytically solvable model of coupled parametric oscillators. The model comprises an infinite set of globally coupled harmonic oscillators whose frequencies are subjected to time-periodic, piecewise-constant modulations with randomly distributed quenched phases. This system exhibits a variety of amplitude instabilities. In addition to the familiar parametric instability of the individual oscillators, two kinds of collective instabilities are identified. In one, the mean amplitude diverges monotonically, while in the other the divergence is oscillatory. The frequencies of the collective oscillatory instabilities bear no simple relation to the natural frequency of the individual oscillators, or to the frequency of the external modulation. A phase diagram is constructed to delineate the extent of the different regimes in the space of the parameters of the perturbation (its amplitude and period) . Some of the features of the collective instabilities in the mean field model are already present in the simple system of just two coupled parametric oscillators with out-of-phase perturbations. It is also shown that the above phenomena are robust, in the sense that they do not depend crucially on the details of the model. Numerical simulations support the theoretical predictions. The second part of the thesis uses the framework of the Landau-Lifshitz fluctuating hydrodynamics in order to study the statistical properties of Kolmogorov flow. A detailed analysis of the linearized fluctuation spectrum is carried out from the near-equilibrium regime up to the vicinity of the first convective instability threshold ( that corresponds to the appearance of rotating convective patterns). It is shown that in the long-time limit the flow behaves as an incompressible fluid, regardless of the value of the Reynolds number. This is not the case for the short-time behavior, where the incompressibility assumption leads in general to an incorrect form of the static correlation functions, except near the instability threshold. However, in this latter region, nonlinear effects have to be taken into account appropriately. We derive the normal form amplitude equation for an incompressible fluid, and construct the velocity field close to, and just above, the threshold. The compressible case is analyzed as well. Using a perturbative technique, it is shown that close to the instability threshold the stochastic dynamics of the system is governed by two coupled nonlinear Langevin equations in Fourier space. The solution of these equations can be cast in the form of the exponential of a Landau-Ginzburg functional, which proves to be identical to the one obtained for the case of an incompressible fluid. The theoretical predictions are confirmed by numerical simulations of the full fluctuating hydrodynamic equations. It is also shown that the results of particle simulations of Kolmogorov flow are vitiated by a spurious diffusion of the center of mass in phase space. The analytical expression for the corresponding diffusion coefficient is derived, using which we show that the effect is negligible in a macroscopic system.