Lyapunov exponent of many-particle systems: testing the stochastic approach

The stochastic approach to the determination of the largest Lyapunov exponent of a many-particle system is tested in the so-called mean-field XY-Hamiltonians. In weakly chaotic regimes, the stochastic approach relates the Lyapunov exponent to a few statistical properties of the Hessian matrix of the interaction, which can be calculated as suitable thermal averages. We have verified that there is a satisfactory quantitative agreement between theory and simulations in the disordered phases of the XY models, either with attractive or repulsive interactions. Part of the success of the theory is due to the possibility of predicting the shape of the required correlation functions, because this permits the calculation of correlation times as thermal averages.Comment: 11 pages including 6 figure

The Brownian Mean Field model

We discuss the dynamics and thermodynamics of the Brownian Mean Field (BMF) model which is a system of N Brownian particles moving on a circle and interacting via a cosine potential. It can be viewed as the canonical version of the Hamiltonian Mean Field (HMF) model. We first complete the description of this system in the mean field approximation. Then, we take fluctuations into account and study the stochastic evolution of the magnetization both in the homogeneous phase and in the inhomogeneous phase. We discuss its behavior close to the critical point

Canonical and non-canonical equilibrium distribution

We address the problem of the dynamical foundation of non-canonical equilibrium. We consider, as a source of divergence from ordinary statistical mechanics, the breakdown of the condition of time scale separation between microscopic and macroscopic dynamics. We show that this breakdown has the effect of producing a significant deviation from the canonical prescription. We also show that, while the canonical equilibrium can be reached with no apparent dependence on dynamics, the specific form of non-canonical equilibrium is, in fact, determined by dynamics. We consider the special case where the thermal reservoir driving the system of interest to equilibrium is a generator of intermittent fluctuations. We assess the form of the non-canonical equilibrium reached by the system in this case. Using both theoretical and numerical arguments we demonstrate that Levy statistics are the best description of the dynamics and that the Levy distribution is the correct basin of attraction. We also show that the correct path to non-canonical equilibrium by means of strictly thermodynamic arguments has not yet been found, and that further research has to be done to establish a connection between dynamics and thermodynamics.Comment: 13 pages, 6 figure

Effects of turbulent mixing on critical behaviour: Renormalization group analysis of the Potts model

Critical behaviour of a system, subjected to strongly anisotropic turbulent mixing, is studied by means of the field theoretic renormalization group. Specifically, relaxational stochastic dynamics of a non-conserved multicomponent order parameter of the Ashkin-Teller-Potts model, coupled to a random velocity field with prescribed statistics, is considered. The velocity is taken Gaussian, white in time, with correlation function of the form $\propto \delta(t-t') /|{\bf k}_{\bot}|^{d-1+\xi}$, where ${\bf k}_{\bot}$ is the component of the wave vector, perpendicular to the distinguished direction ("direction of the flow") --- the $d$-dimensional generalization of the ensemble introduced by Avellaneda and Majda [1990 {\it Commun. Math. Phys.} {\bf 131} 381] within the context of passive scalar advection. This model can describe a rich class of physical situations. It is shown that, depending on the values of parameters that define self-interaction of the order parameter and the relation between the exponent $\xi$ and the space dimension $d$, the system exhibits various types of large-scale scaling behaviour, associated with different infrared attractive fixed points of the renormalization-group equations. In addition to known asymptotic regimes (critical dynamics of the Potts model and passively advected field without self-interaction), existence of a new, non-equilibrium and strongly anisotropic, type of critical behaviour (universality class) is established, and the corresponding critical dimensions are calculated to the leading order of the double expansion in $\xi$ and $\epsilon=6-d$ (one-loop approximation). The scaling appears strongly anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different.Comment: 21 page, LaTeX source, 7 eps figures. arXiv admin note: substantial text overlap with arXiv:cond-mat/060701