Out of equilibrium dynamics of classical and quantum complex systems

Equilibrium is a rather ideal situation, the exception rather than the rule in Nature. Whenever the external or internal parameters of a physical system are varied its subsequent relaxation to equilibrium may be either impossible or take very long times. From the point of view of fundamental physics no generic principle such as the ones of thermodynamics allows us to fully understand their behaviour. The alternative is to treat each case separately. It is illusionary to attempt to give, at least at this stage, a complete description of all non-equilibrium situations. Still, one can try to identify and characterise some concrete but still general features of a class of out of equilibrium problems - yet to be identified - and search for a unified description of these. In this report I briefly describe the behaviour and theory of a set of non-equilibrium systems and I try to highlight common features and some general laws that have emerged in recent years.Comment: 36 pages, to be published in Compte Rendus de l'Academie de Sciences, T. Giamarchi e

Effective temperatures out of equilibrium

We describe some interesting effects observed during the evolution of nonequilibrium systems, using domain growth and glassy systems as examples. We breafly discuss the analytical tools that have been recently used to study the dynamics of these systems. We mainly concentrate on one of the results obtained from this study, the violation of the fluctuation-dissipation theorem and we discuss, in particular, its relation to the definition and measurement of effective temperatures out of equilibrium.Comment: 13 pages, RevTex, 2 figs, to appear in ``Trends in Theoretical Physics II'', eds. H Falomir et al, Am. Inst. Phys. Conf. Proc. of the 1998 Buenos Aires meetin

Coupled logistic maps and non-linear differential equations

We study the continuum space-time limit of a periodic one dimensional array of deterministic logistic maps coupled diffusively. First, we analyse this system in connection with a stochastic one dimensional Kardar-Parisi-Zhang (KPZ) equation for confined surface fluctuations. We compare the large-scale and long-time behaviour of space-time correlations in both systems. The dynamic structure factor of the coupled map lattice (CML) of logistic units in its deep chaotic regime and the usual d=1 KPZ equation have a similar temporal stretched exponential relaxation. Conversely, the spatial scaling and, in particular, the size dependence are very different due to the intrinsic confinement of the fluctuations in the CML. We discuss the range of values of the non-linear parameter in the logistic map elements and the elastic coefficient coupling neighbours on the ring for which the connection with the KPZ-like equation holds. In the same spirit, we derive a continuum partial differential equation governing the evolution of the Lyapunov vector and we confirm that its space-time behaviour becomes the one of KPZ. Finally, we briefly discuss the interpretation of the continuum limit of the CML as a Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) non-linear diffusion equation with an additional KPZ non-linearity and the possibility of developing travelling wave configurations.Comment: 23 page

Aging and effective temperatures in the low temperature mode-coupling equations

The low-temperature generalization of the mode-coupling equations corresponds to the dynamics of mean-field disordered models in the glassy phase. The system never achieves equilibrium, preserving the memory of the time elapsed after the quench throughout its evolution. A concept of effective temperature can be made quite rigorous in this context by considering readings of thermometers in different time-scales and the thermalization of weakly coupled subsystems.Comment: 8 pages, 6 figures, LaTeX, Contribution to YKIS'96, Kyoto, November 1996, to appear in Prog. Theor. Phy

Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz-Thouless phase

We study analytically the distribution of fluctuations of the quantities whose average yield the usual two-point correlation and linear response functions in three unfrustrated models: the random walk, the $d$ dimensional scalar field and the 2d XY model. In particular we consider the time dependence of ratios between composite operators formed with these fluctuating quantities which generalize the largely studied fluctuation-dissipation ratio, allowing us to discuss the relevance of the effective temperature notion beyond linear order. The behavior of fluctuations in the aforementioned solvable cases is compared to numerical simulations of the 2d clock model with $p=6,12$ states.Comment: 27 pages, 3 figure

Out of equilibrium dynamics of the spiral model

We study the relaxation of the bi-dimensional kinetically constrained spiral model. We show that due to the reversibility of the dynamic rules any unblocked state fully decorrelates in finite times irrespectively of the system being in the unjammed or the jammed phase. In consequence, the evolution of any unblocked configuration occurs in a different sector of phase space from the one that includes the equilibrium blocked equilibrium configurations at criticality and in the jammed phase. We argue that such out of equilibrium dynamics share many points in common with coarsening in the one-dimensional Ising model and we identify the coarsening structures that are, basically, lines of vacancies. We provide evidence for this claim by analyzing the behaviour of several observables including the density of particles and vacancies, the spatial correlation function, the time-dependent persistence and the linear response.Comment: 14 pages 12 figure