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We consider the problem of building a continuous stochastic model, i.e., a Langevin or Fokker-Planck equation, through a well-controlled coarse-graining procedure. Such a method usually involves the elimination of the fast degrees of freedom of the “bath” to which the particle is coupled. Specifically, we look into the general case where the bath may be at negative temperatures, as found, for instance, in models and experiments with bounded effective kinetic energy. Here, we generalize previous studies by considering the case in which the coarse graining leads to (i) a renormalization of the potential felt by the particle, and (ii) spatially dependent viscosity and diffusivity. In addition, a particular relevant example is provided, where the bath is a spin system and a sort of phase transition takes place when going from positive to negative temperatures. A Chapman-Enskog-like expansion allows us to rigorously derive the Fokker-Planck equation from the microscopic dynamics. Our theoretical predictions show excellent agreement with numerical simulations

We study a model describing the force-extension curves of modular proteins, nucleic acids, and other biomolecules made out of several single units or modules. At a mesoscopic level of description, the configuration of the system is given by the elongations of each of the units. The system free energy includes a double-well potential for each unit and an elastic nearest neighbor interaction between them. Minimizing the free energy yields the system equilibrium properties whereas its dynamics is given by (overdamped) Langevin equations for the elongations, in which friction and noise amplitude are related by the fluctuation-dissipation theorem. Our results, both for the equilibrium and the dynamical situations, include analytical and numerical descriptions of the system force-extension curves under force or length control, and agree very well with actual experiments in biomolecules. Our conclusions also apply to other physical systems comprising a number of metastable units, such as storage systems or semiconductor superlattices., major revision, accepted for publication in PRE; 20 pages, 13 figures

A simple lattice model is used to study compaction in granular media. As in real experiments, we consider a series of taps separated by large enough waiting times. The relaxation of the density exhibits the characteristic inverse logarithmic law. Moreover, we have been able to identify analytically the relevant time scale, leading to a relaxation law independent of the specific values of the parameters. Also, an expression for the asymptotic density reached in the compaction process has been derived. The theoretical predictions agree fairly well with the results from the Monte Carlo simulation., 15 pages, 4 figures, REVTeX file; no changes except for single-spacing to save paper (previous version 22 pages)

The decay of the spin-spin time correlation functions in a one-dimensional Ising model with Glauber dynamics is studied. In the low-temperature limit, an asymptotically valid continuous space equation is derived. It is a modified diffusion equation with a purely exponential relaxation term. Its solution leads to an exact Cole-Davison behavior of the spin autocorrelation in the frequency domain, while in the time description a KWW function followed by an exponential decay is obtained. The exponent of the KWW, analytically derived, is 1/2, and not 0.63, as has been reported in the literature. España Direccion General de Investigacion Cientifica y Tecnica through Grant No. PB92-0683

Each oscillator in a linear chain (a string) interacts with a local Ising spin in contact with a thermal bath. These spins evolve according to Glauber dynamics. Below a critical temperature, a rippled state in the string is accompanied by a nonzero spin polarization. The system is shown to form ripples in the string which, for slow spin relaxation, vibrates rapidly about quasi-stationary states described as snapshots of a coarse-grained stroboscopic map. For moderate observation times, ripples are observed irrespective of the final thermodynamically stable state (rippled or not)., 5 pages, 2 figures

The long-time limit of the solutions of a master equation with time-dependent transition rates is analyzed, and the existence of a special (normal) solution, that all the other solutions tend to approach, is shown under quite general conditions. In general, the normal solution will be quite different from the time-dependent equilibrium distribution, although for not too fast continuous heating processes, it asymptotically tends to the equilibrium curve. The results seem to be relevant for the explanation of what is observed in real experiments. The general theory is applied to a very simple model, for which the normal solution is exactly found. España Direccion general de Investigacion Cientifica y Tecnica through Cxrant No. PB89-0618