Optimizing the relaxation route with optimal control

We look into the minimization of the connection time between nonequilibrium steady states. As a prototypical example of an intrinsically nonequilibrium system, a driven granular gas is considered. For time-independent driving, its natural time scale for relaxation is characterized from an empirical (the relaxation function) and a theoretical (the recently derived classical speed limits) point of view. Using control theory, we find that bang-bang protocols (comprising two steps, heating with the largest possible value of the driving and cooling with zero driving) minimize the connecting time. The bang-bang time is shorter than both the empirical relaxation time and the classical speed limit: in this sense, the natural time scale for relaxation is beaten. Information theory quantities stemming from the Fisher information are also analyzed over these optimal protocols. The implementation of the bang-bang processes in numerical simulations of the dynamics of the granular gas show an excellent agreement with the theoretical predictions. Moreover, general implications of our results are discussed for a wide class of driven nonequilibrium systems. Specifically, we show that analogous bang-bang protocols, with a number of bangs equal to the number of relevant physical variables, give the minimum connecting time under quite general conditions.FEDER/Ministerio de Ciencia e Innovación–Agencia Estatal de Investigación PGC2018-093998-B-I0

Derivation of a Langevin equation in a system with multiple scales: the case of negative temperatures

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 simulation

Hysteresis in vibrated granular media

Some general dynamical properties of models for compaction of granular media based on master equations are analyzed. In particular, a one-dimensional lattice model with short-ranged dynamical constraints is considered. The stationary state is consistent with Edward's theory of powders. The system is submitted to processes in which the tapping strength is monotonically increased and decreased. In such processes the behavior of the model resembles the reversible–irreversible branches which have been recently observed in experiments. This behavior is understood in terms of the general dynamical properties of the model, and related to the hysteresis cycles exhibited by structural glasses in thermal cycles. The existence of a “normal” solution, i.e., a special solution of the master equation which is monotonically approached by all the other solutions, plays a fundamental role in the understanding of the hysteresis effects.Dirección General de Investigación Científica y Técnica (Spain) through Grant No. PB98-112

Sawtooth patterns in force-extension curves of biomolecules: An equilibrium-statistical-mechanics theory

We analyze the force-extension curve for a general class of systems, which are described at the mesoscopic level by a free energy depending on the extension of its components. Similarly to what is done in real experiments, the total length of the system is the controlled parameter. This imposes a global constraint in the minimization procedure leading to the equilibrium values of the extensions. As a consequence, the force-extension curve has multiple branches in a certain range of forces. The stability of these branches is governed by the free energy: there are a series of first-order phase transitions at certain values of the total length, in which the free energy itself is continuous but its first derivative, the force, has a finite jump. This behavior is completely similar to that observed in real experiments with biomolecules like proteins and with other complex systemsEspaña Ministerio de Economía y Competitividad Grants No. FIS2011- 24460 (A.P.), No. FIS2011-28838-C02-0

Global stability and H theorem in lattice models with nonconservative interactions

In kinetic theory, a system is usually described by its one-particle distribution function f (r,v,t), such that f (r,v,t)drdv is the fraction of particles with positions and velocities in the intervals (r,r + dr) and (v,v + dv), respectively. Therein, global stability and the possible existence of an associated Lyapunov function or H theorem are open problems when nonconservative interactions are present, as in granular fluids. Here, we address this issue in the framework of a lattice model for granularlike velocity fields. For a quite general driving mechanism, including both boundary and bulk driving, we show that the steady state reached by the system in the long-time limit is globally stable. This is done by proving analytically that a certain H functional is nonincreasing in the long-time limit. Moreover, for a quite general energy injection mechanism, we are able to demonstrate that the proposed H functional is nonincreasing for all times. Also, we put forward a proof that clearly illustrates why the “classical” Boltzmann functional HB[f ] = dr dvf (r,v,t) ln f (r,v,t) is inadequate for systems with nonconservative interactions. This is done not only for the simplified kinetic description that holds in the lattice models analyzed here but also for a general kinetic equation, like Boltzmann’s or Enskog’s

Slow logarithmic relaxation in models with hierarchically constrained dynamics

A general kind of model with hierarchically constrained dynamics is shown to exhibit logarithmic anomalous relaxation similar to a variety of complex strongly interacting materials. The logarithmic behavior describes most of the decay of the response functio

Effective dynamics and steady state of an Ising model submitted to tapping processes

A one-dimensional Ising model with nearest neighbor interactions is applied to study compaction processes in granular media. An equivalent particle-hole picture is introduced, with the holes being associated to the domain walls of the Ising model. Trying to mimic the experiments, a series of taps separated by large enough waiting times, for which the system freely relaxes, is considered. The free relaxation of the system corresponds to a T50 dynamics which can be analytically solved. There is an extensive number of metastable states, characterized by all the holes being isolated. In the limit of weak tapping, an effective dynamics connecting the metastable states is obtained. The steady state of this dynamics is analyzed, and the probability distribution function is shown to have the canonical form. Then, the stationary state is described by Edwards thermodynamic granular theory. Spatial correlation functions in the steady state are also studiedEspaña Ministerio de Ciencia y Tecnología Grant No. BFM2002-0030

Normal solutions for master-equations with time-dependent transition rates - application to heating processes

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. PB89-061

Low-temperature relaxation in the one-dimensional Ising model

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. PB92-068

Residual properties of a two-level system

The residual properties, after cooling from a high temperature to T=0 K, of a system of identical and independent two-level systems are analyzed. A general criterion to know whether a given cooling law will lead to a nonvanishing residual population is derived. After defining in a precise way the condition of slow cooling, asymptotic expressions for the residual population and entropy are obtained for a family of cooling laws. It is shown that the expressions for the residual properties, and their own existence, depend quite strongly on the cooling procedure.Dirección General de Investigación Científica y Técnica PB89-06