NONEQUILIBRIUM STATISTICAL PHYSICS TODAY: Proceedings of the 11th Granada Seminar on Computational and Statistical Physics

EDITORS’ PREFACE
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Nonequilibrium Statistical Physics today. Where shall we go from here?
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Three lectures: NEMD, SPAM, and shockwaves
View Description Hide DescriptionWe discuss three related subjects well suited to graduate research. The first, Nonequilibrium molecular dynamics or “NEMD”, makes possible the simulation of atomistic systems driven by external fields, subject to dynamic constraints, and thermostated so as to yield stationary nonequilibrium states. The second subject, Smooth Particle Applied Mechanics or “SPAM”, provides a particle method, resembling molecular dynamics, but designed to solve continuum problems. The numerical work is simplified because the SPAM particles obey ordinary, rather than partial, differential equations. The interpolation method used with SPAM is a powerful interpretive tool converting point particle variables to twice‐differentiable field variables. This interpolation method is vital to the study and understanding of the third research topic we discuss, strong shockwaves in dense fluids. Such shockwaves exhibit stationary far‐from‐equilibrium states obtained with purely reversible Hamiltonian mechanics. The SPAM interpolation method, applied to this molecular dynamics problem, clearly demonstrates both the tensor character of kinetic temperature and the time‐delayed response of stress and heat flux to the strain rate and temperature gradients. The dynamic Lyapunov instability of the shockwave problem can be analyzed in a variety of ways, both with and without symmetry in time. These three subjects suggest many topics suitable for graduate research in nonlinear nonequilibrium problems.

Stochastic thermodynamics: An introduction
View Description Hide DescriptionThese seminar notes contain a brief introduction into the principles of stochastic thermodynamics and some of its recent ramifications from a personal perspective. Thermodynamic concepts like work, exchanged heat and entropy production can consistently be defined on the level of individual fluctuating trajectories taken from either a time‐dependent or a non‐equilibrium steady state ensemble. Fluctuation theorems constrain the probability distributions for these thermody‐namic quantities. For systems containing fast internal degrees of freedom the crucial distinction between internal and free energy for a correct identification of both dissipated heat and system entropy is emphasized. For non‐equilibrium steady states, a generalized fluctuation‐dissipation theorem relates the response to a small perturbation to correlation functions in the steady state involving observables expressing the various contributions to entropy production along the trajectory.

Hydrodynamics from dynamical non‐equilibrium MD
View Description Hide DescriptionWe review a dynamical approach to non‐equilibrium MD (D‐NEMD). We show how, using a proper simulation setup, is possible to treat interesting cases in which the initial condition is a stationary non‐equilibrium state produced by a suitable dynamical system. We then extend the class of non‐equilibrium phenomena that can be studied by atomistic simulations to the case of complex initial conditions consisting in assigning a macroscopic value of a scalar or vector observable or a field. We illustrate the functioning of this method by applying it to the relaxation of an interface between two immiscible liquids. We have shown that our method generate unbiased results while this might not be the case for the often used short time average approach.

Recent progress in fluctuation theorems and free energy recovery
View Description Hide DescriptionIn this note we review recent progress about fluctuation relations and their applicability to free energy recovery in single molecule experiments. We underline the importance of the operational definition for the mechanical work and the non‐invariance of fluctuation relations under Galilean transformations, both aspects currently amenable to experimental test. Finally we describe a generalization of the Crooks fluctuation relation useful to recover free energies of partially equilibrated states and thermodynamic branches.

Universality in equilibrium and away from it: A personal perspective
View Description Hide DescriptionIn this talk/paper I discuss the concept of universality in phase transitions and the question of whether universality classes are more robust in equilibrium than away from it. In both of these situations, the main ingredients determining universality are symmetries, conservation laws, the dimension of the space and of the order‐parameter and the presence of long‐range interactions or quenched disorder. The existence of detailed‐balance and fluctuation‐dissipation theorems imposes severe constraints on equilibrium systems, allowing to define universality classes in a very robust way; instead, non‐equilibrium allows for more variability. Still, quite robust non‐equilibrium universality classes have been identified in the last decades. Here, I discuss some examples in which (i) non‐equilibrium phase transitions are simply controlled by equilibrium critical points, i.e. non‐equilibrium ingredients turn out to be irrelevant in the renormalization group sense and (ii) non‐equilibrium situations in which equilibrium seems to come out of the blue, generating an adequate effective description of intrinsically non‐equilibrium problems. Afterwards, I shall describe different genuinely non‐equilibrium phase transitions in which introducing small, apparently innocuous changes (namely: presence or absence of an underlying lattice, parity conservation in the overall number of particles, existence of an un‐accessible vacuum state, deterministic versus stochastic microscopic rules, presence or absence of a Fermionic constraint), the critical behavior is altered, making the case for lack of robustness. However, it will be argued that in all these examples, there is an underlying good reason (in terms of general principles) for universality to be altered. The final conclusions are that: (i) robust universality classes exist both in equilibrium and non‐equilibrium; (ii) symmetry and conservation principles are crucial in both, (iii) non‐equilibrium allows for more variability (i.e. it is less constrained).

Fourier law, phase transitions, and the Stefan problem
View Description Hide DescriptionWhen hydrodynamic or thermodynamic limits are performed in systems which are in the phase transitions regime we may observe perfectly smooth profiles develop singularities with the appearance of sharp interfaces. I will discuss the phenomenon in stationary non equilibrium states which carry non zero steady currents. The general context is the one where the Fourier law applies, but here it is complemented by a free boundary problem due to the presence of interfaces.
I will specifically consider an Ising system with Kac potentials which evolves under the stochastic Kawasaki dynamics. In a continuum limit the evolution is described by an integro‐differential equation, as proved by Giacomin and Lebowitz in [2], see also [3]. I will then study its stationary solutions with a non zero current (produced by suitable boundary conditions) and derive, in the infinite volume limit, macroscopic profiles with an interface proving that the profiles satisfy a stationary Stefan problem and obey the Fourier law.

Bringing thermodynamics to non‐equilibrium microscopic processes
View Description Hide DescriptionWe present recent developments that extend the applicability of thermodynamic concepts deep into mesoscopic and irreversible regimes. We show how a probabilistic interpretation of thermodynamics together with probability conservation laws can be used to obtain Fokker‐Planck equations for the relevant degrees of freedom. This approach provides a systematic method to obtain the stochastic dynamics of a system directly from the knowledge of its equilibrium properties. A wide variety of situations can be studied in this way, including many that were thought to be out of reach of thermodynamic theories, such as non‐linear transport in the presence of potential barriers, activated processes, slow relaxation phenomena, and basic processes in biomolecules, like translocation and stretching.

Noise‐induced transitions vs. noise‐induced phase transitions
View Description Hide DescriptionI will briefly review the field of noise‐induced phase transitions, emphasizing the main differences with the phase‐induced transitions and showing that they appear in different systems. I will show that a noise‐induced transition can disappear after a suitable change of variables and I will also discuss the breaking of ergodicity and symmetry breaking that occur in noise‐induced phase transitions in the thermodynamic limit, but not in noise‐induced transitions.

On the approach to thermal equilibrium of macroscopic quantum systems
View Description Hide DescriptionIn joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro‐canonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and The thermal equilibrium macro‐state at energy E corresponds to a subspace of H such that is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is “close” to We argue that for “typical” Hamiltonians, all initial state vectors evolve in such a way that is in thermal equilibrium for most times t. This is closely related to von Neumann’s quantum ergodic theorem of 1929.

Temperature, entropy and second law beyond local equilibrium: An illustration
View Description Hide DescriptionLocal‐equilibrium thermodynamics applies to local scale the concepts and methods of equilibrium thermodynamics concerning the meaning of entropy, temperature and equations of state. However, when going beyond local equilibrium, the basic problems avoided by local equilibrium hypothesis arise: how temperature and entropy are defined, how second law is formulated, how macroscopic theory is related to microscopic formulations. Here, we illustrate these topics with phonon hydrodynamics as a model for the description of heat transfer in nanosystems and its corresponding non‐equilibrium thermodynamic potentials, and discuss the limits for the existence of non‐equilibrium thermodynamic potentials in general situations.

Griffiths phases in the contact process on complex networks
View Description Hide DescriptionDynamical processes occurring on top of complex networks have become an exciting area of research. Quenched disorder plays a relevant role in general dynamical processes and phase transitions, but the effect of topological quenched disorder on the dynamics of complex networks has not been systematically studied so far. Here, we provide heuristic and numerical analyses of the contact process defined on some complex networks with topological disorder. We report on Griffiths phases and other rare region effects, leading rather generically to anomalously slow relaxation in generalized small‐world networks. In particular, it is illustrated that Griffiths phases can emerge as the consequence of pure topological heterogeneity if the topological dimension of the network is finite.

Stationary points approach to thermodynamic phase transitions
View Description Hide DescriptionNonanalyticities of thermodynamic functions are studied by adopting an approach based on stationary points of the potential energy. For finite systems, each stationary point is found to cause a nonanalyticity in the microcanonical entropy, and the functional form of this nonanalytic term is derived explicitly. With increasing system size, the order of the nonanalytic term grows, leading to an increasing differentiability of the entropy. It is found that only “asymptotically flat” stationary points may cause a nonanalyticity that survives in the thermodynamic limit, and this property is used to derive an analytic criterion establishing the existence or absence of phase transitions. We sketch how this result can be employed to analytically compute transition energies of classical spin models.

Energy bursts in vibrated shallow granular systems
View Description Hide DescriptionIn a mixture of two species of inelastic spheres of equal size but different mass, placed in a vertically vibrated shallow box (large horizontal dimensions and height comparable to the grains’ size), there is spontaneous segregation. Once the system is at least partly segregated energy bursts recurrently take place: the horizontal kinetic energy of the heavy particles, that normally is small, suddenly increases an order of magnitude. An explanation of these events is provided based on the existence of a fixed point for an isolated particle bouncing with only vertical motion between the top and bottom plates. Energy bursts occur when clusters of heavy particles start a chain reaction of collisions that transfer vertical energy to horizontal energy producing an expansion of the cluster.

Layering and wetting transitions for an interface model
View Description Hide DescriptionWe study the solid‐on‐solid interface model above a horizontal wall in three dimensional space, with an attractive interaction when the interface is in contact with the wall, at low temperatures. The system presents a sequence of layering transitions, whose levels increase with the temperature, before the complete wetting above a certain value of this quantity.

On the role of Galilean invariance in KPZ
View Description Hide DescriptionStarting from a variational formulation of the Kardar‐Parisi‐Zhang (KPZ) equation, we point out some strong constraints and consistency tests, to be fulfilled by real‐space discretization schemes. In the light of these findings, the mainstream opinion on the relevance of Galilean invariance and the fluctuation—dissipation theorem (peculiar of 1D) is challenged.

Anomalous diffusion and basic theorems of statistical mechanics
View Description Hide DescriptionRecent works call attention that basic concepts in statistical mechanics are still under discussion. In particular, we have shown that some of those concepts can be discussed in a direct and analytical way in diffusion.

Large deviations of the current in a two‐dimensional diffusive system
View Description Hide DescriptionIn this notes we study the large deviations of the time‐averaged current in the two‐dimensional (2D) Kipnis‐Marchioro‐Presutti model of energy transport when subject to a boundary gradient. We use the tools of hydrodynamic fluctuation theory, supplemented with an appropriate generalization of the additivity principle. As compared to its one‐dimensional counterpart, which amounts to assume that the optimal profiles responsible of a given current fluctuation are time‐independent, the 2D additivity conjecture requires an extra assumption, i.e. that the optimal, divergence‐free current vector field associated to a given fluctuation of the time‐averaged current is in fact constant across the system. Within this context we show that the current distribution exhibits in general non‐Gaussian tails. The ensuing optimal density profile can be either monotone for small current fluctuations, or non‐monotone with a single maximum for large enough current deviations. Furthermore, this optimal profile remains invariant under arbitrary rotations of the current vector, providing a detailed example of the recently introduced Isometric Fluctuation Relation.