Heat and Fluctuations from Order to Chaos

The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e.existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as $10^{23}$ degrees of freedom systems, {\it i.e.} for simple as well as very complex systems, and reflecting the Hamiltonian nature of the microscopic motion. In Nonequilibrium Thermodynamics theorems of comparable generality do not seem to be available. Yet it is possible to find general, model independent, properties valid even for simple chaotic systems ({\it i.e.} the hyperbolic ones), which acquire special interest for large systems: the Chaotic Hypothesis leads to the Fluctuation Theorem which provides general properties of certain very large fluctuations and reflects the time-reversal symmetry. Implications on Fluids and Quantum systems are briefly hinted. The physical meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation Theorem is discussed in the context of their interpretation and relevance in terms of Coarse Grained Partitions of phase space. This review is written taking some care that each section and appendix is readable either independently of the rest or with only few cross references.Comment: 1) added comment at the end of Sec. 1 to explain the meaning of the title (referee request) 2) added comment at the end of Sec. 17 (i.e. appendix A4) to refer to papers related to the ones already quoted (referee request

Dynamics of Patterns

Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction

Investigating the Relationships Between Disorder, Structure, and Dynamics in Amorphous Systems

In this thesis, we investigate the relationships between the disorder, structure, and deformation in amorphous materials. First, to understand the surprising low-frequency vibrational modes in structural glasses, and how it arises from the microscopic disorder in the system, we study the spectra of a large ensemble of sparse random matrices where disorder is controlled by the distribution of bond weights and network coordination. When there is a finite probability density of infinitesimal bond weights, we find a region in the vibrational density of states that is consistent with the low-frequency behavior in structural glasses. Next, in order to investigate structural properties of active systems, we develop a novel method to generate static, finite packings in an artificial potential that reproduce the packing structures observed in a class of point-of-interest active self-propelled particle simulations. This allows us to compute structural measures, such as the vibrational modes, in an unstable active system. Finally, we evaluate the evolution of structure during strain-induced avalanches in athermal, amorphous systems using numerical simulation of soft spheres. We find that these avalanches can be decomposed into a series of bursts of localized deformations, and we develop an extension of persistent homology to isolate these bursts of localized deformations. Further, we extend existing tools for the structural evaluation of mechanically stable systems to generically unstable systems to identify how soft regions evolve and change throughout an avalanche

Pullback Exponential Attractors for Nonautonomous Klein-Gordon-Schrödinger Equations on Infinite Lattices

This paper proves the existence of the pullback exponential attractor for the process associated to the nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble and discuss its connection with ideas of prethermalization in weakly interacting systems.This work was supported by the Army Research Office [grant number W911NF1410540] (L.D., A.P, and M.R.), the U.S.-Israel Binational Science Foundation [grant number 2010318] (Y.K. and A.P.), the Israel Science Foundation [grant number 1156/13] (Y.K.), the National Science Foundation [grant numbers DMR-1506340 (A.P.)and PHY-1318303 (M.R.)], the Air Force Office of Scientific Research [grant number FA9550-13-1-0039] (A.P.), and the Office of Naval Research [grant number N000141410540] (M.R.). The computations were performed in the Institute for CyberScience at Penn State. (W911NF1410540 - Army Research Office; 2010318 - U.S.-Israel Binational Science Foundation; 1156/13 - Israel Science Foundation; DMR-1506340 - National Science Foundation; PHY-1318303 - National Science Foundation; FA9550-13-1-0039 - Air Force Office of Scientific Research; N000141410540 - Office of Naval Research)Accepted manuscrip