Nonequilibrium Statistical Physics Research Articles

Deformation of Crystals: Connections with Statistical Physics

We give a bird's-eye view of the plastic deformation of crystals aimed at the statistical physics community, as well as a broad introduction to the statistical theories of forced rigid systems aimed at the plasticity community. Memory effects in magnets, spin glasses, charge density waves, and dilute colloidal suspensions are discussed in relation to the onset of plastic yielding in crystals. Dislocation avalanches and complex dislocation tangles are discussed via a brief introduction to the renormalization group and scaling. Analogies to emergent scale invariance in fracture, jamming, coarsening, and a variety of depinning transitions are explored. Dislocation dynamics in crystals challenge nonequilibrium statistical physics. Statistical physics provides both cautionary tales of subtle memory effects in nonequilibrium systems and systematic tools designed to address complex scale-invariant behavior on multiple length scales and timescales.

Unreasonable effectiveness of learning neural networks: From accessible states and robust ensembles to basic algorithmic schemes

In artificial neural networks, learning from data is a computationally demanding task in which a large number of connection weights are iteratively tuned through stochastic-gradient-based heuristic processes over a cost function. It is not well understood how learning occurs in these systems, in particular how they avoid getting trapped in configurations with poor computational performance. Here, we study the difficult case of networks with discrete weights, where the optimization landscape is very rough even for simple architectures, and provide theoretical and numerical evidence of the existence of rare-but extremely dense and accessible-regions of configurations in the network weight space. We define a measure, the robust ensemble (RE), which suppresses trapping by isolated configurations and amplifies the role of these dense regions. We analytically compute the RE in some exactly solvable models and also provide a general algorithmic scheme that is straightforward to implement: define a cost function given by a sum of a finite number of replicas of the original cost function, with a constraint centering the replicas around a driving assignment. To illustrate this, we derive several powerful algorithms, ranging from Markov Chains to message passing to gradient descent processes, where the algorithms target the robust dense states, resulting in substantial improvements in performance. The weak dependence on the number of precision bits of the weights leads us to conjecture that very similar reasoning applies to more conventional neural networks. Analogous algorithmic schemes can also be applied to other optimization problems.

Nonequilibrium statistical mechanics of a solid immersed in a continuum

In the introductory part of this survey, we briefly discuss the problems of nonequilibrium statistical physics that arise in the study of energy transport in solids as well as the results available at the moment. In the main part of the survey, we explain, compare, and generalize results obtained in our previous works. We study the dynamics and energy transport in Hamiltonian systems of particles where each particle is weakly perturbed by the interaction with its own stochastic Langevin thermostat. Such systems can be regarded as models of solids that interact weakly with a continuum.

Cycle symmetry, limit theorems, and fluctuation theorems for diffusion processes on the circle

Cyclic structure and dynamics are of great interest in both the fields of stochastic processes and nonequilibrium statistical physics. In this paper, we find a new symmetry of the Brownian motion named as the quasi-time-reversal invariance. It turns out that such an invariance of the Brownian motion is the key to prove the cycle symmetry for diffusion processes on the circle, which says that the distributions of the forming times of the forward and backward cycles, given that the corresponding cycle is formed earlier than the other, are exactly the same. With the aid of the cycle symmetry, we prove the strong law of large numbers, functional central limit theorem, and large deviation principle for the sample circulations and net circulations of diffusion processes on the circle. The cycle symmetry is further applied to obtain various types of fluctuation theorems for the sample circulations, net circulation, and entropy production rate.

On the Role of Fluctuations in the Modeling of Complex Systems

The study of models is ubiquitous in sciences like physics, chemistry, ecology, biology or sociology. Models are used to explain experimental facts or to make new predictions. For any system, one can distinguish several levels of description. In the simplest mean-field like description the dynamics is described in terms of spatially averaged quantities while in a microscopic approach local properties are taken into account and local fluctuations for the relevant variables are present. The properties predicted by these two different approaches may be drastically different. In a large body of research literature concerning complex systems this problem is often overlooked and simple mean-field like approximation are used without asking the question of the robustness of the corresponding predictions. The goal of this paper is twofold, first to illustrate the importance of the fluctuations in a self-contained and pedagogical way, by revisiting two different classes of problems where thorough investigations have been conducted (equilibrium and non-equilibrium statistical physics). Second, we present our original research on the dynamics of population of annual plants which are competing among themselves for just one resource (water) through a stochastic dynamics. Depending on the observable considered, the mean-field like and microscopic approaches agree or totally disagree. There is not a general criterion allowing to decide a priori when the two approaches will agree.

Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response

We review some approaches to macroscopic irreversibility from reversible microscopic dynamics, introducing the contribution of time dependent perturbations within the framework of recent developments in non-equilibrium statistical physics. We show that situations commonly assumed to violate the time reversal symmetry (presence of magnetic fields, rotating reference frames, and some time dependent perturbations) in reality do not violate this symmetry, and can be treated with standard theories and within standard experimental protocols.

Quantum thermodynamics

Quantum thermodynamics is an emerging research field aiming to extend standard thermodynamics and non-equilibrium statistical physics to ensembles of sizes well below the thermodynamic limit, in non-equilibrium situations and with the full inclusion of quantum effects. Fuelled by experimental advances and the potential of future nanoscale applications, this research effort is pursued by scientists with different backgrounds, including statistical physics, many-body theory, mesoscopic physics and quantum information theory, who bring various tools and methods to the field. A multitude of theoretical questions are being addressed ranging from issues of thermalisation of quantum systems and various definitions of ‘work’ to the efficiency and power of quantum engines. This overview provides a perspective on a selection of these current trends accessible to postgraduate students and researchers alike.

Phase transitions in models of human cooperation

If only the fittest survive, why should one cooperate? Why should one sacrifice personal benefits for the common good? Recent research indicates that a comprehensive answer to such questions requires that we look beyond the individual and focus on the collective behavior that emerges as a result of the interactions among individuals, groups, and societies. Although undoubtedly driven also by culture and cognition, human cooperation is just as well an emergent, collective phenomenon in a complex system. Nonequilibrium statistical physics, in particular the collective behavior of interacting particles near phase transitions, has already been recognized as very valuable for understanding counterintuitive evolutionary outcomes. However, unlike pairwise interactions among particles that typically govern solid-state physics systems, interactions among humans often involve group interactions, and they also involve a larger number of possible states even for the most simplified description of reality. Here we briefly review research done in the realm of the public goods game, and we outline future research directions with an emphasis on merging the most recent advances in the social sciences with methods of nonequilibrium statistical physics. By having a firm theoretical grip on human cooperation, we can hope to engineer better social systems and develop more efficient policies for a sustainable and better future.

Nonergodic complexity management.

Linear response theory, the backbone of nonequilibrium statistical physics, has recently been extended to explain how and why nonergodic renewal processes are insensitive to simple perturbations, such as in habituation. It was established that a permanent correlation results between an external stimulus and the response of a complex system generating nonergodic renewal processes, when the stimulus is a similar nonergodic process. This is the principle of complexity management, whose proof relies on ensemble distribution functions. Herein we extend the proof to the nonergodic case using time averages and a single time series, hence making it usable in real life situations where ensemble averages cannot be performed because of the very nature of the complex systems being studied.

Spatially Constrained Growth Enhances Conversional Meltdown

Cells that mutate or commit to a specialized function (differentiate) often undergo conversions that are effectively irreversible. Slowed growth of converted cells can act as a form of selection, balancing unidirectional conversion to maintain both cell types at a steady-state ratio. However, when one-way conversion is insufficiently counterbalanced by selection, the original cell type will ultimately be lost, often with negative impacts on the population’s overall fitness. The critical balance between selection and conversion needed for preservation of unconverted cells and the steady-state ratio between cell types depends on the spatial circumstances under which cells proliferate. We present experimental data on a yeast strain engineered to undergo irreversible conversion: this synthetic system permits cell-type-specific fluorescent labeling and exogenous variation of the relative growth and conversion rates. We find that populations confined to grow on a flat agar surface are more susceptible than their well-mixed counterparts to fitness loss via a conversion-induced “meltdown.” We then present analytical predictions for growth in several biologically relevant geometries—well-mixed liquid media, radially expanding two-dimensional colonies, and linear fronts in two dimensions—by employing analogies to the directed-percolation transition from nonequilibrium statistical physics. These simplified theories are consistent with the experimental results.

The Free Energy Requirements of Biological Organisms; Implications for Evolution

Recent advances in nonequilibrium statistical physics have provided unprecedented insight into the thermodynamics of dynamic processes. The author recently used these advances to extend Landauer’s semi-formal reasoning concerning the thermodynamics of bit erasure, to derive the minimal free energy required to implement an arbitrary computation. Here, I extend this analysis, deriving the minimal free energy required by an organism to run a given (stochastic) map π from its sensor inputs to its actuator outputs. I use this result to calculate the input-output map π of an organism that optimally trades off the free energy needed to run π with the phenotypic fitness that results from implementing π. I end with a general discussion of the limits imposed on the rate of the terrestrial biosphere’s information processing by the flux of sunlight on the Earth.

Physical biology of bacterial motility

Bacteria form a complex system. It consists of many components that cover broad size scales, including ions, small molecules, DNA, polymers, sub-micrometer sized organelles and compartments, micrometer sized cells, packs of cells in films of a few micrometers in thickness, large swarms or populations spanning plates over several centimeters in diameter, etc. The mechanisms to be explored span a wide range of time scales from micro-second or shorter for molecular interaction, to milli-second or longer times for diffusion and transport, up to minutes and hours for cellular metabolism, growth, and reproduction. An invisible colony of bacteria can grow rapidly and becomes visible to the human eye in several hours. Novel phenomena or behaviors emerge across these broad size and time scales. For example, the rotation direction and speed of a flagella motor, about 50 nm in diameter, are both tightly regulated by a signaling pathway within the cell. The fast rotation of the helical flagellum driven by the rotary motor is a key to explaining the bacterial swimming trajectory, chemo-taxis, accumulation, adhesion, or anchored body rotation near or at a solid surface. The activities of individual bacteria in response to their physicochemical environment give rise to their collective response such as quorum sensing, swarming, and growth of biofilms. The physical biology of bacteria is an interdisciplinary research covering micromechanics, micro-fluidics, non-equilibrium statistical physics, etc. This review covers several aspects of bacterial motility, including flagella motor behavior, bacterial swimming and accumulation near the surface, the self-organized patterns of bacterial swarms, and chemo-taxis regulated by the biochemical signaling network inside bacteria. Instead of presenting each aspect as a separate topic of microbiological study, we emphasize the strong relations among these topics, as well as the multidisciplinary perspective required to appreciate the strong relations among the topics covered. For instance, we point out the relevance of numerous phenomena in thin film fluid physics to bacterial swarming, such as capillary flow, surface tension reduction by surfactant, Marangoni flow, and viscous fingering. Another notable example is a recent application of a statistical mechanical theory called the first passage time theory to account for the intervals between the switches of bacterial motor rotation from clockwise to counter-clockwise, and vice versa. In concluding remarks, we point out a few open questions in the field of bacterial motility and likely advances that might transform the field. The central view conveyed through this review article is that further progress in the field demands interdisciplinary efforts. Therefore, a collaborative approach among those with both in depth knowledge and broad perspectives in biological and physical sciences will prove to be the most successful ones.

What can we learn from noise? - Mesoscopic nonequilibrium statistical physics.

Mesoscopic systems — small electric circuits working in quantum regime — offer us a unique experimental stage to explorer quantum transport in a tunable and precise way. The purpose of this Review is to show how they can contribute to statistical physics. We introduce the significance of fluctuation, or equivalently noise, as noise measurement enables us to address the fundamental aspects of a physical system. The significance of the fluctuation theorem (FT) in statistical physics is noted. We explain what information can be deduced from the current noise measurement in mesoscopic systems. As an important application of the noise measurement to statistical physics, we describe our experimental work on the current and current noise in an electron interferometer, which is the first experimental test of FT in quantum regime. Our attempt will shed new light in the research field of mesoscopic quantum statistical physics.

Role of ergodicity in the transient Fluctuation Relation and a new relation for a dissipative non-chaotic map

Toy model dynamical systems, such as the baker maps, are useful to shed light on some of the conditions verified by deterministic models in non-equilibrium statistical physics. We investigate a 2D dynamical system, enjoying a weak form of reversibility, with peculiar basins of attraction and steady states. In particular, we test the conditions required for the validity of the transient Fluctuation Relation. Our analysis illustrates by means of concrete examples why ergodicity of the equilibrium dynamics (also known as “ergodic consistency”) seems to be a necessary condition for the transient Fluctuation Relation. This investigation then leads to the numerical verification of a kind of transient relation which, differently from the usual transient Fluctuation Relation, holds only asymptotically. At the same time, this relation is not a steady state Fluctuation Relation, because the steady state is a fixed point without fluctuations.

Non-equilibrium statistical approach to friction models

A geometric approach to the friction phenomena is presented. It is based on the holographic view which has recently been popular in the theoretical physics community. We see the system in one-dimension-higher space. The heat-producing phenomena are most widely treated by using the non-equilibrium statistical physics. We take 2 models of the earthquake. The dissipative systems are here formulated from the geometric standpoint. The statistical fluctuation is taken into account by using the (generalized) Feynman׳s path-integral.

Asymptotics of the Entropy Production Rate for d-Dimensional Ornstein–Uhlenbeck Processes

In the context of non-equilibrium statistical physics, the entropy production rate is an important concept to describe how far a specific state of a system is from its equilibrium state. In this paper, we establish a central limit theorem and a moderate deviation principle for the entropy production rate of $d$-dimensional Ornstein-Uhlenbeck processes, by the techniques of functional inequalities such as Poincar\'e inequality and log-Sobolev inequality. As an application, we obtain a law of iterated logarithm for the entropy production rate.

Thermodynamic Limits of Spatial Resolution in Active Thermography.

Thermal waves are caused by pure diffusion: their amplitude is decreased by more than a factor of 500 within a propagation distance of one wavelength. The diffusion equation, which describes the temperature as a function of space and time, is linear. For every linear equation the superposition principle is valid, which is known as Huygens principle for optical or mechanical wave fields. This limits the spatial resolution, like the Abbe diffraction limit in optics. The resolution is the minimal size of a structure which can be detected at a certain depth. If an embedded structure at a certain depth in a sample is suddenly heated, e.g., by eddy current or absorbed light, an image of the structure can be reconstructed from the measured temperature at the sample surface. To get the resolution the image reconstruction can be considered as the time reversal of the thermal wave. This inverse problem is ill-conditioned and therefore regularization methods have to be used taking additional assumptions like smoothness of the solutions into account. In the present work for the first time, methods of non-equilibrium statistical physics are used to solve this inverse problem without the need of such additional assumptions and without the necessity to choose a regularization parameter. For reconstructing such an embedded structure by thermal waves the resolution turns out to be proportional to the depth and inversely proportional to the natural logarithm of the signal-to-noise ratio. This result could be derived from the diffusion equation by using a delta-source at a certain depth and setting the entropy production caused by thermal diffusion equal to the information loss. No specific model about the stochastic process of the fluctuations and about the distribution densities around the mean values was necessary to get this result.

Spin effects induced by thermal perturbation in a normal metal/magnetic insulator system

Using one of the methods of quantum nonequilibrium statistical physics, we have investigated the spin transport transverse to the normal metal/ferromagnetic insulator interface in hybrid nanostructures. An approximation of the effective parameters, when each of the interacting subsystems (electron spin, magnon, and phonon) is characterized by its own effective temperature, has been considered. The generalized Bloch equations which describe the spin-wave current propagation in the dielectric have been derived. Finally, two sides of the spin transport ``coin'' have been revealed: the diffusive nature of the magnon motion and magnon relaxation processes, responsible for the spin pumping, and the spin-torque effect.

Work relations connecting nonequilibrium steady states without detailed balance

Bridging equilibrium and nonequilibrium statistical physics attracts sustained interest. Hallmarks of nonequilibrium systems include a breakdown of detailed balance, and an absence of a priori potential function corresponding to the Boltzmann-Gibbs distribution, without which classical equilibrium thermodynamical quantities could not be defined. Here, we construct dynamically the potential function through decomposing the system into a dissipative part and a conservative part, and develop a nonequilibrium theory by defining thermodynamical quantities based on the potential function. Concepts for equilibrium can thus be naturally extended to nonequilibrium steady state. We elucidate this procedure explicitly in a class of time-dependent linear diffusive systems without mathematical ambiguity. We further obtain the exact work distribution for an arbitrary control parameter, and work equalities connecting nonequilibrium steady states. Our results provide a direct generalization on Jarzynski equality and Crooks fluctuation theorem to systems without detailed balance.

Special issue on mathematics for the fluid Earth

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to mathematics for the fluid Earth. The fluid Earth is an excellent example of a forced, dissipative non-equilibrium system dominated by nonlinear processes featuring multi-scale interactions. The full spectrum will be considered: the issue will consist of reviews and original research articles from mathematical papers addressing general questions, articles considering these questions for more realistic problems on the level of rigour in theoretical physics and articles that apply new mathematical tools to concrete physical situations with application to the Fluid Earth system. The objective is to catch the complexity of this system and its phenomena as a statistical manifestation of large-scale systems in the asymptotic regimes far from the equilibrium where loss of memory and recurrence prevail. Through excellent scientific contributions, we aim to present the interplay between theoretical, numerical and computational techniques that is necessary to unravel the structure of Planet Earth. Editorial policy The guest editors of this issue are Tobias Kuna and Sandro Vaienti. The areas and topics of issue include, but are not limited to: • PDEs of geodynamical fluids • dynamical systems and equilibrium and non-equilibrium statistical physics • random systems and their statistics • extremes and tipping points • mathematics in atmospheric and oceanic flow, hydrology, and climate science All contributions will be refereed and processed according to the usual procedures of the journal.