Statistical mechanics and thermodynamics of small systems

In this thesis many aspects of the statistical mechanics and thermodynamics of small systems are studied. The very same possibility of defining a thermodynamics for this class of systems, for which the usual properties of the thermodynamic limit do not apply, is discussed by means of general considerations and specific examples. We show that it is possible to preserve most of the features of thermodynamics for a specific class of systems which are, at the same time, far enough from the infinite-N limit to be small, but large enough to be studied with a statistical approach. A review of the necessary mathematical and physical tools to study this particular class of systems is included. Eventually, a specific system is studied, both from an equilibrium and a non- equilibrium perspective: it is found that this system, composed by a gas included in a container with a moving wall (the piston), has an highly non-trivial dynamics caused by the interplay of the different degrees of freedom of the system, which cannot be easily reproduced by means of coarse-grained equations. At the same time, the smallness of the system is responsible for large fluctuations that strongly characterize the system. We show that this system reproduces the behavior of an heat engine, when the external parameters vary in time: in particular we show that different working regimes (engine, refrigerator, heat pump) can be obtained depending upon the total time of a cycle of the external parameters. We also derive some analytical results reproducing, with a fair degree of approximation, the behavior of the system

Kovacs-like memory effect in athermal systems: linear response analysis

We analyse the emergence of Kovacs-like memory effects in athermal systems within the linear response regime. This is done by starting from both the master equation for the probability distribution and the equations for the physically relevant moments. The general results are applied to a general class of models with conserved momentum and non-conserved energy. Our theoretical predictions, obtained within the first Sonine approximation, show an excellent agreement with the numerical results.Comment: 18 pages, 6 figures; submitted to the special issue of the journal Entropy on "Thermodynamics and Statistical Mechanics of Small Systems

Statistical Entropy in General Equilibrium Theory

This essay seeks to develop an integrated account of the workings of statistical mechanics and thermodynamics as a theory of economic equilibrium. It begins with a probabilistic description of general systems (made out of numerous elements), based on the practice of statistical physics and the work of E. T. Jaynes, and a self-contained overview of the arguments that lead to the concept of statistical entropy as a measure of uncertainty or disorder and the maximum statistical entropy principle . This provides the conceptual setting for developing a statistical mechanical model of general equilibrium in pure exchange economies, inspired by the statistical theory of markets of Duncan K. Foley. Emphasis is placed in the derivation of the properties of the entropy function of an economy—the maximized statistical entropy as a function of the amounts of resources in that economy. We then show that the statistical equilibrium theory of pure exchange economies gives rise to a phenomenological or ‘macro’ theory of resource allocation in the image of classical thermodynamics (and the generalized thermodynamics of L. I. Rozonoer). We thus establish the fundamental principle of the phenomenological theory—the maximum entropy principle—and illustrate its use for the study of isolated and small open economies.statistical entropy, thermodynamics, general equilibrium, physics

Maximum one-shot dissipated work from Renyi divergences

Thermodynamics describes large-scale, slowly evolving systems. Two modern approaches generalize thermodynamics: fluctuation theorems, which concern finite-time nonequilibrium processes, and one-shot statistical mechanics, which concerns small scales and finite numbers of trials. Combining these approaches, we calculate a one-shot analog of the average dissipated work defined in fluctuation contexts: the cost of performing a protocol in finite time instead of quasistatically. The average dissipated work has been shown to be proportional to a relative entropy between phase-space densities, to a relative entropy between quantum states, and to a relative entropy between probability distributions over possible values of work. We derive one-shot analogs of all three equations, demonstrating that the order-infinity Renyi divergence is proportional to the maximum possible dissipated work in each case. These one-shot analogs of fluctuation-theorem results contribute to the unification of these two toolkits for small-scale, nonequilibrium statistical physics.Comment: 8 pages. Close to published versio

Temperature in and out of equilibrium: a review of concepts, tools and attempts

We review the general aspects of the concept of temperature in equilibrium and non-equilibrium statistical mechanics. Although temperature is an old and well-established notion, it still presents controversial facets. After a short historical survey of the key role of temperature in thermodynamics and statistical mechanics, we tackle a series of issues which have been recently reconsidered. In particular, we discuss different definitions and their relevance for energy fluctuations. The interest in such a topic has been triggered by the recent observation of negative temperatures in condensed matter experiments. Moreover, the ability to manipulate systems at the micro and nano-scale urges to understand and clarify some aspects related to the statistical properties of small systems (as the issue of temperature's "fluctuations"). We also discuss the notion of temperature in a dynamical context, within the theory of linear response for Hamiltonian systems at equilibrium and stochastic models with detailed balance, and the generalised fluctuation-response relations, which provide a hint for an extension of the definition of temperature in far-from-equilibrium systems. To conclude we consider non-Hamiltonian systems, such as granular materials, turbulence and active matter, where a general theoretical framework is still lacking.Comment: Review article, 137 pages, 12 figure

Thermodynamics of natural images

The scale invariance of natural images suggests an analogy to the statistical mechanics of physical systems at a critical point. Here we examine the distribution of pixels in small image patches and show how to construct the corresponding thermodynamics. We find evidence for criticality in a diverging specific heat, which corresponds to large fluctuations in how "surprising" we find individual images, and in the quantitative form of the entropy vs. energy. The energy landscape derived from our thermodynamic framework identifies special image configurations that have intrinsic error correcting properties, and neurons which could detect these features have a strong resemblance to the cells found in primary visual cortex

On the nonequilibrium entropy of large and small systems

Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in and out of equilibrium. Statistical mechanics aims to derive this behavior from the dynamics and statistics of the atoms and molecules making up these systems. A key element in this derivation is the large number of microscopic degrees of freedom of macroscopic systems. Therefore, the extension of thermodynamic concepts, such as entropy, to small (nano) systems raises many questions. Here we shall reexamine various definitions of entropy for nonequilibrium systems, large and small. These include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies. We shall argue that, despite its common use, the last is not an appropriate physical entropy for such systems, either isolated or in contact with thermal reservoirs: physical entropies should depend on the microstate of the system, not on a subjective probability distribution. To square this point of view with experimental results of Bechhoefer we shall argue that the Gibbs-Shannon entropy of a nano particle in a thermal fluid should be interpreted as the Boltzmann entropy of a dilute gas of Brownian particles in the fluid

Thermodynamic Limit in Statistical Physics

The thermodynamic limit in statistical thermodynamics of many-particle systems is an important but often overlooked issue in the various applied studies of condensed matter physics. To settle this issue, we review tersely the past and present disposition of thermodynamic limiting procedure in the structure of the contemporary statistical mechanics and our current understanding of this problem. We pick out the ingenious approach by N. N. Bogoliubov, who developed a general formalism for establishing of the limiting distribution functions in the form of formal series in powers of the density. In that study he outlined the method of justification of the thermodynamic limit when he derived the generalized Boltzmann equations. To enrich and to weave our discussion, we take this opportunity to give a brief survey of the closely related problems, such as the equipartition of energy and the equivalence and nonequivalence of statistical ensembles. The validity of the equipartition of energy permits one to decide what are the boundaries of applicability of statistical mechanics. The major aim of this work is to provide a better qualitative understanding of the physical significance of the thermodynamic limit in modern statistical physics of the infinite and "small" many-particle systems.Comment: 28 pages, Refs.180. arXiv admin note: text overlap with arXiv:1011.2981, arXiv:0812.0943 by other author