We demonstrate the equilibration of isolated macroscopic quantum systems, prepared in non-equilibrium mixed states with significant population of many energy levels, and observed by instruments with a reasonably bound working range compared to the resolution limit. Both properties are fulfilled under many, if not all, experimentally realistic conditions. At equilibrium, the predictions and limitations of Statistical Mechanics are recovered.
This is an extensive review of recent work on the foundations of statistical mechanics. Subject matters discussed include: interpretation of probability, typicality, recurrence, reversibility, ergodicity, mixing, coarse graining, past hypothesis, reductionism, phase average, thermodynamic limit, interventionism, entropy.
The first part of the paper is devoted to the foundations, that is the mathematical and physical justification, of equilibrium statistical mechanics. It is a pedagogical attempt, mostly based on Khinchin's presentation, which purpose is to clarify some aspects of the development of statistical mechanics. In the second part, we discuss some recent developments that appeared out of equilibrium, such as fluctuation theorem and Jarzynski equality.
In this article, it is suggested that a pedagogical point of departure in the teaching of classical mechanics is the Liouville theorem. The theorem is interpreted to define the condition that describe the conservation of information in classical mechanics. The Hamilton equations and the Hamilton principle of least action are derived from the Liouville theorem.
We discuss the condition for the validity of equilibrium quantum statistical mechanics in the light of recent developments in the understanding of classical and quantum chaotic motion. In particular, the ergodicity parameter is shown to provide the conditions under which quantum statistical distributions can be derived from the quantum dynamics of a classical ergodic Hamiltonian system.
Traditionally the exponential canonical distributions of Gibbsian statistical mechanics are given theoretical justification in at least four different ways: steepest descent method, counting method, Khinchin's method based on te central limit theorem, and maximum entropy principle of Jaynes. Equally ubiquitous power-law canonical distributions are shown to be given similar justification by appropriately adopting these formulations.
On the base of years of experience of working on the problem of the physical foundation of quantum mechanics the author offers principles of solving it. Under certain pressure of mathematical formalism there has raised a hypothesis of complexity of space and time by Minkovsky, being significant mainly for quantum objects. In this eight-dimensional space and time with six space and two time dimensions all the problems and peculiarities of quantum mechanical formalism disappear, the reasons of their appearance become clear, and there comes a clear and physically transparent picture of the founda…
Besides the growing interest in old concepts like temperature and entropy at the nanoscale, theories of relaxation and transport have recently regained a lot of attention. With the electronic circuits and computer chips getting smaller and smaller, a fresh look should be appropriate on the equilibrium and nonequilibrium thermodynamics at small length scales far below the thermodynamic limit, i.e. on the theoretical understanding of original macroscopic processes, e.g. transport of energy, heat, charge, mass, magnetization etc. Only from the foundations of a theory its limits of applicability m…
The present paper is meant to give a simple introduction to the problem of the connection between microscopic dynamics and statistical laws. For sake of simplicity, we mostly refer to non-dissipative dynamics, since dissipation adds technical difficulties to the conceptual issues, although part of our discussion extends beyond this limit. In particular, the relevance of chaos and ergodicity is here confronted with that of the large number of degrees of freedom. In Section 2, we review the microscopic connection, along the lines of Boltzmann's approach, and of its further developments. In Secti…
The derivation of Student's pdf from superstatistics is a mere coincidence due to the choice of the $χ^2$ distribution for the inverse temperature which is actually the Maxwell distribution for the speed. The difference between the estimator and the variance introduces a fluctuating temperature that is generally different than the temperature of the variance. Fluctuations in the energy of a composite system are transferred to fluctuations in the temperature via Lévy's transform. The randomization of the operational parameter leading to a subordinated process requires the same number of degrees…
On the basis of new, concise foundations, this paper establishes the four laws of thermodynamics, the Maxwell relations, and the stability requirements for response functions, in a form applicable to global (homogeneous), local (hydrodynamic) and microlocal (kinetic) equilibrium. The present, self-contained treatment needs very little formal machinery and stays very close to the formulas as they are applied by the practicing physicist, chemist, or engineer. From a few basic assumptions, the full structure of phenomenological thermodynamics and of classical and quantum statistical mechanics is …
We construct classes of stochastic differential equations with fluctuating friction forces that generate a dynamics correctly described by Tsallis statistics and nonextensive statistical mechanics. These systems generalize the way in which ordinary Langevin equations underly ordinary statistical mechanics to the more general nonextensive case. As a main example, we construct a dynamical model of velocity fluctuations in a turbulent flow, which generates probability densities that very well fit experimentally measured probability densities in Eulerian and Lagrangian turbulence. Our approach pro…
We consider an alternative approach to the foundations of statistical mechanics, in which subjective randomness, ensemble-averaging or time-averaging are not required. Instead, the universe (i.e. the system together with a sufficiently large environment) is in a quantum pure state subject to a global constraint, and thermalisation results from entanglement between system and environment. We formulate and prove a "General Canonical Principle", which states that the system will be thermalised for almost all pure states of the universe, and provide rigorous quantitative bounds using Levy's Lemma.
An equation describing the irreversible evolution of the local density of a continuous medium without involving any statistical hypotheses and assumptions is derived. The derivation is based on the smoothing of the microscopic dynamic characteristics of a many-body system, taking into account the retardation of the interactions between them. The resulting equation generalizes the classical equation of motion for fluids. Several particular cases of the resulting equation, as well as its connection with the dynamic density functional theory, are considered.
In 1916 Einstein introduced the first rules for a quantum theory of electromagnetic radiation, and he applied them to a model of matter in thermal equilibrium with radiation to derive Planck's black-body formula. Einstein's treatment is extended here to time-dependent stochastic variables, which leads to a master equation for the probability distribution that describes the irreversible approach of Einstein's model towards thermal equilibrium, and elucidates aspects of the foundation of statistical mechanics. An analytic solution of this equation is obtained in the Fokker-Planck approximation w…
The foundations of the Boltzmann-Gibbs (BG) distributions for describing equilibrium statistical mechanics of systems are examined. Broadly, they fall into: (i) probabilistic paaroaches based on the principle of equal a priori probability (counting technique and method of steepest descents), law of large numbers, or the state density considerations and (ii) a variational scheme -- maximum entropy principle (due to Gibbs and Jaynes) subject to certain constraints. A minimum set of requirements on each of these methods are briefly pointed out: in the first approach, the function space and the co…
Boltzmann's ergodic hypothesis furnishes a possible explanation for the emergence of statistical mechanics in the framework of classical physics. In quantum mechanics, the Eigenstate Thermalization Hypothesis (ETH) is instead generally considered as a possible route to thermalization. This is because the notion of ergodicity itself is vague in the quantum world and it is often simply taken as a synonym for thermalization. Here we show, in an elementary way, that when quantum ergodicity is properly defined, it is, in fact, equivalent to ETH. In turn, ergodicity is equivalent to thermalization, …
Focusing on isolated macroscopic systems, described either in terms of a quantum mechanical or a classical model, our two key questions are: In how far does an initial ensemble (usually far from equilibrium and largely unknown in detail) evolve towards a stationary long-time behavior ("equilibration")? In how far is this steady state in agreement with the microcanonical ensemble as predicted by Statistical Mechanics ("thermalization")? In the first part of the paper, a recently developed quantum mechanical treatment of the problem is briefly summarized, putting particular emphasis on the reali…
Simple models have played an important role in the discussion of foundational issues in statistical mechanics. Among them the spin--echo system is of particular interest since it can be realized experimentally. This has led to inferences being drawn about approaches to the foundations of statistical mechanics, particularly with respect to the use of coarse-graining. We examine these claims with the help of computer simulations.
Even though the phenomenological relations between perception and stimulus have been firmly established, a theoretical argument for Weber's and Fechner's law in terms of relevant models or from statistical physics is largely missing. We present such a discussion in terms of response theory for nonequilibrium systems, where the induced displacement or current, which stands for the perceived stimulus, crucially depends on the change in time-symmetric reactivities. Stationary nonequilibria may indeed generate extra currents by changing the dynamical activity. The argument finishes by understandin…
We examine the fundamental aspects of statistical mechanics, dividing the problem into a discussion purely about probability, which we analyse from a Bayesian standpoint. We argue that the existence of a unique maximising probability distribution $\{p(j\vert K)\}$ for states labelled by $j$ given data $K$ implies that the corresponding maximal value of the information entropy $σ(\{(p_j\vert K)\}) = -\sum_j (p_j \vert K)\ln{(p_j\vert K)}$ depends explicitly on the data at equilibrium and on the Hamiltonian of the system. As such, it is a direct measure of our uncertainty about the exact state o…
Envariance -- entanglement assisted invariance -- is a recently discovered symmetry of composite quantum systems. We show that thermodynamic equilibrium states are fully characterized by their envariance. In particular, the microcanonical equilibrium of a system $\mathcal{S} $ with Hamiltonian $H_\mathcal{S}$ is a fully energetically degenerate quantum state envariant under every unitary transformation. The representation of the canonical equilibrium then follows from simply counting degenerate energy states. Our conceptually novel approach is free of mathematically ambiguous notions such as e…
In a recent series of papers [1--3], a statistical model that accounts for correlations between topological and geometrical properties of a two-dimensional shuffled foam has been proposed and compared with experimental and numerical data. Here, the various assumptions on which the model is based are exposed and justified: the equiprobability hypothesis of the foam configurations is argued. The range of correlations between bubbles is discussed, and the mean field approximation that is used in the model is detailed. The two self-consistency equations associated with this mean field description …
The Feynman quantum-classical isomorphism between classical statistical mechanics in 3+1 dimensions and quantum statistical mechanics in 3 dimensions is used to connect classical polymer self-consistent field theory with quantum time-dependent density functional theory. This allows the theorems of density functional theory to relate non-relativistic quantum mechanics back to a classical statistical mechanical derivation of polymer self-consistent field theory for ring polymers in a 4 dimensional thermal-space. One dynamic postulate is added to two static postulates which allows for a complete …
The project concerns the interplay among quantum mechanics, statistical mechanics and thermodynamics, in isolated quantum systems. The underlying goal is to improve our understanding of the concept of thermal equilibrium in quantum systems. First, I investigated the role played by observables and measurements in the emergence of thermal behaviour. This led to a new notion of thermal equilibrium which is specific for a given observable, rather than for the whole state of the system. The equilibrium picture that emerges is a generalization of statistical mechanics in which we are not interested …
We propose a statistical mechanics model for DNA melting in which base stacking and pairing are explicitly introduced as distinct degrees of freedom. Unlike previous approaches, this model describes thermal denaturation of DNA secondary structure in the whole experimentally accessible temperature range. Base pairing is described through a zipper model, base stacking through an Ising model. We present experimental data on the unstacking transition, obtained exploiting the observation that at moderately low pH this transition is moved down to experimentally accessible temperatures. These measure…
We analyze how the thermal history of the universe is influenced by the statistical description, assuming a deviation from the usual Bose-Einstein, Fermi-Dirac and Boltzmann-Gibbs distribution functions. These deviations represent the possible appearance of non-extensive effects related with the existence of long range forces, memory effects, or evolution in fractal or multi-fractal space. In the early universe, it is usually assumed that the distribution functions are the standard ones. Then, considering the evolution in a larger theoretical framework will allow to test this assumption and to…
A zero mode, or floppy mode, is a non-trivial coupling of mechanical components yielding a degree of freedom with no resistance to deformation. Engineered zero modes have the potential to act as microscopic motors or memory devices, but this requires an internal actuation mechanism that can overcome unwanted fluctuations in other modes and the dissipation inherent in real systems. In this work, we show theoretically and experimentally that complex zero modes in mechanical networks can be selectively mobilized by non-equilibrium activity. We find that a correlated active bath actuates an infini…
