Foundations Of Statistical Physics Research Articles

Entropy, irreversibility and inference at the foundations of statistical physics

Entropy, irreversibility and inference at the foundations of statistical physics

The Stokes-Einstein-Sutherland Equationat the Nanoscale Revisited.

The Stokes-Einstein-Sutherland (SES) equationis at the foundation of statistical physics, relating a particle's diffusion coefficient and size with the fluid viscosity, temperature, and the boundary condition for the particle-solvent interface. It is assumed that it relies on the separation of scales between the particle and the solvent, hence it is expected to break down for diffusive transport on the molecular scale. This assumption is however challenged by a number of experimental studies showing a remarkably small, if any, violation, while simulations systematically report the opposite. To understand these discrepancies, analytical ultracentrifugation experiments are combined with molecular simulations, both performed at unprecedented accuracies, to study the transport of buckminsterfullerene C60 in toluene at infinite dilution. This system is demonstrated to clearly violate the conditions of slow momentum relaxation. Yet, through a linear response to a constant force, the SES equationcan be recovered in the long time limit with no more than 4% uncertainty both in experiments and in simulations. This nonetheless requires partial slip on the particle interface, extracted consistently from all the data. These results, thus, resolve a long-standing discussion on the validity and limits of the SES equationat the molecularscale.

Foundations of statistical mechanics for unstable interactions.

Traditional Boltzmann-Gibbs statistical mechanics does not apply to systems with unstable interactions, because for such systems the conventional thermodynamic limit does not exist. In unstable systems the ground state energy does not have an additive lower bound, i.e., no lower bound linearly proportional to the number N of particles or degrees of freedom. In this article unstable systems are studied whose ground state energy is bounded below by a regularly varying function of N with index σ≥1. The index σ≥1 of regular variation introduces a classification with respect to stability. Stable interactions correspond to σ=1. A simple example for an unstable system with σ=2 is an ideal gas with a nonvanishing constant two-body potential. The foundations of statistical physics are revisited, and generalized ensembles are introduced for unstable interactions in such a way that the thermodynamic limit exists. The extended ensembles are derived by identifying and postulating three basic properties as extended foundations for statistical mechanics: first, extensivity of thermodynamic systems, second, divisibility of equilibrium states, and third, statistical independence of isolated systems. The traditional Boltzmann-Gibbs postulate, resp. the hypothesis of equal a priori probabilities, is identified as a special case of the extended ensembles. Systems with unstable interactions are found to be thermodynamically normal and extensive. The formalism is applied to ideal gases with constant many-body potentials. The results show that, contrary to claims in the literature, stability of the interaction is not a necessary condition for the existence of a thermodynamic limit. As a second example the formalism is applied to the Curie-Weiss-Ising model with strong coupling. This model has index of stability σ=2. Its thermodynamic potentials [originally obtained in R. Hilfer, Physica A 320, 429 (2003)10.1016/S0378-4371(02)01585-6] are confirmed up to a trivial energy shift. The strong coupling model shows a thermodynamic phase transition of order 1 representing a novel mean-field universality class. The disordered high temperature phase collapses into the ground state of the system. The metastable extension of the high temperature free energy to low temperatures ends at absolute zero in a phase transition of order 1/2. Between absolute zero and the critical temperature of the first order transition all fluctuations are absent.

A Phenomenological Epidemic Model Based On the Spatio-Temporal Evolution of a Gaussian Probability Density Function

A novel phenomenological epidemic model is proposed to characterize the state of infectious diseases and predict their behaviors. This model is given by a new stochastic partial differential equation that is derived from foundations of statistical physics. The analytical solution of this equation describes the spatio-temporal evolution of a Gaussian probability density function. Our proposal can be applied to several epidemic variables such as infected, deaths, or admitted-to-the-Intensive Care Unit (ICU). To measure model performance, we quantify the error of the model fit to real time-series datasets and generate forecasts for all the phases of the COVID-19, Ebola, and Zika epidemics. All parameters and model uncertainties are numerically quantified. The new model is compared with other phenomenological models such as Logistic Grow, Original, and Generalized Richards Growth models. When the models are used to describe epidemic trajectories that register infected individuals, this comparison shows that the median RMSE error and standard deviation of the residuals of the new model fit to the data are lower than the best of these growing models by, on average, 19.6% and 35.7%, respectively. Using three forecasting experiments for the COVID-19 outbreak, the median RMSE error and standard deviation of residuals are improved by the performance of our model, on average by 31.0% and 27.9%, respectively, concerning the best performance of the growth models.

Lee-Yang Zeros and Critical Times in Decoherence of a Probe Spin Coupled to a Bath

Phase transitions are not usually seen in the time domain. Here, we report on the finding of critical times at which a physical observable, in the thermodynamic limit, becomes nonanalytic as a function of time. We find that the coherence of a probe spin coupled to a many-body system vanishes at times in one-to-one correspondence to the Lee-Yang zeros of the partition function of the many-body system. In the thermodynamic limit, the Lee-Yang zeros form a continuum cut in the complex plane of fugacity and the probe spin coherence presents sudden death and birth at the critical times corresponding to the Yang-Lee singularities. These results provide new experimental possibilities in many-body physics.

Analysis on theory of selective model

Selective model provided a way to explain the generation of index law and power law on the basis of preferential probability analysis. A transformation formula between the selective scope rule and selective distribution was deduced through the research on preferential trials. It was named as preferential transformation, which offered a mean to study the rule of monotone distributions. A corresponding continuous transformation can be deduced by the preferential transformation for analyzing probability density function. Compared with the discrete distribution, we defined a comparison function for continuous distribution, and pointed out the relationship between the discrete distribution and continuous distribution. Through selective model, we simulated and analyzed the power law phenomena. Selective theory was important to the research on huge complex system and social behavior. It could also expand the foundation of statistical physics and probability theory.

Caratheodory and the Foundations of Thermodynamics and Statistical Physics

Constantin Caratheodory offered the first systematic and contradiction free formulation of thermodynamics on the basis of his mathematical work on Pfaff forms. Moreover, his work on measure theory provided the basis for later improved formulations of thermodynamics and physics of continua where extensive variables are measures and intensive variables are densities. Caratheodory was the first to see that measure theory and not topology is the natural tool to understand the difficulties (ergodicity, approach to equilibrium, irreversibility) in the Foundations of Statistical Physics. He gave a measure-theoretic proof of Poincare's recurrence theorem in 1919. This work paved the way for Birkhoff to identify later ergodicity as metric transitivity and for Koopman and von Neumann to introduce spectral analysis of dynamical systems in Hilbert spaces. Mixing provided an explanation of the approach to equilibrium but not of irreversibility. The recent extension of spectral theory of dynamical systems to locally convex spaces, achieved by the Brussels–Austin groups, gives new nontrivial time asymmetric spectral decompositions for unstable and/or non-integrable systems. In this way irreversibility is resolved in a natural way.

SCALING THEORY AND THE CLASSIFICATION OF PHASE TRANSITIONS

The recent classification theory for phase transitions (R. Hilfer, Physica Scripta 44, 321 (1991)) and its relation with the foundations of statistical physics is reviewed. First it is outlined how Ehrenfests classification scheme can be generalized into a general thermodynamic classification theory for phase transitions. The classification theory implies scaling and multiscaling thereby eliminating the need to postulate the scaling hypothesis as a fourth law of thermodynamics. The new classification has also led to the discovery and distinction of nonequilibrium transitions within equilibrium statistical physics. Nonequilibrium phase transitions are distinguished from equilibrium transitions by orders less than unity and by the fact that equilibrium thermodynamics and statistical mechanics become inapplicable at the critical point. The latter fact requires a change in the Gibbs assumption underlying the canonical and grandcanonical ensembles in order to recover the thermodynamic description in the critical limit.

Works on the Foundations of Statistical Physics.

Initially published in Moscow in 1950 following the author's death, this book contains the first chapters of a large monograph Krylov planned entitled foundations of physical statistics, his doctoral thesis on The processes of relaxation of statistical systems and the criterion of mechanical instability, and a small paper entitled On the description of exhaustively complete experiments.Originally published in 1980.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Book reviews

Abstract Abstract Microwaves. By K. C. Gupta. (Wiley Eastern Ltd, 1979.) [Pp. xii + 242.] £3·50. Level: Honours/postgraduate. Nuclear Spectroscopy of Fission Products. Conference Series No. 51 Edited by T. van Egidy. (The Institute of Physics, 1980.) [Pp. x + 334.] £29·00; $75·00. Scope: Experimental and theoretical nuclear spectroscopy. Level: Specialist. Physics of Modern Materials. Volume 2. Lectures presented at an International Course organized by the International Centre for Theoretical Physics, Trieste. (International Atomic Energy Authority, 1980.) [Pp. 690.] Austrian Schillings, 980. Scope: Treatise/Textbook; Survey. Level; Undergraduate. Electron Microscopy and Analysis, 1979. Conference Series No. 52. Edited by T. Mulvey. (The Institute of Physics, 1980.) [Pp. xvi + 472.] £23·00; $57·50. Scope: Conference proceedings. Level: Postgraduate, but of interest to others than specialists in the field. Introduction to Solid State Electronics. By F. F. Y. Wang. (North Holland, 1980.) [Pp. xii + 265.] Level: Undergraduate, postgraduate. Thermal Expansion of Solids. By R. S. Krishnan, R. Srinvasan and S. Devanarayanan. (Pergamon Press, 1979.) [Pp. x + 305.] £15·00. Scope: Textbook/Survey. Level: Specialist/Postgraduate. Non-Linear Optics of Free Atoms and Molecules. By D. C. Hanna, M. A. Yuratich and D. Cotter. (Springer-Verlag, 1980.) [Pp. ix + 351.] DM79; $44·30. Molecular Beam Epitaxy. Edited by B. R. Pamplin. (Pergamon Press, 1980.) [Pp. v + 174.] £15·75. The Monte Carlo Methods in Atmospheric Optics. By G. I. Marchuk, G. A. Mikhailov, M. A. Nazarliev, R. A. Darbinjan, B. A. Kargin and B. S. Elepov. (Springer-Verlag, 1980.) [Pp. viii + 208.] DM59; $34·90 approx. Scope: Computational methods and results. Level: Very advanced. Physics of the Expanding Universe. Edited by M. Demianski. (Springer-Verlag, 1979.) [Pp. viii + 332.] DM 48; $26·90. Cosmic X-ray Astronomy. By D. J. Adams. (Adam Hilger, 1980.) [Pp. x + 150.] £13·50. Scope: Introductory text. Level: Postgraduate, undergraduate, serious amateurs. Charge-coupled Devices. Edited by D. F. Barbe. (Springer-Verlag, 1980.) [Pp. xi + 180.] DM69; $40·80. Modern Trends in the Theory of Condensed Matter. Edited by A. Pekalski and J. A. Przystawa. (Springer-Verlag, 1980.) [Pp. ix + 597.] DM68; $40·20. Scope: Libraries. Level: Specialists (Postgraduate upwards). Holography and Coherent Optics. By L. M. Soroko. (Plenum Press, 1980.) [Pp. xvi + 818.] $59·50. Level: Postgraduate. Computer Processing of Electron Microscope Images. Edited by P. W. Hawkes. (Springer-Verlag, 1980.) [Pp. xiv + 296.] DM 65; $36·50. Scope: State of the art review. Level: Postgraduate; research. The Earth's Variable Rotation. By Kurt Lambeck. (Cambridge University Press, 1980.) [Pp. xi + 449.] £35. Scope: Treatise; Survey—for the library. Level: Postgraduate. The Microprocessor and its Applications: An Advanced Course. Edited by D. Aspinall. (Cambridge University Press, 1980.) [Pp. xviii + 383.] £6·95. Polar Dielectrics and their Applications. By J. C. Burfoot and G. W. Taylor. (Macmillan, 1979.) [Pp. viii + 465.] £20. Models of High-Energy Processes. By J. C. Polkinghorne. (Cambridge University Press, 1980.) [Pp. xiv + 129.] £11·00. Works on the Foundations of Statistical Physics. By N. S. Krylov. (Princeton University Press, 1980.) [Pp. xxviii + 283.] £11·80; paper, £4·50.