Methods Of Statistical Physics Research Articles

Concepts on statistical physics - fluctuations in equilibrium

This essay corresponds to the content of three lectures about statistical physics delivered to the audience of the 2014 section of the R. A. Salmeron School of Physics, at the UnB. Our starting point was very simple statistical models (lattice gas, spin-1/2 ferromagnet), used as illustrations of the competencies and methods in statistical physics. Thus we introduce the Gibbs ensembles, defining a connection with thermodynamics and discussing the role played by fluctuations and large numbers. We present phenomenological aspects of phase transitions and critical phenomena in simple fluids and in uniaxial ferromagnets, emphasizing the universal character of the critical exponents. We describe the phenomenological van der Waals and Curie-Weiss theories and the Landau expansion, which are present-day relevant methods, despite the fact that such theories give rise to critical exponents in disagreement with experiments. We present then the paradigmatic Ising model, which points us to a way to overcome the phenomenological results. A brief presentation of the scale phenomenological methods and the contemporaneous renormalization group are considered at the end of these lectures.

Analysis of propagation dynamics in complex dynamical network based on disturbance propagation model

The paper regards the complex dynamical network (CDN) as a static network with temporal characteristics so as to consider its dynamic behavior. The influence factor and dynamics laws in CDN are explored by using the methods of simulation and statistical physics. It is found that both the dynamic behavior of the nodes and the number of initial disturbed nodes can accelerate the failure propagation and enlarge the influence scope. There is a phase transformation point of the repair ratio with the increase of repair time τ for the homogeneous network (the value of τ for all the nodes is the same). Beyond the threshold, the entire network will collapse; otherwise, the network can be back to normal after a period of time. There is an inflection point of the repair rate with the simulation step and the recovery rate in heterogeneous network is much slower than that in homogeneous network. The results can provide technical support for the fault management and network optimization in CDN which will promote the development of corresponding theory in CDN.

Bethe free-energy approximations for disordered quantum systems.

Given a locally consistent set of reduced density matrices, we construct approximate density matrices which are globally consistent with the local density matrices we started from when the trial density matrix has a tree structure. We employ the cavity method of statistical physics to find the optimal density matrix representation by slowly decreasing the temperature in an annealing algorithm, or by minimizing an approximate Bethe free energy depending on the reduced density matrices and some cavity messages originated from the Bethe approximation of the entropy. We obtain the classical Bethe expression for the entropy within a naive (mean-field) approximation of the cavity messages, which is expected to work well at high temperatures. In the next order of the approximation, we obtain another expression for the Bethe entropy depending only on the diagonal elements of the reduced density matrices. In principle, we can improve the entropy approximation by considering more accurate cavity messages in the Bethe approximation of the entropy. We compare the annealing algorithm and the naive approximation of the Bethe entropy with exact and approximate numerical simulations for small and large samples of the random transverse Ising model on random regular graphs.

On some thermodynamic effects in small molecular systems

The behavior of a gas-liquid system in a small volume has been studied. In the absence of the thermodynamic limit, the problem has been examined by methods of statistical physics. This allows the use of new mathematical methods previously applied in nuclear physics. Within the modified droplet model of a real gas sandwiched between two extended plates, an additional component of the pressure, which was previously discussed and exhibits characteristic oscillations at change in the volume, has been found. Qualitative coincidence with the previously reported molecular dynamics simulations has been found.

Gibbs random fields on a lattice: Definitions, existence, uniqueness, and phase transitions (a review of proceedings of the seminar on statistical physics at the faculty of mechanics and mathematics of the Moscow state university in 1962–1994)

The Moscow Grand Seminar on Statistical Physics is one of the first seminars in the world devoted to rigorous methods in statistical physics. The achievements of the seminar can be appreciated in full measure from publications by the leaders of the seminar (R.L. Dobrushin, V.A. Malyshev, R.A. Minlos, Ya.G. Sinai). This paper presents a review (far from complete) of the proceedings of the seminar. The emphasis is made on the fundamental definitions and most important results.

Remarks on mechanical approach to observable Universe

We consider the Universe deep inside the cell of uniformity. At these scales, the Universe is filled with inhomogeneously distributed discrete structures (galaxies, groups and clusters of galaxies), which perturb the background Friedmann model. Here, the mechanical approach (Eingorn & Zhuk, 2012) is the most appropriate to describe the dynamics of the inhomogeneities which is defined, on the one hand, by gravitational potentials of inhomogeneities and, on the other hand, by the cosmological expansion of the Universe. In this paper, we present additional arguments in favor of this approach. First, we estimate the size of the cell of uniformity. With the help of the standard methods of statistical physics and for the galaxies of the type of the Milky Way and Andromeda, we get that it is of the order of 190 Mpc which is rather close to observations. Then, we show that the nonrelativistic approximation (with respect to the peculiar velocities) is valid for z≲10, i.e. approximately for 13 billion years from the present moment. We consider scalar perturbations and, within the ΛCDM model, justify the main equations. Moreover, we demonstrate that radiation can be naturally incorporated into our scheme. This emphasizes the viability of our approach. This approach gives a possibility to analyze different cosmological models and compare them with the observable Universe. For example, we indicate some problematic aspects of the spatially flat models. Such models require a rather specific distribution of the inhomogeneities to get a finite potential at any points outside gravitating masses. We also criticize the application of the Schwarzschild-de Sitter solution to the description of the motion of test bodies on the cosmological background.

Historical Review on Analytic, Monte Carlo, and Renormalization Group Approaches to Critical Phenomena of Some Lattice Models

Analytic calculations, Monte Carlo (MC) simulations [N. Metropolis and S. Ulam, J. Am. Stat. Asso 44, 335 (1949)] and renormalization group (RG) theory [K. G. Wilson, Phys. Rev. B 4, 3174 and 3184 (1971)] are important methods of statistical physics to study phase transitions and critical phenomena. This paper gives a brief historical review on analytic, MC, and RG approaches to some lattice critical systems in memory of late Professors Shang-keng Ma (1940-1983) and Yu-Ming Shih (1942-2005), who played some key roles in early developments of statistical physics of critical phenomena in Taiwan. The paper first introduces some developments in the study of critical phenomena from late 19th century to early 70s in the 20th century when renormalization group theory was formulated. The paper then reviews some topics on analytic, Monte Carlo, and RG approach to some lattice models, including variational RG approach to the Ising model, slow relaxation of a spin glass model at low temperatures, iterative method for quantum spin models, percolation theory of critical phenomena of Ising-like models and hard-core particle model, histogram Monte Carlo method and histogram Monte Carlo RG method, universal finite-size scaling functions of lattice models, critical slowing down of the Ising model, finite-size corrections for critical lattice models, and self-organized critical systems. Such topics are closely related to the author's own research experience. Finally, the paper outlines some very recent developments, summarizes main results of this paper, and points out some interesting problems for further studies, e.g. why a biological system can maintain in a non-equilibrium state for a very long time.

The role of fluctuating modes of autocorrelation in crude oil prices

Autocorrelation exists in the crude oil price due to price inertia, the cobweb theorem, model errors, etc. Many researchers have studied the fluctuation of the crude oil price, but few have focused on the autocorrelation fluctuation in crude oil prices. Exploring the fluctuating rules of autocorrelation can aid in understanding the fluctuating mechanism of crude oil prices. To study the role of fluctuating modes of autocorrelation in crude oil prices, which have time series characteristics, this study selected international crude oil spot prices as sample data to employ the methods of statistical physics. The fluctuating modes of autocorrelation were defined by the autocorrelation coefficient, symbolization, and a coarse-graining process. We set the modes as nodes and the transformation between modes as edges; the fluctuating mode weight network of autocorrelation was then built. Thus, the study of autocorrelation fluctuation was transformed to a network study. Then, certain aspects, such as the statistical properties, the “small-world” behavior, and the transmission medium in the network, could be analyzed using complex network theory and analytical methods. The periodicity of the fluctuation was calculated using a spectral analysis method. This study not only describes the fluctuation of the time series more precisely than other methods but also provides ideas for methods of studying the fluctuation of univariate autocorrelations.

Probing the Statistical Properties of Unknown Texts: Application to the Voynich Manuscript

While the use of statistical physics methods to analyze large corpora has been useful to unveil many patterns in texts, no comprehensive investigation has been performed on the interdependence between syntactic and semantic factors. In this study we propose a framework for determining whether a text (e.g., written in an unknown alphabet) is compatible with a natural language and to which language it could belong. The approach is based on three types of statistical measurements, i.e. obtained from first-order statistics of word properties in a text, from the topology of complex networks representing texts, and from intermittency concepts where text is treated as a time series. Comparative experiments were performed with the New Testament in 15 different languages and with distinct books in English and Portuguese in order to quantify the dependency of the different measurements on the language and on the story being told in the book. The metrics found to be informative in distinguishing real texts from their shuffled versions include assortativity, degree and selectivity of words. As an illustration, we analyze an undeciphered medieval manuscript known as the Voynich Manuscript. We show that it is mostly compatible with natural languages and incompatible with random texts. We also obtain candidates for keywords of the Voynich Manuscript which could be helpful in the effort of deciphering it. Because we were able to identify statistical measurements that are more dependent on the syntax than on the semantics, the framework may also serve for text analysis in language-dependent applications.

THERMAL RELAXATION IN MAGNETIC MATERIALS: A SURVEY

This paper presents a survey of the methods of statistical physics which are applied to the problem of thermal agitation in magnetic materials. The main focus of the work is the stochastic dynamics described by the Landau-Lifshitz-Gilbert equation for which most analytic results are known, and which has been most commonly used in numerical simulations. We also present the much more recent Landau–Lifshitz–Bloch equation and the numerical calculations describing magnetization dynamics close to the Curie point. The paper is concluded by a description of the newly introduced jump-noise and of the Barkhausen jumps.

Low-temperature excitations within the Bethe approximation

We propose the variational quantum cavity method to construct a minimal energy subspace of wavevectors that are used to obtain some upper bounds for the energy cost of the low-temperature excitations. Given a trial wavefunction we use the cavity method of statistical physics to estimate the Hamiltonian expectation and to find the optimal variational parameters in the subspace of wavevectors orthogonal to the lower-energy wavefunctions. To this end, we write the overlap between two wavefunctions within the Bethe approximation, which allows us to replace the global orthogonality constraint with some local constraints on the variational parameters. The method is applied to the transverse Ising model and different levels of approximations are compared with the exact numerical solutions for small systems.

Evolutionary dynamics of group interactions on structured populations: a review.

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.

Applying modern methods of statistical physics to describe fluctuations of soil moisture reserve

The application of methods of modern theory of Brownian motion to the calculation of fluctua� tions of soil moisture reserve is considered. Analytical dependences of the correlation time of soil moisture reserve fluctuations and its mean on the characteristics of external stochastic and deterministic impacts are obtained for a nonlinear hydrological model of moisture reserve formation based on Langevin equation and generalized Langevin relationships. Variations of the lowfrequency part of the spectrum of fast synoptic vari� ables (the difference between precipitation and evaporation) because of interaction with moisture reserve fluctuations is examined. A new effect—an increase in the spectrum in the intermediate domain of time intervals between characteristic times—is explained by the time of variations of synoptic fluctuations and the time of variations of fluctuations of soil moisture reserve. The problem of determining a nonsteady response of the mean soil moisture reserve to fluctuation of moistening regime is used to demonstrate the potentialities of a new apparatus of statistical physics—fluctuation theorems.

Crises and Collective Socio-Economic Phenomena: Simple Models and Challenges

Financial and economic history is strewn with bubbles and crashes, booms and busts, crises and upheavals of all sorts. Understanding the origin of these events is arguably one of the most important problems in economic theory. In this paper, we review recent efforts to include heterogeneities and interactions in models of decision. We argue that the Random Field Ising model (RFIM) indeed provides a unifying framework to account for many collective socio-economic phenomena that lead to sudden ruptures and crises. We discuss different models that can capture potentially destabilising self-referential feedback loops, induced either by herding, i.e. reference to peers, or trending, i.e. reference to the past, and account for some of the phenomenology missing in the standard models. We discuss some empirically testable predictions of these models, for example robust signatures of RFIM-like herding effects, or the logarithmic decay of spatial correlations of voting patterns. One of the most striking result, inspired by statistical physics methods, is that Adam Smith's invisible hand can badly fail at solving simple coordination problems. We also insist on the issue of time-scales, that can be extremely long in some cases, and prevent socially optimal equilibria to be reached. As a theoretical challenge, the study of so-called "detailed-balance" violating decision rules is needed to decide whether conclusions based on current models (that all assume detailed-balance) are indeed robust and generic.

Collisionless kinetic theory of rolling molecules

We derive a collisionless kinetic theory for an ensemble of molecules undergoing nonholonomic rolling dynamics. We demonstrate that the existence of nonholonomic constraints leads to problems in generalizing the standard methods of statistical physics. In particular, we show that even though the energy of the system is conserved, and the system is closed in the thermodynamic sense, some fundamental features of statistical physics such as invariant measure do not hold for such nonholonomic systems.Nevertheless, we are able to construct a consistent kinetic theory using Hamilton's variational principle in Lagrangian variables, by regarding the kinetic solution as being concentrated on the constraint distribution. A cold fluid closure for the kinetic system is also presented, along with a particular class of exact solutions of the kinetic equations.

Percolation in an ultrametric space

We study percolation in the hierarchical lattice of order N where the probability of connection between two points separated by distance k is of the form c k / N k ( 1+δ) , δ>-1. Since the distance is an ultrametric , there are significant differences with percolation in the Euclidean lattice. We consider three regimes: δ 1 , where it does not occur and δ=1 which is the critical case corresponding to the phase transition. In the critical case we use an approach in the spirit of the renormalization group method of statistical physics, and connectivity results of Erdős-Renyi random graphs play a key role. We find sufficient conditions on c k such that percolation occurs, or that it does not occur. An intermediate situation called pre-percolation, which is necessary for percolation, is also considered. In the cases of percolation we prove uniqueness of the constructed percolation clusters. In a previous paper we studied percolation in the N→∞ limit (mean field percolation), which provided a simplification that allowed finding a necessary and sufficient condition for percolation. For fixed N there are open questions, in particular regarding the behaviour at the critical values of parameters in the definition of c k . Those questions suggest the need to study ultrametric random graphs .

Spectral properties of the Jacobi ensembles via the Coulomb gas approach

Using the Coulomb gas method and standard methods of statistical physics, we compute analytically the joint cumulative probability distribution of the extreme eigenvalues of the Jacobi–MANOVA ensemble of random matrices in the limit of large matrices. This allows us to derive the rate functions for the large fluctuations to the left and the right of the expected values of the smallest and largest eigenvalues analytically. Our findings are compared with some available known exact results as well as with numerical simulations finding good agreement.

From the Newton version of the Kolmogorov–Arnol’d–Moser theory to a weakly dissipative version

In this paper, we present results of the numerical study of statistical macroscopic parameters in dynamics of a test particle (or ensembles of particles) in the potential field of an anti-harmonic oscillator with weakly dissipative perturbation starting from the Bogdanov–Tackens bifurcation. This is the most well-studied example of perturbation of Hamiltonian dynamics in the class of all (smooth) dynamical systems. It is possible to characterize stochastic and regular regimes of the test particle dynamics and also interaction of particles by using methods of statistical mechanics and statistical physics.

On the use of topological features and hierarchical characterization for disambiguating names in collaborative networks

Many features of complex systems can now be unveiled by applying statistical physics methods to treat them as social networks. The power of the analysis may be limited, however, by the presence of ambiguity in names, e.g., caused by homonymy in collaborative networks. In this paper we show that the ability to distinguish between homonymous authors is enhanced when longer-distance connections are considered, rather than looking at only the immediate neighbors of a node in the collaborative network. Optimized results were obtained upon using the 3rd hierarchy in connections. Furthermore, reasonable distinction among authors could also be achieved upon using pattern recognition strategies for the data generated from the topology of the collaborative network. These results were obtained with a network from papers in the arXiv repository, into which homonymy was deliberately introduced to test the methods with a controlled, reliable dataset. In all cases, several methods of supervised and unsupervised machine learning were used, leading to the same overall results. The suitability of using deeper hierarchies and network topology was confirmed with a real database of movie actors, with the additional finding that the distinguishing ability can be further enhanced by combining topology features and long-range connections in the collaborative network.

Web Search Queries Can Predict Stock Market Volumes

We live in a computerized and networked society where many of our actions leave a digital trace and affect other people’s actions. This has lead to the emergence of a new data-driven research field: mathematical methods of computer science, statistical physics and sociometry provide insights on a wide range of disciplines ranging from social science to human mobility. A recent important discovery is that search engine traffic (i.e., the number of requests submitted by users to search engines on the www) can be used to track and, in some cases, to anticipate the dynamics of social phenomena. Successful examples include unemployment levels, car and home sales, and epidemics spreading. Few recent works applied this approach to stock prices and market sentiment. However, it remains unclear if trends in financial markets can be anticipated by the collective wisdom of on-line users on the web. Here we show that daily trading volumes of stocks traded in NASDAQ-100 are correlated with daily volumes of queries related to the same stocks. In particular, query volumes anticipate in many cases peaks of trading by one day or more. Our analysis is carried out on a unique dataset of queries, submitted to an important web search engine, which enable us to investigate also the user behavior. We show that the query volume dynamics emerges from the collective but seemingly uncoordinated activity of many users. These findings contribute to the debate on the identification of early warnings of financial systemic risk, based on the activity of users of the www.