Problems In Statistical Physics Research Articles

Boxed plane partitions as an exactly solvable boson model

Plane partitions naturally appear in many problems of statistical physics and quantum field theory, for instance, in the theory of faceted crystals and of topological strings on Calabi-Yau threefolds. In this paper a connection is made between the exactly solvable model with the boson dynamical variables and a problem of enumeration of boxed plane partitions - three dimensional Young diagrams placed into a box of a finite size. The correlation functions of the boson model may be considered as the generating functionals of the Young diagrams with the fixed heights of its certain columns. The evaluation of the correlation functions is based on the Yang-Baxter algebra. The analytical answers are obtained in terms of determinants and they can also be expressed through the Schur functions.

Constructing Free-Energy Approximations and Generalized Belief Propagation Algorithms

Important inference problems in statistical physics, computer vision, error-correcting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree, but only approximate when the factor graph has cycles. We show that BP fixed points correspond to the stationary points of the approximation of the free energy for a factor graph. We explain how to obtain region-based free energy approximations that improve the approximation, and corresponding generalized belief propagation (GBP) algorithms. We emphasize the conditions a free energy approximation must satisfy in order to be a or maxent-normal approximation. We describe the relationship between four different methods that can be used to generate valid approximations: the Bethe method, the junction graph method, the cluster variation method, and the region graph method. Finally, we explain how to tell whether a region-based approximation, and its corresponding GBP algorithm, is likely to be accurate, and describe empirical results showing that GBP can significantly outperform BP.

A dynamical evolution model on the black-hole horizon

This paper demonstrates a dynamical evolution model of the black-hole (BH) horizon. The result indicates that a kinetic area-cell model of the BH horizon can model the evolution of the BH due to the Hawking radiation, and this area-cell system can be considered as an interacting geometrical particle system. Thus, the evolution turns into a problem of statistical physics. In the present work, this problem is treated in the framework of non-equilibrium statistics. It is proposed that each area cell possesses energy like a microscopic black hole and has gravitational interaction with the other area cells. We consider both a non-interaction ideal system and a system with small nearest-neighbour interactions, and obtain an analytic expression of the expected value of the horizon area of a dynamical BH. We find that, after a long enough evolution, a dynamical BH with the Hawking radiation can be in equilibrium with a finite-temperature radiation field. However, we also find that the system has a critical point, and when the temperature of the radiation field surrounding the BH approaches the critical temperature of the BH, a critical slowing-down phenomenon occurs.

Colloids as model systems for problems in statistical physics

Owing to their mesoscopic length scales, colloidal suspensions provide ideal model systems suitable for addressing many problems in the field of statistical physics. Exemplarily, we highlight the versatile nature of such systems by discussing experiments with stochastic resonance and a practical realization of a recently proposed ratchet cellular automaton.

From quantum mechanics to classical statistical physics: Generalized Rokhsar–Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition

Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with stochastic classical systems described by a Master equation of the matrix type, hence their name. It then follows that the equilibrium partition function of the stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.

Statistical physics of grain-boundary engineering

Percolation theory is now standard in the analysis of polycrystalline materials where the grain boundaries can be divided into two distinct classes, namely "good" boundaries that have favorable properties and "bad" boundaries that seriously degrade the material performance. Grain-boundary engineering (GBE) strives to improve material behavior by engineering the volume fraction c and arrangement of good grain boundaries. Two key percolative processes in GBE materials are the onset of percolation of a strongly connected aggregate of grains, and the onset of a connected path of weak grain boundaries. Using realistic polycrystalline microstructures, we find that in two dimensions the threshold for strong aggregate percolation c(SAP) and the threshold for weak boundary percolation c(WBP) are equivalent and have the value c(SAP) = c(WBP) =0.38 (1) , which is slightly higher than the threshold found for regular hexagonal grain structures, c(RH) =2 sin (pi/18) =0.347... . In three dimensions strong aggregate percolation and weak boundary percolation occur at different locations and we find c(SAP) =0.12 (3) and c(WBP) =0.77 (3) . The critical current in high T(c) materials and the cohesive energy in structural systems are related to the critical manifold problem in statistical physics. We develop a theory of critical manifolds in GBE materials, which has three distinct regimes: (i) low concentrations, where random manifold theory applies, (ii) critical concentrations where percolative scaling theory applies, and (iii) high concentrations, c> c(SAP) , where the theory of periodic elastic media applies. Regime (iii) is perhaps most important practically and is characterized by a critical length L(c) , which is the size of cleavage regions on the critical manifold. In the limit of high contrast epsilon-->0 , we find that in two dimensions L(c) proportional, gc/ (1-c) , while in three dimensions L(c) proportional, g exp [ b(0) c/ (1-c) ] / [c (1-c) ](1/2) , where g is the average grain size, epsilon is the ratio of the bonding energy of the weak boundaries to that of the strong boundaries, and b(0) is a constant which is of order 1. Many of the properties of GBE materials can be related to L(c) , which diverges algebraically on approach to c=1 in two dimensions, but diverges exponentially in that limit in three dimensions. We emphasize that GBE percolation processes and critical manifold behavior are very different in two dimensions as compared to three dimensions. For this reason, the use of two dimensional models to understand the behavior of bulk GBE materials can be misleading.

New Directions in Statistical Physics: Econophysics, Bioinformatics, and Pattern Recognition

New Directions in Statistical Physics: Econophysics, Bioinformatics, and Pattern Recognition

Rasprostranenie elektromagnitnykh voln v sluchaino-neodnorodnoi srede kak zadacha statisticheskoi matematicheskoi fiziki

Rasprostranenie elektromagnitnykh voln v sluchaino-neodnorodnoi srede kak zadacha statisticheskoi matematicheskoi fiziki

Fundamental problems in statistical physics of jammed packings

For packed i.e., “jammed”, hard and rough objects kinetic energy is a minor and ignorable quantity, as is elastic strain. Hence in the static case, the stress equations need supplementing by “missing equations” depending solely on configurations. A different pathway of analysis is the calculation of the probability distribution of interparticle forces. This paper presents the mini-review of recently obtained results in this field and poses a number of fundamental problems which are yet to be solved.

Generalized Kramers–Wannier duality for spin systems with non-commutative symmetry

In 1941 H Kramers and G Wannier discovered a special symmetry which relates low-temperature and high-temperature phases in the planar Ising model. The corresponding transformation, the Kramers–Wannier transform, is a special non-local substitution in the partition function. The existence of such transformations is a general property of lattice spin systems. Generalization of the KW transform to spin systems with non-Abelian symmetry is essential for many problems in statistical physics and field theory. This problem is very difficult and cannot be carried out by classical methods (like Fourier transform in the commutative case). We present new results which solve this problem for finite non-Abelian groups.

Experimental Characterization of Transition to Chaos in the Presence of Noise

Transition to chaos in the presence of noise is an important problem in nonlinear and statistical physics. Recently, a scaling law has been theoretically predicted which relates the Lyapunov exponent to the noise variation near the transition. Here we present experimental observation of noise-induced chaos in an electronic circuit and obtain the fundamental scaling law characterizing the transition. The experimentally obtained scaling exponent agrees very well with that predicted by theory.

Deterministic patterns of noise and the control of chaos

Real systems in physics, chemistry and biology are always subject to fluctuations that change qualitatively the systems' dynamics. In particular, rare large fluctuations are responsible for the nucleation at phase transitions, mutations in DNA sequences, protein transport in cells and failure of electronic devices. In many cases of practical interest systems are away from thermal equilibrium, and understanding the fluctuations in such systems is one of the fundamental problems of statistical physics that has challenged researchers for decades. Recent progress in the solution of this problem is closely related to the emerging understanding of patterns of deterministic trajectories underlying non-equilibrium fluctuations. These trajectories correspond to the Hamilton equations of motion written for the asymptotic solution of the Fokker - Planck equation and were often thought of as a mere mathematical abstraction. The possibility of quantitative experiments could not be entertained until the appropriate statistical quantity (prehistory probability distribution) had been introduced. In this paper it is shown how such trajectories can be measured experimentally in a number of systems and how the knowledge of these trajectories can be used to solve long standing problems in the theory of fluctuations and in the control theory.

Nuclear physics and ideas of quantum chaos

The field nowadays called “many-body quantum chaos” was started in 1939 with the article by I.I. Gurevich studying the regularities of nuclear spectra. The field has been extensively developed recently, both mathematically and in application to mesoscopic systems and quantum fields. We argue that nuclear physics and the theory of quantum chaos are mutually beneficial. Many ideas of quantum chaos grew up from the factual material of nuclear physics; this enrichment still continues to take place. On the other hand, many phenomena in nuclear structure and reactions, as well as the general problem of statistical physics of finite strongly interacting systems, can be understood much deeper with the help of ideas and methods borrowed from the field of quantum chaos. A brief review of the selected topics related to the recent development is presented.

Go with the winners: a general Monte Carlo strategy

We describe a general strategy for sampling configurations from a given distribution, not based on the standard Metropolis (Markov chain) strategy. It uses the fact that nontrivial problems in statistical physics are high dimensional and often close to Markovian. Therefore, configurations are built up in many, usually biased, steps. Due to the bias, each configuration carries its weight which changes at every step. If the bias is close to optimal, all weights are similar and importance sampling is perfect. If not, “population control” is applied by cloning/killing partial configurations with too high/low weight. This is done such that the final (weighted) distribution is unbiased. We apply this method (which is also closely related to diffusion type quantum Monte Carlo) to several problems of polymer statistics, reaction-diffusion models, sequence alignment, and percolation.

Statistical physics of the glass phase

This paper, prepared for the proceedings of the ‘Statphys 21’ conference (Cancun, July 2001) gives an introduction to the statistical physics problems which appear in the study of structural glasses. It is a shortened and updated version of a more detailed review paper which has appeared in Mézard, (Ong, Bhatt (Eds.), More is different, Princeton University Press, Princeton, NJ, 2001).

New progress in the principle of nonequilibrium statistical physics

In recent years, a new fundamental equation of nonequilibrium statistical physics was proposed in place of the Liouville equation. That is the anomalous Langevin equation in Γ space or its equivalent Liouville diffusion equation of time-reversal asymmetry. This equation reflects that the form of motion of particles in statistical thermodynamic systems has the drift-diffusion duality and the law of motion of statistical thermodynamics is stochastic in essence, but does not obey the Newton equation of motion, though it is also constrained by dynamics. The stochastic diffusion of the particles is the microscopic origin of macroscopic irreversibility. Starting from this equation, the BBGKY diffusion equation hierarchy was presented, the hydrodynamic equations, such as the generalized Navier-Stokes equation, the mass drift-diffusion equation and the thermal conductivity equation have been derived succinctly. The unified description of all three level equations of microscopic, kinetic and hydrodynamic was completed. Furthermore, a nonlinear evolution equation of Gibbs and Boltzmann nonequilibrium entropy density was constructed, and the existence of entropy diffusion was predicted. The evolution equation shows that the change of nonequilibrium entropy density originates together from drift, diffusion and source production. Entropy production is manifestations of the law of entropy increase. Entropy diffusion governs the approach to equilibrium. All these derivations and results are unified and rigorous from the new fundamental equation without adding any extra assumption. In this review, an overview on the above main ideas, methods and results is given, and the international new progress in related problems of nonequilibrium statistical physics is summarized.

Implications of a Statistical Physics Approach for Earthquake Hazard Assessment and Forecasting

—There is accumulating evidence that distributed seismicity is a problem in statistical physics. Seismicity is taken to be a type example of self-organized criticality. This association has important implications regarding earthquake hazard assessment and forecasting. A characteristic of a thermodynamic system is that it exhibits a background noise that is self-organized. In the case of a dilute gas, this self-organization is the Maxwell–Boltzmann distribution of molecular velocities. In seismicity, it is the Gutenberg–Richter frequency-magnitude scaling; this scaling is fractal. Observations favor the hypothesis that smaller earthquakes in moderate-sized regions occur at rates that are only weakly dependent on time. Thus, the rate of occurrence of smaller earthquakes can be extrapolated to assess the hazard of larger earthquakes in a region. We obtain the rate of occurrence of earthquakes with m > 4 in 1°× 1° areas from the NEIC catalog. Using only this data we produce global maps of the seismic hazard. Observations also favor the hypothesis that the stress level at which an earthquake occurs is a second-order critical point. As a critical point is approached, correlations extend over increasingly larger distances. In terms of seismicity, the approach to a critical point is associated with an increase in the rate of occurrence of intermediate-sized earthquakes prior to a large earthquake. This precursory activation has been shown to exhibit power-law scaling and to occur over a region about ten times larger than the rupture length of the large earthquake. Analyses of the spinoidal behavior associated with second-order critical points predict the power-law increase in seismic activity prior to a characteristic earthquake. This precursory activation provides the basis for intermediate-range earthquake forecasting.

Persistence exponents of non-Gaussian processes in statistical mechanics

Motivated by certain problems of statistical physics we consider a stationary stochastic process in which deterministic evolution is interrupted at random times by upward jumps of a fixed size. If the evolution consists of linear decay, the sample functions are of the "random sawtooth" type and the level dependent persistence exponent \theta can be calculated exactly. We then develop an expansion method valid for small curvature of the deterministic curve. The curvature parameter g plays the role of the coupling constant of an interacting particle system. The leading order curvature correction to \theta is proportional to g^{2/3}. The expansion applies in particular to exponential decay in the limit of large level, where the curvature correction considerably improves the linear approximation. The Langevin equation, with Gaussian white noise, is recovered as a singular limiting case.

Computation of functional integrals in problems of quantum and statistical physics

A method for computation of characteristics in quantum and statistical physics is described. This method is based on the representation of the matrix element of the evolution operator in Euclidean metrics in the form of a functional integral with a certain measure in the corresponding space and on the use of approximation formulas which we constructed for this kind of integral. No simplifying assumptions such as semiclassical approximation, no introduction of “short-time” propagators are necessary. The method is applied to the numerical study of some potential nuclear models and to investigation of the topological effects in gauge theories. The comparison of results with experimental data and with the values obtained by other authors are presented.

Kinetics of two-dimensional ballistic annihilation: A multiparticle lattice gas study

In many equilibrium or nonequilibrium statistical physics problems, fluctuations play a crucial role. Often those problems are too complex to be solved analytically. Accordingly, numerical algorithms keeping track of the fluctuations are needed. Cellular automata (CA) and Lattice Boltzmann (LB) models are two possible approaches to simulate complex systems. CA models keep track of many-body correlations and provide a description of the fluctuations. However, they lead to a noisy dynamics and impose a restriction on the possible values of the viscosity. On the other hand, LB models are numerically more efficient and offer much more flexibility to adjust the physical parameters, but they neglect the fluctuations. We have developed a new multiparticle lattice model which reconciles both approaches. The main characteristics of this approach are explained, and our model is used to study the kinetics of two-dimensional ballistic annihilation.