Statistical Physics Systems Research Articles

Large-W limit of the knapsack problem.

We formulate the knapsack problem (KP) as a statistical physics system and compute the corresponding partition function as an integral in the complex plane. The introduced formalism allows us to derive three statistical-physics-based algorithms for the KP: one based on the recursive definition of the exact partition function, another based on the large weight limit of that partition function, and a final one based on the zero-temperature limit of the second. Comparing the performances of the algorithms, we find that they do not consistently outperform (in terms of runtime and accuracy) dynamic programming, annealing, or standard greedy algorithms. However, the exact partition function is shown to reproduce the dynamic programming solution to the KP, and the zero-temperature algorithm is shown to produce a greedy solution. Therefore, although dynamic programming and greedy solutions to the KP are conceptually distinct, a statistical physics formalism introduced reveals that the large weight-constraint limit of the former leads to the latter. We conclude by discussing how to extend this formalism in order to obtain more accurate versions of the introduced algorithms and other similar combinatorial optimization problems.

Financial price dynamics and phase transitions in the stock markets.

Price dynamics in stock market is modelled by a statistical physics systems: Ising model. A comparative analysis of the historical dynamics of stock returns between the US, UK, and French markets is given. Since the Ising model requires binary inputs, the effect of binarization is studied. Then, using the TAP approximation method, external fields and coupling strengths are calculated. The fluctuation cycles of coupling strengths have a remarkable corresponding relationship with the important period of the financial market. The highlight of this paper is to verify the phase transition can also occur in the stock market and it reveals the transformation of the market state. The numerical solution in this paper is consistent with the exact solution obtained by Lars Onsager. Our findings can help to discover the economic cycles and provide more possibilities for studying financial markets using physical models.

Weighted averages in population annealing: Analysis and general framework.

Population annealing is a powerful sequential Monte Carlo algorithm designed to study the equilibrium behavior of general systems in statistical physics through massive parallelism. In addition to the remarkable scaling capabilities of the method, it allows for measurements to be enhanced by weighted averaging [J. Machta, Phys. Rev. E 82, 026704 (2010)1539-375510.1103/PhysRevE.82.026704], admitting to reduce both systematic and statistical errors based on independently repeated simulations. We give a self-contained introduction to population annealing with weighted averaging, generalize the method to a wide range of observables such as the specific heat and magnetic susceptibility and rigorously prove that the resulting estimators for finite systems are asymptotically unbiased for essentially arbitrary target distributions. Numerical results based on more than 10^{7} independent population annealing runs of the two-dimensional Ising ferromagnet and the Edwards-Anderson Ising spin glass are presented in depth. In the latter case, we also discuss efficient ways of measuring spin overlaps in population annealing simulations.

Comparison of the magneto-optical properties of the semi-parabolic well with those of the parabolic and rectangular wells under the combined influences of aluminum concentration and hydrostatic pressure

In this paper, we report the comparison of the magneto-optical transport properties of the semi-parabolic potential quantum well with those of the parabolic and rectangular potential quantum wells under the combined influences of the aluminum concentration, the temperature, the hydrostatic pressure, and the Landau levels by using the methods of the quantum field theory for many-particle systems in statistical physics and the profile to calculate analytically the conductivity tensor expression, the magneto-optical absorption power expression of the optically detected magneto-phonon resonance (ODMPR) oscillation, and the full width at half maximum (FWHM) of the ODMPR peak, respectively, in quantum wells with three above different confining potential shapes for typical GaAs/AlxGa1−xAs material. The results obtained from the present study show that (i) the ODMPR peak position undergoes a red-shift as the aluminum concentration and the hydrostatic pressure increase while it experiences a blue-shift as the temperature augments for all three above confining potential shapes; (ii) the FWHM as functions of the barrier's Al-concentration, the system's temperature, and the hydrostatic pressure for all three confining potential shapes is presented to compare; (iii) the dependence of the FWHM on the barrier's Al-concentration and the hydrostatic pressure for the Landau levels for three confining potential shapes also is taken into account in detail; (iv) our present theoretical calculations are found to be consistent with previous experimental calculations.

Lack of phase transitions in staggered magnetic systems. A comparison of uniqueness criteria

We study a ferromagnetic Ising model with a staggered cell-board magnetic field previously proposed for image processing [Maruani et al., Markov Processes Relat. Fields 1, 419–442 (1995)]. We complement previous results on the existence of phase transitions at low temperature [González-Navarrete et al., J. Stat. Phys. 162, 139–161 (2016)] by determining bounds to the region of uniqueness of Gibbs measures. We establish sufficient rigorous uniqueness conditions derived from three different criteria: (1) Dobrushin criterion [R. Dobrushin, Theory Probab. Appl. 13, 197–224 (1968)], (2) disagreement percolation [J. van den Berg and C. Maes, Ann. Probab. 22, 749–763 (1994)], and (3) Dobrushin–Shlosman criteria [R. Dobrushin and S. Shlosman, in Statistical Physics and Dynamical Systems: Rigorous Results, edited by J. Fritz, A. Jaffe, and D. Szasz (Birkhauser, Basel, 1985)]. These conditions are subsequently solved numerically and the resulting uniqueness regions are compared.

Relevance in the Renormalization Group and in Information Theory.

The analysis of complex physical systems hinges on the ability to extract the relevant degrees of freedom from among the many others. Though much hope is placed in machine learning, it also brings challenges, chief of which is interpretability. It is often unclear what relation, if any, the architecture- and training-dependent learned "relevant" features bear to standard objects of physical theory. Here we report on theoretical results which may help to systematically address this issue: we establish equivalence between the field-theoretic relevance of the renormalization group, and an information-theoretic notion of relevance we define using the information bottleneck (IB) formalism of compression theory. We show analytically that for statistical physical systems described by a field theory the relevant degrees of freedom found using IB compression indeed correspond to operators with the lowest scaling dimensions. We confirm our field theoretic predictions numerically. We study dependence of the IB solutions on the physical symmetries of the data. Our findings provide a dictionary connecting two distinct theoretical toolboxes, and an example of constructively incorporating physical interpretability in applications of deep learning in physics.

Monte Carlo examination of first-order phase transitions in a system with many independent order parameters: Three-dimensional Ashkin-Teller model.

A Monte Carlo (MC) computer experiment for the analysis of first-order temperature-driven phase transitions in a system with one or many independently behaving order parameters is presented using the example of the three-dimensional (3D) Ashkin-Teller model, one of the important reference systems in statistical physics showing a rich and complex phase diagram. The properties of a number of quantities, such as magnetization, three types of cumulants, the internal energy, and its histogram, are exploited. The Lee and Kosterlitz concept proposed for strong first-order phase transitions in systems with one independent order parameter is significantly expanded to obtain results with comparable error bars in reasonable computation times at an arbitrary amount of latent heat. The proposed computer MC experiment uses parallel processing and both the Metropolis and recently formulated cluster algorithms. Arbitrarily weak to strong first-order phase transitions in the phase diagram region with ferromagnetic interactions are investigated and the latent heat associated with individual degrees of freedom is carefully computed. In the discussion of results, the behavior of our 3D system between that of the mean-field and that of the 2D one is bracketed and the role of the Potts point is clarified.

Multiscale entropy and asynchronies of percolation-based price model and Chinese stock market

In recent years, the concept of entropy is widely used to measure the degree of uncertainty or complexity of dynamics system. In our work, we utilize the composite multiscale entropy (CMSE) and the composite multiscale cross-sample entropy (CMSCE) which are two modified algorithms of SampEn and Cross-SampEn by considering multiscale factors, to, respectively, investigate the multiscale complexities and asynchronies (correlations) in the Chinese stock market (SSZ, SZSE and HSI) as well as in our established financial stock price model. The price model is given based on a greatly important statistical physics system, the two-dimensional continuum percolation system. In the model, the fluctuations of stock price changes are assumed to be attributed to the market information interactions among the traders, and the percolation cluster is taken to represent the traders holding the same investment attitude. The empirical CMSE and CMSCE results display and meanwhile make comparisons of abundant complexities and correlations properties about Chinese stock indices and simulation data on both overall and the upwards, downwards trend of their returns.

Nonlinear fluctuation behaviors of complex voter financial price dynamics on small-world network

To simulate the price fluctuation dynamics of financial markets, a novel financial price model is developed by the voter dynamic system on the Watts-Strogtz small-world network and the random jump process. The voter system is a classical statistical physics system, which describes the dynamics of voters’ attitudes towards a certain topic in the mutual influence. The Watts-Strogtz small-world network is a special kind of complex networks, which can be used to study the transmission dynamics of different things in complex and real-world systems. The paper first attempts to use the voter dynamic system on the small-world network to reproduce the micro-mechanism of price fluctuations caused by the interaction among different investors in financial markets, where investors can potentially disseminate information and interact via additional long-distance contacts. Moreover, considering that external macro environments have the impact on price fluctuations in financial markets, this paper introduces the random jump process in the price model. The effectiveness of the proposed model can be verified by comparing price returns generated by the model with returns of several important stock indexes in terms of nonlinear fluctuation behaviors. First, some statistical behaviors of the fluctuation dynamics are explored, including distribution characteristics and autocorrelation. Moreover, based on the ensemble empirical mode decomposition method, multifractal behaviors and complexity behaviors of returns and the first three intrinsic mode functions are investigated. The empirical results show that the dynamical model can well simulate these nonlinear fluctuation behaviors of real markets.

Fluctuation behavior analysis of stochastic exclusion financial dynamics with random jump

A novel financial price model combining stochastic exclusion system and compound Poisson process with lattice fractal Sierpinski carpet percolation jumps is established to reflect fluctuation behaviors of energy markets. The exclusion system (one of statistical physics systems) is applied in implementing interaction behaviors of traders, and the percolation theory is used to implement gusty jumps of price volatilities. Further, the actual market data and the simulation data are combined to analyze statistical properties and complex characteristics by applying autocorrelation analysis, detrended fluctuation analysis method and fractional fuzzy entropy. Moreover, a new synchronization measurement method named the cross CID FuzzyEn, which is composed of cross-fuzzy entropy and complexity-invariant distance, is firstly proposed in this work. Empirical results show that the analog returns and the actual returns have parallel characteristics and strong synchronicity, which displays that the proposed model is rational and can effectively reflect the internal mechanism of real energy markets to some extent.

Heat capacities of thermally manipulated mechanical oscillator at strong coupling

Coherent quantum oscillators are basic physical systems both in quantum statistical physics and quantum thermodynamics. Their realizations in lab often involve solid-state devices sensitive to changes in ambient temperature. We represent states of the solid-state optomechanical oscillator with temperature-dependent frequency by equivalent states of the mechanical oscillator with temperature-dependent energy levels. We interpret the temperature dependence as a consequence of strong coupling between the oscillator and the heat bath. We explore parameter regimes corresponding to anomalous behavior of mechanical and thermodynamic characteristics as a consequence of the strong coupling: (i) The localization and the purification induced by heating, and (ii) the negativity of two generalized heat capacities. The capacities can be used to witness non-linearity in the temperature dependency of the energy levels. Our phenomenological experimentally-oriented approach can stimulate development of new optomechanical and thermomechanical experiments exploring basic concepts of strong coupling thermodynamics.

Multiscale statistical behaviors for Ising financial dynamics with continuum percolation jump

A financial dynamics of interaction and jump is developed and investigated by two statistical physics systems — Ising model and continuum percolation. This proposed model aiming at understanding price fluctuations needs to define a mechanism for the formation of the price, in an attempt to describe interacting micromechanism and sudden jump for stock price changes. Further, the corresponding fluctuation behaviors and various complexity properties of logarithmic returns for the financial model are investigated by some statistical and complex analyses. Then p-order multiscale autocorrelation function and q-order multiscale entropy are also introduced to study the financial model with scaled analysis methods. Moreover, the real stock market indexes are used to compare with the simulation returns from the financial model. The empirical research shows that the simulation time series exhibits the similar fluctuation patterns with real time series.

Nonlinear monotonic volatility duration and matching energy of random exclusion financial dynamics

Financial markets have been known to exhibit plentiful nonlinear complex volatility behaviors. In order to reproduce the volatility dynamics of financial price changes, the agent-based financial model is established by stochastic finite-range exclusion process. The exclusion process is a kind of statistical physics system, which is considered as modeling particle Markov motion with conserved number of particles. To measure the volatility of financial return series, a novel statistic called maximum monotonic volatility rate is put forward to measure the speed of monotonic volatility of returns. Meanwhile, average monotonic volatility duration of returns is also investigated, which can reflect the average volatility level. For verifying the rationality of the model, matching energy analysis that can detect chaos and complexity in nonlinear time series is applied to study the new statistics. Further, empirical mode decomposition and multifractal are employed to study the behaviors of monotonic volatility duration. The model has similar complexity behaviors with real markets in terms of monotonic volatility with matching energy analysis, and the proposed financial model and real markets both show multifractal and anti-correlation for average monotonic volatility series by MFDFA method. The results display that the model is feasible with respect to above volatility analyses.

Volatility aggregation intensity energy futures series on stochastic finite-range exclusion dynamics

The stochastic finite-range exclusion process, one of statistical physics systems, is introduced to construct a new agent-based financial price model to study the mechanism of market dynamics. A novel volatility aggregation intensity (VAI) time series, describing the fluctuation aggregation intensity of the volatility series, is developed for further investigating the nonlinear volatility behaviors in energy markets. For studying the proposed VAI series and the proposed price model, the New York Mercantile Exchange energy futures data is selected and analyzed. Further, cross-correlation analysis, auto-correlation analysis with multiscale, and multifractal detrended fluctuation analysis are applied to analyze the correlation, volatility-clustering and multifractal natures of the VAI time series. The empirical results show that the proposed model has the parallel behaviors with the authentic markets, indicating that it is rational and available. The new concept of VAI series is of great value and can enrich the study of volatility behaviors in energy markets to some extent.

Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants

We present two new data structures for computing values of an n-variate polynomial P of degree at most d over a finite field of q elements. Assuming that d divides q-1, our first data structure relies on (d+1)^{n+2} tabulated values of P to produce the value of P at any of the q^n points using O(nqd^2) arithmetic operations in the finite field. Assuming that s divides d and d / s divides q-1, our second data structure assumes that P satisfies a degree-separability condition and relies on (d/s+1)^{n+s} tabulated values to produce the value of P at any point using Oleft( nq^ssqright) arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (Duke Math J 121(1):35–74, 2004), Saraf and Sudan (Anal PDE 1(3):375–379, 2008) and Dvir (Incidence theorems and their applications, 2012. arXiv:1208.5073) for Kakeya sets in finite vector spaces from linear to higher-degree polynomial curves. As an application we show that the new data structures enable a faster algorithm for computing integer-valued fermionants, a family of self-reducible polynomial functions introduced by Chandrasekharan and Wiese (Partition functions of strongly correlated electron systems as fermionants, 2011. arXiv:1108.2461v1) that captures numerous fundamental algebraic and combinatorial functions such as the determinant, the permanent, the number of Hamiltonian cycles in a directed multigraph, as well as certain partition functions of strongly correlated electron systems in statistical physics. In particular, a corollary of our main theorem for fermionants is that the permanent of an mtimes m integer matrix with entries bounded in absolute value by a constant can be computed in time 2^{m-Omega left( sqrt{m/log log m}right) }, improving an earlier algorithm of Björklund (in: Proceedings of the 15th SWAT, vol 17, pp 1–11, 2016) that runs in time 2^{m-Omega left( sqrt{m/log m}right) }.

Permutation glass.

The field of disordered systems in statistical physics provides many simple models in which the competing influences of thermal and nonthermal disorder lead to new phases and nontrivial thermal behavior of order parameters. In this paper, we add a model to the subject by considering a disordered system where the state space consists of various orderings of a list. As in spin glasses, the disorder of such "permutation glasses" arises from a parameter in the Hamiltonian being drawn from a distribution of possible values, thus allowing nominally "incorrect orderings" to have lower energies than "correct orderings" in the space of permutations. We analyze a Gaussian, uniform, and symmetric Bernoulli distribution of energy costs, and, by employing Jensen's inequality, derive a simple condition requiring the permutation glass to always transition to the correctly ordered state at a temperature lower than that of the nondisordered system, provided that this correctly ordered state is accessible. We in turn find that in order for the correctly ordered state to be accessible, the probability that an incorrectly ordered component is energetically favored must be less than the inverse of the number of components in the system. We show that all of these results are consistent with a replica symmetric ansatz of the system. We conclude by arguing that there is no distinct permutation glass phase for the simplest model considered here and by discussing how to extend the analysis to more complex Hamiltonians capable of novel phase behavior and replica symmetry breaking. Finally, we outline an apparent correspondence between the presented system and a discrete-energy-level fermion gas. In all, the investigation introduces a class of exactly soluble models into statistical mechanics and provides a fertile ground to investigate statistical models of disorder.

Application of Neural Networks in Probabilistic Forecasting of Earthquakes in the Southern California Region

During the last few decades, many statistical physicists have devoted re-search efforts to the study of the problem of earthquakes. The purpose of this work is to apply methods of Statistical Physics and network systems based on “neurons” in the study of seismological events. Data from the Advanced National Seismic System (ANSS) of Southern California were used to verify the relationship between time differences between consecutive seismic events with magnitudes greater than 3.0, 3.5, 4.0 and 4.5 through the modeling of neural networks. The problem we are analyzing is time differences between seismological events and how these data can be adopted as a time series with non linear characteristic. We are therefore using the multilayer perceptron neural network system with a backpropagation learning algorithm, because its characteristics allow for the analysis of non-linear data in order to obtain statistical results regarding the probabilistic forecast of tremor occurrence.

Statistical physics of the symmetric group.

Ordered chains (such as chains of amino acids) are ubiquitous in biological cells, and these chains perform specific functions contingent on the sequence of their components. Using the existence and general properties of such sequences as a theoretical motivation, we study the statistical physics of systems whose state space is defined by the possible permutations of an ordered list, i.e., the symmetric group, and whose energy is a function of how certain permutations deviate from some chosen correct ordering. Such a nonfactorizable state space is quite different from the state spaces typically considered in statistical physics systems and consequently has novel behavior in systems with interacting and even noninteracting Hamiltonians. Various parameter choices of a mean-field model reveal the system to contain five different physical regimes defined by two transition temperatures, a triple point, and a quadruple point. Finally, we conclude by discussing how the general analysis can be extended to state spaces with more complex combinatorial properties and to other standard questions of statistical mechanics models.

Multiscale volatility duration characteristics on financial multi-continuum percolation dynamics

A random stock price model based on the multi-continuum percolation system is developed to investigate the nonlinear dynamics of stock price volatility duration, in an attempt to explain various statistical facts found in financial data, and have a deeper understanding of mechanisms in the financial market. The continuum percolation system is usually referred to be a random coverage process or a Boolean model, it is a member of a class of statistical physics systems. In this paper, the multi-continuum percolation (with different values of radius) is employed to model and reproduce the dispersal of information among the investors. To testify the rationality of the proposed model, the nonlinear analyses of return volatility duration series are preformed by multifractal detrending moving average analysis and Zipf analysis. The comparison empirical results indicate the similar nonlinear behaviors for the proposed model and the actual Chinese stock market.

Reflection positive doubles

Here we introduce reflection positive doubles, a general framework for reflection positivity, covering a wide variety of systems in statistical physics and quantum field theory. These systems may be bosonic, fermionic, or parafermionic in nature. Within the framework of reflection positive doubles, we give necessary and sufficient conditions for reflection positivity. We use a reflection-invariant cone to implement our construction. Our characterization allows for a direct interpretation in terms of coupling constants, making it easy to check in concrete situations. We illustrate our methods with numerous examples.