Methods Of Statistical Mechanics Research Articles

Binary non-Markovian theory of bulk associative–dissociative reaction $$ A + A \leftrightarrow C $$ A + A ↔ C . I: many-particle aspects of the theory

It is shown that reversible reaction may be considered as irreversible one on restricted time interval. However, the time range of the applicability of the law of mass action for the case of many-particle derivation of binary kinetic equations which is valid at small density parameters also has time restrictions. If the range of binary description is narrower than that of irreversible approximation of the kinetics, then this approximation is valid over the entire time range of the applicability of the law of mass action. The necessary conditions are found for which hierarchies for reduced distribution functions (RDFs) for the reaction at hand are constructed that satisfy the required hierarchy condition, and have all the properties inherent in BBGKY hierarchies of non-equilibrium statistical mechanics. This allows one to use the advanced methods of non-equilibrium statistical mechanics in many-particle derivation of kinetic equations of chemical reactions in solutions. It is shown that such initial conditions are possible at which for complete evolution no closed kinetic equation exists. Development of RDFs for reaction systems opens the way to the employment of advanced methods of non-equilibrium statistical mechanics in many-particle derivation of kinetic equations of chemical reactions in solutions.

Resolving phase stability in the Ti-O binary with first-principles statistical mechanics methods

The Ti-O system consists of a multitude of stable and metastable oxides that are used in wide ranging applications. In this work we investigate phase stability in the Ti-O binary from first principles. We perform a systematic search for ground state structures as a function of oxygen concentration by considering oxygen-vacancy and/or titanium-vacancy orderings over four parent crystal structures: (i) hcp Ti, (ii) $\ensuremath{\omega}$-Ti, (iii) rocksalt, and (iv) hcp oxygen containing interstitial titanium. We explore phase stability at finite temperature using cluster expansion Hamiltonians and Monte Carlo simulations. The calculations predict a high oxygen solubility in hcp Ti and the stability of suboxide phases that undergo order-disorder transitions upon heating. Vacancy ordered rocksalt phases are also predicted at low temperature that disorder to form an extended solid solution at high temperatures. Predicted stable and metastable phase diagrams are qualitatively consistent with experimental observations, however, important discrepancies are revealed between first-principles density functional theory predictions of phase stability and the current understanding of phase stability in this system.

The structure factor of primes

Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes using statistical–mechanical methods, particularly the structure factor in an interval with M large, and L/M smaller than unity. We show that the structure factor of the prime-number configurations in such intervals exhibits well-defined Bragg-like peaks along with a small ‘diffuse’ contribution. This indicates that primes are appreciably more correlated and ordered than previously thought. Our numerical results definitively suggest an explicit formula for the locations and heights of the peaks. This formula predicts infinitely many peaks in any non-zero interval, similar to the behavior of quasicrystals. However, primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational. We also show numerically that the diffuse part decays slowly as M and L increases. This suggests that the diffuse part vanishes in an appropriate infinite-system-size limit.

Quantum Gravity Effects in Statistical Mechanics with Modified Dispersion Relation

Planck scale inspired theories which are also often accompanied with maximum energy and/or momentum scale predict deformed dispersion relations compared to ordinary special relativity and quantum mechanics. In this paper, we resort to the methods of statistical mechanics in order to determine the effects of a deformed dispersion relation along with an upper bound in the partition function that maximum energy and/or momentum scale can have on the thermodynamics of photon gas. We also analyzed two distinct quantum gravity models in this paper.

Mechanism of protein-RNA recognition: analysis based on the statistical mechanics of hydration.

We investigate the RBD1-r(GUAGU) binding as a case study using all-atom models for the biomolecules, molecular models for water, and the currently most reliable statistical-mechanical method. RBD1 is one of the RNA-binding domains of mammalian Musashi1 (Msi1), and r(GUAGU) contains the minimum recognition sequence for Msi1, r(GUAG). We show that the binding is driven by a large gain of configurational entropy of water in the entire system. It is larger than the sum of conformational-entropy losses for RBD1 and r(GUAGU). The decrease in RBD1-r(GUAGU) interaction energy upon binding is largely cancelled out by the increase in the sum of RBD1-water, r(GUAGU)-water, and water-water interaction energies. We refer to this increase as "energetic dehydration". The decrease is larger than the increase for the van der Waals component, whereas the opposite is true for the electrostatic component. We give a novel reason for the empirically known fact that protein residues possessing side chains with positive charges and with flat moieties frequently appear within protein-RNA binding interfaces. A physical picture of the general protein-RNA binding mechanism is then presented. To achieve a sufficiently large water-entropy gain, shape complementarity at the atomic level needs to be constructed by utilizing the stacking and sandwiching of flat moieties (aromatic rings of the protein and nucleobases of RNA) as fundamental motifs. To compensate for electrostatic energetic dehydration, charge complementarity becomes crucial within the binding interface. We argue the reason why the RNA recognition motif (RRM) is the most ubiquitous RNA binding domain.

Statistical Mechanics of Coherent Ising Machine — The Case of Ferromagnetic and Finite-Loading Hopfield Models —

The coherent Ising machine (CIM) has attracted attention as one of the most effective Ising computing architectures for solving large scale optimization problems because of its scalability and high-speed computational ability. However, it is difficult to implement the Ising computation in the CIM because the theories and techniques of classical thermodynamic equilibrium Ising spin systems cannot be directly applied to the CIM. This means we have to adapt these theories and techniques to the CIM. Here we focus on a ferromagnetic model and a finite loading Hopfield model, which are canonical models sharing a common mathematical structure with almost all other Ising models. We derive macroscopic equations to capture nonequilibrium phase transitions in these models. The statistical mechanical methods developed here constitute a basis for constructing evaluation methods for other Ising computation models.

Melting temperature of iron at high pressure: statistical moment method approach

The pressure effects on melting temperatures of iron have been studied based on the combination of the modified Lindemann criterion with statistical moment method in quantum statistical mechanics. Numerical calculations have been performed up to pressure 150 GPa. Our results are in good and reasonable agreements with available experimental data. This approach gives us a relatively simple method for qualitatively calculating high-pressure melting temperature. Moreover, it can be used to verify future experimental and theoretical works. This research proposes the potential of the combination of statistical moment method and the modified Lindemann criterion on predicting high-pressure melting of materials.

Detailed kinetic modeling of dimethoxymethane. Part I: Ab initio thermochemistry and kinetics predictions for key reactions

Despite the great interest in oxygenated methyl ethers as diesel fuel additives and as fuels themselves, the influence of their methylenedioxy group(s) (O–CH2–O) has never been quantified using ab initio methods. In this study we elucidate the kinetics and thermochemistry of dimethoxymethane using high-level ab initio (CCSD(T)/aug-cc-pV(D+T)Z//B2PLYPD3BJ/6-311++g(d,p)) and statistical mechanics methods. We model torsional modes as hindered rotors which has a large influence on the description of the thermal behavior. Rate constants for hydrogen abstraction by Ḣ and ĊH3 are computed and show that abstraction from the methylenedioxy group is favored over abstraction from the terminal methyl groups. β-scission and isomerization of the radicals are computed using master equations. The effect of rovibrationally excited radicals from preceding hydrogen abstraction reactions on subsequent hot β-scission is computed and has large influence on the decomposition of the formed dimethylether radical. The quantification of the effect of the dominant methylenedioxy group using ab initio methods can guide modeling of oxygenated methyl ethers that contain that group several times.

Structural and thermomechanical properties of the zinc-blende AlX (X = P, As, Sb) compounds

The structural and thermomechanical properties of zinc-blende aluminum class of III–V compounds have been studied based on the statistical moment method (SMM) in quantum statistical mechanics. Within the SMM scheme, we derived the analytical expressions of the nearest-neighbor distance, thermal expansion coefficient, atomic mean-square displacement and elastic moduli (Young’s modulus, bulk modulus and shear modulus). Numerical calculations have been performed for zinc-blende AlX (X = As, P, Sb) at ambient conditions up to the temperature of 1000 K. Our results are in good and reasonable agreements with earlier measurements and can provide useful references for future experimental and theoretical works. This research presents a systematic approach to investigate the thermodynamic and mechanical properties of materials.

Stochastic level-set method for shape optimisation

We present a new method for stochastic shape optimisation of engineering structures. The method generalises an existing deterministic scheme, in which the structure is represented and evolved by a level-set method coupled with mathematical programming. The stochastic element of the algorithm is built on the methods of statistical mechanics and is designed so that the system explores a Boltzmann–Gibbs distribution of structures. In non-convex optimisation problems, the deterministic algorithm can get trapped in local optima: the stochastic generalisation enables sampling of multiple local optima, which aids the search for the globally-optimal structure. The method is demonstrated for several simple geometrical problems, and a proof-of-principle calculation is shown for a simple engineering structure.

Statistical-Mechanical Analysis Connecting Supervised Learning and Semi-Supervised Learning

The generalization performance of semi-supervised learning is analyzed in the framework of online learning using the statistical-mechanical method. We derive deterministically formed simultaneous differential equations that describe the dynamical behaviors of order parameters using the self-averaging property under the thermodynamic limit. By generalizing the number of labeled data, the derived theory connects supervised learning and semi-supervised learning.

A theoretical study on the optical absorption of green fluorescent protein chromophore in solutions

Ethyl 4-(4-hydroxyphenyl) methylidene- 2-methyl-5-oxoimidazolacetate (HBMIA) is a model chromophore of green fluorescent protein. The electronic structure of HBMIA in aqueous solution phase is studied using a hybrid method of quantum chemistry and statistical mechanics, RISM-SCF-SEDD. The solvatochromic shift is correctly reproduced by the present computations.

Probabilistic inverse design for self-assembling materials

One emerging approach for the fabrication of complex architectures on the nanoscale is to utilize particles customized to intrinsically self-assemble into a desired structure. Inverse methods of statistical mechanics have proven particularly effective for the discovery of interparticle interactions suitable for this aim. Here we evaluate the generality and robustness of a recently introduced inverse design strategy [B. A. Lindquist et al., J. Chem. Phys. 145, 111101 (2016)] by applying this simulation-based machine learning method to optimize for interparticle interactions that self-assemble particles into a variety of complex microstructures as follows: cluster fluids, porous mesophases, and crystalline lattices. Using the method, we discover isotropic pair interactions that lead to the self-assembly of each of the desired morphologies, including several types of potentials that were not previously understood to be capable of stabilizing such systems. One such pair potential led to the assembly of the highly asymmetric truncated trihexagonal lattice and another produced a fluid containing spherical voids, or pores, of designed size via purely repulsive interactions. Through these examples, we demonstrate several advantages inherent to this particular design approach including the use of a parametrized functional form for the optimized interparticle interactions, the ability to constrain the range of said parameters, and compatibility of the inverse design strategy with a variety of simulation protocols (e.g., positional restraints).

Pressure effects on the thermodynamic and mechanical properties of zinc-blende ZnTe compound

In this paper, the moment method in statistical mechanics has been employed to study the pressure effects on thermodynamic and mechanical properties of zinc-blende zinc telluride using many-body potential. We have derived the analytical expressions of the pressure-dependent lattice parameter, volume compression as well as mean-square displacement of zinc-blende type compound. Numerical calculations performed for ZnTe compound up to 12 GPa are found to be in good and reasonable agreement with available experimental data as well as with previous theoretical studies. These results have been used to evaluate the bulk modulus and its first pressure derivative of ZnTe. The present moment method has taken into account the quantum zero-point vibrations at low temperature and the higher-order anharmonic terms in the atomic displacements. This research shows the advantage of moment method on extensively studying thermo-mechanical properties of materials under high pressures.

A quest toward a mathematical theory of the dynamics of swarms

This paper addresses some preliminary steps toward the modeling and qualitative analysis of swarms viewed as living complex systems. The approach is based on the methods of kinetic theory and statistical mechanics, where interactions at the microscopic scale are nonlocal, nonlinearly additive and modeled by theoretical tools of stochastic game theory. Collective learning theory can play an important role in the modeling approach. We present a kinetic equation incorporating the Cucker–Smale flocking force and stochastic game theoretic interactions in collision operators. We also present a sufficient framework leading to the asymptotic velocity alignment and global existence of smooth solutions for the proposed kinetic model with a special kernel. Analytic results on the global existence and flocking dynamics are presented, while the last part of the paper looks ahead to research perspectives.

Conditions for the Existence and Stability of the Continuous Attractor in the Classical XY Model with an Associative-Memory-Type Interaction

We analyze the structure of attractors in the classical XY model with an associative-memory-type interaction by the statistical mechanical method. Previously, it was found that when patterns are uncorrelated, points on a path connecting two memory patterns in the space of the order parameters are solutions of the saddle point equations (SPEs) in the case that p is \(\mathcal{O}(1)\) irrespective of N and N ≫ 1, where p and N are the numbers of patterns and spins, respectively. This state is called the continuous attractor (CA). In this paper, we clarify the conditions for the existence and stability of the CA with and without the correlation a (0 ≤ a < 1) between any two patterns in the case that N ≫ 1 and the self-averaging property holds. We find that the CA exists for any p ≥ 2 when a = 0, but it exists only for p = 2 when 0 < a < 1 and for p = 3 when a < 1/3. For p = 2 and 3, and for a < 1, we analyze the SPEs and find all solutions and study their stabilities. We perform Markov chain Monte Carlo simu...

Fluctuations, response, and resonances in a simple atmospheric model

We study the response of a simple quasi-geostrophic barotropic model of the atmosphere to various classes of perturbations affecting its forcing and its dissipation using the formalism of the Ruelle response theory. We investigate the geometry of such perturbations by constructing the covariant Lyapunov vectors of the unperturbed system and discover in one specific case–orographic forcing–a substantial projection of the forcing onto the stable directions of the flow. This results into a resonant response shaped as a Rossby-like wave that has no resemblance to the unforced variability in the same range of spatial and temporal scales. Such a climatic surprise corresponds to a violation of the fluctuation–dissipation theorem, in agreement with the basic tenets of nonequilibrium statistical mechanics. The resonance can be attributed to a specific group of rarely visited unstable periodic orbits of the unperturbed system. Our results reinforce the idea of using basic methods of nonequilibrium statistical mechanics and high-dimensional chaotic dynamical systems to approach the problem of understanding climate dynamics.

Statistical-Mechanical Analysis of LMS Algorithm for Time-Varying Unknown System

We analyze the behaviors of the least-mean-square algorithm for a time-varying unknown system using a statistical-mechanical method. Cross-correlations between the elements of a primary path and those of an adaptive filter and autocorrelations of the elements of the adaptive filter are treated as macroscopic variables. We obtain simultaneous differential equations that describe the dynamical behaviors of the macroscopic variables under conditions in which the tapped delay line is sufficiently long. We analytically show the existence of an optimal step size. This result is supporting evidence of Widrow et al.’s pioneering work that clarified the trade-off between the noise misadjustment and the lag misadjustment. Furthermore, we obtain the exact solution of the optimal step size in the case of a white reference signal. The derived theory includes the behaviors for a time-constant unknown system as a special case.

经济力学与经济热力学

本文用力学和统计力学观点和方法，分析经济现象，提出了经济力学和经济热力学等交叉学科概念，对于推动和深化经济学的研究和发展具有重要意义。 Using the viewpoint and method in mechanics and statistical mechanics to analyze economic phe-nomena, this paper proposed such interdisciplinary concepts as economic mechanics and economic thermodynamics, so as to promote and deepen the research and development of economics.

Estimation of microstructural failure probability based on restoration of the field distributions laws in components of heterogenous media

One of the common approaches in the micromechanics of structurally inhomogeneous media with a random structure is connected with the methods of statistical mechanics for many-particle systems. The approach to the analysis of the distribution of stress and strain fields in representative volumes of heterogeneous media introduced in this paper is based on reconstruction of the distribution laws for each component. Methods for finding parameters of distribution laws are described, different types of distribution laws were compared. The technique of analysis of failure probability of microstructural elements is given. Some numerical results for particular case studies are presented.