Methods Of Statistical Mechanics Research Articles

Reliable emulation of complex functionals by active learning with error control.

A statistical emulator can be used as a surrogate of complex physics-based calculations to drastically reduce the computational cost. Its successful implementation hinges on an accurate representation of the nonlinear response surface with a high-dimensional input space. Conventional "space-filling" designs, including random sampling and Latin hypercube sampling, become inefficient as the dimensionality of the input variables increases, and the predictive accuracy of the emulator can degrade substantially for a test input distant from the training input set. To address this fundamental challenge, we develop a reliable emulator for predicting complex functionals by active learning with error control (ALEC). The algorithm is applicable to infinite-dimensional mapping with high-fidelity predictions and a controlled predictive error. The computational efficiency has been demonstrated by emulating the classical density functional theory (cDFT) calculations, a statistical-mechanical method widely used in modeling the equilibrium properties of complex molecular systems. We show that ALEC is much more accurate than conventional emulators based on the Gaussian processes with "space-filling" designs and alternative active learning methods. In addition, it is computationally more efficient than direct cDFT calculations. ALEC can be a reliable building block for emulating expensive functionals owing to its minimal computational cost, controllable predictive error, and fully automatic features.

On the Correspondence between Subshifts of Finite Type and Statistical Mechanics Models.

Several classical problems in symbolic dynamics concern the characterization of the simplex of measures of maximal entropy. For subshifts of finite type in higher dimensions, methods of statistical mechanics are ideal for dealing with these problems. R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics-a correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it and present some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way the possibilities of this correspondence.

Heterogeneous consecutive reaction kinetics of direct oxidation of H2 to H2O2: Effect and regulation of confined mass transfer

Porous catalysts in heterogeneous reactions have played an important role in the modern chemical industry, but it is still challenging to quantitatively describe mass transfer and surface reaction behaviors of reactants in nano-confined space. Direct synthesis of hydrogen peroxide (H2O2) is considered as an attractive alternative to anthraquinone oxidation process, while the confined mass transfer of H2O2 in porous catalysts limits the reactivity. In this work, taking the consecutive reaction of H2O2 synthesis as an example, a quantitative method in modeling the effects of confined mass transfer on the reactivity was studied. More specifically, calorimetry was developed to characterize the confined structures of porous carbon experimentally, the linear nonequilibrium thermodynamics and the statistical mechanics method were further combined. Then, the heterogeneous consecutive reaction kinetics and the Thiele modulus influenced by confined mass transfer were modeled. Consequently, regulation strategies were proposed with the help of theoretical models. The optimized catalyst with biological skeleton carbon support and 0.5 wt% palladium loading shows an excellent catalytic performance. Lastly, for the mesoscience in heterogeneous reaction, the resistance was explored as a quantitative descriptor to compromise in the competition between mass transfer and surface reaction. The mesoscale structures were considered as the dynamic spatiotemporal distribution of substance concentrations, and the resistance minimization multi-scale (RMMS) model was proposed.

Second- to first-order phase transition: Coevolutionary versus structural balance

In social networks, the balance theory has been studied by considering either the triple interactions between the links (structural balance) or the triple interaction of nodes and links (coevolutionary balance). In the structural balance theory, the links are not independent from each other, implying a global effect of this term and it leads to a discontinuous phase transition in the system's balanced states as a function of temperature. However, in the coevolutionary balance the links only connect two local nodes and a continuous phase transition emerges. In this paper, we consider a combination of both to understand which of these types of interactions will identify the stability of the network. We are interested to see how adjusting the robustness of each term versus the other might affect the system to reach a balanced state. We use statistical mechanics methods and the mean-field theory and also the Monte Carlo numerical simulations to investigate the behavior of the order parameters and the total energy of the system. We find the phase diagram of the system which demonstrates the competition of these two terms at different ratios against each other and different temperatures. The system shows a tricritical point above which the phase transition switches from continuous to discrete. Also the superiority of the local perspective is observed at low temperatures and the global view will be the dominant term in determining the stability of the system at higher temperatures.

Mean-field theory of the spin-1/2 transverse field Ising model with a negative thermal expansion

A generalized spin-1/2 transverse field Ising model with a negative thermal expansion of the lattice is introduced and investigated using standard methods of statistical mechanics. Besides the volume-dependent magnetic energy, the static lattice energy, and anharmonic Einstein phonon energy are also considered in calculations. Analytic relations for the Gibbs free energy, magnetic moments, and equations of state are obtained, taking into account a simple volume dependence of all energy contributions. The ground-state and finite-temperature phase diagrams are discussed in detail for the strong and weak magneto-elastic coupling. It is clearly demonstrated that the generalized spin-1/2 transverse field Ising model exhibits a novel critical behavior, due to the strong negative expansion of the lattice, which is controlled by the strength of magneto-elastic coupling. The presented approach can be easily modified to study also other magnetic and non-magnetic crystalline models of solids.

Revisiting the melting curves of vanadium and niobium metals under pressure

The Lindemann criterion is adjusted to revisit melting curves of vanadium and niobium metals under pressure thanks to the statistical moment method (SMM) in statistical mechanics. The equation-of-state and atomic mean-square displacement in SMM approach are applied to implement numerical calculations for vanadium and niobium metals up to 250 GPa and 150 GPa, respectively. The obtained theoretical melting curves are compared with results of previous experiments and molecular dynamics (MD) simulations where possible to check the theory. We expose that our theoretical melting line of vanadium is overall in good agreement with previous results determined by in situ X-ray diffraction (XRD) and diffuse scattering signals using synchrotron XRD together with shock experiments up to pressure 250GPa. For niobium, our melting curve follows very well with those of recent experimental XRD data as well as ab initio Z method up to 124 GPa. However, we also observe the difference of classical MD simulation results from SMM calculations at pressure below 2.5 GPa. This research enriches the high-pressure melting point database as well as provides a relatively simple approach with low computational cost to estimate high-pressure melting points of materials.

Macroscopic Analysis of Vector Approximate Message Passing in a Model-Mismatched Setting

In this study, macroscopic properties of the vector approximate message passing (VAMP) algorithm for inference of generalized linear models are investigated using a non-rigorous heuristic method of statistical mechanics when the true posterior cannot be used and the measurement matrix is a sample from rotation-invariant random matrix ensembles. The focus is on the correspondence between the non-rigorous replica analysis of statistical mechanics and the performance assessment of VAMP in the model-mismatched setting. The correspondence of this kind is well-known when the measurement matrix has independent and identically distributed entries. However, when the measurement matrix follows a general rotation-invariant matrix ensemble, the correspondence has been validated only under limited cases, such as the Bayes optimal inference or the convex empirical risk minimization. The result presented in this paper is to extend the scope of such correspondence. Herein, we heuristically derive the explicit formula of state-evolution equations, which macroscopically describe VAMP dynamics for the current model-mismatched case, and show that their fixed point is generally consistent with the replica symmetric solution obtained by the replica method of statistical mechanics. We also show that the fixed point of VAMP can exhibit a microscopic instability, which indicates that message variables continue to move by VAMP while their macroscopically summarized quantities converge to fixed values. The critical condition the for microscopic instability agrees with that for breaking the replica symmetry that is derived within the non-rigorous replica analysis. The results of the numerical experiments cross-check our findings.

Energy spectrum of interacting gas: Cluster expansion method

In this paper, we calculate the energy spectrum of interacting gases by converting the cluster expansion method in statistical mechanics into a method of solving energy eigenvalues. We obtain an explicit expression of the energy eigenvalue, by which we can calculate the eigenvalue of an interacting gas from the interparticle potential directly. As an example, we calculate the energy spectrum for an interacting gas with soft-sphere potentials.

Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism

Explaining the emergence of stochastic irreversible macroscopic dynamics from time-reversible deterministic microscopic dynamics is one of the key problems in philosophy of physics. The Mori-Zwanzig (MZ) projection operator formalism, which is one of the most important methods of modern nonequilibrium statistical mechanics, allows for a systematic derivation of irreversible transport equations from reversible microdynamics and thus provides a useful framework for understanding this issue. However, discussions of the MZ formalism in philosophy of physics tend to focus on simple variants rather than on the more sophisticated ones used in modern physical research. In this work, I will close this gap by studying the problems of probability and irreversibility using the example of Grabert’s time-dependent projection operator formalism. This allows to better understand how general proposals for understanding probability in statistical mechanics, namely (a) quantum approaches and (b) almost-objective probabilities, can be accomodated in the MZ formalism. Moreover, I will provide a detailed physical analysis, based on the MZ formalism, of various proposals from the philosophical literature, such as (a) Robertson’s theory of justifying coarse-graining via autonomous macrodynamics, (b) Myrvold’s problem of explaining autonomous macrodynamics, and (c) Wallace’s simple dynamical conjecture.

Theoretical prediction of melting curves of gold and silver up to pressure 150 GPa

We investigate melting curves of gold and silver thanks to the statistical moment method in quantum statistical mechanics and the modified Lindemann melting criterion. Numerical calculations have been performed for face-centered cubic (fcc) gold and silver metals up to pressure 150 GPa. Our results are compared with available experimental data and ab initio molecular dynamics simulations where possible. We show that theoretical melting line of gold goes along with recent X-ray diffraction measurements reported by Weck et al. up to pressure more than 110GPa. For silver, our work virtually coincides with the quantum molecular dynamics melting curve which is a combination of fcc and body-centered cubic (bcc) segments freshly implemented by Baty et al. At the pressure 94.2 GPa of the predicted fcc–bcc-liquid triple point of silver, our calculated melting temperature is 4302 K that is only about 2% difference from the prediction of Baty et al. This research proposes a relatively simple approach made from the modified Lindemann criterion and the moment method for predicting melting points of materials at high pressure.

Microscopic density-functional approach to nonlinear elasticity theory

Starting from a general classical model of many interacting particles we present a well defined step by step procedure to derive the continuum-mechanics equations of nonlinear elasticity theory with fluctuations which describe the macroscopic phenomena of a solid crystal. As the relevant variables we specify the coarse-grained densities of the conserved quantities and a properly defined displacement field which describes the local translations, rotations, and deformations. In order to stay within the framework of the conventional density-functional theory we first and mainly consider the isothermal case and omit the effects of heat transport and warming by friction where later we extend our theory to the general case and include these effects. We proceed in two steps. First, we apply the concept of local thermodynamic equilibrium and minimize the free energy functional under the constraints that the macroscopic relevant variables are fixed. As results we obtain the local free energy density and we derive explicit formulas for the elastic constants which are exact within the framework of density-functional theory. Second, we apply the methods of nonequilibrium statistical mechanics with projection-operator techniques. We extend the projection operators in order to include the effects of coarse-graining and the displacement field. As a result we obtain the time-evolution equations for the relevant variables with three kinds of terms on the right-hand sides: reversible, dissipative, and fluctuating terms. We find explicit formulas for the transport coefficients which are exact in the limit of continuum mechanics if the projection operators are properly defined. By construction the theory allows the diffusion of particles in terms of point defects where, however, in a normal crystal this diffusion is suppressed.

First Principles Calculation of Protein-Protein Dimer Affinities of ALS-Associated SOD1 Mutants.

Cu,Zn superoxide dismutase (SOD1) is a 32 kDa homodimer that converts toxic oxygen radicals in neurons to less harmful species. The dimerization of SOD1 is essential to the stability of the protein. Monomerization increases the likelihood of SOD1 misfolding into conformations associated with aggregation, cellular toxicity, and neuronal death in familial amyotrophic lateral sclerosis (fALS). The ubiquity of disease-associated mutations throughout the primary sequence of SOD1 suggests an important role of physicochemical processes, including monomerization of SOD1, in the pathology of the disease. Herein, we use a first-principles statistical mechanics method to systematically calculate the free energy of dimer binding for SOD1 using molecular dynamics, which involves sequentially computing conformational, orientational, and separation distance contributions to the binding free energy. We consider the effects of two ALS-associated mutations in SOD1 protein on dimer stability, A4V and D101N, as well as the role of metal binding and disulfide bond formation. We find that the penalty for dimer formation arising from the conformational entropy of disordered loops in SOD1 is significantly larger than that for other protein–protein interactions previously considered. In the case of the disulfide-reduced protein, this leads to a bound complex whose formation is energetically disfavored. Somewhat surprisingly, the loop free energy penalty upon dimerization is still significant for the holoprotein, despite the increased structural order induced by the bound metal cations. This resulted in a surprisingly modest increase in dimer binding free energy of only about 1.5 kcal/mol upon metalation of the protein, suggesting that the most significant stabilizing effects of metalation are on folding stability rather than dimer binding stability. The mutant A4V has an unstable dimer due to weakened monomer-monomer interactions, which are manifested in the calculation by a separation free energy surface with a lower barrier. The mutant D101N has a stable dimer partially due to an unusually rigid β-barrel in the free monomer. D101N also exhibits anticooperativity in loop folding upon dimerization. These computational calculations are, to our knowledge, the most quantitatively accurate calculations of dimer binding stability in SOD1 to date.

Molecular Modeling and Phenotypic Description of a Patient with a Novel Exonic Deletion of GALNS with Resultant Morquio Syndrome with Two Successful Pregnancies

In this report, we describe phenotypic features of a patient with mucopolysaccharidosis type IVA (Morquio syndrome) harboring a novel exon 1 deletion in GALNS with enzymatic confirmation consistent with Morquio syndrome. To our knowledge, this is the first reported case of this variant. Additionally, we protein modelled wild-type GALNS and the pathogenic variant with an exon 1 deletion for comparative analysis using statistical mechanics methods described herein. We demonstrate that, even when the protein is translated, the mutation would affect protein stability and function via homodimer interaction modifications. Lastly, given the patient’s 2 successful pregnancies, data about the management of pregnancies in mucopolysaccharidoses are reviewed, and we discuss the management of pregnancy in patients with Morquio syndrome.

Statistical-Mechanical Analysis of Adaptive Filter With Clipping Saturation-Type Nonlinearity

In most practical adaptive signal processing systems, e.g., active noise control, active vibration control, and acoustic echo cancellation, substantial nonlinearities that cannot be neglected exist. In this paper, we analyze the behaviors of an adaptive system in which the output of the adaptive filter has the clipping saturation-type nonlinearity by a statistical-mechanical method. We discuss the dynamical and steady-state behaviors of the adaptive system by asymptotic analysis, steady-state analysis, and numerical calculation. As a result, it has become clear that the saturation value has the critical point at which the system's mean-square stability and instability switch. The obtained theory well explains the strange behaviors around the critical point observed in the computer simulation. Finally, the exact value of the critical point is also derived.

Statistical theory of individual ionic activity coefficients of electrolytes with multiple – Charged ions including seawater

Based on methods of statistical mechanics for the calculation of individual activity coefficients for electrolytic solutions we present analytical nonlinear extensions of the standard Debye - Hückel and Mean-Spherical approximations. These extensions, so - called DHX and MSX approximations, respectively, generalize the exact results from cluster theory and are given for the model of charged hard spheres with non - additive contact distances. The flexibility of the fully analytical model which is able to reflect e.g. cationic solvation effects and allows the treatment of ionic activities in a 6-component seawater model. Denoting by e the elementary charge, DHX and MSX include in the third order, e6, the Poirier effects of asymmetry of ion charges, and in the fourth order, e8, as well as in higher orders, weak association effects which are relevant for electrolytes with multivalent ions. Following the Justice method, weak association effects are treated in a semi - chemical description avoiding mass - action laws. An extrapolation of the theory of individual activities to a description of individual conductivities is proposed.

Role of Electronic Structure in Li Ordering and Chemical Strain in the Fast Charging Wadsley–Roth Phase PNb9O25

Wadsley–Roth crystallographic shear phases are a family of transition-metal oxides that show tremendous promise as electrode materials in Li-ion batteries. Despite their ability to intercalate lithium at high rates, little is known about their structural, thermodynamic, and electronic properties as a function of Li concentration. In this study, we use first-principles statistical mechanics methods to explore the lithium-site preference, lithiation strain, and electronic structure of PNb9O25, a Wadsley–Roth phase that has been shown to reversibly cycle at a rate of 60 C and that can accommodate more than one Li per Nb. We find that Li ions can occupy five symmetrically distinct interstitial sites within the PNb9O25 crystal structure, with three being pyramidal sites coordinated by five oxygen and two being window sites with square-planar oxygen coordination. The insertion of Li into PNb9O25 leads to a complex site filling sequence, with pyramidal sites preferred at low Li concentrations, followed by the filling of window sites at higher Li concentrations. Our findings are aided by neutron diffraction where pyramidal sites are found to be filled at low compositions. The order in which sites are filled is strongly influenced by the chemical strain due to Li insertion. The strain arises from the delocalization of donated electrons over the d orbitals of the structure’s edge-sharing niobium, which leads to a tetragonal distortion along the c-axis, thereby making vertical window sites favorable for Li occupancy at intermediate to high Li concentrations. Given the crystallographic similarities among different shear phases, we expect that the results of this study will also shed light on the electrochemical properties of other Wadsley–Roth chemistries.

Analysis of the Hopfield Model with Discrete Coupling

Growing demand for high-speed Ising-computing-specific hardware has prompted a need for determining how the accuracy depends on a hardware implementation with physically limited resources. For instance, in digital hardware such as field-programmable gate arrays, as the number of bits representing the coupling strength is reduced, the density of integrated Ising spins and the speed of computing can be increased while the calculation accuracy becomes lower. To optimize the accuracy-efficiency trade-off, we have to estimate the change in performance of the Ising computing machine depending on the number of bits representing the coupling strength. In this study, we tackle this issue by focusing on the Hopfield model with discrete coupling. The Hopfield model is a canonical Ising computing model. Previous studies have analyzed the effect of a few nonlinear functions (e.g. sign) for mapping the coupling strength on the Hopfield model with statistical mechanics methods, but not the effect of discretization of the coupling strength in detail. Here, we derived the order parameter equations of the Hopfield model with discrete coupling by using the replica method and clarified the relationship between the number of bits representing the coupling strength and the critical memory capacity. In this paper, we used the replica method for the Hopfield model with general nonlinear coupling (Sompolinsky (1986)) to analyze the model with a multi-bit discrete coupling strength, and we novelly derived the de Almeida-Thouless line of the model with general nonlinear coupling.

Understanding protein-complex assembly through grand canonical maximum entropy modeling

Inside a cell, heterotypic proteins assemble in inhomogeneous, crowded systems where the abundance of these proteins vary with cell types. While some protein complexes form putative structures that can be visualized with imaging, there are far more protein complexes that are yet to be solved because of their dynamic associations with one another. Nevertheless, it is possible to infer these protein complexes through a physical model. However, it is often not clear to physicists what kind of data from biology is necessary for such a modeling endeavor. Here, we aim to model these clusters of coarse-grained protein assemblies from multiple subunits through the constraints of interactions among the subunits and the chemical potential of each subunit. We obtained the constraints on the interactions among subunits from the known protein structures. We inferred the chemical potential that dictates the particle number distribution of each protein subunit from the knowledge of protein abundance from experimental data. Guided by the maximum entropy principle, we formulated an inverse statistical mechanical method to infer the distribution of particle numbers from the data of protein abundance as chemical potentials for a grand canonical multicomponent mixture. Using grand canonical Monte Carlo simulations, we captured a distribution of high-order clusters in a protein complex of succinate dehydrogenase with four known subunits. The complexity of hierarchical clusters varies with the relative protein abundance of each subunit in distinctive cell types such as lung, heart, and brain. When the crowding content increases, we observed that crowding stabilizes emergent clusters that do not exist in dilute conditions. We, therefore, proposed a testable hypothesis that the hierarchical complexity of protein clusters on a molecular scale is a plausible biomarker of predicting the phenotypes of a cell.

Simulation of a generalized asset exchange model with economic growth and wealth distribution.

The agent-based yard-sale model of wealth inequality is generalized to incorporate exponential economic growth and its distribution. The distribution of economic growth is nonuniform and is determined by the wealth of each agent and a parameter λ. Our numerical results indicate that the model has a critical point at λ=1 between a phase for λ<1 with economic mobility and exponentially growing wealth of all agents and a nonstationary phase for λ≥1 with wealth condensation and no mobility. We define the energy of the system and show that the system can be considered to be in thermodynamic equilibrium for λ<1. Our estimates of various critical exponents are consistent with a mean-field theory [see W. Klein et al., following paper, Phys. Rev. E 104, 014151 (2021)10.1103/PhysRevE.104.014151]. The exponents do not obey the usual scaling laws unless a combination of parameters that we refer to as the Ginzburg parameter is held fixed as the phase transition is approached. The model illustrates that both poorer and richer agents benefit from economic growth if its distribution does not favor the richer agents too strongly. This work and the following theoretical paper contribute to our understanding of whether the methods of equilibrium statistical mechanics can be applied to economic systems.

Mechanical problems for the long-term stability of rocks surrounding deep level underground tunnels

Based on the research of Academician Tan Tjong-Kie on the long-term stability mechanics of underground tunnel and considering Academician Sadovsky’s structural hierarchy theories on complex geological rock masses, the inherent statistical mechanical properties of inhomogeneous structure and closed stress in deep rock masses were investigated. Two mechanical problems were mainly studied, i.e. characteristics of inherent non-uniform deformation and closed stress of rock masses, and long-term stability of deep tunnels. The quantitative mathematical characterization of inherent non-uniform deformation and closed stress of rock masses were given using the method of statistical mechanics. Based on the law of mass conservation, a general calculation method of long-term stability and deformation of rock masses surrounding deep level tunnels was proposed. The Maxwell model was used to calculate the threshold of splitting. A post-peak failure model of rock was established to estimate the in-situ stress at which rock undergoes post-peak failure. With the help of the theory of the hirerachical structure of rock masses, the maximum value of rock displacement due to splitting dilatation was obtained. A dimensionless energy factor was introduced to define the extent of zonal disintegration of surrounding rock masses. It was concluded that the splitting and dilatancy deformation is the main part of the deformation of surrounding rocks. The reasons why unloading splitting failure is more likely to take place in rock masses surrounding deep level tunnels were explained. The split evolution pattern of rock masses surrounding deep level tunnels and the calculation method of dilatancy displacement were obtained. The calculation results of the range of loosening zone, the location of the rupture zone in rock masses surrounding deep level tunnels, and the displacement of the sidewall were compared with the existing monitoring data in the underground powerhouse of the Jinping Ⅰ Hydropower Station, and the agreement is good.