Statistical Mechanics Approach Research Articles

Mixed Alkali Effect in Silicate Glass Structure: Viewpoint of 29Si Nuclear Magnetic Resonance and Statistical Mechanics

The mixed alkali effect in glasses is the deviation from linear property changes when alkali cations are mixed. The extent of this effect and its structural origin remain topics of interest. In this work, we use a statistical mechanics approach to predict the composition-structure relationship in mixed modifier Na2O-K2O-SiO2 glasses. This is achieved by accounting for the enthalpy between each pairwise alkali ion and silicate unit interaction. The initial enthalpy parameters are obtained based on experimental structural data for binary Na2O-SiO2 and K2O-SiO2 glasses, which can be transferred to predict the short-range order structure of mixed modifier glasses without additional free parameters. To this end, we have performed 29Si magic angle spinning NMR spectroscopy measurements on y(xNa2O-(1 - x)K2O)-(100 - y)SiO2 glasses with x = 0, 0.25, 0.5, 0.75, and 1 and y = 34, 42, and 50. Good agreement between experimental data and model predictions are observed. Finally, we use this information to discuss the relative entropic and enthalpic contributions to the mixed modifier effect in silicate glass structure.

Angiogenesis and vessel co-option in a mathematical model of diffusive tumor growth: The role of chemotaxis

This work considers the propagation of a tumor from the stage of a small avascular sphere in a host tissue and the progressive onset of a tumor neovasculature stimulated by a pro-angiogenic factor secreted by hypoxic cells. The way new vessels are formed involves cell sprouting from pre-existing vessels and following a trail via a chemotactic mechanism (CM). Namely, it is first proposed a detailed general family of models of the CM, based on a statistical mechanics approach. The key hypothesis is that the CM is composed by two components: i) the well–known bias induced by the angiogenic factor gradient; ii) the presence of stochastic changes of the velocity direction, thus giving rise to a diffusive component. Then, some further assumptions and simplifications are applied in order to derive a specific model to be used in the simulations. The tumor progression is favored by its acidic aggression towards the healthy cells. The model includes the evolution of many biological and chemical species. Numerical simulations show the onset of a traveling wave eventually replacing the host tissue with a fully vascularized tumor. The results of simulations agree with experimental measures of the vasculature density in tumors, even in the case of particularly hypoxic tumors.

On a Statistical Mechanics Approach to Some Problems of the Social Sciences

This work is a survey of some results on a statistical mechanics approach to the social sciences emerged in the last two decades. The pioneering work of Daniel McFadden, known as discrete choice theory, is interpreted in terms of a non-interacting model and extended along the lines of the Brock and Durlauf interacting systems. The generalization to the multi-populated model is presented and two specific case studies are reviewed with their phenomenological and theoretical analysis.

Microscopic approach to the macrodynamics of matter with broken symmetries

A unified set of hydrodynamic equations describing condensed phases of matter with broken continuous symmetries is derived using a generalization of the statistical-mechanical approach based on the local equilibrium distribution. The dissipativeless and dissipative parts of the current densities and the entropy production are systematically deduced in this approach by expanding in powers of the gradients of the macrofields. Green–Kubo formulas are obtained for all the linear transport coefficients. The consequences of microreversibility and spatial symmetries are investigated, leading to the prediction of cross effects resulting from Onsager–Casimir reciprocal relations. Crystalline solids and liquid crystals are potential examples of application. The approach is clarifying the links between the microscopic Hamiltonian dynamics and the thermodynamic and transport properties at the macroscale.

Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking

<p style='text-indent:20px;'>Behavior that a customer who has just arrived at a crowded queueing system leaves without joining the queue is known as the phenomenon of balking. Queueing systems with balking have been studied continually as one of significant subjects. In this paper, the theoretical approach for the steady-state analysis of the Markovian queueing systems with balking is considered based on the concept of the statistical mechanics. Here, it can be easily seen that the strength of balking is not constant but various in each queueing systems. Note that the strength of balking means how degree a customer who has just arrived at a crowded queueing system leaves without joining the queue. In our approach, under considering the difference of the strength of balking for each queueing systems, we have proposed a statistical mechanics model for analyzing the M/M/<inline-formula><tex-math id="M1">\begin{document}$ s $\end{document}</tex-math></inline-formula> queueing system with balking by introducing a parameter influencing the strength of balking. Further, we define a procedure for estimating the model parameter influencing the strength of balking. In addition, we consider a method of improving the performance of the M/M/<inline-formula><tex-math id="M2">\begin{document}$ s $\end{document}</tex-math></inline-formula> queueing system with balking by utilizing the statistical mechanics approach.</p>

Control of Transcription Initiation by Biased Thermal Fluctuations on Repetitive Genomic Sequences.

In the process of transcription initiation by RNA polymerase, promoter DNA sequences affect multiple reaction pathways determining the productivity of transcription. However, the question of how the molecular mechanism of transcription initiation depends on the sequence properties of promoter DNA remains poorly understood. Here, combining the statistical mechanical approach with high-throughput sequencing results, we characterize abortive transcription and pausing during transcription initiation by Escherichia coli RNA polymerase at a genome-wide level. Our results suggest that initially transcribed sequences, when enriched with thymine bases, contain the signal for inducing abortive transcription, whereas certain repetitive sequence elements embedded in promoter regions constitute the signal for inducing pausing. Both signals decrease the productivity of transcription initiation. Based on solution NMR and in vitro transcription measurements, we suggest that repetitive sequence elements within the promoter DNA modulate the nonlocal base pair stability of its double-stranded form. This stability profoundly influences the reaction coordinates of the productive initiation via pausing.

On the interaction of a solute atmosphere with a circular void in two-dimensional elasticity: A statistical mechanics approach

Solid solutions are of critical importance to many technological applications. While continuum frameworks have been developed and invoked in the past for studying behavior of solid solutions, recent studies point to the importance of image stresses in finite systems, and in particular, suggest that the homogenized self-stress field of the solutes, which enter the chemical potential in the prevalent continuum models, should be excluded in a correct treatment. The present work is concerned with the study of a solute atmosphere in the vicinity of a void in two-dimensional elasticity wherein the solute atmosphere is treated as a distribution of solute rods extended indefinitely normal to the plane of analysis. A statistical mechanics framework is adopted based on the Gibbs canonical ensemble, and the well-established path-independent integrals of solid mechanics are used for examining configurational forces on the void induced by the presence of a solute atmosphere. Numerical Monte Carlo simulations fully accounting for the interaction energy between the solute rods are used for calculation of the ensemble averages of the energy release rates associated with void translation and expansion, and semi-analytical solutions are further developed in the limit of dilute solute concentrations and large temperatures. Effect of interaction between the stress fields of a solute atmosphere and an external loading on the configurational forces is also investigated through both numerical and analytical calculations. It is shown that the interaction between the image stress field of a given concentration of solute rods and the stress field of an equi-biaxial loading results in an added contribution to the energy release rate of void expansion which does not fade away even when the solid dimensions grow indefinitely. Numerical simulations are further used to examine the effect of loading-induced solute redistribution on the energetic driving force for void expansion.

On the Solvation Thermodynamics Involving Species with Large Intermolecular Asymmetries: A Rigorous Molecular-Based Approach to Simple Systems with Unconventionally Complex Behaviors.

This work deals with the solvation thermodynamics of systems containing extreme intermolecular interaction asymmetries, such as binary mixtures comprising ideal gas species plus other real species, and analyzes the composition dependence of relevant solvation properties from a rigorous molecular-based theoretical viewpoint. It focuses on the effect of large intermolecular interaction asymmetries, involving simple model species for which we have available rigorous results, to advance our understanding of the true molecular-based origin of their unconventional solvation behavior. This effort involves a statistical mechanics approach to identify the thermodynamic constraints and microstructural manifestation underlying the nonmonotonous composition behavior of the species' partial molar properties, as well as to describe formally the explicit link between the intermolecular asymmetries and the observed anomalous behavior. The outcome of the analysis provides sound support to the depiction of the solvation of a weakly interacting solute in a real solvent, a scenario that facilitates the molecular thermodynamic interpretation of gas solvation characterized by low solubility and thermodynamic stability gaps. Moreover, this work illustrates direct comparisons between real aqueous systems and the ideal gas-aqueous system counterparts to highlight and discuss the usefulness of invoking the ideal gas species as a reference solute to gain understanding of the solvation of real gas species.

Statistical description of the thermodynamics of glass-forming liquids

The design of new glass products centers around connecting the structural features of a network to their macroscopic properties. Since glass-forming liquids lack long-range order, their local structure is described through statistical mechanical means. Though it is possible to determine the distribution of probable microstates through stochastic sampling, such techniques may be computationally inefficient. In this paper, we introduce a statistical mechanical approach to calculate the probability and degeneracy of each possible microstate for a given composition and temperature. These probability distributions are used to calculate the corresponding thermodynamic properties, including configurational entropy, configurational enthalpy, and configurational Gibbs free energy, for an arbitrary glass-forming liquid.

Statistics of geometric clusters in Potts model: statistical mechanics approach.

The percolation of Potts spins with equal values in Potts model on graphs (networks) is considered. The general method for finding the Potts clusters' size distributions is developed. It allows full description of percolation transition when a giant cluster of equal-valued Potts spins appears. The method is applied to the short-ranged q-state ferromagnetic Potts model on the Bethe lattices with the arbitrary coordination number z. The analytical results for the field-temperature percolation phase diagram of geometric spin clusters and their size distribution are obtained. The last appears to be proportional to that of the classical non-correlated bond percolation with the bond probability, which depends on temperature and Potts model parameters.

Energetics and kinetics of substrate analog-coupled staphylococcal nuclease folding revealed by a statistical mechanical approach

Protein conformational changes associated with ligand binding, especially those involving intrinsically disordered proteins, are mediated by tightly coupled intra- and intermolecular events. Such reactions are often discussed in terms of two limiting kinetic mechanisms, conformational selection (CS), where folding precedes binding, and induced fit (IF), where binding precedes folding. It has been shown that coupled folding/binding reactions can proceed along both CS and IF pathways with the flux ratio depending on conditions such as ligand concentration. However, the structural and energetic basis of such complex reactions remains poorly understood. Therefore, we used experimental, theoretical, and computational approaches to explore structural and energetic aspects of the coupled-folding/binding reaction of staphylococcal nuclease in the presence of the substrate analog adenosine-3',5'-diphosphate. Optically monitored equilibrium and kinetic data, combined with a statistical mechanical model, gave deeper insight into the relative importance of specific and Coulombic protein-ligand interactions in governing the reaction mechanism. We also investigated structural aspects of the reaction at the residue level using NMR and all-atom replica-permutation molecular dynamics simulations. Both approaches yielded clear evidence for accumulation of a transient protein-ligand encounter complex early in the reaction under IF-dominant conditions. Quantitative analysis of the equilibrium/kinetic folding revealed that the ligand-dependent CS-to-IF shift resulted from stabilization of the compact transition state primarily by weakly ligand-dependent Coulombic interactions with smaller contributions from specific binding energies. At a more macroscopic level, the CS-to-IF shift was represented as a displacement of the reaction "route" on the free energy surface, which was consistent with a flux analysis.

Fluctuation theory of immune response: A statistical mechanical approach to understand pathogen induced T-cell population dynamics.

In this period of intense interest in human immunity, we attempt here to quantify the immune response against pathogen invasion through T-cell population dynamics. Borrowing concepts from equilibrium statistical mechanics, we introduce a new description of the immune response function (IMRF) in terms of fluctuations in the population number of relevant biological cells (effector and regulatory T-cells). We use a coarse-grained chemical reaction network model (CG-CRNM) to calculate the number fluctuations and show that the response function derived as such can, indeed, capture the crossover observed in a T-cell driven immune response. We employ the network model to learn the effect of vitamin-D as an immunomodulator. We solve our CG-CRNM using a stochastic Gillespie algorithm. Depending on the effector T-cell concentration, we can classify immune regulation regimes into three categories: weak, strong, and moderate. The IMRF is found to behave differently in these three regimes. A damped cross-regulatory behavior found in the dynamics of effector and regulatory T-cell concentration in the diseased states correlates well with the same found in a cohort of patients with specific malignancies and autoimmune diseases. Importantly, the crossover from the weakly regulated steady state to the other (the strongly regulated) is accompanied by a divergence-like growth in the fluctuation of both the effector and the regulatory T-cell concentration, characteristic of a dynamic phase transition. We believe such steady-state IMRF analyses could help not only to phase-separate different immune stages but also aid in the valuable connection between autoimmunity, optimal vitamin-D, and consequences of immunosuppressive stress and malignancy.

A tight binding model of the bond dissociation mechanism in thermochromic leuco dyes

It has long been known that heat, UV light, or protonization will cause the bond in Leuco dye's to break. The underlying microscopic mechanism is a still a matter of discussion. It stands to reason that a local model should correctly predict the temperature behavior of the heat capacity at the thermochromic phase transition. Here we develop a tight binding (TB) model for the variation of the energy of the bond as it breaks as a function of temperature. We use a statistical mechanical approach to incorporate this variation in a partition function, calculate a free energy, and obtain the heat capacity of the phase transition. A distribution of such heat capacities is shown to fit the data for the leuco dye. The heat capacity distribution is compared to the experimentally determined heat capacity by differential scanning calorimetry (DSC).

Equilibrium solute segregation to matrix- θ′ precipitate interfaces in Al-Cu alloys from first principles

Particular combinations of solute atoms segregated to the interface of the Al and ${\ensuremath{\theta}}^{\ensuremath{'}}\text{\ensuremath{-}}{\mathrm{Al}}_{2}\mathrm{Cu}$ in Al-Cu alloys can help stabilize ${\ensuremath{\theta}}^{\ensuremath{'}}$ precipitates at high temperatures. Stabilization of such precipitates is determined by a combination of thermodynamics (including driving forces for coarsening and transformation and solute segregation tendencies) and kinetic effects (including solute diffusion and interfacial mobility in the presence of interfacial solute segregation). For some alloys such as recent Al-Cu-based alloys, multiple solutes segregate in significant quantities to interfaces, and solute-solute interactions at the interface are important, with multiple types of solutes competing for similar interfacial sites. To treat this situation, we develop and apply a statistical mechanics approach to calculate the temperature-dependent equilibrium solute atoms distribution near the coherent and semicoherent interfaces between the Al matrix and the ${\ensuremath{\theta}}^{\ensuremath{'}}\text{\ensuremath{-}}{\mathrm{Al}}_{2}\mathrm{Cu}$ precipitates. The developed approach is applied to the investigation of Si, Mn, and Zr segregation at the interface, as particular combinations of these elements affect the thermal stability of the ${\ensuremath{\theta}}^{\ensuremath{'}}$ precipitates. We demonstrate that because Si and Mn atoms segregate on the same semicoherent interface, the presence of Si reduces the concentration of Mn solutes at the interface. Si atoms preferably occupy the first layer of the interface and compete with Mn atoms for one type of particular sites in the layer. Mn atoms preferably occupy the second layer of the semicoherent interface, and the Mn-Mn interaction plays an important role in their distribution. Zr atoms mostly segregate on one of the two nonequivalent sites of the second layer of the coherent interface. Due to symmetry properties of the coherent interface, the calculations show that the segregation Zr of atoms to this interface will likely lead to the formation of ${\mathrm{L}1}_{2}$ ordered ${\mathrm{Al}}_{3}\mathrm{Zr}$ layer.

Maximum entropy approach to multivariate time series randomization

Natural and social multivariate systems are commonly studied through sets of simultaneous and time-spaced measurements of the observables that drive their dynamics, i.e., through sets of time series. Typically, this is done via hypothesis testing: the statistical properties of the empirical time series are tested against those expected under a suitable null hypothesis. This is a very challenging task in complex interacting systems, where statistical stability is often poor due to lack of stationarity and ergodicity. Here, we describe an unsupervised, data-driven framework to perform hypothesis testing in such situations. This consists of a statistical mechanical approach—analogous to the configuration model for networked systems—for ensembles of time series designed to preserve, on average, some of the statistical properties observed on an empirical set of time series. We showcase its possible applications with a case study on financial portfolio selection.

Statistical quantum mechanical approach to diatom-diatom capture dynamics and application to ultracold KRb + KRb reaction.

A general and rigorous quantum method is proposed for studying capture dynamics between two diatomic molecules in full dimensionality. By solving the time-independent Schrödinger equation with proper boundary conditions, this method is ideally suited for studying quantum dynamics of cold and ultracold reactions. To illustrate its applicability, the capture dynamics between ultracold KRb molecules is characterized in full six dimensions for the first time using a first-principles based long-range interaction potential. The calculated capture rates for collisions involving distinguishable and indistinguishable 40K87Rb molecules are in good agreement with the experiment and exhibit clear Wigner threshold behaviors. Predictions for ultracold collisions between internally excited 40K87Rb suggest minor changes in the loss rate, consistent with experimental observations in similar systems.

A Statistical Mechanics Approach to Systemic Risk under Bilateral Netting Agreements

We propose a statistical mechanics approach to the problem of financial network reconstruction and systemic risk when participants benefit from bilateral netting agreements. We apply physical reasoning to directly estimate individual financial liabilities from data on both total gross and net positions of total liabilities and total assets. We map this constrained network reconstruction problem to an energy-minimization one, to which we apply a Markov Chain Monte Carlo algorithmic to sample from this restricted space. These samples are then used to evaluate the impact of bilateral netting strategies to individual defaults and contagion. As an application, we employ this method to derivative networks and derive individual probabilities of default of banks. The comparison against popular alternative methods underlines the importance of restricting sample solutions to those compatible with the netting strategies of banks. We also provide an R package implementing our methodology.

A Statistical Mechanics Approach to the Study of Energy Use Behaviour

This paper proposes a statistical mechanical model for the energy conservation behaviour of energy users. The model was inspired by the multipopulation Curie-Weiss model and the discrete choice with social interaction model. We demonstrate that the model integrates the sociopsychological, the economic, and the energy technologist viewpoints to energy management, allowing us to subject these perspectives to an empirical study.

From the Kinetic Theory of Gases to the Kinetics of Rate Processes: On the Verge of the Thermodynamic and Kinetic Limits.

A variety of current experiments and molecular dynamics computations are expanding our understanding of rate processes occurring in extreme environments, especially at low temperatures, where deviations from linearity of Arrhenius plots are revealed. The thermodynamic behavior of molecular systems is determined at a specific temperature within conditions on large volume and number of particles at a given density (the thermodynamic limit): on the other side, kinetic features are intuitively perceived as defined in a range between the extreme temperatures, which limit the existence of each specific phase. In this paper, extending the statistical mechanics approach due to Fowler and collaborators, ensembles and partition functions are defined to evaluate initial state averages and activation energies involved in the kinetics of rate processes. A key step is delayed access to the thermodynamic limit when conditions on a large volume and number of particles are not fulfilled: the involved mathematical analysis requires consideration of the role of the succession for the exponential function due to Euler, precursor to the Poisson and Boltzmann classical distributions, recently discussed. Arguments are presented to demonstrate that a universal feature emerges: Convex Arrhenius plots (super-Arrhenius behavior) as temperature decreases are amply documented in progressively wider contexts, such as viscosity and glass transitions, biological processes, enzymatic catalysis, plasma catalysis, geochemical fluidity, and chemical reactions involving collective phenomena. The treatment expands the classical Tolman’s theorem formulated quantally by Fowler and Guggenheim: the activation energy of processes is related to the averages of microscopic energies. We previously introduced the concept of “transitivity”, a function that compactly accounts for the development of heuristic formulas and suggests the search for universal behavior. The velocity distribution function far from the thermodynamic limit is illustrated; the fraction of molecules with energy in excess of a certain threshold for the description of the kinetics of low-temperature transitions and of non-equilibrium reaction rates is derived. Uniform extension beyond the classical case to include quantum tunneling (leading to the concavity of plots, sub-Arrhenius behavior) and to Fermi and Bose statistics has been considered elsewhere. A companion paper presents a computational code permitting applications to a variety of phenomena and provides further examples.

From complex to simple : hierarchical free-energy landscape renormalized in deep neural networks

We develop a statistical mechanical approach based on the replica method to study the design space of deep and wide neural networks constrained to meet a large number of training data. Specifically, we analyze the configuration space of the synaptic weights and neurons in the hidden layers in a simple feed-forward perceptron network for two scenarios: a setting with random inputs/outputs and a teacher-student setting. By increasing the strength of constraints,~i.e. increasing the number of training data, successive 2nd order glass transition (random inputs/outputs) or 2nd order crystalline transition (teacher-student setting) take place layer-by-layer starting next to the inputs/outputs boundaries going deeper into the bulk with the thickness of the solid phase growing logarithmically with the data size. This implies the typical storage capacity of the network grows exponentially fast with the depth. In a deep enough network, the central part remains in the liquid phase. We argue that in systems of finite width N, the weak bias field can remain in the center and plays the role of a symmetry-breaking field that connects the opposite sides of the system. The successive glass transitions bring about a hierarchical free-energy landscape with ultrametricity, which evolves in space: it is most complex close to the boundaries but becomes renormalized into progressively simpler ones in deeper layers. These observations provide clues to understand why deep neural networks operate efficiently. Finally, we present some numerical simulations of learning which reveal spatially heterogeneous glassy dynamics truncated by a finite width N effect.