Framework Of Statistical Mechanics Research Articles

Signals of detailed balance violation in nonequilibrium stationary states: subtle, manifest, and extraordinary

The evolution of physical systems are often modeled by simple Markovian processes. When settled into stationary states, the probability distributions of such systems are time independent, by definition. However, they do not necessarily fall within the framework of equilibrium statistical mechanics. Instead, they may be non-equilibrium steady states (NESS). One distinguishing feature of NESS is the presence of time reversal asymmetry (TRA) and persistent probability current loops. These loops lead naturally to the notion of probability angular momenta, which play a role on the same footing as the noise covariance in stochastic processes. Illustrating with simulations of simple models and physical data, we present ways to detect these signals of TRA, from the subtle to the prominent.

Quantum versus classical quenches and the broadening of wave packets

The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.

Bayesian Mechanistic Inference, Statistical Mechanics, and a New Era for Monte Carlo.

On the one hand, much of computational chemistry is concerned with "bottom-up" calculations which elucidate observable behavior starting from exact or approximated physical laws, a paradigm exemplified by typical quantum mechanical calculations and molecular dynamics simulations. On the other hand, "top down" computations aiming to formulate mathematical models consistent with observed data, e.g., parametrizing force fields, binding or kinetic models, have been of interest for decades but recently have grown in sophistication with the use of Bayesian inference (BI). Standard BI provides an estimation of parameter values, uncertainties, and correlations among parameters. Used for "model selection," BI can also distinguish between model structures such as the presence or absence of individual states and transitions. Fortunately for physical scientists, BI can be formulated within a statistical mechanics framework, and indeed, BI has led to a resurgence of interest in Monte Carlo (MC) algorithms, many of which have been directly adapted from or inspired by physical strategies. Certain MC algorithms─notably procedures using an "infinite temperature" reference state─can be successful in a 5-20 parameter BI context which would be unworkable in molecular spaces of 103 coordinates and more. This Review provides a pedagogical introduction to BI and reviews key aspects of BI through a physical lens, setting the computations in terms of energy landscapes and free energy calculations and describing promising sampling algorithms. Statistical mechanics and basic probability theory also provide a reference for understanding intrinsic limitations of Bayesian inference with regard to model selection and the choice of priors.

A statistical mechanics framework for Bayesian deep neural networks beyond the infinite-width limit

A statistical mechanics framework for Bayesian deep neural networks beyond the infinite-width limit

A statistical mechanics framework for constructing nonequilibrium thermodynamic models

Abstract Far-from-equilibrium phenomena are critical to all natural and engineered systems, and essential to biological processes responsible for life. For over a century and a half, since Carnot, Clausius, Maxwell, Boltzmann, and Gibbs, among many others, laid the foundation for our understanding of equilibrium processes, scientists and engineers have dreamed of an analogous treatment of nonequilibrium systems. But despite tremendous efforts, a universal theory of nonequilibrium behavior akin to equilibrium statistical mechanics and thermodynamics has evaded description. Several methodologies have proved their ability to accurately describe complex nonequilibrium systems at the macroscopic scale, but their accuracy and predictive capacity is predicated on either phenomenological kinetic equations fit to microscopic data or on running concurrent simulations at the particle level. Instead, we provide a novel framework for deriving stand-alone macroscopic thermodynamic models directly from microscopic physics without fitting in overdamped Langevin systems. The only necessary ingredient is a functional form for a parameterized, approximate density of states, in analogy to the assumption of a uniform density of states in the equilibrium microcanonical ensemble. We highlight this framework’s effectiveness by deriving analytical approximations for evolving mechanical and thermodynamic quantities in a model of coiled-coil proteins and double-stranded DNA, thus producing, to the authors’ knowledge, the first derivation of the governing equations for a phase propagating system under general loading conditions without appeal to phenomenology. The generality of our treatment allows for application to any system described by Langevin dynamics with arbitrary interaction energies and external driving, including colloidal macromolecules, hydrogels, and biopolymers.

Ergodic observables in non-ergodic systems: The example of the harmonic chain

In the framework of statistical mechanics the properties of macroscopic systems are deduced starting from the laws of their microscopic dynamics. One of the key assumptions in this procedure is the ergodic property, namely the equivalence between time averages and ensemble averages. This property can be proved only for a limited number of systems; however, as proved by Khinchin (1949), weak forms of it hold even in systems that are not ergodic at the microscopic scale, provided that extensive observables are considered.Here we show in a pedagogical way the validity of the ergodic hypothesis, at a practical level, in the paradigmatic case of a chain of harmonic oscillators. By using analytical results and numerical computations, we provide evidence that this non-chaotic integrable system shows ergodic behavior in the limit of many degrees of freedom. In particular, the Maxwell–Boltzmann distribution turns out to fairly describe the statistics of the single particle velocity. A study of the typical time-scales for relaxation is also provided.

Optimal intercellular competition in senescence and cancer

Effective multicellularity requires both cooperation and competition between constituent cells. Cooperation involves sacrificing individual fitness in favour of that of the community, but excessive cooperation makes the community susceptible to senescence and ageing. Competition eliminates unfit senescent cells via natural selection and thus slows down ageing, but excessive competition makes the community susceptible to cheaters, as exemplified by cancer and cancer-like phenomena. These observations suggest that an optimal level of intercellular competition in a multicellular organism maximizes organismal vitality by delaying the inevitability of ageing. We quantify this idea using a statistical mechanical framework that leads to a generalized replicator dynamical system for the population of cells that changes their vitality and cooperation due to somatic mutations that make them susceptible to ageing and/or cancer. By accounting for the cost of cooperation and strength of competition in a minimal setting, we show that our model predicts an optimal value of competition that maximizes vitality and delays the inevitability of senescence or cancer. The results have implications for the design of strategies aimed at delaying ageing in biological, technical and social systems that exhibit similar processes.

Quantum field corrections to the equation of state of freely streaming matter in the Friedman–Lemaître–Robertson–Walker space-time

We calculate the energy density and pressure of a scalar field after its decoupling from a thermal bath in the spatially flat Friedman–Lemaître–Robertson–Walker space-time, within the framework of quantum statistical mechanics. By using the density operator determined by the condition of local thermodynamic equilibrium, we calculate the mean value of the stress-energy tensor of a real scalar field by subtracting the vacuum expectation value at the time of the decoupling. The obtained expressions of energy density and pressure involve corrections with respect to the classical free-streaming solution of the relativistic Boltzmann equation, which may become relevant even at long times.

A statistical mechanics framework for polymer chain scission, based on the concepts of distorted bond potential and asymptotic matching

To design increasingly tough, resilient, and fatigue-resistant elastomers and hydrogels, the relationship between controllable network parameters at the molecular level (bond type, non-uniform chain length, entanglement density, etc.) to macroscopic quantities that govern damage and failure must be established. Many of the most successful constitutive models for elastomers have been rooted in statistical mechanical treatments of polymer chains. Typically, such constitutive models have used variants of the freely jointed chain model with rigid links. However, since the free energy state of a polymer chain is dominated by enthalpic bond distortion effects as the chain approaches its rupture point, bond extensibility ought to be accounted for if the model is intended to capture chain rupture. To that end, a new bond potential is supplemented to the freely jointed chain model (as derived in the uFJC framework of Buche and Silberstein (2021) and Buche et al. (2022)), which we have extended to yield a tractable, closed-form model of single chain behavior that should be amenable to continuum-level constitutive model development. Inspired by the asymptotically matched uFJC model response in both the low/intermediate chain force and high chain force regimes, a simple, quasi-polynomial bond potential energy function is derived. This bond potential exhibits harmonic behavior near the equilibrium state and anharmonic behavior for large bond stretches tending to a characteristic energy plateau (akin to the Lennard-Jones and Morse bond potentials). Using this bond potential, approximate yet highly-accurate analytical functions for bond stretch and chain force dependent upon chain stretch are established. Then, using this polymer chain model, a stochastic thermal fluctuation-driven chain rupture framework is developed. This framework is based upon a force-modified tilted bond potential that accounts for distortional bond potential energy, allowing for the derivation and subsequent calculation of the dissipated chain scission energy. The cases of rate-dependent and rate-independent scission are accounted for throughout the rupture framework. The impact of Kuhn segment number on chain rupture behavior is also investigated. The model is fit to single chain mechanical response data collected from atomic force microscopy tensile tests for validation and to glean deeper insight into the molecular physics taking place. Due to their analytical nature, this polymer chain model and the associated rupture framework can in the future be implemented in finite element models accounting for fracture and fatigue in polydisperse elastomer networks.

Exact magnetic properties of two-dimensional spin-1 Cooper pairs

Exact magnetic properties of two-dimensional (2D) spin-1 Cooper pairs are investigated within the framework of quantum statistical mechanics. We derive an exact analytical expression of the magnetization of 2D spin-1 Cooper pairs, which involves a q-digamma function and a q-Pochhammer symbol in mathematics. The magnetization density m(T) is a function of temperature T, Landé factor g, number density σ, and magnetic field B. Firstly, we examine the evolution of the magnetization density m(T) with the Landé factor g. m(T) is negative when g < 1, and changes gradually to positive after g c . Here, is the critical value of the Landé factor g, at which . Secondly, we examine the evolution of the magnetization density m(T) with the temperature T. When g < 1, there are the two asymptotic lines for the magnetization density m(T). The low-temperature asymptotic line is nC s−1 and the high-temperature asymptotic line is . Finally, we examine the evolution of the magnetization density m(T) with the number density σ. No matter how large the Landé factor g is, there is a lower cutoff σ c of the number density σ. The lower cutoff number density is about cm−2. If , the Cooper pair system does not support a solution of the reduced chemical potential x. As the number density σ is increased, the Bose–Einstein condensation of 2D Cooper pairs is strengthened.

A predictive model for the thermomechanical melting transition of double stranded DNA.

By extending the classical Peyrard-Bishop model, we are able to obtain a fully analytical description for the mechanical response of DNA under stretching at variable values of temperature, number of base pairs and intrachains and interchains bonds stiffness. In order to compare elasticity and temperature effects, we first analyze the system in the zero temperature mechanical limit, important to describe several experimental effects including possible hysteresis. We then analyze temperature effects in the framework of equilibrium Statistical Mechanics. In particular, we obtain an analytical expression for the temperature-dependent melting force and unzipping assigned displacement in the thermodynamical limit, also depending on the relative stability of intra vs. inter molecular bonds. Such results coincide with the purely mechanical model in the limit of zero temperature and with the denaturation temperature that we obtain with the classical transfer integral method. Based on our analytical results, we obtain explicitly phase diagrams and cooperativity parameters, where also discreteness effect can be accounted for. The obtained results are successfully applied in reproducing the thermomechanical experimental melting of DNA and the response of DNA hairpins. Due to the generality of the model, exemplified in the proposed analysis of both overstretching and unzipping experiments, we argue that the proposed approach can be extended to other thermomechanically induced molecular melting phenomena. STATEMENT OF SIGNIFICANCE: We obtain a fully analytical description of the complex wiggly energy landscape of two stranded macromolecules under unzipping loading. Based on Equilibrium Statistical Mechanics, we describe the combined thermomechanical effects and the melting transition of double stranded molecules such as nucleic acids. This is proved by quantitatively predicting the experimental behavior of both melting of DNA and DNA hairpins opening. While analytical results have been previously attained under special conditions on the relative stiffness of the covalent vs. non-covalent bonds of the base pairs, our model is completely general in this respect, thus representing a tool in the perspective of the design at the molecular scale. We show that the obtained model can be fully inscribed in the theory of phase transitions giving a new interpretation of the thermomechanical behavior of double stranded molecules.

Thermodynamic architecture and conformational plasticity of GPCRs

G-protein-coupled receptors (GPCRs) are ubiquitous integral membrane proteins involved in diverse cellular signaling processes. Here, we carry out a large-scale ensemble thermodynamic study of 45 ligand-free GPCRs employing a structure-based statistical mechanical framework. We find that multiple partially structured states co-exist in the GPCR native ensemble, with the TM helices 1, 6 and 7 displaying varied folding status, and shaping the conformational landscape. Strongly coupled residues are anisotropically distributed, accounting for only 13% of the residues, illustrating that a large number of residues are inherently dynamic. Active-state GPCRs are characterized by reduced conformational heterogeneity with altered coupling-patterns distributed throughout the structural scaffold. In silico alanine-scanning mutagenesis reveals that extra- and intra-cellular faces of GPCRs are coupled thermodynamically, highlighting an exquisite structural specialization and the fluid nature of the intramolecular interaction network. The ensemble-based perturbation methodology presented here lays the foundation for understanding allosteric mechanisms and the effects of disease-causing mutations in GCPRs.

Network Analysis of Molecular Dynamics Sectors in the p53 Protein.

Design of allosteric regulators is an emergent field in the area of drug discovery holding promise for currently untreated diseases. Allosteric regulators bind to a protein in one location and affect a distant site. The ubiquitous presence of allosteric effectors in biology and the success of serendipitously identified allosteric compounds point to the potential they hold. Although the mechanism of transmission of an allosteric signal is not unequivocally determined, one hypothesis suggests that groups of evolutionarily covarying residues within a protein, termed sectors, are conduits. A long-term goal of our lab is to allosterically modulate the activity of proteins by binding small molecules at points of allosteric control. However, methods to consistently identify such points remain unclear. Sector residues on the surfaces of proteins are a promising source of allosteric targets. Recently, we introduced molecular dynamics (MD)-based sectors; MD sectors capitalize on covariance of motion, in place of evolutionary covariance. By focusing on motional covariance, MD sectors tap into the framework of statistical mechanics afforded by the Boltzmann ensemble of structural conformations comprising the underlying data set. We hypothesized that the method of MD sectors can be used to identify a cohesive network of motionally covarying residues capable of transmitting an allosteric signal in a protein. While our initial qualitative results showed promise for the method to predict sectors, that a network of cohesively covarying residues had been produced remained an untested assumption. In this work, we apply network theory to rigorously analyze MD sectors, allowing us to quantitatively assess the biologically relevant property of network cohesiveness of sectors in the context of the tumor suppressor protein, p53. We revised the methodology for assessing and improving MD sectors. Specifically, we introduce a metric to calculate the cohesive properties of the network. Our new approach separates residues into two categories: sector residues and non-sector residues. The relatedness within each respective group is computed with a distance metric. Cohesive sector networks are identified as those that have high relatedness among the sector residues which exceeds the relatedness of the residues to the non-sector residues in terms of the correlation of motions. Our major finding was that the revised means of obtaining sectors was more efficacious than previous iterations, as evidenced by the greater cohesion of the networks. These results are discussed in the context of the development of allosteric regulators of p53 in particular and the expected applicability of the method to the drug design field in general.

A statistical mechanics framework for immiscible and incompressible two-phase flow in porous media

We construct a statistical mechanics for immiscible and incompressible two-phase flow in porous media under local steady-state conditions based on the Jaynes maximum entropy principle. A cluster entropy is assigned to our lack of knowledge of, and control over, the fluid and flow configurations in the pore space. As a consequence, two new variables describing the flow emerge: The agiture, which describes the level of agitation of the two fluids, and the flow derivative, which is conjugate to the saturation. Agiture and flow derivative are the analogs of temperature and chemical potential in standard (thermal) statistical mechanics. The associated thermodynamics-like formalism reveals a number of hitherto unknown relations between the variables that describe the flow, including fluctuations. The formalism opens for new approaches to characterize porous media with respect to multi-phase flow for practical applications, replacing the simplistic relative permeability theory while still keeping the number of variables tractable.

Computing Thermodynamic Properties of Fluids Augmented by Nanoconfinement: Application to Pressurized Methane.

Nanoconfined fluids exhibit remarkably different thermodynamic behavior compared to the bulk phase. These confinement effects render predictions of thermodynamic quantities of nanoconfined fluids challenging. In particular, confinement creates a spatially varying density profile near the wall that is primarily responsible for adsorption and capillary condensation behavior. Significant fluctuations in thermodynamic quantities, inherent in such nanoscale systems, coupled to strong fluid-wall interactions give rise to this near-wall density profile. Empirical models have been proposed to explain and model these effects, yet no first-principles based formulation has been developed. We present a statistical mechanics framework that embeds such a coupling to describe the effect of the fluid-wall interaction in amplifying the near-wall density behavior for compressible gases at elevated pressures such as pressurized methane in confinement. We show that the proposed theory predicts accurately the adsorbed layer thickness as obtained with small-angle neutron scattering measurements. Furthermore, the predictions of density under confinement from the proposed theory are shown to be in excellent agreement with available experimental and atomistic simulations data for a range of temperatures for nanoconfined methane. While the framework is presented for evaluating the near-wall density, owing to its rigorous foundation in statistical mechanics, the proposed theory can also be generalized for predicting phase-transition and nonequilibrium transport of nanoconfined fluids.

Thermodynamic model of twisted bilayer graphene: Entropy matters

Twisted bilayer materials have attracted tremendous attention due to their unique and novel properties. In this work, we derive a thermodynamic model for twisted bilayer graphene (tBLG) within the framework of the classical statistical mechanics. The effect of interlayer twist is introduced by the Moiré-dependent out-of-plane deformation, based on which the twist associated Helmholtz free energy is quantified. Furthermore, the configuration entropy, reflecting the number of micro-states in Moiré unit-cells, is directly derived from both the Helmholtz free energy and the Boltzmann entropy equation with a clear physical interpretation. We show the configuration entropy of a tBLG relative to the AB-stacked bilayer graphene is proportional to the logarithmic function of the ratio of Moiré period ( a m ) and the lattice constant ( a ), i.e., S tBLG − S AB = 12 k B ln ( a m / a ) . Finally, based on the observation that the out-of-plane deformation follows the evolution of Moiré patterns, a possible dissipation mechanism in the interlayer sliding of tBLG is discussed. This work provides a theoretical guidance for studying the Moiré effect of incommensurate contact interfaces such as tribology.

Bose–Einstein condensation of two-dimensional spin-1 Cooper pairs in a uniform magnetic field

Bose–Einstein condensation of two-dimensional spin-1 Cooper pairs in a uniform magnetic field is investigated within the framework of quantum statistical mechanics. We derive an exact analytical solution of two-dimensional spin-1 Cooper pairs in a uniform magnetic field, which involves a q-digamma function in mathematics. In two dimensions, it is found that the critical temperature and condensate fraction are monotonically increasing functions of magnetic field B, Landé factor g, and number density σ. Our investigation in two dimensions provides an example of the magnetic-field-tuned insulator-superconductor quantum phase transition.

Temporal variations of the probability distribution of voronoi cells generated by earthquake epicenters

The area of the cells of Voronoi tessellations has been modelled through different probability distributions among which the most promising are the generalized gamma and tapered Pareto distributions. In particular the latter has been used to model times and distances between successive earthquakes besides area and perimeter of cells generated by earthquake epicenters. In the framework of nonextensive statistical mechanics applied in geophysics, variables like seismic moment, inter-event time or Euclidean distance between successive earthquakes or length of faults in a given region have been studied through the so-calledq-exponential distributions obtained by maximizing the Tsallis entropy under suitable conditions. These distributions take also the name of generalized Pareto distributions in the context of the limit distributions of excesses. In this work we consider the spatial distribution of a set of earthquakes and its temporal variations by modelling the area of Voronoi cells generated by the epicenters through a generalized Pareto distribution. Following the Bayesian paradigm we analyze the recent seismicity of the central Italy and we compare the posterior marginal likelihood of the aforementioned distributions in shifting time windows. We point out that the best fitting distribution varies over time and the trend of all three distributions converges to that of the exponential distribution in the preparatory phase for the mainshock.

Cell reorientation on a cyclically strained substrate.

Cyclic strain avoidance, the phenomenon of cell and cytoskeleton alignment perpendicular to the direction of cyclic strain of the underlying 2D substrate, is an important characteristic of the adherent cell organization. This alignment has typically been attributed to the stress-fiber reorganization although observations clearly show that stress-fiber reorganization under cyclic loading is closely coupled to cell morphology and reorientation of the cells. Here, we develop a statistical mechanics framework that couples the cytoskeletal stress-fiber organization with cell morphology under imposed cyclic straining and make quantitative comparisons with observations. The framework accurately predicts that cyclic strain avoidance stems primarily from cell reorientation away from the cyclic straining rather than cytoskeletal reorganization within the cell. The reorientation of the cell is a consequence of the cell lowering its free energy by largely avoiding the imposed cyclic straining. Furthermore, we investigate the kinetics of the cyclic strain avoidance mechanism and demonstrate that it emerges primarily due to the rigid body rotation of the cell rather than via a trajectory involving cell straining. Our results provide clear physical insights into the coupled dynamics of cell morphology and stress-fibers, which ultimately leads to cellular organization in cyclically strained tissues.

Gradient dynamics in reinforcement learning.

Despite the success achieved by the analysis of supervised learning algorithms in the framework of statistical mechanics, reinforcement learning has remained largely untouched by physicists. Here we move towards closing the gap by analyzing the dynamics of the policy gradient algorithm. For a convex problem, namely the k-armed bandit, we show that the learning dynamics obeys a drift-diffusion motion described by a Langevin equation, the coefficients of which can be tuned by the learning rate. We explore the striking similarity between our Langevin equationand the Kimura equation, describing genotypes evolution. Furthermore, we propose a mapping between a nonconvex reinforcement learning setting describing multiple joints of a robotic arm and a disordered system, namely a p-spin glass. This mapping enables us to show how the learning rate acts as an effective temperature and thus is capable of smoothing rough landscapes, corroborating what is displayed by the drift-diffusive description and paving the way for physics-inspired algorithmic optimization based on annealing procedures in disordered systems.