Framework Of Statistical Mechanics Research Articles

Bose-Einstein condensation of ideal photons in a one-dimensional barrel cavity

Our experimental scheme is based on a barrel optical microresonator filled with a dye solution. The barrel mirror provides a confining potential, a chemical potential, and an effective mass for a photon, making the system formally equivalent to a one-dimensional gas of harmonically trapped, number-conserving, and massive bosons. Within the framework of quantum statistical mechanics, we propose an exact analytical solution to the problem of Bose-Einstein condensation in harmonically trapped, one-dimensional, and ideal photons. It is found that the photon number of vapor is characterized by an analytical function, which involves a $q$-digamma function in mathematics. The numerical calculation of the analytical solution gives many interesting results. In the thermodynamic limit, the analytical expressions of the critical temperature and the condensate fraction are derived. We find that the spectral radiance of a one-dimensional barrel cavity has a sharp peak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature.

Statistical Mechanical Framework for Predicting Cellular Responses from Single-Cell Data

Developments in single-cell analysis techniques have provided detailed resolution on cellular component copy number and variation within a cell population. These data provide a probability distribution for all possible states of the cell, as determined by the measured component copy number per cell. We have developed a statistical mechanical framework that uses single-cell protein population data to model the effective interactions of the measured proteins. This method enables us to predict ensemble changes in cellular components under a variety of perturbations. This framework uses Bayesian inference to compare given models or descriptions of the system, as well as determine the uncertainty in the parameter estimations given the data. We have applied this methodology to study the impact of changes in oxygen partial pressure on the behavior of glioblastoma multiforme (GBM) cancer cells. Our framework shows how we can determine the coupling of oxygen concentration changes to the measured proteins and their underlying effective interactions.

Statistical Physics of the Causality and Energetics in Allosteric Communication

Allostery is one of the pervasive mechanisms through which proteins in living systems carry out enzymatic activity, cell signaling, and metabolism control. The effective modeling of protein function regulation requires a synthesis of the thermodynamic and structural views of allostery. We present here a structure-based statistical mechanical model of allostery, allowing to observe causality of communication between regulatory and functional sites and to estimate per-residue free energy changes. Based on the notions of ligand free and ligand occupied systems in the context of protein harmonic representation, corresponding sets of characteristic normal modes are obtained and used as inputs for the allosteric potential. The latter quantifies mean elastic work exerted on a residue because of the local motion of its neighbors. Subsequently, in a statistical mechanical framework the entropic contribution to allosteric free energy of a residue is directly calculated from the comparison of the conformational ensembles in the ligand free and occupied systems. The method is a systematic approach for analyzing the energetics of allosteric communication based on a single structure, and its feasibility was tested on a variety of allosteric proteins particularly heterogeneous in terms of size, topology, and degree of oligomerization. The allosteric free energy calculations show diversity of ways and complexity of scenarios existing in the phenomenology of allosteric causality and communication. Achieving the discriminative power between agonist and antagonists and developing computational techniques for on-demand design of desired allosteric effectors are among the major goals of future theoretical studies towards allosteric drug design.

Equilibrium Fluctuation Relations for Gating Charge in Voltage Sensitive Membrane Proteins: A Microscopic Capacitor Model Approach

We present a novel statistical mechanical framework for the fundamental free energy relationships in the voltage gated ion channels, which have been used to quantify thermodynamics of voltage activation. The framework is based on the equilibrium fluctuation relation that makes a connection between the two forward and backward probability densities of voltage activation. The application of linear response approximation to the protein/membrane/electrolyte system yields the equilibrium fluctuation-response relation for gating charge in a way that best reflects voltage sensitivity in ion channels. The theoretical framework is applied to modeling Ci-VSD activation, using our coarse-grained model of voltage coupling that include electrode, electrolyte, membrane and protein. Our CG model reproduces the observed gating charge of Ci-VSD activation in several different perspectives, offering a unique viewpoint on the nature of gating charge. The energetics of CG model yields capacitor-like voltage dependent free energy parabolas, providing the equilibrium free energy difference and the free energy barrier between the resting and the activated states of Ci-VSD. Significantly, the voltage dependent free energies allow for a direct evaluation of the Q-V curve (along with the voltage dependent capacitance) that is in a good agreement with the observed measurement of Ci-VSD voltage activation. Overall, our framework bring new perspectives to the thermodynamic models of voltage activation in voltage-sensitive membrane proteins, offering an intuitive and quantitative framework for a better understating of the structure-function correlations of voltage gating in ion channels as well as electrogenic phenomena in ion pumps and transporters. Our framework also provides an explicit connection between the CG model of voltage coupling and the macroscopic continuum model, offering a new microscopic perspective to the capacitor model of the treatment of ion channels and transporters.

Canonical ensemble in non-extensive statistical mechanics

The framework of non-extensive statistical mechanics, proposed by Tsallis, has been used to describe a variety of systems. The non-extensive statistical mechanics is usually introduced in a formal way, using the maximization of entropy. In this paper we investigate the canonical ensemble in the non-extensive statistical mechanics using a more traditional way, by considering a small system interacting with a large reservoir via short-range forces. The reservoir is characterized by generalized entropy instead of the Boltzmann–Gibbs entropy. Assuming equal probabilities for all available microstates we derive the equations of the non-extensive statistical mechanics. Such a procedure can provide deeper insight into applicability of the non-extensive statistics.

Estimation of Solvation Entropy and Enthalpy via Analysis of Water Oxygen-Hydrogen Correlations.

A statistical-mechanical framework for estimation of solvation entropies and enthalpies is proposed, which is based on the analysis of water as a mixture of correlated water oxygens and water hydrogens. Entropic contributions of increasing order are cast in terms of a Mutual Information Expansion that is evaluated to pairwise interactions. In turn, the enthalpy is computed directly from a distance-based hydrogen bonding energy algorithm. The resulting expressions are employed for grid-based analyses of Molecular Dynamics simulations. In this first assessment of the methodology, we obtained global estimates of the excess entropy and enthalpy of water that are in good agreement with experiment and examined the method's ability to enable detailed elucidation of solvation thermodynamic structures, which can provide valuable knowledge toward molecular design.

Accelerating molecular simulations of proteins using Bayesian inference on weak information

Atomistic molecular dynamics (MD) simulations of protein molecules are too computationally expensive to predict most native structures from amino acid sequences. Here, we integrate "weak" external knowledge into folding simulations to predict protein structures, given their sequence. For example, we instruct the computer "to form a hydrophobic core," "to form good secondary structures," or "to seek a compact state." This kind of information has been too combinatoric, nonspecific, and vague to help guide MD simulations before. Within atomistic replica-exchange molecular dynamics (REMD), we develop a statistical mechanical framework, modeling using limited data with coarse physical insight(s) (MELD + CPI), for harnessing weak information. As a test, we apply MELD + CPI to predict the native structures of 20 small proteins. MELD + CPI samples to within less than 3.2 Å from native for all 20 and correctly chooses the native structures (<4 Å) for 15 of them, including ubiquitin, a millisecond folder. MELD + CPI is up to five orders of magnitude faster than brute-force MD, satisfies detailed balance, and should scale well to larger proteins. MELD + CPI may be useful where physics-based simulations are needed to study protein mechanisms and populations and where we have some heuristic or coarse physical knowledge about states of interest.

Exact analytical study of ideal Bose atoms in a one-dimensional harmonic trap

.Within the framework of quantum statistical mechanics, we have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) in harmonically trapped, one-dimensional and ideal atoms. It is found that the number of atoms of vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. The results of the numerical calculation of the analytical solution agree completely with the previous experimental results in the BEC of harmonically trapped, one-dimensional and ideal atoms. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit.

Creep rupture of fiber bundles: A molecular dynamics investigation.

The creep deformation and eventual breaking of polymeric samples under a constant tensile load F is investigated by molecular dynamics based on a particle representation of the fiber bundle model. The results of the virtual testing of fibrous samples consisting of 40000 particles arranged on Nc=400 chains reproduce characteristic stages seen in the experimental investigations of creep in polymeric materials. A logarithmic plot of the bundle lifetime τ versus load F displays a marked curvature, ruling out a simple power-law dependence of τ on F. A power law τ∼F-4, however, is recovered at high load. We discuss the role of reversible bond breaking and formation on the eventual fate of the sample and simulate a different type of creep testing, imposing a constant stress rate on the sample up to its breaking point. Our simulations, relying on a coarse-grained representation of the polymer structure, introduce new features into the standard fiber bundle model, such as real-time dynamics, inertia, and entropy, and open the way to more detailed models, aiming at material science aspects of polymeric fibers, investigated within a sound statistical mechanics framework.

Enhanced Sampling of an Atomic Model with Hybrid Nonequilibrium Molecular Dynamics-Monte Carlo Simulations Guided by a Coarse-Grained Model.

Molecular dynamics (MD) trajectories based on a classical equation of motion provide a straightforward, albeit somewhat inefficient approach, to explore and sample the configurational space of a complex molecular system. While a broad range of techniques can be used to accelerate and enhance the sampling efficiency of classical simulations, only algorithms that are consistent with the Boltzmann equilibrium distribution yield a proper statistical mechanical computational framework. Here, a multiscale hybrid algorithm relying simultaneously on all-atom fine-grained (FG) and coarse-grained (CG) representations of a system is designed to improve sampling efficiency by combining the strength of nonequilibrium molecular dynamics (neMD) and Metropolis Monte Carlo (MC). This CG-guided hybrid neMD-MC algorithm comprises six steps: (1) a FG configuration of an atomic system is dynamically propagated for some period of time using equilibrium MD; (2) the resulting FG configuration is mapped onto a simplified CG model; (3) the CG model is propagated for a brief time interval to yield a new CG configuration; (4) the resulting CG configuration is used as a target to guide the evolution of the FG system; (5) the FG configuration (from step 1) is driven via a nonequilibrium MD (neMD) simulation toward the CG target; (6) the resulting FG configuration at the end of the neMD trajectory is then accepted or rejected according to a Metropolis criterion before returning to step 1. A symmetric two-ends momentum reversal prescription is used for the neMD trajectories of the FG system to guarantee that the CG-guided hybrid neMD-MC algorithm obeys microscopic detailed balance and rigorously yields the equilibrium Boltzmann distribution. The enhanced sampling achieved with the method is illustrated with a model system with hindered diffusion and explicit-solvent peptide simulations. Illustrative tests indicate that the method can yield a speedup of about 80 times for the model system and up to 21 times for polyalanine and (AAQAA)3 in water.

The Generalized Reaction Rate Theories in the Systems with Power-Law Distributions

The calculations of reaction rate are important in studying the different processes in physics, chemistry, biology, engineering etc. There exist many theories of reaction rate, such as transition state theory, collision theory, and unimolecular reaction theory. One key assumption in these theories is that thermodynamic equilibrium prevails throughout the entire system studied for all degrees of freedom. According to the Boltzmann-Gibbs statistical mechanics, a Maxwell-Boltzmann distribution whose form is exponential law holds in the whole time. This exponential law form of reaction rate is usually called Arrhenius behavior in chemistry. However, plenty of experimental results show non-Arrhenius behavior, such as power-law behavior. Besides, in reacting systems, the processes of evolution from one meta-stable state to another neighboring metastable state happen all the time, therefore the equilibrium assumption would be quite far-fetched. In these cases, current theories are no longer valid, and they can not even serve as a conceptual guide for understanding the critical factors that determine rates. Therefore, providing corresponding theoretical description for non-Arrhenius behavior or power-law behavior becomes urgent. In this general paper, generalized reaction rate theories with power-law distributions in the framework of nonextensive statistical mechanics were discussed based on experimental and observing facts.

A statistical mechanics framework for the large-scale structure of turbulent von Kármán flows

In the present paper, recent experimental results on large-scale coherent steady states observed in experimental von Kármán flows are revisited from a statistical mechanics perspective. The latter is rooted on two levels of description. We first argue that the coherent steady states may be described as the equilibrium states of well-chosen lattice models, which can be used to define global properties of von Kármán flows, such as their temperatures. The equilibrium description is then enlarged, in order to reinterpret a series of results about the stability of those steady states and their susceptibility to symmetry breaking, in the light of a deep analogy with the statistical theory of ferro-magnetism.

Peridynamic theory of solids from the perspective of classical statistical mechanics

In this paper the classical statistical mechanics has been explored in order to develop statistical mechanical framework for peridynamics. Peridynamic equation of motion is known as upscaled Newton’s equation. The peridynamic system consists of finite number of nonlocally interacting particles at nano and meso scales. This particle representation of peridynamics can be treated in terms of classical statistical mechanics. Hence, in this work the phase space is constructed based on the PD particle from their evolving momentum pi and positions xi. The statistical ensembles are derived by defining appropriate partition functions. The algorithms for NVE and NPH implemented in the classical molecular dynamics are revisited for equilibrium peridynamic models. The current work introduces Langevin dynamics to the peridynamic theory through fluctuation–dissipation principle. This introduces a heat bath to the peridynamic system which eliminates the ambiguity with the role of temperature in a peridynamic system. Finally, it was seen that the homogenization of a peridynamic model with finite number of particles approaches to a conventional continuum model. The upscaled non-equilibrium peridynamics has potential applications in modeling wide variety of multiscale–multiphysics problems from nano to macro scale or vice versa.

Identities for Entropy Change Associated with the Time-Evolution of an Open System

A general relation between entropy and an evolutionary superoperator is derived based on the theory of the real-time formulation. The formulation establishing the relation relies only on the framework of quantum statistical mechanics and the standard definition of the von Neumann entropy. Applying the theory of the imaginary-time formulation, a similar relation is obtained for the entropy change due to the change in reservoir temperatures. To show the usefulness of these formulas, we derived the expression for the entropy production induced by some dissipation in an open quantum system as the exemplary model system.

Experimental observation of a generalized Gibbs ensemble.

The description of the non-equilibrium dynamics of isolated quantum many-body systems within the framework of statistical mechanics is a fundamental open question. Conventional thermodynamical ensembles fail to describe the large class of systems that exhibit nontrivial conserved quantities, and generalized ensembles have been predicted to maximize entropy in these systems. We show experimentally that a degenerate one-dimensional Bose gas relaxes to a state that can be described by such a generalized ensemble. This is verified through a detailed study of correlation functions up to 10th order. The applicability of the generalized ensemble description for isolated quantum many-body systems points to a natural emergence of classical statistical properties from the microscopic unitary quantum evolution.

Statistical physical theory of mode-locking laser generation with a frequency comb

A study of the Mode-locking lasing pulse formation in closed cavities is presented within a statistical mechanical framework where the onset of laser coincides with a thermodynamic phase transition driven by the optical power pumped into the system. Electromagnetic modes are represented by classical degrees of freedom of a Hamiltonian model at equilibrium in an effective ensemble corresponding to the stationary laser regime. By means of optimized Monte Carlo numerical simulations, the system properties are analyzed varying mode interaction dilution, gain profile and number of modes. Novel properties of the resulting mode-locking laser phase are presented, not observable by previous mean-field approaches. For strong dilution of the nonlinear interaction network, power condensation occurs as the whole optical intensity is taken by a few electromagnetic modes, whose number does not depend on the size of the system. For all reported cases laser thresholds, intensity spectra, and ultra-fast electromagnetic pulses are computed.

Finite-temperature non-equilibrium quasi-continuum analysis of nanovoid growth in copper at low and high strain rates

We study dynamic nanovoid growth in copper single crystals under prescribed volumetric strain rates ranging from moderate (∊̇=105s-1) to high (∊̇=1010s-1). We gain access to lower strain rates by accounting for thermal vibrations in an entropic sense within the framework of maximum-entropy non-equilibrium statistical mechanics. We additionally account for heat conduction by means of empirical atomic-level kinetic laws. The resulting mean trajectories of the atoms are smooth and can be integrated implicitly using large time steps, greatly in excess of those required by molecular dynamics. We also gain access to large computational cells by means of spatial coarse-graining using the quasicontinuum method. On this basis, we identify a transition, somewhere between 107 and 108s−1, between two regimes: a quasistatic regime characterized by nearly isothermal behavior and low dislocation velocities; and a dynamic regime characterized by nearly adiabatic conditions and high dislocation velocities. We also elucidate the precise mechanisms underlying dislocation emission from the nanovoids during cavitation. We additionally investigate the sensitivity of the results of the analysis to the choice of interatomic potential by comparing two EAM-type potentials.

Introduction to special section on Origins and Properties of Kappa Distributions: Statistical Background and Properties of Kappa Distributions in Space Plasmas

AbstractEmpirical kappa distributions provide a straightforward replacement of the Maxwell distribution for systems out of thermal equilibrium such as space plasmas. Kappa distributions have become increasingly widespread across space physics with the number of relevant publications following, remarkably, an exponential growth rate. However, a breakthrough in the field came with the connection of kappa distributions with the framework of nonextensive statistical mechanics. This introductory paper clarifies fundamental physical concepts and provides mathematical formulations of the theory of kappa distributions, which are a consequence of the connection of kappa distributions with a solid statistical background. Among others, the paper presents the existence of a consistent definition of temperature in systems out of thermal equilibrium described by kappa distributions, the physical meaning of the kappa index, and the formulation of the kappa distribution of a Hamiltonian. In addition, the paper examines the most frequent values of kappa indices in space plasmas. Statistical analysis reveals trends between the characteristic values of density, temperature, and kappa index of space plasmas. Finally, understanding the kinetic interpretation of the temperature as the mean kinetic energy, and of the kappa index as the correlation of kinetic energies, helps to develop all the possible formulations of isotropic/anisotropic kappa distributions.

Thermodynamic Insights in the Separation of Cellulose/Hemicellulose Components from Lignocellulosic Biomass Using Ionic Liquids

This work reports on the ionic liquid (IL) based separation of cellulose/hemicellulose from lignocellulosic biomass, by means of macroscale predictions using the COnductor like Screening MOdel for Real Solvents (COSMO-RS) model that is based on a statistical mechanical framework. For the benchmarking studies the experimental infinite dilution activity coefficient values for 13 components were predicted in 1-alkyl-3-methylimidazolium bis{(trifluoromethyl)sulphonyl}imide with an average absolute deviation (%AAD) of 15 %. Further, the solid–liquid equilibria of glucose, fructose, galactose, and xylose in [EMIM][EtSO4] and Aliquat®336 were predicted successfully with 15 % root-mean-square deviation. Afterwards, the selectivities were predicted at infinite dilution for cellulose/hemicellulose in 1,156 ILs with a combination of 34 cations and 34 anions. Based on these values, the ammonium-based ILs 1-methyl-4-aminotriazolium hexafluorophosphate [14MATAZ][PF6] and 4-aminotriazolium bis(oxalato(2)borate) [4ATAZ][BOB] were found to be good candidates for cellulose and hemicellulose extraction, respectively. The ILs selected using the COSMO-RS methodology were then studied qualitatively in terms of interaction energies and HOMO–LUMO energy gap. The binding energy for [1-methyl-4-aminotriazolium][PF6] + cellulose and [4-aminotriazolium][BOB] + cellulose systems are higher as compared with IL + cellulose + hemicellulose, indicating greater stability.

Devising a protocol-related statistical mechanics framework for granular materials.

Devising a statistical mechanics framework for jammed granular materials is a challenging task as those systems do not share some important properties required to characterize them with statistical thermodynamics tools. In a recent paper [Asenjo et al. Phys. Rev. Lett. 112, 098002 (2014)], a new definition of a granular entropy, which puts the protocol used to generate the packings at its roots, has been proposed. Following up these results, it is shown that the protocol used in Asenjo et al. can be recast as a canonical ensemble with a particular value of the temperature. Signature of gaussianity for large system sizes strongly suggests an asymptotic equivalence with a corresponding microcanonical ensemble where jammed states with certain basin volumes are sampled uniformly. We argue that this microcanonical ensemble is not Edwards's microcanonical ensemble and generalize this argument to other protocols.