Statistical Mechanical Theory Research Articles

The effect of magnetic field on the orientational ordering of easy magnetization axes in superparamagnetic nanoparticles

The active use of nanoparticles in modern biomedical applications and technologies requires reliable theoretical models capable of predicting physical properties. The effect of a static magnetic field on the orientational ordering of the easy magnetization axes in an ensemble of superparamagnetic nanoparticles (SNPs) is studied using a combination of statistical-mechanical theory and computer simulations. The magnetic dipole-dipole interaction between particles is taken into account within the framework of the modified mean-field approach. The alignment distributions of the easy magnetization axes of the SNPs and the static magnetic characteristics of the ferrocomposite are analyzed. The presence of even weak interparticle magneto-dipole interactions results in a notable enhancement of the magnetic susceptibility of the ferrocomposite.

Addendum on ‘Entropy uniqueness determines temperature’

We discuss how to apply the proposed minimum entropy deviation criterion when there is a measurable property P for which there is a statistical mechanical theory as a function of temperature T. When P m can be measured more accurately than heat flow dQ, then P(T) can be inverted to obtain accurate values for T(P); following this, the minimum rms entropy deviation criterion using T(P) then can be used to refine measurements of dQ. When dQ is more accurately measurable than P, then measurement of dQ and the minimum rms entropy deviation criterion provides a means to obtain T, which can then be used to obtain more accurate values of P(T).

Theoretical study of kinetic arrest, shear elastic modulus, and yielding in simple biphasic colloidal mixtures.

Based on integrating microscopic statistical mechanical theories for structure and ideal kinetic arrest at the naive mode coupling level, we study dynamic localization, the linear elastic shear modulus, applied stress induced modulus softening, and the absolute yielding of simple biphasic binary mixtures composed of equal diameter hard and attractive spheres. The kinetic arrest map is a rich function of total packing fraction, strength of attraction, and mixture composition. The gel to attractive ideal glass transition, the degree of glass melting re-entrancy, and the crossover boundary separating repulsive glasses from attractive glasses vary with the mixture composition. Exponential and/or apparent (high) power law dependences of the elastic shear modulus on the total packing fraction are predicted with effective exponents or exponential prefactors that are sensitive to mixture composition and location in the kinetic arrest map. An analysis of the effective mean square force on a tagged particle that induces dynamic localization reveals a compensation effect between structural correlations and degree of particle localization, resulting in the emergence of a weaker dependence of the shear modulus on mixture composition at very high attraction strengths. Based on a microrheologically inspired formulation of how external stress weakens particle localization and the shear modulus, we analyze mechanical-induced modulus softening and absolute yielding, defined as a discontinuous solid-to-fluid stress-induced transition that can occur in either one or two steps. Estimates of the corresponding yield strains predict that the binary mixture becomes more brittle with increasing sticky particle composition and/or attraction strength.

Entropy of the Canonical Occupancy (Macro) State in the Quantum Measurement Theory.

The paper analyzes the probability distribution of the occupancy numbers and the entropy of a system at the equilibrium composed by an arbitrary number of non-interacting bosons. The probability distribution is obtained through two approaches: one involves tracing out the environment from a bosonic eigenstate of the combined environment and system of interest (the empirical approach), while the other involves tracing out the environment from the mixed state of the combined environment and system of interest (the Bayesian approach). In the thermodynamic limit, the two coincide and are equal to the multinomial distribution. Furthermore, the paper proposes to identify the physical entropy of the bosonic system with the Shannon entropy of the occupancy numbers, fixing certain contradictions that arise in the classical analysis of thermodynamic entropy. Finally, by leveraging an information-theoretic inequality between the entropy of the multinomial distribution and the entropy of the multivariate hypergeometric distribution, Bayesianism of information theory and empiricism of statistical mechanics are integrated into a common "infomechanical" framework.

Cage occupancies of CH4, CO2, and Xe hydrates: Mean field theory and grandcanonical Monte Carlo simulations.

We propose a statistical mechanical theory for the thermodynamic stability of clathrate hydrates, considering the influence of the guest-guest interaction on the occupancies of the cages. A mean field approximation is developed to examine the magnitude of the influence. Our new method works remarkably well, which is manifested by two sorts of grandcanonical Monte Carlo (GCMC) simulations. One is full GCMC, and the other is designed in the present study for clathrate hydrates, called lattice-GCMC, in which each guest can be adsorbed at one of the centers of the cage. In the latter simulation, only the guest-guest interaction is explicitly treated, incorporating the host-guest interaction into the free energy of the cage occupation without other guests. Critical phenomena for guest species, such as large density fluctuations, are observed when the temperature is low or the guest-guest interaction is strong.

Harmonic models and molecular dynamics simulations of isomorph behavior of Lennard-Jones fluids: Excess entropy and high temperature limiting behavior

Henchman’s approximate harmonic model of liquids is extended to predict the thermodynamic behavior along lines of constant excess entropy (“isomorphs”) in the liquid and supercritical fluid regimes of the Lennard-Jones (LJ) potential phase diagram. Simple analytic expressions based on harmonic cell models of fluids are derived for the isomorph lines, one accurate version of which only requires as input parameters the average repulsive and attractive parts of the potential energy per particle at a single reference state point on the isomorph. The new harmonic cell routes for generating the isomorph lines are compared with those predicted by the literature molecular dynamics (MD) methods, the small step MD method giving typically the best agreement over a wide density and temperature range. Four routes to calculate the excess entropy in the MD simulations are compared, which includes employing Henchman’s formulation, Widom’s particle insertion method, thermodynamic integration, and parameterized LJ equations of state. The thermodynamic integration method proves to be the most computationally efficient. The excess entropy is resolved into contributions from the repulsive and attractive parts of the potential. The repulsive and attractive components of the potential energy, excess Helmholtz free energy, and excess entropy along a fluid isomorph are predicted to vary as ∼T−1/2 in the high temperature limit by an extension of classical inverse power potential perturbation theory statistical mechanics, trends that are confirmed by the MD simulations.

Perturbation theory in a microcanonical ensemble

The microcanonical ensemble is a natural starting point of statistical mechanics. However, when it comes to perturbation theory in statistical mechanics, mainly the canonical and grand canonical ensembles have been used. In this article we show how the microcanonical ensemble can be directly used to carry out perturbation theory for both non-interacting and interacting systems. We obtain the first non-trivial order answers for the specific heat of anharmonic oscillators and for the virial expansion in real gases. They are in exact agreement with the results obtained from the canonical ensemble. In addition, we show how crossover functions for the specific heat of anharmonic oscillators can be constructed using a microcanonical ensemble and also how the subsequent terms of the virial expansion can be obtained. We find that if we consider quantum free particles in a one-dimensional box of extension L, then both the ensembles give an unusual result for the first correction to the specific heat in the high temperature limit.

Inelastic mechanical behaviour of an additively manufactured titanium alloy: a statistical continuum mechanics theory perspective

Statistical continuum mechanics theory was used to simulate the inelastic stress of polycrystalline materials using two-point statistics. For the experimental part, the Electron beam melting (EBM) technique (Arcam EBM Q10 additive machine) was used to fabricate cylindrical rods of Ti-6Al-4V both in horizontal and vertical directions. Electron backscatter diffraction (EBSD) technique was employed to achieve statistically reliable orientation maps of vertically and horizontally printed samples. In this study, high strain rate compression tests at six different strain rates were performed, and the stress–strain curves were generated. This work is amongst the first attempts to model the microstructure of additively manufactured hexagonal alloys under compressive loadings using the statistical continuum mechanics theory. The model is capable of simulating reasonably large microstructures (statistically representative) with a practical computational cost and accuracy, unlike numerical models that require a high computational cost. It should be noted that in additive manufacturing, due to large grains and high anisotropy, microstructures used in the simulations should be large enough to include sufficient information from the material’s structure. Therefore, using finite element models would be very challenging here. On the other hand, the statistical continuum mechanics theory uses the statistical representation of the material’s characteristics for solving the governing equations with Green’s function that enables this methodology to use more microstructure characteristic information without having a noticeable change to the computational cost. The proposed model in this study uses different microstructure characteristics such as crystal grain orientation, total slip systems, active slip systems, gain morphology, and chemical phases that are obtained from EBSD images for simulating the inelastic mechanical behavior of polycrystalline materials. Although this model simulates polycrystalline materials by considering various crystal and grain information, unlike numerical methods, it doesn’t simulate the grain interactions well and we cannot study local deformation and crack nucleation sites. This model works very well for simulating the overall behavior of material instead of each individual grain and failure analysis. This model has shown a good combination of computational cost and accuracy in which the error between the simulated and experimental strength for vertical and horizontal samples was 6.21% and 8.07%, respectively.

Efficiency and energy balance for substitution of CH4 in clathrate hydrates with CO2 under multiple-phase coexisting conditions.

Many experimental and theoretical studies on CH4-CO2 hydrates have been performed aiming at the extraction of CH4 as a relatively clean energy resource and concurrent sequestration of CO2. However, vague or insufficient characterization of the environmental conditions prevents us from a comprehensive understanding of even equilibrium properties of CH4-CO2 hydrates for this substitution. We propose possible reaction schemes for the substitution, paying special attention to the coexisting phases, the aqueous and/or the fluid, where CO2 is supplied from and CH4 is transferred to. We address the two schemes for the substitution operating in three-phase and two-phase coexistence. Advantages and efficiencies of extracting CH4 in the individual scheme are estimated from the chemical potentials of all the components in all the phases involved in the substitution on the basis of a statistical mechanical theory developed recently. It is found that although substitution is feasible in the three-phase coexistence, its working window in temperature-pressure space is much narrower compared to the two-phase coexistence condition. Despite that the substitution normally generates only a small amount of heat, a large endothermic substitution is suggested in the medium pressure range, caused by the vaporization of liquid CO2 due to mixing with a small amount of the released CH4. This study provides the first theoretical framework toward the practical use of hydrates replacing CH4 with CO2 and serves as a basis for quantitative planning.

Development of an analytical exponential-6 equation of state through Monte Carlo simulations.

The exponential-6 (exp-6) potential is commonly used to model fluids at high densities. In this paper, I propose a new equation of state (EOS) in the form of an analytical expression for the excess Helmholtz free energy of an exp-6 fluid. The EOS is based on extensive Monte Carlo simulations and therefore combines the excellent accuracy of the simulations with the numerical efficiency of a polynomial expression. The mean relative error in compressibility factor and internal energy is 0.14% and 0.25% respectively, which is a significant improvement over statistical mechanical theories. The EOS was implemented into a thermochemical code in order to optimize gas parameters and evaluate its performance on pure gas data, shock compression and detonation properties. Predicted gas densities, heat capacities and speed of sound for pure gases were generally within experimental uncertainties at pressures up to 1GPa and temperatures above 300K. For polar molecules, a simple free energy correction was introduced which greatly improved accuracy at low temperature. Calculated shock Hugoniots showed excellent agreement with experimental values up to 150GPa and 10 000K, and the detonation performance was accurately predicted for a number of different types of explosives.

The God of Chance and Purpose: Divine Involvement in a Secular Evolutionary World

The God of Chance and Purpose: Divine Involvement in a Secular Evolutionary World

Character Theory and Categorification

Over a hundred years after the work of Frobenius and Schur, the sheer enormity of what is not known about the character theory of symmetric and alternating groups continues to surprise and awe the uninitiated. How does one decompose the tensor product of a pair of complex characters? Or the restriction of a complex character to a Sylow or wreath product subgroup? Can we understand the vanishing sets of complex characters? What about the asymptotic behaviour of complex characters? What are the dimensions of the modular characters? These questions have been hailed as some of the definitive open problems in representation theory and algebraic combinatorics, they have deep connections with Lie theory, group theory, local-global conjectures in representation theory of finite groups, symplectic geometry, complexity theory, statistical mechanics and quantum information theory. The overarching theme of this proposal is the use of hidden, richer representation theoretic structures arising in modular, local-global, and categorical representation theory in order to prove and disprove conjectures concerning characters of symmetric and alternating groups.

On the phase behaviors of CH4-CO2 binary clathrate hydrates: Two-phase and three-phase coexistences.

We develop a statistical mechanical theory on clathrate hydrates in order to explore the phase behaviors of clathrate hydrates containing two kinds of guest species and apply it to CH4-CO2 binary hydrates. The two boundaries separating water and hydrate and hydrate and guest fluid mixtures are estimated, which are extended to the lower temperature and the higher pressure region far distant from the three-phase coexisting conditions. The chemical potentials of individual guest components can be calculated from free energies of cage occupations, which are available from intermolecular interactions between host water and guest molecules. This allows us to derive all thermodynamic properties pertinent to the phase behaviors in the whole space of thermodynamic variables of temperature, pressure, and guest compositions. It is found that the phase boundaries of CH4-CO2 binary hydrates with water and with fluid mixtures locate between simple CH4 and CO2 hydrates, but the composition ratios of CH4 guests in hydrates are disproportional to those in fluid mixtures. Such differences arise from the affinities of each guest species to the large and small cages of CS-I hydrates and significantly affect occupation of each cage type, which results in a deviation of the guest composition in hydrates from that in fluid on the two-phase equilibrium conditions. The present method provides a basis for the evaluation of the efficiency of the guest CH4 replacement to CO2 at the thermodynamic limit.

Unique solvent effect of water in radical cyclization reaction

The radical cyclization reaction in the aqueous environment by Yorimitsu et al. is revisited using the RISM-SCF-cSED method, a hybrid of quantum chemistry and statistical mechanics theory for molecular liquids. The difference between the barrier height of the forward reaction from the intermediate E-rot, the cyclization step, and that of the backward reaction is crucial to the reaction yield. Considering the effect of hydrogen bonding through the RISM theory, we found that the barrier height for the forward reaction is lower, especially in water. In other words, considering microscopic solvation effects clearly shows the difference between water and DMSO solvents, explaining the remarkable acceleration of the reaction in the aqueous environment.

Self-consistent hopping theory of activated relaxation and diffusion of dilute penetrants in dense crosslinked polymer networks.

We generalize a microscopic statistical mechanical theory of the activated dynamics of dilute spherical penetrants in glass-forming liquids to study the influence of crosslinking in polymer networks on the penetrant relaxation time and diffusivity over a wide range of temperature and crosslink fraction (fn). Our calculations are relevant to recent experimental studies of a nm-sized molecule diffusing in poly-(n-butyl methacrylate) networks. The theory predicts the penetrant relaxation time increases exponentially with the glass transition temperature, Tg(fn), which grows roughly linearly with the square root of fn due to the coupling of local hopping to longer-range collective elasticity. Moreover, Tg is also found to be proportional to a geometric confinement parameter defined as the ratio of the penetrant diameter to the mean network mesh size. The decoupling ratio of the penetrant and Kuhn segment alpha times displays a complex non-monotonic dependence on fn and temperature that is well collapsed based on the variable Tg(fn)/T. A model for the penetrant diffusion constant that combines activated relaxation and entropic mesh confinement is proposed, which results in a significantly stronger suppression of mass transport with degree of effective supercooling than predicted for the penetrant alpha time. This behavior corresponds to a new network-based type of "decoupling" of diffusion and relaxation. In contrast to the diffusion of larger nanoparticles in high temperature rubbery networks, our analysis in the supercooled regime suggests that for the penetrants studied the mesh confinement effects are of secondary importance relative to the consequences of crosslink-induced slowing down of activated hopping of glassy physics origin.

The Sensitivity of Convective Cloud Ensemble Statistics to Horizontal Grid Spacing in Idealized RCE Simulations

Abstract In this work, cloud ensemble statistics are extracted from idealized radiative–convective equilibrium simulations performed at horizontal grid spacings Δ ranging from 2 km to 125 m. At the coarsest resolution, convection remains randomly distributed in space such that the equilibrium statistical mechanics theory proposed by Craig and Cohen in 2006 (CC06; assumes Poisson distributed clouds and exponential mass flux distributions) remains valid. Using classical organization metrics, clustering is already observed at Δ = 1 km, but substantial deviations between the simulated cloud ensemble statistics and CC06 are only observed for grid spacings Δ < 500 m. At these resolutions, the cloud mass flux distributions exhibit heavy tails and cloud counts become overdispersed (higher variance than a Poisson distribution). These changes in ensemble statistics are accompanied by a shift in subcloud organization patterns as well as with the fact that individual cloudy updrafts start to be resolved. Consequently, a horizontal grid spacing no larger than 250 m is recommended, not only to properly resolve the dynamics of individual convective clouds, but also to capture the mesoscale organization of the cloud ensemble. Finally, it is shown that the CC06 theory and our high-resolution results including mesoscale organization may be reconciled if one considers 1) areas smaller than approximately 2 km in size, corresponding roughly to the narrow bands along which clouds develop almost randomly; and 2) individual cloud cores instead of cloud objects, core mass fluxes being shown to generally follow exponential distributions.

Microscopic activated dynamics theory of the shear rheology and stress overshoot in ultradense glass-forming fluids and colloidal suspensions

We formulate a particle and force level, activated dynamics-based statistical mechanical theory for the continuous startup nonlinear shear rheology of ultradense glass-forming hard sphere fluids and colloidal suspensions in the context of the elastically collective nonlinear Langevin equation approach and a generalized Maxwell model constitutive equation. Activated structural relaxation is described as a coupled local-nonlocal event involving caging and longer range collective elasticity which controls the characteristic stress relaxation time. Theoretical predictions for the deformation-induced enhancement of mobility, the onset of relaxation acceleration at remarkably low values of stress, strain, or shear rate, apparent power law thinning of the steady-state structural relaxation time and viscosity, a nonvanishing activation barrier in the shear thinning regime, an apparent Herschel–Buckley form of the shear rate dependence of the steady-state shear stress, exponential growth of different measures of a yield or flow stress with packing fraction, and reduced fragility and dynamic heterogeneity under deformation were previously shown to be in good agreement with experiments. The central new question we address here is the defining feature of the transient response—the stress overshoot. In contrast to the steady-state flow regime, understanding the transient response requires an explicit treatment of the coupled nonequilibrium evolution of structure, elastic modulus, and stress relaxation time. We formulate a new quantitative model for this aspect in a physically motivated and computationally tractable manner. Theoretical predictions for the stress overshoot are shown to be in good agreement with experimental observations in the metastable ultradense regime of hard sphere colloidal suspensions as a function of shear rate and packing fraction, and accounting for deformation-assisted activated motion appears to be crucial for both the transient and steady-state responses.

Statistical Mechanical Design Principles for Coarse-Grained Interactions across Different Conformational Free Energy Surfaces.

Systematic bottom-up coarse-graining (CG) of molecular systems provides a means to explore different coupled length and time scales while treating the molecular-scale physics at a reduced level. However, the configuration dependence of CG interactions often results in CG models with limited applicability for exploring the parametrized configurations. We propose a statistical mechanical theory to design CG interactions across different configurations and conditions. In order to span wide ranges of conformational space, distinct classical CG free energy surfaces for characteristic configurations are identified using molecular collective variables. The coupling interaction between different CG free energy surfaces can then be systematically determined by analogy to quantum mechanical approaches describing coupled states. The present theory can accurately capture the underlying many-body potentials of mean force in the CG variables for various order parameters applied to liquids, interfaces, and in principle proteins, uncovering the complex nature underlying the coupling interaction and imparting a new protocol for the design of predictive multiscale models.

Formalism for stochastic perturbations and analysis in relativistic stars

Perturbed Einstein’s equations with a linear response relation and a stochastic source, applicable to a relativistic star model are worked out. These perturbations which are stochastic in nature, are of significance for building a non-equilibrium statistical mechanics theory in connections with relativistic astrophysics. A fluctuation dissipation relation for a spherically symmetric star in its simplest form is obtained. The FD relation shows how the random velocity fluctuations in the background of the unperturbed star can dissipate into Lagrangian displacement of fluid trajectories of the dense matter. Interestingly in a simple way, a constant (in time) coefficient of dissipation is obtained without a delta correlated noise. This formalism is also extended for perturbed TOV equations which have a stochastic contribution, and show up in terms of the effective or root mean square pressure perturbations. Such contributions can shed light on new ways of analysing the equation of state for dense matter. One may obtain contributions of first and second order in the equation of state using this stochastic approach.

Interfacial microstructure of neutral and charged polymer brushes: A density functional theory study.

Polymer density functional theory (PDFT) is a computationally efficient and robust statistical mechanics theory for capturing the interfacial microstructure of grafted polymer brushes (PBs). Undoubtedly, the intramolecular and intermolecular interactions in PDFT (e.g., hard-core interactions and direct Coulomb interactions) are greatly affected by the grafting behavior of PBs. However, the combination of these interactions with the physical constraints on grafting behavior remains unclear and there is a remarkable difference in the density profile of grafted PB between PDFT and simulation. Herein, we propose a PDFT to study neutral and charged grafted PBs by incorporating the physical constraints of end-grafted PBs into the excess free energies due to intramolecular and intermolecular interactions. This PDFT has been successfully validated where the density distributions of neutral and weakly charged PBs predicted by the PDFT are in excellent agreement with the results of the Monte Carlo and molecular dynamics simulations. In addition, the significant contribution of grafting behavior to the free energy of PB systems is presented. Consequently, this work provides a powerful and accurate theoretical method to reveal the interfacial microstructure of grafted PBs.