Statistical Mechanical Theory Research Articles

Curvature of fluctuation geometry and its implications on Riemannian fluctuation theory

Fluctuation geometry was recently proposed as a counterpart approach of the Riemannian geometry of inference theory (widely known as information geometry). This theory describes the geometric features of the statistical manifold of random events that are described by a family of continuous distributions dp(x|θ). A main goal of this work is to clarify the statistical relevance of the Levi-Civita curvature tensor Rijkl(x|θ) of the statistical manifold . For this purpose, the notion of irreducible statistical correlations is introduced. Specifically, a distribution dp(x|θ) exhibits irreducible statistical correlations if every distribution obtained from dp(x|θ) by considering a coordinate change cannot be factorized into independent distributions as . It is shown that the curvature tensor Rijkl(x|θ) arises as a direct indicator about the existence of irreducible statistical correlations. Moreover, the curvature scalar R(x|θ) allows us to introduce a criterium for the applicability of the Gaussian approximation of a given distribution function. This type of asymptotic result is obtained in the framework of the second-order geometric expansion of the distribution family dp(x|θ), which appears as a counterpart development of the high-order asymptotic theory of statistical estimation. In physics, fluctuation geometry represents the mathematical apparatus of a Riemannian extension for Einstein’s fluctuation theory of statistical mechanics. Some exact results of fluctuation geometry are now employed to derive the invariant fluctuation theorems. Moreover, the curvature scalar allows us to express some asymptotic formulae that account for the system fluctuating behavior beyond the Gaussian approximation, e.g.: it appears as a second-order correction of the Legendre transformation between thermodynamic potentials, .

Thermodynamic and surface properties of liquid Ge–Si alloys

In the present work, the surface tension of liquid Si and Ge has been measured by the pendant/sessile drop combined method over the temperature range of 1723–1908 K and 1233–1313 K, respectively. The new surface tension data, the molar volumes and the melting temperatures of silicon and germanium as well as the excess Gibbs energy data of the Ge–Si liquid phase are the inputs for Calphad type modelling to study the mixing behaviour in alloy melts. The energetics of mixing in liquid Ge–Si system has been analysed through the study of the concentration dependence of various thermodynamic (activity, enthalpy of mixing, Gibbs energy of mixing), surface (surface tension and surface composition) and transport (diffusivity) properties as well as the microscopic functions (concentration fluctuations in the long-wavelength limit and chemical short-range order parameter) in the framework of statistical mechanical theory in conjunction with the Quasi-Lattice Theory (QLT).

Extension of Tao-Mason Equation of State to Heavy n-Alkanes

In our previous paper we extended the Tao and Mason equation of state (TM EOS) to refrigerant fluids, using the speed of sound data. This is a continuation for evaluating TM EOS in predicting PVT properties of heavy n-alkanes. Liquid density of long-chain n-alkane systems from C9 to C20 have been calculated using an analytical equation of state based on the statistical-mechanical perturbation theory. The second virial coefficients of these n-alkanes are scarce and there is no accurate potential energy function for their theoretical calculation. In this work the second virial coefficients are calculated using a corresponding state correlation based on surface tension and liquid density at the freezing point. The deviation of calculated densities of these alkanes is within 0.5% from experimental data. The densities of n-alkanes obtained from the TM EOS are compared with those calculated from Ihm-Song-Mason equation of state and the corresponding-states liquid densities (COSTALD). Our results are in favor of the preference of the TM EOS over other two equations of state.

Kinetic exchange models: From molecular physics to social science

We discuss several multi-agent models that have their origin in the kinetic exchange theory of statistical mechanics and have been recently applied to a variety of problems in the social sciences. This class of models can be easily adapted for simulations in areas other than physics, such as the modeling of income and wealth distributions in economics and opinion dynamics in sociology.

Population Based Reweighting of Scaled Molecular Dynamics

Molecular dynamics simulation using enhanced sampling methods is one of the powerful computational tools used to explore protein conformations and free energy landscapes. Enhanced sampling methods often employ either an increase in temperature or a flattening of the potential energy surface to rapidly sample phase space, and a corresponding reweighting algorithm is used to recover the Boltzmann statistics. However, potential energies of complex biomolecules usually involve large fluctuations on a magnitude of hundreds of kcal/mol despite minimal structural changes during simulation. This leads to noisy reweighting statistics and complicates the obtainment of accurate final results. To overcome this common issue in enhanced conformational sampling, we propose a scaled molecular dynamics method, which modifies the biomolecular potential energy surface and employs a reweighting scheme based on configurational populations. Statistical mechanical theory is applied to derive the reweighting formula, and the canonical ensemble of simulated structures is recovered accordingly. Test simulations on alanine dipeptide and the fast folding polypeptide Chignolin exhibit sufficiently enhanced conformational sampling and accurate recovery of free energy surfaces and thermodynamic properties. The results are comparable to long conventional molecular dynamics simulations and exhibit better recovery of canonical statistics over methods which employ a potential energy term in reweighting.

Equation of state for mercury: revisited

An analytical equation of state by Song and Mason is developed to calculate the PVT properties of mercury. The equation of state is based on the statistical-mechanical perturbation theory of hard convex bodies and can be written as a fifth-order polynomial in the density. There exists three temperature-dependent parameters in the equation of state; the second virial coefficient, an effective molecular volume, and a scaling factor for the average contact pair distribution function of hard convex bodies. The temperature-dependant parameters have been calculated using corresponding-states correlations based on the surface tension and the liquid density at the normal boiling point. Employing the present equation of state, we have calculated the PVT properties of mercury over a wide range of temperatures and pressures. The average absolute deviation for the calculated density of mercury in the saturation and compressed states is 1.09%.

Calculation of Thermal Pressure Coefficient of R11, R13, R14, R22, R23, R32, R41, and R113 Refrigerants by pVT Data

For thermodynamic performance to be optimized particular attention must be paid to the fluid’s thermal pressure coefficients and thermodynamic properties. A new analytical expression based on the statistical mechanics is derived for R11, R13, R14, R22, R23, R32, R41, and R113 refrigerants, using the intermolecular forces theory. In this paper, temperature dependency of the parameters of R11, R13, R14, R22, R23, R32, R41, and R113 refrigerants to calculate thermal pressure coefficients in the form of first order has been developed to second and third orders and their temperature derivatives of new parameters are used to calculate thermal pressure coefficients. These problems have led us to try to establish a function for the accurate calculation of the thermal pressure coefficients of R11, R13, R14, R22, R23, R32, R41, and R113 refrigerants based on statistical-mechanics theory for different refrigerants.

Crowding induced entropy-enthalpy compensation in protein association equilibria.

A statistical mechanical theory is presented to predict the effects of macromolecular crowding on protein association equilibria, accounting for both excluded volume and attractive interactions between proteins and crowding molecules. Predicted binding free energies are in excellent agreement with simulation data over a wide range of crowder sizes and packing fractions. It is shown that attractive interactions between proteins and crowding agents counteract the stabilizing effects of excluded volume interactions. A critical attraction strength, for which there is no net effect of crowding, is approximately independent of the crowder packing fraction.

Visualization of hydration layers on muscovite mica in aqueous solution by frequency-modulation atomic force microscopy

A three-dimensional interaction force mapping experiment was carried out on a muscovite mica surface in an aqueous solution using a high-resolution and low-thermal drift frequency-modulation atomic force microscope. By collecting oscillatory frequency shift versus distance curves at the mica∕solution interface, complicated hydration structures on the mica surface were visualized. Reconstructed two-dimensional frequency shift maps showed dot-like or honeycomb-like patterns at different tip-sample distances with a separation of 0.2 nm with each other, which agree well to the water molecule density maps predicted by a statistical-mechanical theory. Moreover, site-specific force versus distance curves showed a good agreement with theoretically calculated site-specific force curves by a molecular dynamics simulation. It is found that the first and second hydration layers give honeycomb-like and dot-like patterns in the two-dimensional frequency shift images, respectively, corresponding to the lateral distribution function in each layer.

Research on active particle distribution in the crop stalk plasma reaction under the condition of streaming and discharging micro electric current

In order to further study sugar-production mechanism in the crop stalk plasma reaction under the condition of streaming and discharging micro electric current with N2 and H2O as the electro-discharging gases, the active particles distribution is researched and analyzed. Through the study of methods and theories of statistical mechanics and statistica thermodynamics, the ionization process of the mixed discharging gas and the distribution features of the active particles changing with temperature in the plasma reactor are discussed. The constant pressure form of Saha equation that applies to this experiment is deduced. And the active particles generated in the dense ionization discharging process and the crop stalk conversion processes are also analyzed. The experimental results show that N2 and H2 can ionize synchronously. And it is found that during the crop stalk conversion process, while one gas is ionised, the other gas will not take the energy away, which will not affect the conversion. Among the discharging gases consisting of water vapour and nitrogen, the main components are ions of H2O+, H+, N2+ and molecules of N2, H2O, which are all beneficial to a high sugar yield.

Molecular structure and elastic properties of thermotropic liquid crystals: Integrated molecular dynamics—Statistical mechanical theory vs molecular field approach

The connection between the molecular structure of liquid crystals and their elastic properties, which control the director deformations relevant for electro-optic applications, remains a challenging objective for theories and computations. Here, we compare two methods that have been proposed to this purpose, both characterized by a detailed molecular level description. One is an integrated molecular dynamics-statistical mechanical approach, where the bulk elastic constants of nematics are calculated from the direct correlation function (DCFs) and the single molecule orientational distribution function [D. A. McQuarrie, Statistical Mechanics (Harper & Row, New York, 1973)]. The latter is obtained from atomistic molecular dynamics trajectories, together with the radial distribution function, from which the DCF is then determined by solving the Ornstein-Zernike equation. The other approach is based on a molecular field theory, where the potential of mean torque experienced by a mesogen in the liquid crystal phase is parameterized according to its molecular surface. In this case, the calculation of elastic constants is combined with the Monte Carlo sampling of single molecule conformations. Using these different approaches, but the same description, at the level of molecular geometry and torsional potentials, we have investigated the elastic properties of the nematic phase of two typical mesogens, 4'-n-pentyloxy-4-cyanobiphenyl and 4'-n-heptyloxy-4-cyanobiphenyl. Both methods yield K3(bend) >K1 (splay) >K2 (twist), although there are some discrepancies in the average elastic constants and in their anisotropy. These are interpreted in terms of the different approximations and the different ways of accounting for the structural properties of molecules in the two approaches. In general, the results point to the role of the molecular shape, which is modulated by the conformational freedom and cannot be fully accounted for by a single descriptor such as the aspect ratio.

Electrochemical deposition of platinum within nanopores on silicon: Drastic acceleration originating from surface-induced phase transition

An electrochemical reaction within nanopores is remarkably decelerated once a diffusion-limited condition is reached due to the difficulty in supply of reactants from the bulk. Here, we report a powerful method of overcoming this problem for electrochemical deposition of platinum within nanopores formed on silicon. We made the pore wall surface of the silicon electrode hydrophobic by covering it with organic molecules and adopted platinum complex ions with sufficiently large sizes. Such ions, which are only weakly hydrated, are excluded from the bulk aqueous electrolyte solution to the surface and rather hydrophobic in this sense. When the ion concentration in the bulk was gradually increased, at a threshold the deposition behavior exhibited a sudden change, leading to drastic acceleration of the electrochemical deposition. Using our statistical-mechanical theory for confined molecular liquids, we show that this change originates from a surface-induced phase transition: The space within nanopores is abruptly filled with the second phase within which the ion concentration is orders of magnitude higher. When the affinity of the surface with water was gradually reduced with fixing the ion concentration, qualitatively the same transition phenomenon was observed, which can also be elucidated by our theory. The utilization of the surface-induced phase transition sheds new light on the design and control of a chemical reaction in nanospace.

Cleansing dynamics of oily soil using nanofluids

We explored the technological concept of the nanoparticle structuring in the wedge film with regards to its application to the oily soil removal phenomena. The experimental and theoretical investigations on the cleansing of canola oil from a glass substrate using commercially available nanofluids were pursued. Five commercially-available nanofluids with pHs varying from 9.3 to 9.9 were used in the experiments. Experimental results clearly indicated that the time to separate the oily soil from the glass surfaces by nanofluids was much shorter than that for the reference alkaline solution at the same pH. The positive contributions of the nanoparticles to the soil cleaning performance were rationalized in terms of the decrease in the contact angle and the interfacial tension, positive second virial coefficient, and high osmotic pressure of the nanofluid. The effective nanoparticle diameter and the effective volume (i.e., concentration) of the nanoparticles were determined using our novel capillary force balance technique in conjunction with the microinterferometric method. Using the experimentally measured values of the effective particle diameter, effective volume, and the osmotic pressure, the structural disjoining pressure in the wedge film was calculated from a theoretical model based on the statistical mechanics theory. The experimental data for the oil cleaning performance correlated well with the calculated values of the disjoining pressure, the spreading coefficient, and the film tension. We used the drop profile analysis based on the Laplace equation augmented with the extra term of the disjoining pressure to theoretically analyze the nanofluid spreading and wetting phenomena, and the detachment of the oil drop from the solid surface. These results confirm the novel mechanism of detergency using nanofluids based on the normal force (i.e., structural disjoining pressure) arising from the ordered nanoparticle structure formation in the confined space between the soil and the solid substrate (i.e., the wedge film).

Sampling microcanonical measures of the 2D Euler equations through Creutz’s algorithm: a phase transition from disorder to order when energy is increased

The 2D Euler equations are basic examples of fluid models for which a microcanonical measure can be constructed from first principles. This measure is defined through finite-dimensional approximations and a limiting procedure. Creutz’s algorithm is a microcanonical generalization of the Metropolis–Hastings algorithm (to sample Gibbs measures, in the canonical ensemble). We prove that Creutz’s algorithm can sample finite-dimensional approximations of the 2D Euler microcanonical measures (incorporating fixed energy and other invariants). This is essential as microcanonical and canonical measures are known to be inequivalent at some values of energy and vorticity distribution. Creutz’s algorithm is used to check predictions from the mean-field statistical mechanics theory of the 2D Euler equations (the Robert–Sommeria–Miller theory). We find full agreement with theory. Three different ways to compute the temperature give consistent results. Using Creutz’s algorithm, a first-order phase transition never observed previously and a situation of statistical ensemble inequivalence are found and studied. Strikingly, and in contrast to the usual statistical mechanics interpretations, this phase transition appears from a disordered phase to an ordered phase (with fewer symmetries) when the energy is increased. We explain this paradox.

Theoretical Equation of State for Highly Anharmonic Solids

Thermodynamic properties of Lennard–Jones (LJ) face-centered cubic crystals are calculated in a wide range of temperatures along sublimation and melting lines on the basis of statistical mechanical theory which accounts for anharmonicity, pair and triple correlations in the displacements of atoms. Results are compared with the most accurate Monte Carlo data available. Excellent agreement was found. Anharmonic and correlation contributions to entropy, internal energy, and the compressibility factor of LJ solids are computed and their relative importance analyzed. An analytical equation of state approximating the theoretical results is developed.

Case study of Rb+(aq), quasi-chemical theory of ion hydration, and the no split occupancies rule

Quasi-chemical theory applied to ion hydration combines statistical mechanical theory, electronic structure calculations, and molecular simulation, disciplines which are individually subjects for specialized professional attention. Because it combines activities which are themselves non-trivial, quasi-chemical theory is typically viewed with surprise. Nevertheless, it provides a fully-considered framework for analysis of ion hydration. Furthermore, the initial calculations are indeed simple, successful, and provide new information to long-standing experimental activities such as neutron diffraction by hydrated ions. Here we review quasi-chemical theory in the context of a challenging application, Rb+(aq).

Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories

Thermodynamic properties of liquids and liquid mixtures play very important role in understanding the nature of molecular interactions occurring in the system. In the present work different thermodynamic properties of 15 pure liquids and 34 equimolar binary liquid mixtures of benzene, toluene, p-xylene, chlorobenzene and 1-chloronaphthalene with linear and branched alkanes have been computed with the help of Flory’s statistical theory (FST), Hard sphere equation of state (HSE) and Hole theory (HT) simultaneously. The calculated values are compared with the experimental findings collected from literature and quite satisfactory results are obtained.

Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories

Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories

Hybrid Dynamical Evolutionary Algorithm and Time Complexity Analysis

The dynamical evolutionary algorithm (DEA) is a new evolutionary algorithm based on the theory of statistical mechanics, however, DEA converges slowly and often converge at local optima for some function optimization problems. In this paper, a hybrid dynamical evolutionary algorithm (HDEA) with multi-parent crossover and differential evolution mutation is proposed for accelerating convergence velocity and easily escaping suboptimal solutions. Moreover, the population of HDEA is initialized by chaos. In order to confirm the effectiveness of our algorithm, HDEA is applied to solve the typical numerical function minimization problems. The computational complexity of HDEA is analyzed, and the experimental results show that HDEA outperforms the DEA in the aspect of convergence velocity and precision, even the two algorithms have the similar time complexity.

Incommensurate phases of a supported nanoparticle film subjected to uniaxial compression.

We investigate experimentally and theoretically the sequence of phases that occurs when a self-assembled monolayer of gold nanoparticles supported on a fluid is compressed uniaxially in a Langmuir trough. Uniaxial compression of the monolayer results in the appearance of lines that have been shown to be regions of trilayer. These lines exhibit complex patterns that depend on the extent of compression. We show that these patterns can be understood in terms of an equilibrium statistical mechanical theory, originally developed in the context of commensurate-incommensurate transitions in krypton monolayers adsorbed on graphite, in which there is an energy cost to line deformations and to line intersections. Even though line intersections are energetically costly, they lower the free energy because they cause the entropy of the system to increase when the density of lines is low enough. Our analytic and Monte Carlo analyses of the model demonstrate that the model exhibits two-phase coexistence. Our experimental observations are qualitatively consistent with the predictions of the model.