Statistical Mechanics Approach Research Articles

Adaptive intermolecular interaction parameters for accurate Mixture Density Functional Theory calculations

The description of fluid mixtures molecular behavior is significant for various industry fields due to the complex composition of fluid found in nature. Statistical mechanics approaches use intermolecular interaction potential to predict fluids behavior on the molecular scale. The paper provides a comparative analysis of mixing rules applications for obtaining intermolecular interaction parameters of mixture components. These parameters are involved in the density functional theory equation of state for mixtures (Mixture DFT EoS) and characterize thermodynamic mixture properties in the bulk. The paper demonstrates that Mixture DFT EoS with proper intermolecular parameters agrees well with experimental mixtures isotherms in bulk: Ar+Ne,CO2+CH4,CO2+C2H6 and CH4+C2H6. However, predictions of vapor–liquid equilibrium (VLE) experimental data for CO2+C4H10 are not successful. Halgren HHG, Waldman – Hagler, and adaptive mixing rules that adjust on the experimental data from the literature are used for the first time to obtain intermolecular interaction parameters for the mixture DFT model. The results obtained provide a base for understanding how to validate the DFT fluid mixture model for calculating thermodynamic properties of fluid mixtures on a micro and macro scale.

Statistical mechanical approach of complex networks with weighted links

Systems that consist of many localized constituents interacting with each other can be represented by complex networks. Consistently, network science has become highly popular in vast fields focusing on natural, artificial and social systems. We numerically analyze the growth of d-dimensional geographic networks (characterized by the index α G ⩾ 0; d = 1, 2, 3, 4) whose links are weighted through a predefined random probability distribution, namely , w being the weight (w c ⩾ 0; τ > 0). In this model, each site has an evolving degree k i and a local energy (i = 1, 2, …, N) that depend on the weights of the links connected to it. Each newly arriving site links to one of the pre-existing ones through preferential attachment given by the probability , where d ij is the Euclidean distance between the sites. Short- and long-range interactions respectively correspond to α A/d > 1 and 0 ⩽ α A/d ⩽ 1; α A/d → ∞ corresponds to interactions between close neighbors, and α A/d → 0 corresponds to infinitely-ranged interactions. The site energy distribution p(ɛ) corresponds to the usual degree distribution p(k) as the particular instance (w c, τ) = (1, 0). We numerically verify that the corresponding connectivity distribution p(ɛ) converges, when α A/d → ∞, to the weight distribution P(w) for infinitely wide distributions (i.e. τ → ∞, ∀w c) as well as for w c → 0, ∀τ. Finally, we show that p(ɛ) is well approached by the q-exponential distribution , which optimizes the nonadditive entropy S q under simple constraints; q depends only on α A/d, thus exhibiting universality.

Atomic structures of ordered monolayer GP zones in Mg-Zn-X (X= Ca, Nd) systems

Atomic structures of ordered monolayer Guinier-Preston (GP) zones in Mg-Zn-X (X = Ca, Nd) systems are predicted based on first-principles calculations combined with the statistical-mechanical approach of cluster expansion. Multiple metastable GP zones with various compositions are identified in both systems. The predicted metastable structures in both systems show striking similarities and are all structurally related to monolayer Mg2X with each X atom 6-fold coordinated by Mg atoms on the HCP basal plane. An increased number of Mg atoms in monolayer Mg2X are replaced by Zn atoms to form ordered GP zones at an increased Zn/X ratio due to strong attractive interactions between Zn and X solutes, which ultimately leads to lower formation energies of the GP zones. Simulated TEM images and diffraction patterns of the predicted structures agree well with prior experimental observations.

An inference problem in a mismatched setting: a spin-glass model with Mattis interaction

The Wigner spiked model in a mismatched setting is studied with the finite temperature Statistical Mechanics approach through its representation as a Sherrington-Kirkpatrick model with added Mattis interaction. The exact solution of the model with Ising spins is rigorously proved to be given by a variational principle on two order parameters, the Parisi overlap distribution and the Mattis magnetization. The latter is identified by an ordinary variational principle and turns out to concentrate in the thermodynamic limit. The solution leads to the computation of the Mean Square Error of the mismatched reconstruction. The Gaussian signal distribution case is investigated and the corresponding phase diagram is identified.

Thermophysical behavior of mercury-lead liquid alloy

Thermophysical properties of compound forming binary liquid mercury-lead alloy at temperature 600 K have been reported as a function of concentration by considering HgPb2 complex using different modelling equations. The thermodynamic properties such as the Gibbs free energy, enthalpy of mixing, chemical activity of each component, and microscopic properties such as concentration fluctuation in long-wavelength limit and Warren-Cowley short range order parameter of the alloy are studied by quasi-chemical approximation. This research paper places additional emphasis on the interaction energy parameters between the atoms of the alloy. The theoretical and experimental data are compared to determine the model’s validity. Compound formation model, statistical mechanical technique, and improved derivation of the Butler equation have all been used to investigate surface tension. The alloy’s viscosity is investigated using the Kozlov-Ronanov-Petrov equation, the Kaptay equation, and the Budai-Benko-Kaptay model. The study depicts a weak interaction of the alloy, and the theoretical thermodynamic data derived at 600 K are in good agreement with the experimental results. The surface tension is slightly different in the compound formation model than in the statistical mechanical approach and the Butler equation at greater bulk concentrations of lead. The estimated viscosities in each of the three models are substantially identical.

Statistical physics of exchangeable sparse simple networks, multiplex networks, and simplicial complexes.

Exchangeability is a desired statistical property of network ensembles requiring their invariance upon relabeling of the nodes. However, combining sparsity of network ensembles with exchangeability is challenging. Here we propose a statistical physics framework and a Metropolis-Hastings algorithm defining exchangeable sparse network ensembles. The model generates networks with heterogeneous degree distributions by enforcing only global constraints while existing (nonexchangeable) exponential random graphs enforce an extensive number of local constraints. This very general theoretical framework to describe exchangeable networks is here first formulated for uncorrelated simple networks and then it is extended to treat simple networks with degree correlations, directed networks, bipartite networks, and generalized network structures including multiplex networks and simplicial complexes. In particular here we formulate and treat both uncorrelated and correlated exchangeable ensembles of simplicial complexes using statistical mechanics approaches.

Exact face-landing probabilities for bouncing objects: Edge probability in the coin toss and the three-sided die problem.

Have you ever taken a disputed decision by tossing a coin and checking its landing side? This ancestral "heads or tails" practice is still widely used when facing undecided alternatives since it relies on the intuitive fairness of equiprobability. However, it critically disregards an interesting third outcome: the possibility of the coin coming at rest on its edge. Provided this evident yet elusive possibility, previous works have already focused on capturing all three landing probabilities of thick coins, but have only succeeded computationally. Hence, an exact analytical solution for the toss of bouncing objects still remains an open problem due to the strongly nonlinear processes induced at each bounce. In this Letter we combine the classical equationsof collisions with a statistical-mechanics approach to derive an exact analytical solution for the outcome probabilities of the toss of a bouncing object, i.e., the coin toss with arbitrarily inelastic bouncing. We validate the theoretical prediction by comparing it to previously reported simulations and experimental data; we discuss the moderate discrepancies arising at the highly inelastic regime; we describe the differences with previous, approximate models; we propose the optimal geometry for the fair cylindrical three-sided die; and we finally discuss the impact of current results within and beyond the coin toss problem.

Mechanics of transient semi-flexible networks: Soft-elasticity, stress relaxation and remodeling

Networks of semi-flexible (or athermal) filaments cross-linked by flexible chains are found in a variety of biopolymers such as soft connective tissues, the cell’s cytoskeleton or the wall of plant cells. They can also be synthetized in the lab to create liquid crystal elastomers-like gels as well as tissue mimetics. While the elasticity of these networks has been explored, the visco-elastic response that originate from the existence of reversible and dynamic cross-links is still poorly understood. We here develop a model for these networks by taking a multiscale, statistical mechanics approach where the network is decomposed into its most basic building blocks: elastic rods (to describe semi-flexible filaments) and the flexible chains used to cross-link them. The topology of this assembly is represented by a hairy rod model for which we express the non-affine kinematics, and evolution equations for both cross-linkers and rods conformation. The mechanical response of this hairy rod is then expressed by an elastic potential that is built as a function of the basic elasticity of its components. The resulting model is able to capture salient features of the mechanics of such networks, including nonlinear elasticity (and in particular a liquid crystal-like soft-elastic response), creep and stress relaxation, as well as rate- and history-dependent network remodeling. The theory can thus be potentially used to better understand the rich response of these complex, yet ubiquitous networks and guide their development in the laboratory.

A model for the size distribution of marine microplastics: A statistical mechanics approach.

The size distribution of marine microplastics provides a fundamental data source for understanding the dispersal, break down, and biotic impacts of the microplastics in the ocean. The observed size distribution at the sea surface generally shows, from large to small sizes, a gradual increase followed by a rapid decrease. This decrease has led to the hypothesis that the smallest fragments are selectively removed by sinking or biological uptake. Here we propose a new model of size distribution, focusing on the fragmentation of marine plastics. The model is inspired by ideas from statistical mechanics. In this model, the original large plastic piece is broken into smaller pieces once by the application of “energy” or work by waves or other processes, under two assumptions, one that fragmentation into smaller pieces requires larger energy and the other that the occurrence probability of the “energy” exponentially decreases toward larger energy values. Our formula well reproduces observed size distributions over wide size ranges from micro- to mesoplastics. According to this model, the smallest fragments are fewer because large “energy” required to produce such small fragments occurs more rarely.

Predicting conductivities of alkali borophosphate glasses based on site energy distributions derived from network former unit concentrations

Abstract For ion transport in network glasses, it is a great challenge to predict conductivities specifically based on structural properties. To this end it is necessary to gain an understanding of the energy landscape where the thermally activated hopping motion of the ions takes place. For alkali borophosphate glasses, a statistical mechanical approach was suggested to predict essential characteristics of the distribution of energies at the residence sites of the mobile alkali ions. The corresponding distribution of site energies was derived from the chemical units forming the glassy network. A hopping model based on the site energy landscape allowed to model the change of conductivity activation energies with the borate to phosphate mixing ratio. Here we refine and extend this general approach to cope with minimal local activation barriers and to calculate dc-conductivities without the need of performing extensive Monte-Carlo simulations. This calculation relies on the mapping of the many-body ion dynamics onto a network of local conductances derived from the elementary jump rates of the mobile ions. Application of the theoretical modelling to three series of alkali borophosphate glasses with the compositions 0.33Li2O–0.67[xB2O3–(1 − x)P2O5], 0.35Na2O–0.65[xB2O3–(1 − x)P2O5] and 0.4Na2O–0.6[xB2O3–(1 − x)P2O5] shows good agreement with experimental data.

Effects of confinement and vaccination on an epidemic outburst: A statistical mechanics approach.

This work describes a simple agent model for the spread of an epidemic outburst, with special emphasis on mobility and geographical considerations, which we characterize via statistical mechanics and numerical simulations. As the mobility is decreased, a percolation phase transition is found separating a free-propagation phase in which the outburst spreads without finding spatial barriers and a localized phase in which the outburst dies off. Interestingly, the number of infected agents is subject to maximal fluctuations at the transition point, building upon the unpredictability of the evolution of an epidemic outburst. Our model also lends itself to testing vaccination schedules. Indeed, it has been suggested that if a vaccine is available but scarce it is convenient to carefully select the vaccination program to maximize the chances of halting the outburst. We discuss and evaluate several schemes, with special interest on how the percolation transition point can be shifted, allowing for higher mobility without epidemiological impact.

A statistical mechanics approach to macroscopic limits of car-following traffic dynamics

We study the derivation of macroscopic traffic models from car-following vehicle dynamics by means of hydrodynamic limits of an Enskog-type kinetic description. We consider the superposition of Follow-the-Leader (FTL) interactions and relaxation towards a traffic-dependent Optimal Velocity (OV) and we show that the resulting macroscopic models depend on the relative frequency between these two microscopic processes. If FTL interactions dominate then one gets an inhomogeneous Aw–Rascle–Zhang model, whose (pseudo) pressure and stability of the uniform flow are precisely defined by some features of the microscopic FTL and OV dynamics. Conversely, if the rate of OV relaxation is comparable to that of FTL interactions then one gets a Lighthill–Whitham–Richards model ruled only by the OV function. We further confirm these findings by means of numerical simulations of the particle system and the macroscopic models. Unlike other formally analogous results, our approach builds the macroscopic models as physical limits of particle dynamics rather than assessing the convergence of microscopic to macroscopic solutions under suitable numerical discretisations.

Complex formation of sodium-mercury alloy at molten state

Thermodynamic properties of compound forming binary liquid sodium-mercury alloy at temperature 673 K have been analyzed as a function of concentration by considering complex using quasi chemical approximation. The surface tension of the alloy has been studied by the compound formation model, Butler equation as improved by Kaptay, and Statistical mechanical approach. The mixing behavior of the alloy is studied in detail with more emphasis on the interaction energy parameters between neighboring atoms of the alloy. The study provides the information of moderately interacting as well as ordering nature on the entire range of concentration of the liquid alloy and the computed theoretical thermodynamic data are in good agreement with the corresponding experimental data at 673 K. The surface tension of the alloy computed predicts deviation from the ideal case.

Mechanical and Electrical Interaction of Biological Membranes with Nanoparticles and Nanostructured Surfaces.

In this review paper, we theoretically explain the origin of electrostatic interactions between lipid bilayers and charged solid surfaces using a statistical mechanics approach, where the orientational degree of freedom of lipid head groups and the orientational ordering of the water dipoles are considered. Within the modified Langevin Poisson–Boltzmann model of an electric double layer, we derived an analytical expression for the osmotic pressure between the planar zwitterionic lipid bilayer and charged solid planar surface. We also show that the electrostatic interaction between the zwitterionic lipid head groups of the proximal leaflet and the negatively charged solid surface is accompanied with a more perpendicular average orientation of the lipid head-groups. We further highlight the important role of the surfaces’ nanostructured topography in their interactions with biological material. As an example of nanostructured surfaces, we describe the synthesis of TiO2 nanotubular and octahedral surfaces by using the electrochemical anodization method and hydrothermal method, respectively. The physical and chemical properties of these nanostructured surfaces are described in order to elucidate the influence of the surface topography and other physical properties on the behavior of human cells adhered to TiO2 nanostructured surfaces. In the last part of the paper, we theoretically explain the interplay of elastic and adhesive contributions to the adsorption of lipid vesicles on the solid surfaces. We show the numerically predicted shapes of adhered lipid vesicles corresponding to the minimum of the membrane free energy to describe the influence of the vesicle size, bending modulus, and adhesion strength on the adhesion of lipid vesicles on solid charged surfaces.

Conjugate Acid-Base Interaction Driven Phase Transition at a 2D Air-Water Interface.

A lattice model is described to explain a recent striking Sum Frequency Generation (SFG) observation of a cooperative surface adsorption effect for an organic acid system at an air-water interface. The reported anomalous pH-dependent enhancement in p-methylbenzoic acid (pmBA) arises from an interaction between the acid (HA) and its conjugate base anion (A-), which competes with strong Coulombic repulsion between the conjugate bases (A--A -). Using a statistical mechanical approach, this lattice gas model reveals an analogy to well-studied magnetic systems in which the attraction between the two different molecular species leads to a phase transition to a two-dimensional checkerboard phase consisting of a network of anion-acid complexes formed at the low-dielectric air-water interface. Cooperative acid-anion interactions that control partitioning at solution and aerosol interfaces are of interest to fields ranging from oceanic and atmospheric chemistry, pharmacology, and chemical engineering.

Introducing Memory in Coarse-Grained Molecular Simulations.

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori–Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

Simultaneous Identification of Multiple Binding Sites in Proteins: A Statistical Mechanics Approach.

We present an extension of the Poisson-Boltzmann model in which the solute of interest is immersed in an assembly of self-orienting Langevin water dipoles, anions, cations, and hydrophobic molecules, all of variable densities. Interactions between charges are controlled by electrostatics, while hydrophobic interactions are modeled with a Yukawa potential. We impose steric constraints by assuming that the system is represented on a cubic lattice. We also assume incompressibility; i.e., all sites of the lattice are occupied. This model, which we refer to as the Hydrophobic Dipolar Poisson-Boltzmann Langevin (HDPBL) model, leads to a system of two equations whose solutions give the water dipole, salt, and hydrophobic molecule densities, all of them in the presence of the others in a self-consistent way. We use those to study the organization of the ions, cosolvent, and solvent molecules around proteins. In particular, peaks of densities are expected to reveal, simultaneously, the presence of compatible binding sites of different kinds on a protein. We have tested and validated the ability of HDPBL to detect pockets in proteins that bind to hydrophobic ligands, polar ligands, and charged small probes as well as to characterize the binding sites of lipids for membrane proteins.

Effect of Temperature and Concentration on the Physical Properties of Cholesteryl Pelargonate Near Transition Temperatures

Thermodynamic and acoustical study of cholesteric liquid crystal (CLC) was carried out near phase transition temperatures using an ultrasonic interferometer. The study of ultrasonic waves and related thermo-acoustical parameters provides information in understanding the nature of molecular interactions associated with liquid crystals. A statistical mechanical approach is used to calculate the viscosity of samples. The characteristic textures and phase transition temperatures are obtained with Polarizing Microscopy (POM). Fabry Perot scattering studies (FPSS) are used to ascertain the phase transition temperatures of CLC. The thermo-acoustical parameters have been calculated using experimental data with well-known formulae. The acoustical parameters show variation with temperature as well as with concentration of CLC. Anomalous behavior of these parameters is noticed at the clearing temperature of CLC. The variation of the parameters is explained in terms of solute-solvent molecular interactions in the solution. The molecular behavior, structural changes, and intermolecular interactions between CLC and solvent are discussed in detail.

The impulsive swirl of a gas

Abstract

κ-generalised Gutenberg–Richter law and the self-similarity of earthquakes

• Generalised Gutenberg–Richter law appropriately describes the earthquakes in a wide range of magnitudes. • Self-organised criticality of earthquakes and the fractal nature of the geological faults are introduced in the context of Kaniadakis statistics. • The entropic index is a universal parameter in the analysis of more than 150,000 earthquakes. • The earthquakes are well described by the same physics, for all scales of magnitude. The earthquakes statistical analysis are essential for understanding the seismic activity of a region and consequently, in seismic hazard studies. Nowadays, the statistics that describe the relationship between the earthquake magnitude and the total number of quakes in a given region has been dominated by the Gutenberg–Richter (GR) power-law. However, the GR law is only valid from a threshold magnitude (local scale invariance). To mitigate this limitation, we introduce in this work a statistical mechanics approach to appropriately describe the earthquakes in a wide range of magnitudes. In this way, we formulate a generalisation of the GR law based on a κ -probability function, which is a distribution linked to Kaniadakis statistics (or κ -statistics). We have tested the viability of our proposal using real data sets recorded in 6 different regions of the Earth by considering more than 150,000 earthquakes. The results show that the κ -generalised GR law describes appropriately enough the energy distribution in an ample range of magnitudes, especially for low magnitude earthquakes where the standard GR law fails. Besides, the entropic index of the Kaniadakis entropy is a universal parameter with a value close to 1. Furthermore, the results reveal the fractal nature of the fragments filling the space between tectonic plates. They also show the self-similarity of earthquakes regardless of their magnitude.