Statistical Mechanical Model Research Articles

A statistical mechanical model of supercooled water based on minimal clusters of correlated molecules.

In this paper, we apply a theoretical model for fluid state thermodynamics to investigate simulated water in supercooled conditions. This model, which we recently proposed and applied to sub- and super-critical fluid water [Zanetti-Polzi et al., J. Chem. Phys. 156(4), 44506 (2022)], is based on a combination of the moment-generating functions of the enthalpy and volume fluctuations as provided by two gamma distributions and provides the free energy of the system as well as other relevant thermodynamic quantities. The application we make here provides a thermodynamic description of supercooled water fully consistent with that expected by crossing the liquid-liquid Widom line, indicating the presence of two distinct liquid states. In particular, the present model accurately reproduces the Widom line temperatures estimated with other two-state models and well describes the heat capacity anomalies. Differently from previous models, according to our description, a cluster of molecules that extends beyond the first hydration shell is necessary to discriminate between the statistical fluctuation regimes typical of the two liquid states.

A stroll around the critical Potts model

Over the past decade or so, a broad research programme spearheaded by H. Duminil-Copin and his collaborators has vastly increased our understanding of a number of critical or near-critical statistical mechanics models. Most prominently, these include the q q -state Potts models and, essentially equivalently, the FK cluster models. In this short review, we present a small selection of recent results from this research area.

Fusion of Centroid-Based Clustering With Graph Clustering: An Expectation-Maximization-Based Hybrid Clustering.

This article extends the expectation-maximization (EM) formulation for the Gaussian mixture model (GMM) with a novel weighted dissimilarity loss. This extension results in the fusion of two different clustering methods, namely, centroid-based clustering and graph clustering in the same framework in order to leverage their advantages. The fusion of centroid-based clustering and graph clustering results in a simple "soft" asynchronous hybrid clustering method. The proposed algorithm may start as a pure centroid-based clustering algorithm (e.g., k -means), and as the time evolves, it may eventually and gradually turn into a pure graph clustering algorithm [e.g., basic greedy asynchronous distributed interference avoidance (GADIA) (Babadi and Tarokh, 2010)] as the algorithm converges and vice versa. The "hard" version of the proposed hybrid algorithm includes the standard Hopfield neural networks (and, thus, Bruck's Ln algorithm by (Bruck, 1990) and the Ising model in statistical mechanics), Babadi and Tarokh's basic GADIA in 2010, and the standard k -means (Steinhaus, 1956), (MacQueen, 1967) [i.e., the Lloyd algorithm (Lloyd, 1957, 1982)] as its special cases. We call the "hard version" of the proposed clustering as "hybrid-nongreedy asynchronous clustering (H-NAC)." We apply the H-NAC to various clustering problems using well-known benchmark datasets. The computer simulations confirm the superior performance of the H-NAC compared to the k -means clustering, k -GADIA, spectral clustering, and a very recent clustering algorithm structured graph learning (SGL) by Kang et al. (2021), which represents one of the state-of-the-art clustering algorithms.

Improvement of predictive ability of self-association model to segregation of liquid Zn−Bi alloys

The self-association model is an effective theoretical method for determining the thermodynamic parameters of immiscible molten binary alloys. However, two important parameters (the ratios of numbers of two atoms in the cluster n and the interchange energy W) in the model had confusing temperature dependences, which prevented the statistical mechanical model from accurately predicting the physical and chemical properties at various temperatures. The ratios of the two atoms in the cluster and the interchange energy were proposed in this study to be linear with temperature, respectively, to analyze the segregation of liquid Zn–Bi alloys. The hypothesis about two key parameters is helpful to predict the segregating nature of liquid Zn–Bi alloys, as evidenced by the good accordance of the predicted thermophysical properties (activity, free energy of mixing, miscibility gap, entropy of mixing, partial molar entropy, and concentration fluctuation) with the existing experimental data.

The effects of functionalization on graphene oxide for organic dye adsorption: An experimental-theoretical study using electronic structure calculations and statistical mechanical modeling

In this work, an alternative theoretical–experimental approach was combined to investigate the influence of the degree of functionalization of reduced graphene oxides (rGO) on the capacity and adsorption mechanisms of two model organic dyes. The experimental adsorption equilibrium data for methylene blue (MB) and indigo carmine (IC) were adjusted and studied using an efficient Multi-Layer Finite Model (MLFM) based on statistical mechanics. Additionally, density functional calculations (DFTB and DFT) were carried out to study the molecular interaction geometry and adsorption energies. With this approach, a greater amount of information about the adsorption process can be obtained in relation to commonly used methodologies for adsorption at a solid–liquid interface. Herein, kinetic and adsorption equilibrium experiments were conducted on rGOs prepared via thermal reduction at different temperatures (300, 700, and 1000 °C) in order to obtain very different contents of oxygenated functional groups. The highest adsorbed amounts are obtained for rGO1000, reaching 208 and 320 mg g−1 for MB and IC, respectively. The MLFM results revealed that the rGO-dye interaction occurs preferentially via π-stacking, and with the formation of up to two adsorption layers. The elimination of functional groups from the surface of rGOs is also revealed to be favorable, decreasing the adsorption energy and increasing the amount and fraction of molecules adsorbed in parallel orientation.

Spectral Analysis of Discrete Metastable Diffusions

We consider a discrete Schrödinger operator H_varepsilon = -varepsilon ^2Delta _varepsilon + V_varepsilon on ell ^2(varepsilon mathbb {Z}^d), where varepsilon >0 is a small parameter and the potential V_varepsilon is defined in terms of a multiwell energy landscape f on mathbb {R}^d. This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of mathbb {R}^d. It is unitarily equivalent to the generator of a diffusion on varepsilon mathbb {Z}^d, satisfying the detailed balance condition with respect to the Boltzmann weight exp {(-f/varepsilon )}. These type of diffusions exhibit metastable behavior and arise in the context of disordered mean field models in Statistical Mechanics. We analyze the bottom of the spectrum of H_varepsilon in the semiclassical regime varepsilon ll 1 and show that there is a one-to-one correspondence between exponentially small eigenvalues and local minima of f. Then we analyze in more detail the bistable case and compute the precise asymptotic splitting between the two exponentially small eigenvalues. Through this purely spectral-theoretical analysis of the discrete Witten Laplacian we recover in a self-contained way the Eyring–Kramers formula for the metastable tunneling time of the underlying stochastic process.

Thermo-aqueous coupling behaviors for shape memory hydrogels: A statistical–mechanical model and simulations faced with experiments

The shape memory behavior of hydrogels endows themselves with the capacities of shape programming and shape recovering, which have great potential in many fields, such as drug release and soft robotics. In thermally responsive hydrogel, particularly, there are two critical factors, temperature and water, that can induce the shape memory effect(SME). Temperature-induced SME has been extensively studied, while water-induced SME has been rarely investigated. In our work, the combined effect of water and temperature on shape memory behavior of hydrogels is experimentally studied and deeply analyzed from a physical perspective. We found that the shape fixed ratio of shape memory hydrogels (SMHs) increases monotonically with temperature, but first increases and then decreases with water content. Moreover, it exits a nonlinear coupling between the water and temperature effects on the SME. On the basis of these findings, a constitutive model is developed to capture this thermo-aqueous coupling using the statistical theory of transient networks, from which we obtain an explicit evolution equation of deformation gradient. In the proposed theory, the temperature-induced SME is investigated through the temperature-water responsive association and dissociation of dynamic bonds while the water-induced SME is explained by the variation of the glass transition temperature according to the Fox–Flory model. Finally, and to provide a final test of the theoretical work, it is implemented into a user element (UEL) subroutine which provide simulations in agreement with our experiments. Our work is expected to provide guidance for future application of SMHs.

Statistical Mechanical Model of the Giant Electrocaloric Effect in Ferroelectric Polymers.

The development of highly efficient cooling technologies has been identified as a key strategy to address the mitigation of global warming. Especially, electrocaloric materials have emerged as promising candidates for cooling applications, owing to their potential to provide high cooling capacity with low energy consumption. To advance the development of electrocaloric materials with a significant electrocaloric effect (ECE), a thorough understanding of the underlying mechanisms is required. Previous studies have estimated the maximum ECE temperature change by calculating the entropy change between two assumed states of a dipole model, assuming polarization saturation with a sufficiently large electric field. However, it is more relevant to assess the ECE under continuously changing electric fields as this is more reflective of real-world conditions. To this end, we establish a continuous transition between the complete disorder state and the polarization saturation state using the partition function to derive the entropy change. Our results demonstrate excellent agreement with experimental data, and our analysis of energy items within the partition function attributes the increase in the ECE entropy change with decreasing crystal size to interfacial effects. This statistical mechanical model reveals the in-depth ferroelectric polymers producing the ECE and offers significant potential for predicting the ECE in ferroelectric polymers and thus guides the design of high-performance ECE materials.

Spin ½ model in statistical mechanics and relation to a truncation of the Riemann ξ function in the Riemann Hypothesis

In the search for a solution to the Riemann Hypothesis, we have studied an approach that connects the Riemann Problem with physical modeling that refers to statistical mechanics. Thus, we study the relation between a truncation of the Riemann ξ function in the variable z = 1-1/s, where s is the usual complex variable (s=Re(s)+i·Im(s) = ρ + i·t) and the partition function of a ferromagnetic spin 1/2 model on a circle C with long-range interaction, to give a concrete look at a strategy for a possible proof of the Riemann Hypothesis.

What quantum matter tells about quantisation of gravity in a statistical mechanics context

We propose a “guide” towards quantisation of gravity based on quantum matter in a statistical mechanics context. On one hand, a statistical mechanics model naturally arises from the thermodynamic interpretation of horizons in Rindler space. On the other hand, the path integral formulation of quantum field theory can be interpreted from the point of view of statistical mechanics. From these perspectives, gravity and matter are related to each other in the same way as a gas and its chemical potential are. This statistical mechanics interpretation of gravity and matter suggests that gravity should be quantised in a precise way which is determined by the quantisation of matter. Although, in a first step, quantisation of gravity applies for small perturbations of the metric with respect to the vacuum, the most central and general features of quantisation (quantisation prescription, quantum space dimensions) are supported from statistical mechanics and remain valid non-perturbatively.

Theory and Simulations of Ionic Liquids in Nanoconfinement.

Room-temperature ionic liquids (RTILs) have exciting properties such as nonvolatility, large electrochemical windows, and remarkable variety, drawing much interest in energy storage, gating, electrocatalysis, tunable lubrication, and other applications. Confined RTILs appear in various situations, for instance, in pores of nanostructured electrodes of supercapacitors and batteries, as such electrodes increase the contact area with RTILs and enhance the total capacitance and stored energy, between crossed cylinders in surface force balance experiments, between a tip and a sample in atomic force microscopy, and between sliding surfaces in tribology experiments, where RTILs act as lubricants. The properties and functioning of RTILs in confinement, especially nanoconfinement, result in fascinating structural and dynamic phenomena, including layering, overscreening and crowding, nanoscale capillary freezing, quantized and electrotunable friction, and superionic state. This review offers a comprehensive analysis of the fundamental physical phenomena controlling the properties of such systems and the current state-of-the-art theoretical and simulation approaches developed for their description. We discuss these approaches sequentially by increasing atomistic complexity, paying particular attention to new physical phenomena emerging in nanoscale confinement. This review covers theoretical models, most of which are based on mapping the problems on pertinent statistical mechanics models with exact analytical solutions, allowing systematic analysis and new physical insights to develop more easily. We also describe a classical density functional theory, which offers a reliable and computationally inexpensive tool to account for some microscopic details and correlations that simplified models often fail to consider. Molecular simulations play a vital role in studying confined ionic liquids, enabling deep microscopic insights otherwise unavailable to researchers. We describe the basics of various simulation approaches and discuss their challenges and applicability to specific problems, focusing on RTIL structure in cylindrical and slit confinement and how it relates to friction and capacitive and dynamic properties of confined ions.

Simple rose model of water in constant electric field.

A simple two-dimensional statistical mechanical water model, called the rose model, was used in this work. We studied how a homogeneous constant electric field affects the properties of water. The rose model is a very simple model that helps explain the anomalous properties of water. Rose water molecules are represented as two-dimensional Lennard-Jones disks with potentials for orientation-dependent pairwise interactions mimicking formations of hydrogen bonds. The original model is modified by addition of charges for interaction with the electric field. We studied what kind of influence the electric field strength has on the model's properties. To determine the structure and thermodynamics of the rose model under the influence of the electric field we used Monte Carlo simulations. Under the influence of a weak electric field the anomalous properties and phase transitions of the water do not change. On the other hand, the strong fields shift the phase transition points as well as the position of the density maximum.

A statistical mechanical model for non-perturbative regimes

Thanks to Feynman graphons, as mathematical tools in dealing with Dyson–Schwinger equations, we formulate a new statistical mechanical model for the study of equilibrium states and observables associated with the solution space of quantum motions in a strongly coupled gauge field theory underlying the evolution of running coupling constants, time and temperature.

P$$ p $$‐Adic phase transitions for countable state Potts model

The statistical mechanical models with finitely many spin values have several applications in many areas of natural science. One of the most studied models is the Potts model, which has a rich structure of physical phenomena. However, countable analog of the mentioned model has not been deeply studied yet. In the present paper, we are going to construct generalized ‐adic Gibbs measures (for the countable state ‐adic Potts model on a Cayley tree) by investigating non‐linear ‐adic dynamical systems on the sequence spaces. The provided study offers the existence of the strong phase transition for the model.

Protein Stability─Analysis of Heat and Cold Denaturation without and with Unfolding Models.

Protein stability is important in many areas of life sciences. Thermal protein unfolding is investigated extensively with various spectroscopic techniques. The extraction of thermodynamic properties from these measurements requires the application of models. Differential scanning calorimetry (DSC) is less common, but is unique as it measures directly a thermodynamic property, that is, the heat capacity Cp(T). The analysis of Cp(T) is usually performed with the chemical equilibrium two-state model. This is not necessary and leads to incorrect thermodynamic consequences. Here we demonstrate a straightforward model-independent evaluation of heat capacity experiments in terms of protein unfolding enthalpy ΔH(T), entropy ΔS(T), and free energy ΔG(T)). This now allows the comparison of the experimental thermodynamic data with the predictions of different models. We critically examined the standard chemical equilibrium two-state model, which predicts a positive free energy for the native protein, and diverges distinctly from the experimental temperature profiles. We propose two new models which are equally applicable to spectroscopy and calorimetry. The ΘU(T)-weighted chemical equilibrium model and the statistical-mechanical two-state model provide excellent fits of the experimental data. They predict sigmoidal temperature profiles for enthalpy and entropy, and a trapezoidal temperature profile for the free energy. This is illustrated with experimental examples for heat and cold denaturation of lysozyme and β-lactoglobulin. We then show that the free energy is not a good criterion to judge protein stability. More useful parameters are discussed, including protein cooperativity. The new parameters are embedded in a well-defined thermodynamic context and are amenable to molecular dynamics calculations.

Sequence-dependent model of allosteric communication.

The omnipresence and diversity of allosteric regulation in proteins and protein associations complemented by the potential for the design of allosterically acting biologics and drugs call for the development of a new generation of computational models for the analysis of allostery and rational engineering/design of desired signaling and effector molecules determining it. One of the most important challenges is the consideration of the role of amino acid sequence in forming the protein's allosteric communication, including the mode and strength of the allosteric signal that is communicated to the regulated functional site. Here, we present the network-based model with a sequence dependence added in consideration of allosteric communication by combining the structure-based statistical mechanical model of allostery with the Miyazawa-Jernigan residue-residue potential. Applying the model in the analysis of five classical allosteric proteins, we found that it is necessary to consider the following two major determinants: (i) the free energy exerted by the allosteric site on the regulated one and (ii) the background (average) change in dynamics of the overall structure. We show that working together these two components determine the allosteric modulation, calling one to study their dependence on structures, oligomerization states, and sequence divergence in different proteins.

Entanglement Domain Walls in Monitored Quantum Circuits and the Directed Polymer in a Random Environment

Monitored quantum dynamics reveal quantum state trajectories which exhibit a rich phenomenology of entanglement structures, including a transition from a weakly-monitored volume law entangled phase to a strongly-monitored area law phase. For one-dimensional hybrid circuits with both random unitary dynamics and interspersed measurements, we combine analytic mappings to an effective statistical mechanics model with extensive numerical simulations on hybrid Clifford circuits to demonstrate that the universal entanglement properties of the volume law phase can be quantitatively described by a fluctuating entanglement domain wall that is equivalent to a "directed polymer in a random environment" (DPRE). This relationship improves upon a qualitative "mean-field" statistical mechanics of the volume-law-entangled phase [1, 2]. For the Clifford circuit in various geometries, we obtain agreement between the subleading entanglement entropies and error correcting properties of the volume-law phase (which quantify its stability to projective measurements) with predictions of the DPRE. We further demonstrate that depolarizing noise in the hybrid dynamics near the final circuit time can drive a continuous phase transition to a non-error correcting volume law phase that is not immune to the disentangling action of projective measurements. We observe this transition in hybrid Clifford dynamics, and obtain quantitative agreement with critical exponents for a "pinning" phase transition of the DPRE in the presence of an attractive interface.

Crystallization of Amorphous Nimesulide: The Relationship between Crystal Growth Kinetics and Liquid Dynamics.

Understanding crystallization and its correlations with liquid dynamics is relevant for developing robust amorphous pharmaceutical solids. Herein, nimesulide, a classical anti-inflammatory agent, was used as a model system for studying the correlations between crystallization kinetics and molecular dynamics. Kinetic parts of crystal growth (ukin) of nimesulide exhibited a power law dependence upon the liquid viscosity (η) as ukin~η-0.61. Bulk molecular diffusivities (DBulk) of nimesulide were predicted by a force-level statistical-mechanical model from the α-relaxation times, which revealed the relationship as ukin~Dbulk0.65. Bulk crystal growth kinetics of nimesulide in deeply supercooled liquid exhibited a fragility-dependent decoupling from τα. The correlations between growth kinetics and α-relaxation times predicted by the Adam-Gibbs-Vogel equation in a glassy state were also explored, for both the freshly made and fully equilibrated glass. These findings are relevant for the in-depth understanding and prediction of the physical stability of amorphous pharmaceutical solids.

Graph coverings and twisted operators

Given a graph and a representation of its fundamental group, there is a naturally associated twisted adjacency operator, uniquely defined up to conjugacy. The main result of this article is the fact that this operator behaves in a controlled way under graph covering maps. When such an operator can be used to enumerate objects, or compute a partition function, this has concrete implications on the corresponding enumeration problem, or statistical mechanics model. For example, we show that if Γ ˜ is a finite covering graph of a connected graph Γ endowed with edge-weights x={x e } e , then the spanning tree partition function of Γ divides the one of Γ ˜ in the ring ℤ[x]. Several other consequences are obtained, some known, others new.

Modified Macdonald polynomials and the multispecies zero-range process: I

In this paper we prove a new combinatorial formula for the modified Macdonald polynomials H ˜ λ (X;q,t), motivated by connections to the theory of interacting particle systems from statistical mechanics. The formula involves a new statistic called queue inversions on fillings of tableaux. This statistic is closely related to the multiline queues which were recently used to give a formula for the Macdonald polynomials P λ (X;q,t). In the case q=1 and X=(1,1,⋯,1), that formula had also been shown to compute stationary probabilities for a particle system known as the multispecies ASEP on a ring, and it is natural to ask whether a similar connection exists between the modified Macdonald polynomials and a suitable statistical mechanics model. In a sequel to this work, we demonstrate such a connection, showing that the stationary probabilities of the multispecies totally asymmetric zero-range process (mTAZRP) on a ring can be computed using tableaux formulas with the queue inversion statistic. This connection extends to arbitrary X=(x 1 ,⋯,x n ); the x i play the role of site-dependent jump rates for the mTAZRP.