Techniques Of Statistical Mechanics Research Articles

Can one hear the shape of a crystal?

Isospectrality is a general fundamental concept often involving whether various operators can have identical spectra, i.e., the same set of eigenvalues. In the context of the Laplacian operator, the famous question “Can one hear the shape of a drum?” concerns whether different shaped drums can have the same vibrational modes. The isospectrality of a lattice in d-dimensional Euclidean space Rd is a tantamount to whether it is uniquely determined by its theta series, i.e., the radial distribution function g2(r). While much is known about the isospectrality of Bravais lattices across dimensions, little is known about this question of more general crystal (periodic) structures with an n-particle basis (n≥2). Here, we ask what is nmin(d), the minimum value of n for inequivalent (i.e., unrelated by isometric symmetries) crystals with the same theta function in space dimension d? To answer these questions, we use rigorous methods as well as a precise numerical algorithm that enables us to determine the minimum multiparticle basis of inequivalent isospectral crystals. Our algorithm identifies isospectral four-, three- and two-particle bases in one, two, and three spatial dimensions, respectively. For many of these isospectral crystals, we rigorously show that they indeed possess identically the same g2(r)'s for all values of r. Based on our analyses, we conjecture that nmin(d)=4, 3, 2 for d=1, 2, 3, respectively. The identification of isospectral crystals enables one to study the degeneracy of the ground-state under the action of isotropic pair potentials. Indeed, using inverse statistical-mechanical techniques, we find an isotropic pair potential whose low-temperature configurations in two dimensions obtained via simulated annealing can lead to both of two isospectral crystal structures with n=3, the proportion of which can be controlled by the cooling rate. Our findings provide general insights into the structural and ground-state degeneracies of crystal structures as determined by radial pair information. Published by the American Physical Society 2024

Storage properties of a quantum perceptron.

Driven by growing computational power and algorithmic developments, machine learning methods have become valuable tools for analyzing vast amounts of data. Simultaneously, the fast technological progress of quantum information processing suggests employing quantum hardware for machine learning purposes. Recent works discuss different architectures of quantum perceptrons, but the abilities of such quantum devices remain debated. Here, we investigate the storage capacity of a particular quantum perceptron architecture by using statistical mechanics techniques and connect our analysis to the theory of classical spin glasses. Specifically, we focus on one concrete quantum perceptron model and explore its storage properties in the limit of a large number of inputs.

Exact equilibrium properties of square-well and square-shoulder disks in single-file confinement.

This study investigates the (longitudinal) thermodynamic and structural characteristics of single-file confined square-well and square-shoulder disks by employing a mapping technique that transforms the original system into a one-dimensional polydisperse mixture of nonadditive rods. Leveraging standard statistical-mechanical techniques, exact results are derived for key properties, including the equationof state, internal energy, radial distribution function, and structure factor. The asymptotic behavior of the radial distribution function is explored, revealing structural changes in the spatial correlations. Additionally, exact analytical expressions for the second virial coefficient are presented. The theoretical results for the thermodynamic and structural properties are validated by our Monte Carlo simulations.

Synchronization of four unalike local coupled phase oscillators in a ring: Analytic formulas for the critical coupling and phase differences

There is no universal technique in statistical mechanics to analytically solve the problem of dissimilar adjacent coupled phase oscillators in a ring at the switch to synchrony. The solution to the four oscillators′ case presents a first step to constructing a regular method based on unscrambling the sync dynamics of a few oscillators. The local interaction between the adjacent oscillators and the periodic boundary conditions introduce finer points to the synchronicity. We notice that the synchronized state has different noticeable characteristics depending on the arrangements of the primary frequencies. This, in turn, yields an involvedness in obtaining an analytic solution. To overcome the difficulty, we classify the twenty-four possible arrangements for the four local coupled phase oscillators in a ring into three classes. Consequently, we introduce a method to rearrange the initial frequencies into consecutive pairs of nearest-neighbor elements for all three classes. Thus, we can define the effective parameters (in terms of the sequential couples of adjacent oscillators) that rule the solution for the oscillators in a ring. Furthermore, the unison behavior resulting from coupling emerges when a clearly defined phase lock condition develops. We utilize the phase condition and trigonometric identities to find a soluble-effortless relation. Consequently, we derive an analytic expression that predicts the threshold coupling for each class when the oscillators synchronize. The formula for the critical factor allows us to obtain usable expressions to determine the phase differences at the transition into a sync stage. Therefore, we figure out the sync mechanism a priori, knowing the initial frequencies.

First-principles thermodynamics of precipitation in aluminum-containing refractory alloys

Materials for high-temperature environments are actively being investigated for deployment in aerospace and nuclear applications. This study uses computational approaches to unravel the crystallography and thermodynamics of a promising class of refractory alloys containing aluminum. Accurate first-principles calculations, cluster expansion models, and statistical mechanics techniques are employed to rigorously analyze precipitation in a prototypical senary Al–Nb–Ta–Ti–V–Zr alloy. Finite-temperature calculations reveal a strong tendency for aluminum to segregate to a single sublattice at elevated temperatures. Precipitate and matrix compositions computed with our ab-initio model are in excellent agreement with previous experimental measurements (Soni et al., 2020). Surprisingly, conventional B2-like orderings are found to be both thermodynamically and mechanically unstable in this alloy system. Complex anti-site defects are essential to forming a stable ordered precipitate. Our calculations reveal that the instability of B2 compounds can be related to a simple electron counting rule across all binary alloys formed by elements in groups 4,5, and 6. The results of this study provide viable routes toward designing high-temperature materials for deployment in extreme environments.

Replica analysis of overfitting in regression models for time to event data: the impact of censoring

We use statistical mechanics techniques, viz. the replica method, to model the effect of censoring on overfitting in Cox’s proportional hazards model, the dominant regression method for time-to-event data. In the overfitting regime, Maximum Likelihood (ML) parameter estimators are known to be biased already for small values of the ratio of the number of covariates over the number of samples. The inclusion of censoring was avoided in previous overfitting analyses for mathematical convenience, but is vital to make any theory applicable to real-world medical data, where censoring is ubiquitous. Upon constructing efficient algorithms for solving the new (and more complex) Replica Symmetric (RS) equations and comparing the solutions with numerical simulation data, we find excellent agreement, even for large censoring rates. We then address the practical problem of using the theory to correct the biased ML estimators without knowledge of the data-generating distribution. This is achieved via a novel numerical algorithm that self-consistently approximates all relevant parameters of the data generating distribution while simultaneously solving the RS equations. We investigate numerically the statistics of the corrected estimators, and show that the proposed new algorithm indeed succeeds in removing the bias of the ML estimators, for both the association parameters and for the cumulative hazard.

On the capacity of a quantum perceptron for storing biased patterns

Although different architectures of quantum perceptrons have been recently put forward, the capabilities of such quantum devices versus their classical counterparts remain debated. Here, we consider random patterns and targets independently distributed with biased probabilities and investigate the storage capacity of a continuous quantum perceptron model that admits a classical limit, thus facilitating the comparison of performances. Such a more general context extends a previous study of the quantum storage capacity where using statistical mechanics techniques in the limit of a large number of inputs, it was proved that no quantum advantages are to be expected concerning the storage properties. This outcome is due to the fuzziness inevitably introduced by the intrinsic stochasticity of quantum devices. We strengthen such an indication by showing that the possibility of indefinitely enhancing the storage capacity for highly correlated patterns, as it occurs in a classical setting, is instead prevented at the quantum level.

Unusual Aspects of Charge Regulation in Flexible Weak Polyelectrolytes.

This article reviews the state of the art of the studies on charge regulation (CR) effects in flexible weak polyelectrolytes (FWPE). The characteristic of FWPE is the strong coupling of ionization and conformational degrees of freedom. After introducing the necessary fundamental concepts, some unconventional aspects of the the physical chemistry of FWPE are discussed. These aspects are: (i) the extension of statistical mechanics techniques to include ionization equilibria and, in particular, the use of the recently proposed Site Binding-Rotational Isomeric State (SBRIS) model, which allows the calculation of ionization and conformational properties on the same foot; (ii) the recent progresses in the inclusion of proton equilibria in computer simulations; (iii) the possibility of mechanically induced CR in the stretching of FWPE; (iv) the non-trivial adsorption of FWPE on ionized surfaces with the same charge sign as the PE (the so-called "wrong side" of the isoelectric point); (v) the influence of macromolecular crowding on CR.

Semigrand-canonical Monte-Carlo simulation methods for charge-decorated cluster expansions

Monte-Carlo sampling of lattice model Hamiltonians is a well-established technique in statistical mechanics for studying the configurational entropy of crystalline materials. When the species to be distributed on the lattice carry charge, the charge balance constraint on the overall system prohibits single-site Metropolis exchanges in MC. In this article, we propose two methods to perform MC sampling in the semigrand-canonical ensemble in the presence of a charge-balance constraint. The table-exchange method (TE) constructs small charge-conserving excitations, and the square-charge bias method (SCB) allows the system to temporarily drift away from charge neutrality. We illustrate the effect of internal hyper-parameters on the efficiency of these algorithms and suggest practical strategies on how to employ these algorithms in real applications.

Entanglement and Charge-Sharpening Transitions in U(1) Symmetric Monitored Quantum Circuits

Monitored quantum circuits can exhibit an entanglement transition as a function of the rate of measurements, stemming from the competition between scrambling unitary dynamics and disentangling projective measurements. We study how entanglement dynamics in non-unitary quantum circuits can be enriched in the presence of charge conservation, using a combination of exact numerics and a mapping onto a statistical mechanics model of constrained hard-core random walkers. We uncover a charge-sharpening transition that separates different scrambling phases with volume-law scaling of entanglement, distinguished by whether measurements can efficiently reveal the total charge of the system. We find that while R\'enyi entropies grow sub-ballistically as $\sqrt{t}$ in the absence of measurement, for even an infinitesimal rate of measurements, all average R\'enyi entropies grow ballistically with time $\sim t$. We study numerically the critical behavior of the charge-sharpening and entanglement transitions in U(1) circuits, and show that they exhibit emergent Lorentz invariance and can also be diagnosed using scalable local ancilla probes. Our statistical mechanical mapping technique readily generalizes to arbitrary Abelian groups, and offers a general framework for studying dissipatively-stabilized symmetry-breaking and topological orders.

TOWARDS A MEASURE OF HARMONIC COMPLEXITY IN WESTERN CLASSICAL MUSIC

We recently introduced the concept of dynamical score network to represent the harmonic progressions in any composition. Through a process of chord slicing, we obtain a representation of the score as a complex network, where every chord is a node and each progression (voice leading) links successive chords. In this paper, we use this representation to extract quantitative information about harmonic complexity from the analysis of the topology of these networks using state-of-the-art statistical mechanics techniques. Since complex networks support the communication of information by encoding the structure of allowed messages, we can quantify the information associated with locating specific addresses through the measure of the entropy of such network. In doing so, we then characterize properties of network topology, such as the degree distribution of a graph or the shortest paths between couples of nodes. Here, we report on two different evaluations of network entropy, diffusion entropy analysis (DEA) and the Kullback–Leibler divergence applied to the conditional degree matrix, and the measurements of complexity they provide, when applied to an extensive corpus of scores spanning 500 years of western classical music. Although the analysis is limited in scope, our results already provide quantitative evidence of an increase of such measures of harmonic complexity over the corpora we have analyzed.

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.

A new microscopic look into pressure fields and local buoyancy

Defining local pressures is pivotal to understanding phenomena such as buoyancy and atmospheric physics, which always require space-varying pressure fields to be considered. Such fields have precise definitions within the frameworks of hydro-thermodynamics and the kinetic theory of gases, but a pedagogical approach based on standard statistical mechanics techniques is still lacking. In this paper, we propose a new microscopic definition of local pressure inside a classical fluid, relying upon a local barometer potential that is built into the many-particle Hamiltonian. This setup allows pressure to be locally defined as an ensemble average of the radial force exerted by the barometer potential on the gas' particles. Moreover, this same framework is also employed in a microscopic derivation of buoyancy of immersed bodies in the presence of arbitrary external fields. As instructive examples, buoyant forces are calculated for ideal fluids in the presence of a uniform force field, in the presence of a spherical harmonic confinement field and in rotation with a constant angular velocity.

Theoretical insights into non-Arrhenius behaviors of thermal vacancies in anharmonic crystals.

Vacancies are prevalent point defects in crystals, but their thermal responses are elusive. Herein, we formulate a simple theoretical model to shed light on the vacancy evolution during heating. Vibrational excitations are thoroughly investigated via moment recurrence techniques in quantum statistical mechanics. On that basis, we carry out numerical analyses for Ag, Cu, and Ni with the Sutton-Chen many-body potential. Our results reveal that the well-known Arrhenius law is insufficient to describe the proliferation of vacancies. Specifically, anharmonic effects lead to a strong nonlinearity in the Gibbs energy of vacancy formation. Our physical picture is well supported by previous simulations and experiments.

Generalized theory of clustering of extended galaxies with core halos

We discuss the clustering of core halo-galaxies of radius predicted by gravitational lensing. We appeal to a blend of recent (1) mathematical results and (2) statistical mechanics techniques. The exact configurational integrals of extended galaxies is evaluated together with the pertinent partition functions, the corresponding thermodynamics, and the distribution function. We are motivated by data regarding observational deflections and measured Einstein radius in gravitational microlensing and macrolensing. The halo size comparable with that of the Einstein radius or the Schwarzschild one, is taken as the lower limit of the configurational integral. The work may provide new insight with regards to gravitational lensing. Giant arcs produced by gravitational lensing, as observed from clusters of galaxies, seem to require the inclusion of substantial quadruple and higher multipole moments in the cluster potential. This opens a possibility of ”modified gravities” , in clustering studies that compare data associated to gravitational lensing.

Partition function and coherent states for the quantum multiverse

Abstract Motivated by a wider acceptance of the quantum cosmology and the idea of multiverse, we follow the second and third quantization formalism for the conceptual comprehension of quantum multiverse. The other motivating feature to pursue the present study is that at the different levels of quantization the equation of quantum state is similar to that of harmonic oscillator, which encourages us to give a glancing taste of the coherent states in the domain of quantum cosmology. Third quantization enables one to model the universe as a particle in a quantum field theory of the minisuperspace and the wave description prompts one to recognize interactions among the universes of the multiverse. This picture of the system further instigates us to investigate the partition function and the thermodynamic equations of the system. The path integral formalism with proper mathematical extensions are employed to find the partition function followed by the usual techniques of statistical mechanics to examine the thermodynamic properties.

Efficient analytical approach for high-pressure melting properties of iron

Abstract Iron represents the principal constituent of the Earth's core, but its high-pressure melting diagram remains ambiguous. Here we present a simple analytical approach to predict the melting properties of iron under deep-Earth conditions. In our model, anharmonic free energies of the solid phase are directly determined by the moment expansion technique in quantum statistical mechanics. This basis associated with the Lindemann criterion for a vibrational instability can deduce the melting temperature. Moreover, we correlate the thermal expansion process with the shear response to explain a discontinuity of atomic volume, enthalpy, and entropy upon melting. Our numerical calculations are quantitatively consistent with recent experiments and simulations. The obtained results would improve understanding of the Earth's structure, dynamics, and evolution.

Does 'Dry Hit' vaping of vitamin E acetate contribute to EVALI? Simulating toxic ketene formation during e-cigarette use.

Vitamin E acetate (VEA) is strongly linked to the outbreak of electronic-cigarette or vaping product use-associated lung injury (EVALI). It has been proposed that VEA decomposition to ketene–a respiratory poison that damages lungs at low ppm levels–may play a role in EVALI. However, there is no information available on the temperature at which VEA decomposes and how this correlates with the vaping process. We have studied the temperature-dependent kinetics of VEA decomposition using quantum chemical and statistical mechanical modelling techniques, developing a chemical kinetic model of the vaping process. This model predicts that, under typical vaping conditions, the use of VEA contaminated e-cigarette products is unlikely to produce ketene at harmful levels. However, at the high temperatures encountered at low e-cigarette product levels, which produce ‘dry hits’, ketene concentrations are predicted to reach acutely toxic levels in the lungs (as high as 30 ppm). We therefore hypothesize that dry hit vaping of e-cigarette products containing VEA contributes to EVALI.

Crystallography, thermodynamics and phase transitions in refractory binary alloys

We investigate phase stability in all binary alloys comprised of elements from groups 4 (Ti, Zr, Hf), 5 (V, Nb, Ta) and 6 (Cr, Mo, W) of the periodic table. First-principles calculations of the energy landscapes along crystallographic pathways that connect bcc to hcp and bcc to ω show that group 4 elements are very distinct from group 5 and 6 elements. While group 5 and 6 elements are stable in bcc, group 4 elements favor hcp and ω and are predicted to be dynamically unstable in bcc. A comprehensive first-principles investigation of the 36 refractory binary systems using statistical mechanics techniques reveals six distinct classes of alloys, each with a unique phase diagram topology. The predictions of this study are in excellent agreement with previous experimental work. One exception is a class of refractory alloys with high temperature miscibility gaps that are not predicted with the methods used in this work. Our calculations predict the stability of a low-temperature Laves phase in the Nb-V binary that has yet to be observed experimentally. The relationships between alloy chemistry and high-temperature phase stability revealed in this study provide a basis for the systematic design of multicomponent disordered refractory alloys.

Generic theory of the dynamic magnetic response of ferrofluids.

Ferrofluids belong to an important class of highly functional soft matter, benefiting from their magnetically controllable physical properties. Therefore, it is of central importance to quantitatively predict the dynamic magnetic response of ferrofluids. Traditional dynamic theories, however, are often restricted to the near-equilibrium regime and/or only apply to nearly ideal ferrofluids that are monodisperse, dilute enough, and weakly interacting. In this paper I develop a self-consistent and nonperturbative dynamical mean field theory for typical ferrofluids which are often polydisperse, concentrated, and strongly interacting, possibly driven far from equilibrium. I obtain a general nonperturbative expression for the dynamic magnetic susceptibility, quantitatively agreeing with the spectra obtained from Brownian Dynamics simulations on both mono- and bidisperse samples. Furthermore, I derive a generic magnetization relaxation equation (MRE) for both mono- and polydisperse ferrofluids by employing the projection operator technique in nonequlibrium statistical mechanics. This MRE is in simple closed form and independent of which model is employed to approximate the equilibrium magnetization curve. Existing models can be recovered as low-order approximations of my generic and nonperturbative MRE. My theory can play a key role in studying the dynamics of ferrofluids and other polar fluids. It may also have substantial and immediate consequences to various ferrofluid applications.