Statistical Mechanics Approach Research Articles

Investigating zero-point vibrations of solid hydrogen via statistical moment method

Zero-point vibrations of solid hydrogen are investigated by analyzing the molecular mean-squared displacement (MSD) and mean-squared relative displacement functions within the statistical moment method approach in statistical mechanics. Numerical computations of these thermodynamic properties were conducted for solid hydrogen from 0 K to its phase transition temperature using the Wigner-Kirkwood mean-field potential derived from the Buckingham exp-6 potential. We have shown that the quantum-mechanical zero-point vibrations play an important role at low temperature. And these thermodynamic quantities increase with temperature, suggesting that both thermal and quantum effects play a significant role near the liquid-solid phase transition. The favorable consistency between our findings and the recent experimental inelastic neutron scattering measurements of MSD attests to the potential of SMM as a novel approach for determining the atomic vibrations of solid hydrogen. This approach allows us to study these effects including the anharmonicity of lattice vibrations.

Spontaneous Brain Activity Emerges from Pairwise Interactions in the Larval Zebrafish Brain

Brain activity is characterized by brainwide spatiotemporal patterns that emerge from synapse-mediated interactions between individual neurons. Calcium imaging provides access to recordings of whole-brain activity at single-neuron resolution and, therefore, allows the study of how large-scale brain dynamics emerge from local activity. In this study, we use a statistical mechanics approach—the pairwise maximum entropy model—to infer microscopic network features from collective patterns of activity in the larval zebrafish brain and relate these features to the emergence of observed whole-brain dynamics. Our findings indicate that the pairwise interactions between neural populations and their intrinsic activity states are sufficient to explain observed whole-brain dynamics. In fact, the pairwise relationships between neuronal populations estimated with the maximum entropy model strongly correspond to observed structural connectivity patterns. Model simulations also demonstrated how tuning pairwise neuronal interactions drives transitions between observed physiological regimes and pathologically hyperexcitable whole-brain regimes. Finally, we use virtual resection to identify the brain structures that are important for maintaining the brain in a physiological dynamic regime. Together, our results indicate that whole-brain activity emerges from a complex dynamical system that transitions between basins of attraction whose strength and topology depend on the connectivity between brain areas. Published by the American Physical Society 2024

Confinement Effects on Reorientation Dynamics of Water Confined within Graphite Nanoslits.

Molecular dynamics simulations were used to investigate the reorientation dynamics of water confined within graphite nanoslits of size less than 2 nm, where molecules formed inner and interfacial layers parallel to the confining walls. Significantly related to molecular reorientations, the hydrogen-bond (HB) network of nanoconfined water therein was scrutinized by HB configuration fractions compared to those of bulk water and the influences on interfacial-molecule orientations relative to a nearby C atom plate. The second-rank orientation time correlation functions (OTCFs) of nanoconfined water were calculated and found to follow stretched-exponential, power-law, and power-law decays in a time series. To understand this naïve behavior of reorientation relaxation, the approach of statistical mechanics was adopted in our studies. In terms of the orientation Van Hove function (OVHF), an alternative meaning was given to the second-rank OTCF, which is a measure of the deviation of the OVHF between a molecular system and free molecules in random orientations. Indicated by the OVHFs at related time scales, the stretched-exponential decay of the second-rank OTCF resulted from molecules evacuating out of HB cages formed by their neighbors. After the evacuations, the inner molecules relaxed at relatively fast rates toward random orientations, but the interfacial molecules reoriented at slow rates due to restrictions by hydrophobic interactions with graphite walls. The first power-law decay of the second-rank OTCF was attributed to the distinct relaxation rates of inner and interfacial molecules within a graphite nanoslit. When the inner molecules were completely random in orientation, the second-rank OTCFs changed to another power law decay with a power smaller than the first one.

Accessing activity and viscoelastic properties of artificial and living systems from passive measurement.

Living systems are complex dynamic entities that operate far from thermodynamic equilibrium. Their active, non-equilibrium behaviour requires energy to drive cellular organization and dynamics. Unfortunately, most statistical mechanics approaches are not valid in non-equilibrium situations, forcing researchers to use intricate and often invasive methods to study living processes. Here we experimentally demonstrate that an observable termed mean back relaxation quantifies the active mechanics of living cells from passively observed particle trajectories. The mean back relaxation represents the average trajectory of a particle after a recent motion and is calculated from three-point probabilities. We show that this parameter allows the detection of broken detailed balance in confined systems. We experimentally observe that it provides access to the non-equilibrium generating energy and viscoelastic properties of artificial bulk materials and living cells. These findings suggest that the mean back relaxation can function as a marker of non-equilibrium dynamics and is a non-invasive avenue to determine viscoelastic material properties from passive measurements.

Mathematical Models for the Large Spread of a Contact-Based Infection: A Statistical Mechanics Approach

In this work, we derive a system of Boltzmann-type equations to describe the spread of contact-based infections, such as SARS-CoV-2 virus, at the microscopic scale, that is, by modeling the human-to-human mechanisms of transmission. To this end, we consider two populations, characterized by specific distribution functions, made up of individuals without symptoms (population 1) and infected people with symptoms (population 2). The Boltzmann operators model the interactions between individuals within the same population and among different populations with a probability of transition from one to the other due to contagion or, vice versa, to recovery. In addition, the influence of innate and adaptive immune systems is taken into account. Then, starting from the Boltzmann microscopic description we derive a set of evolution equations for the size and mean state of each population considered. Mathematical properties of such macroscopic equations, as equilibria and their stability, are investigated, and some numerical simulations are performed in order to analyze the ability of our model to reproduce the characteristic features of Covid-19 type pandemics.

Nucleosomes play a dual role in regulating transcription dynamics

Transcription has a mechanical component, as the translocation of the transcription machinery or RNA polymerase (RNAP) on DNA or chromatin is dynamically coupled to the chromatin torsion. This posits chromatin mechanics as a possible regulator of eukaryotic transcription, however, the modes and mechanisms of this regulation are elusive. Here, we first take a statistical mechanics approach to model the torsional response of topology-constrained chromatin. Our model recapitulates the experimentally observed weaker torsional stiffness of chromatin compared to bare DNA and proposes structural transitions of nucleosomes into chirally distinct states as the driver of the contrasting torsional mechanics. Coupling chromatin mechanics with RNAP translocation in stochastic simulations, we reveal a complex interplay of DNA supercoiling and nucleosome dynamics in governing RNAP velocity. Nucleosomes play a dual role in controlling the transcription dynamics. The steric barrier aspect of nucleosomes in the gene body counteracts transcription via hindering RNAP motion, whereas the chiral transitions facilitate RNAP motion via driving a low restoring torque upon twisting the DNA. While nucleosomes with low dissociation rates are typically transcriptionally repressive, highly dynamic nucleosomes offer less of a steric barrier and enhance the transcription elongation dynamics of weakly transcribed genes via buffering DNA twist. We use the model to predict transcription-dependent levels of DNA supercoiling in segments of the budding yeast genome that are in accord with available experimental data. The model unveils a paradigm of DNA supercoiling-mediated interaction between genes and makes testable predictions that will guide experimental design.

Enhancing the description of multi-time-scale geophysical phenomena: Incorporating finite time Scale separation and feedback, illustrated with the case of a 1D variable of interest.

Stochastic approaches play a vital role in weather, climate, and, more in general, geophysics systems, addressing processes and scales beyond the resolution of deterministic models. Similar to equilibrium/non-equilibrium thermodynamics, intricate fast and local dynamics may not always be the primary focus. Practical applications often prioritize observables capturing phenomena at dominant temporal and spatial scales. Developing models for these "large-scale" observables, resulting from averaging fast and local contributions, can be simplified into Low Order Models (LOMs) with reduced degrees of freedom described by ordinary differential equations. Unresolved degrees of freedom are introduced as stochastic components, exhibiting either Markovian or non-Markovian characteristics. The challenge lies in deriving dependable stochastic differential equations representing the statistics of real large-scale, slow features in the climate/ocean system. While paralleling material physics, it is crucial to recognize that direct transfer of tools and outcomes is hindered by the non-Hamiltonian nature of climate/geophysical LOMs and the impracticality of a Markovian treatment of noise due to wide-ranging time scales. A critical examination of the conventional statistical mechanics approach, customized for such LOMs, becomes essential. To this end, we propose utilizing an approach based on the operator cumulant method, which has been recently revisited and generalized, along with the linear response method in a non-Hamiltonian setting. Formal results are then derived, and applications to some typical classes of examples are presented to clarify this approach.

Theoretical study of S + SH reaction on its ground state HS2(X2 A ″) potential energy surface

ABSTRACT In this study, we explore the initial state selected dynamics of the S + SH ( X 2 Π ) → S 2 ( X 3 ∑ ) + H reaction through the lens of time-dependent quantum dynamics (QD), quasi-classical trajectory (QCT), and statistical quantum mechanical (SQM) approaches, utilising the electronic ground state HS 2 (X 2 A ′′ ) potential energy surface (PES). We present total reaction probabilities and integral cross sections (ICSs) as part of our findings. Our results indicate that at low and moderate collision energies, the reaction predominantly follows an indirect pathway, forming meta-stable quasi-bound complexes. Additionally, we observe that the rotational excitation of the SH reagent initially leads to a decrease and then an increase in reactivity. We conduct a comprehensive analysis and comparison of the outcomes from QD, QCT, and SQM methodologies.

Modeling Charged Defects Using the Defect Energy Formalism With a Charge Sublattice Within a CALPHAD Framework

AbstractThe non‐stoichiometry of material compounds significantly influences their functional properties. In semiconductors, point defects determine whether a material is n‐type or p‐type and the concentration of charge carriers. Even in structural alloys and ion conductors used in batteries and fuel cells, non‐stoichiometry and dilute defects play an integral role in their function. The design of metal alloys has advanced rapidly with the aid of thermodynamic modeling using calculation of phase diagrams (CALPHAD) to predict the phases produced during different processing conditions. While thermodynamic modeling has been done previously by fitting experimental data, the advent of first‐principle techniques has enabled the computational prediction of material properties. The defect energy formalism (DEF) is described for modeling charged and uncharged defects in compounds, showing that it accurately predicts non‐stoichiometry using density functional theory (DFT), without fitting experiments. The model reproduces the expected defect and free‐charge carrier concentrations using the statistical mechanics approach commonly used in most DFT defect studies. Finally, the model is used to accurately predict the single‐phase region of PbTe with no fitting to measured defect concentrations. This method can revolutionize materials development of insulators, semiconductors, and even metals by allowing rapid DFT calculations to replace laborious experiments when dilute defects are involved.

On a conjecture concerning the Fisher–Widom line and the line of vanishing excess isothermal compressibility in simple fluids

In the statistical mechanics approach to liquid-state theory, understanding the role of the intermolecular potential in determining thermodynamic and structural properties is crucial. The Fisher–Widom (FW) line, which separates regions in the temperature vs density plane where the decay of the total correlation function is monotonic or oscillatory, provides insights into the dominance of the attractive or repulsive part of the interactions. Stopper et al. have recently conjectured [J. Chem. Phys. 151, 014501 (2019)] that the line of vanishing excess isothermal compressibility approximates the FW line in simple fluids. Here, we investigate this conjecture using the Jagla potential and also explore the line of vanishing excess pressure. We employ theoretical approximations and Monte Carlo simulations to study one-dimensional and three-dimensional systems. While exact results for the one-dimensional case do not support the conjecture, our Monte Carlo simulations for the three-dimensional fluid validate it. Our findings not only contribute to the understanding of the relationship between the three transition lines but also provide valuable insights into the thermodynamic and structural behaviour of simple fluids.

Statistical Mechanics Approaches for Studying Temperature and Rate Effects in Multistable Systems

Systems with a multistable energy landscape are widespread in physics, biophysics, technology, and materials science. They are strongly influenced by thermal fluctuations and external mechanical actions that can be applied at different rates, moving the system from equilibrium to non-equilibrium regimes. In this paper, we focus on a simple system involving a single breaking phenomenon to describe the various theoretical approaches used to study these problems. To begin with, we propose the exact solution at thermodynamic equilibrium based on the calculation of the partition function without approximations. We then introduce the technique of spin variables, which is able to simplify the treatment even for systems with a large number of coordinates. We then analyze the energy balance of the system to better understand its underlying physics. Finally, we introduce a technique based on transition state theory useful for studying the non-equilibrium dynamical regimes of these systems. This method is appropriate for the evaluation of rate effects and hysteresis loops. These approaches are developed for both the Helmholtz ensemble (prescribed extension) and the Gibbs ensemble (applied force) of statistical mechanics. The symmetry and duality of these two ensembles is discussed in depth. While these techniques are used here for a simple system with theoretical purposes, they can be applied to complex systems of interest for several physical, biophysical, and technological applications.

Inverse design of a pyrochlore lattice of DNA origami through model-driven experiments.

Sophisticated statistical mechanics approaches and human intuition have demonstrated the possibility of self-assembling complex lattices or finite-size constructs. However, attempts so far have mostly only been successful in silico and often fail in experiment because of unpredicted traps associated with kinetic slowing down (gelation, glass transition) and competing ordered structures. Theoretical predictions also face the difficulty of encoding the desired interparticle interaction potential with the experimentally available nano- and micrometer-sized particles. To overcome these issues, we combine SAT assembly (a patchy-particle interaction design algorithm based on constrained optimization) with coarse-grained simulations of DNA nanotechnology to experimentally realize trap-free self-assembly pathways. We use this approach to assemble a pyrochlore three-dimensional lattice, coveted for its promise in the construction of optical metamaterials, and characterize it with small-angle x-ray scattering and scanning electron microscopy visualization.

Assessment of Anharmonicities in Clusters: Developing and Validating a Minimum-Information Partition Function.

Precise thermodynamic calculations are essential for understanding the dynamics of cluster systems and new particle formation. However, the widely employed harmonic statistical mechanical approach often falls short in terms of accuracy. In this study, we present an improved statistical model that incorporates vibrational anharmonicity via a novel partition function that requires only one additional system-specific input parameter. In addition to considering vibrational aspects, we also account for anharmonicity related to the configurational space. The role of anharmonicities is thoroughly examined in the case of general clusters, where the complete sets of conformers, mechanically stable spatial arrangements, are known up to clusters composed of 14 monomers. By performing consistent Monte Carlo simulations on these systems, we benchmark the statistical model's efficacy in reproducing key thermodynamic properties (formation free energy and potential energy) in the classical limit. The model exhibits exceptional alignment with simulations, accurately reproducing free energies within a precision of 2kBT and reliably capturing cluster melting temperatures. Furthermore, we demonstrate the significance and applicability of the model by reproducing thermodynamic barriers in homogeneous gas-phase nucleation of larger clusters. The transferability of our developed approach extends to more complex molecular systems and bears relevance for atmospheric multicomponent clusters, in particular.

Effects of self-avoidance on the packing of stiff rods on ellipsoids.

Using a statistical-mechanics approach, we study the effects of geometry and self-avoidance on the ordering of slender filaments inside nonisotropic containers, considering cortical microtubules in plant cells, and packing of genetic material inside viral capsids as concrete examples. Within a mean-field approximation, we show analytically how the shape of the container, together with self-avoidance, affects the ordering of the stiff rods. We find that the strength of the self-avoiding interaction plays a significant role in the preferred packing orientation, leading to a first-order transition for oblate cells, where the preferred orientation changes from azimuthal, along the equator, to a polar one, when self-avoidance is strong enough. While for prolate spheroids the ground state is always a polar-like order, strong self-avoidance results with a deep metastable state along the equator. We compute the critical surface describing the transition between azimuthal and polar ordering in the three-dimensional parameter space (persistence length, eccentricity, and self-avoidance) and show that the critical behavior of this system is in fact related to the butterfly catastrophe model. We calculate the pressure and shear stress applied by the filament on the surface, and the injection force needed to be applied on the filament in order to insert it into the volume. We compare these results to the pure mechanical study where self-avoidance is ignored, and discuss similarities and differences.

Predicting Thermochemical Equilibria with Interacting Defects: Sr1−xCexMnO3−δ Alloys for Water Splitting

Solar thermochemical hydrogen is one of the few potential routes towards direct fuel production from renewable energy sources, but the thermodynamic boundary conditions for efficient and economic energy conversion are challenging. Success or failure of a given oxide working material depends on the subtle balance between enthalpy and entropy contributions in the redox processes. Developing a mechanistic understanding of the behavior of materials on the basis of atomistic models and first-principles calculations is an important part of advancing the technology. One challenge is to quantitatively predict thermochemical equilibria at high concentrations when the redox-active defects start to interact with each other, thereby impeding the formation of additional defects. This problem is of more general importance to applications that rely on high levels of off-stoichiometry or doping, including, for example, batteries, thermoelectrics, and ceramic fuel cells. To account for such repulsive defect interactions, we introduce a statistical mechanics approach, defining an expression for the free energy of defect interaction based on limited sampling of defect configurations in density functional theory supercell calculations. The parameterization of this energy contribution as a function of defect concentration and temperature allows on-the-fly simulation of thermochemical equilibria. The approach consistently incorporates finite temperature effects by including the leading contributions to the temperature-dependent free energy for the case at hand, i.e., the ideal gas and configurational enthalpies and entropies. We demonstrate the capability and utility of the approach by simulating the water splitting redox processes for Sr1−xCexMnO3−δ alloys. Published by the American Physical Society 2024

Point-vortex statistical mechanics applied to turbulence without vortex stretching

In turbulent systems with inverse cascades, energy will pile up at large scales if no large-scale sink is present. We observe that in forced-dissipative three-dimensional turbulence from which vortex-stretching is removed, such condensation is observed, associated with an inverse cascade of helicity. The large-scale structure of this condensate is characterized by a hyperbolic sine relation between vorticity and velocity, analogous to the sinh relation between vorticity and stream function observed in freely decaying 2D turbulence in periodic domains. We generalize a 2D point-vortex statistical mechanics approach to our 3D system. It is shown that the predictions of this approach are in agreement with observations of both the forced-dissipative system, after appropriate averaging, and of the freely decaying system.

Non-thermal particle acceleration and power-law tails via relaxation to universal Lynden-Bell equilibria

Collisionless and weakly collisional plasmas often exhibit non-thermal quasi-equilibria. Among these quasi-equilibria, distributions with power-law tails are ubiquitous. It is shown that the statistical-mechanical approach originally suggested by Lynden-Bell (Mon. Not. R. Astron. Soc., vol. 136, 1967, p. 101) can easily recover such power-law tails. Moreover, we show that, despite the apparent diversity of Lynden-Bell equilibria, a generic form of the equilibrium distribution at high energies is a ‘hard’ power-law tail$\propto \varepsilon ^{-2}$, where$\varepsilon$is the particle energy. The shape of the ‘core’ of the distribution, located at low energies, retains some dependence on the initial condition but it is the tail (or ‘halo’) that contains most of the energy. Thus, a degree of universality exists in collisionless plasmas.

Detection of Transmission State of Multiple Wireless Sources: A Statistical Mechanics Approach

Consider a random network of static primary wireless sources and a co-located network of secondary wireless devices. The channel coefficients between the two networks are assumed to be known to the secondary users (SUs), e.g., using radio environment maps (REM). However, the operational state of the sources is unknown due to intermittency. In this paper, we study the performance of primary source detection by SUs using a message-passing algorithm. Additionally, we employ methods from statistical mechanics, in particular, the Replica approach, to obtain analytic results for the performance of such networks in the large system-size limit. We test the results through a large-scale simulation analysis, obtaining good agreement. The proposed method provides a simple way to evaluate the performance of the system and assess how it depends on the macroscopic parameters that characterize it, such as the average density of SUs and sources and the signal-to-noise ratio. The main contribution of this paper is the application of an algorithm that quantitatively predicts the parameter value region for which accurate and reliable detection of the operational state of the primary sources can be achieved in a fast and decentralized manner.

Surrogate Modeling of the Relative Entropy for Inverse Design Using Smolyak Sparse Grids.

Relative entropy minimization, a statistical-mechanics approach for finding potential energy functions that produce target structural ensembles, has proven to be a powerful strategy for the inverse design of nanoparticle self-assembly. For a given target structure, the gradient of the relative entropy with respect to the adjustable parameters of the potential energy function is computed by performing a simulation, and then these parameters are updated using iterative gradient-based optimization. Small parameter updates per iteration and many iterations can be required for numerical stability, but this incurs considerable computational expense because a new simulation must be performed to reevaluate the gradient at each iteration. Here, we investigate the use of surrogate modeling to decouple the process of minimizing the relative entropy from the computationally demanding process of determining its gradient. We approximate the relative-entropy gradient using Chebyshev polynomial interpolation on Smolyak sparse grids. Our approach potentially increases the robustness and computational efficiency of using the relative entropy for inverse design, primarily for physically informed potential energy functions that have a small number of adjustable parameters.

Ar+ ArH+ Reactive Collisions of Astrophysical Interest: The Case of 36 Ar.

The reactive collision between 36 Ar and the 36 ArH+ species has been investigated by means of quantum mechanical (QM), quasiclassical trajectories (QCT) and statistical quantum mechanical (SQM) approaches. Reaction probabilities, cross sections as a function of the energy and rate constants in terms of the temperature have been obtained. Cumulative distributions as a function of the collision time and the inspection of selected QCT corresponding to specific dynamical mechanisms have been analysed. Predictions by means of the SQM method are in good agreement with the QM results, thus supporting the complex-forming nature of the process.