Search Results (36)

This work demonstrates that the four laws of classical thermodynamics apply to the statistics of symmetric observation distributions, and provides examples of how this can be exploited in uncertainty assessments. First, an expression for the partition function Z is derived. In contrast with general classical thermodynamics, however, this can be performed without the need for variational calculus, while Z also equals the number of observations N directly. Apart from the partition function $Z\equiv N$ as a scaling factor, three state variables m, n, and $ϵ$ fully statistically characterize the observation distribution, corresponding to its expectation value, degrees of freedom, and random error, respectively. Each term in the first law of thermodynamics is then shown to be a variation on $\delta m^2=\delta {(nϵ)}^2$ for both canonical (constant n and $ϵ$) and macro-canonical (constant $ϵ$) observation ensembles, while micro-canonical ensembles correspond to a single observation result bin having $\delta m^2=0$. This view enables the improved fitting and combining of observation distributions, capturing both measurand variability and measurement precision. Full article

This paper concerns interactive Monte Carlo simulations for adiabatic ensembles and a genetic algorithm to research and educational contexts. In the Introduction, we discuss some concepts of thermodynamics, statistical mechanics and ensembles relevant to molecular simulations. The second and third sections of the paper comprise two programs in JavaScript regarding (i) argon in the grand-isobaric ensemble focusing on the direct calculation of entropy, vapor–liquid equilibria and radial distribution functions and (ii) an ideal system of quantized harmonic oscillators in the microcanonical ensemble for the determination of the entropy and Boltzmann distribution, also including the definition of Boltzmann and Gibbs entropies relative to classical systems. The fourth section is concerned with a genetic algorithm program in Java, as a pedagogical alternative to introduce the Second Law of Thermodynamics, which summarizes artificial intelligence methods and the cumulative selection process in biogenesis. Full article

Resorting to microcanonical ensemble Monte Carlo simulations, we study the geometric and topological properties of the state space of a model of a network glass-former. This model, a Lennard-Jones binary mixture, does not crystallize due to frustration. We have found two peaks in specific heat at equilibrium and at low energy, corresponding to important changes in local ordering. These singularities were accompanied by inflection points in geometrical markers of the potential energy level sets—namely, the mean curvature, the dispersion of the principal curvatures, and the variance of the scalar curvature. Pinkall’s and Overholt’s theorems closely relate these quantities to the topological properties of the accessible state-space manifold. Thus, our analysis provides strong indications that the glass transition is associated with major changes in the topology of the energy level sets. This important result suggests that this phase transition can be understood through the topological theory of phase transitions. Full article

Nanothermites as high-energy-density and high-reaction-rate materials have important applications in civil and military fields. Nevertheless, it is difficult to detect all intermediates and products using conventional experimental methods. In this work, the reaction process of core-shell SiO₂@Al nanoparticles under adiabatic conditions was investigated through molecular dynamics simulations using a reactive force field (ReaxFF). In the microcanonical ensemble, the redox reaction of SiO₂@Al nanothermite becomes explosive due to the huge energy release during Al-O bond formation. The gaseous products are mainly the intermediate products Al₅O and Al₄O as well as the final products Al₂O, AlO, Si and Al. Analyses of the steric charge distributions and evolution show that the Coulomb effect causes the number of intermediates Al₅O (0.32|e|) to increase to the maximum, then slowly decrease and remain stable. But the tetrahedral Al₄O cluster is almost charge-neutral, at −0.05|e|, and the number remained almost constant. This work is expected to provide deeper insights into the complex reaction mechanism of nanothermite. Full article

The first part of this work provides a review of recent research on generalised entropies and their origin, as well as its application to black hole thermodynamics. To start, it is shown that the Hawking temperature and the Bekenstein–Hawking entropy are, respectively, the only possible thermodynamical temperature and entropy of the Schwarzschild black hole. Moreover, it is investigated if the other known generalised entropies, which include Rényi’s entropy, Tsallis entropy, and the four- and five-parameter generalised entropies, could correctly yield the Hawking temperature and the ADM mass. The possibility that generalised entropies could describe hairy black hole thermodynamics is also considered, both for the Reissner–Nordström black hole and for Einstein’s gravity coupled with two scalar fields. Two possibilities are investigated, namely, the case when the ADM mass does not yield the Bekenstein–Hawking entropy, and the case in which the effective mass expressing the energy inside the horizon does not yield the Hawking temperature. For the model with two scalar fields, the radii of the photon sphere and of the black hole shadow are calculated, which gives constraints on the BH parameters. These constraints are seen to be consistent, provided that the black hole is of the Schwarzschild type. Subsequently, the origin of the generalised entropies is investigated, by using their microscopic particle descriptions in the frameworks of a microcanonical ensemble and canonical ensemble, respectively. Finally, the McLaughlin expansion for the generalised entropies is used to derive, in each case, the microscopic interpretation of the generalised entropies, via the canonical and the grand canonical ensembles. Full article

Recent studies have shown correlations between the microbiota’s composition and various health conditions. Machine learning (ML) techniques are essential for analyzing complex biological data, particularly in microbiome research. ML methods help analyze large datasets to uncover microbiota patterns and understand how these patterns affect human health. This study introduces a novel approach combining statistical physics with the Monte Carlo (MC) methods to characterize bacterial species in the human microbiota. We assess the significance of bacterial species in different age groups by using notions of statistical distances to evaluate species prevalence and abundance across age groups and employing MC simulations based on statistical mechanics principles. Our findings show that the microbiota composition experiences a significant transition from early childhood to adulthood. Species such as Bifidobacterium breve and Veillonella parvula decrease with age, while others like Agathobaculum butyriciproducens and Eubacterium rectale increase. Additionally, low-prevalence species may hold significant importance in characterizing age groups. Finally, we propose an overall species ranking by integrating the methods proposed here in a multicriteria classification strategy. Our research provides a comprehensive tool for microbiota analysis using statistical notions, ML techniques, and MC simulations. Full article

After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a “binary star” consisting of two particles of size a in gravitational interaction in a box of radius R. The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a “dilute” state, where the particles are well separated, and a “condensed” state, where the particles are bound together. We show that the evolution of the system is given by a Fokker–Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for N Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star ($N=2$), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit $N\to +\infty$. Full article

When writing down a Langevin equation for the time evolution of a “system” in contact with a thermal bath, one typically makes the implicit (and often tacit) assumption that the thermal environment is in equilibrium at all times. Here, we take this assumption as a starting point to formulate the problem of a system evolving in contact with a thermal bath from the perspective of the bath, which, since it is in equilibrium, can be described by the microcanonical ensemble. We show that the microcanonical ensemble of the bath, together with the Hamiltonian equations of motion for all the constituents of the bath and system together, give rise to a Langevin equation for the system evolution alone. The friction coefficient turns out to be given in terms of auto-correlation functions of the interaction forces between the bath particles and the system, and the Einstein relation is recovered. Moreover, the connection to the Fokker–Planck equation is established. Full article

Machine learning potentials, particularly the deep potential (DP) model, have revolutionized molecular dynamics (MD) simulations, striking a balance between accuracy and computational efficiency. To facilitate the DP model’s integration with the popular MD engine OpenMM, we have developed a versatile OpenMM plugin. This plugin supports a range of applications, from conventional MD simulations to alchemical free energy calculations and hybrid DP/MM simulations. Our extensive validation tests encompassed energy conservation in microcanonical ensemble simulations, fidelity in canonical ensemble generation, and the evaluation of the structural, transport, and thermodynamic properties of bulk water. The introduction of this plugin is expected to significantly expand the application scope of DP models within the MD simulation community, representing a major advancement in the field. Full article

The vibrational dynamics of rutile (TiO₂) as a function of temperature has been studied by combining molecular dynamics (MD) simulations in conjunction with the generalized two-dimensional correlation spectroscopy analysis (2D COS) technique. Molecular dynamics simulations within the microcanonical ensemble were performed with the self-consistent charge density functional tight binding formalism at a series of different temperatures. To validate the DFTB MD results against the experimental data, the method of atomic pair distribution functions (PDFs) was used. IR absorption spectra were calculated implementing the time correlation function formalism. Subsequently, the generalized two-dimensional correlation approach was applied to obtain further insights into the temperature-dependent vibrational dynamics. The static DFTB calculations of the vibrational force field of the rutile reproduced excellently the experimental data and allowed for more exact assignments of the corresponding experimental IR/Raman spectral bands. Through the detailed analysis of the synchronous and asynchronous 2D spectra computed on the basis of MD-generated anharmonic spectra, we have provided new insights into the couplings between the modes in the studied system, as well as into the sequential character of the temperature-induced changes in the vibrational force field. The sensitivity of IR active modes to the temperature-induced perturbation of the system decreases in the order 685 cm⁻¹E_u mode > 370 cm⁻¹E_u mode > 982 cm⁻¹A_2u mode. The results presented in this study clearly demonstrate the usefulness of the combination of periodic SCC DFTB MD simulations coupled to the 2D COS analysis techniques in solid-state vibrational spectroscopy. Full article

On the capital market, price movements of stock corporations can be observed independent of overall market developments as a result of company-specific news, which suggests the occurrence of a sudden risk event. In recent years, numerous concepts from statistical physics have been transferred to econometrics to model these effects and other issues, e.g., in socioeconomics. Like other studies, we extend the approaches based on the “buy” and “sell” positions of agents (investors’ stance) with a third “hold” position. We develop the corresponding theory within the framework of the microcanonical and canonical ensembles for an ideal agent system and apply it to a capital market example. We thereby design a procedure to estimate the required model parameters from time series on the capital market. The aim is the appropriate modeling and the one-step-ahead assessment of the effect of a sudden risk event. From a one-step-ahead performance comparison with selected benchmark approaches, we infer that the model is well-specified and the model parameters are well determined. Full article

We introduce fractal Tsallis entropy and show that it satisfies Shannon–Khinchin axioms. Analogously to Tsallis divergence (or Tsallis relative entropy, according to some authors), fractal Tsallis divergence is defined and some properties of it are studied. Within this framework, Lesche stability is verified and an example concerning the microcanonical ensemble is given. We generalize the LMC complexity measure (LMC is Lopez-Ruiz, Mancini and Calbert), apply it to a two-level system and define the statistical complexity by using the Euclidean and Wootters’ distance measures in order to analyze it for two-level systems. Full article

Non-standard thermostatistical formalisms derived from generalizations of the Boltzmann–Gibbs entropy have attracted considerable attention recently. Among the various proposals, the one that has been most intensively studied, and most successfully applied to concrete problems in physics and other areas, is the one associated with the $S_q$ non-additive entropies. The $S_q$-based thermostatistics exhibits a number of peculiar features that distinguish it from other generalizations of the Boltzmann–Gibbs theory. In particular, there is a close connection between the $S_q$-canonical distributions and the micro-canonical ensemble. The connection, first pointed out in 1994, has been subsequently explored by several researchers, who elaborated this facet of the $S_q$-thermo-statistics in a number of interesting directions. In the present work, we provide a brief review of some highlights within this line of inquiry, focusing on micro-canonical scenarios leading to $S_q$-canonical distributions. We consider works on the micro-canonical ensemble, including historical ones, where the $S_q$-canonical distributions, although present, were not identified as such, and also more resent works by researchers who explicitly investigated the $S_q$-micro-canonical connection. Full article

We study the set of integer partitions as a probability space that generates distributions and, in the asymptotic limit, obeys thermodynamics. We view ordered integer partition as a configuration of cluster masses and associate them with the distribution of masses it contains. We organized the set of ordered partitions into a table that forms a microcanonical ensemble and whose columns form a set of canonical ensembles. We define a functional of the distribution (selection functional) that establishes a probability measure on the distributions of the ensemble, study the combinatorial properties of this space, define its partition functions, and show that, in the asymptotic limit, this space obeys thermodynamics. We construct a stochastic process that we call exchange reaction and used it to sample the mean distribution by Mote Carlo simulation. We demonstrated that, with appropriate choice of the selection functional, we can obtain any distribution as the equilibrium distribution of the ensemble. Full article

We consider the H-theorem in an isolated quantum harmonic oscillator through the time-dependent Schrödinger equation. The effect of potential in producing entropy is investigated in detail, and we found that including a barrier potential into a harmonic trap would lead to the thermalization of the system, while a harmonic trap alone would not thermalize the system. During thermalization, Shannon entropy increases, which shows that a microscopic quantum system still obeys the macroscopic thermodynamics law. Meanwhile, initial coherent mechanical energy transforms to incoherent thermal energy during thermalization, which exhibiting the decoherence of an oscillating wave packet featured by a large decreasing of autocorrelation length. When reaching thermal equilibrium, the wave packet comes to a halt, with the density distributions both in position and momentum spaces well-fitted by a microcanonical ensemble of statistical mechanics. Full article