Formulation Of Statistical Mechanics Research Articles

Quantum statistical mechanics from a Bohmian perspective

We develop a general formulation of quantum statistical mechanics in terms of probability currents that satisfy continuity equations in the multi-particle position space, for closed and open systems with a fixed number of particles. The continuity equation for any closed or open system suggests a natural Bohmian interpretation in terms of microscopic particle trajectories, that make the same measurable predictions as standard quantum theory. The microscopic trajectories are not directly observable, but provide a general, simple and intuitive microscopic interpretation of macroscopic phenomena in quantum statistical mechanics. In particular, we discuss how various notions of entropy, proper and improper mixtures, and thermodynamics are understood from the Bohmian perspective.

Amino acid chiral amplification using Monte Carlo dynamic.

This study investigates the stability of chiral-molecule solution phases, with a specific focus on amino acids. The model framework is based on a two-dimensional square lattice model, where individual sites may be occupied by oriented chiral molecules or structureless solvent particles. Utilizing the Glauber dynamics and statistical mechanical formalism, as previously introduced and examined by Lombardo et al., we explore the influence of temperature, amino acid concentration, enantiomeric excess, and homochiral interaction strength on nucleation mechanisms, equilibrium phase behavior, and crystal composition. Our findings offer thermodynamic insights into the chiral amplification process of amino acids, contributing to a deeper understanding of the underlying processes.

Forecasting in Stock Markets Using the Formalism of Statistical Mechanics

The possibility and expediency of forecasting in the stock markets are analyzed analytically using the methods and approaches of statistical mechanics. The apparatus of statistical mechanics is used to analyze and forecast one of the most important indicators of the market – the distribution of its logarithmic profitability. The Lotka-Volterra model used in ecology to describe systems of the "predator-prey" type was used as the initial model. It approximates market dynamics adequately. In the article, its Hamiltonian property is used, which makes it possible to apply the apparatus of statistical mechanics. The apparatus of statistical mechanics (using the principle of maximum entropy) makes it possible to implement a probabilistic approach that is adapted to the conditions of stock market uncertainty. The canonical variables of the Hamiltonian are presented as logarithms of stock and bond prices, the joint probability distribution function of stock and bond prices is obtained as a Gibbs distribution. The Boltzmann factor, included in the Gibbs distribution, allows us to estimate the probability of the occurrence of certain stock and bond prices and obtain an analytical expression for calculating the logarithmic return, which gives more accurate results than the widely used normal (Gaussian) distribution. According to its characteristics, the resulting distribution resembles the Laplace distribution. The main characteristics of the resulting distribution are calculated – the mean value, variance, asymmetry, and kurtosis. Mathematical results are presented graphically. An explanation is given of the cause-and-effect mechanism that causes a change in the profitability of the market. For this, the idea of Theodore Modis about the competition between stocks and bonds for the attention and money of investors is developed (by analogy with the turnover of biomass in models of the "predator-prey" type in biology). The results of the study are of interest to investors, theorists, and practitioners of the stock market. They allow us to make thoughtful and balanced investment decisions due to a more realistic idea of the expected return and a more adequate assessment of investment risk.

Almost Everywhere Ergodicity in Quantum Lattice Models

We rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the C^* algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on C^* algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.

Constitutive relationships for osteonal microcracking in human cortical bone using statistical mechanics

The post-elastic mechanical behavior of cortical bone, which is represented by extensive microcracking once the elastic regime is exceeded, has been characterized by a nonlinear constitutive relationship for osteonal microcracking. The relationship/model is based on the formalism of Statistical Mechanics, allowing the degree of irreversibility to be calculated using the increase in entropy associated with the progression of microcracking. Specific tensile and bending tests were conducted to compare theoretical predictions of constitutive relationships to empirical curves. In addition, the tests were utilized to determine the model’s parameters, whose values were used to explicitly calculate the entropy increase. A large sample was used: 51 cortical bone coupons (dog-bone-shaped specimens) were extracted from the 4th ribs of numerous individuals and subjected to uniaxial tensile testing. Additionally, fifteen complete 4th ribs were used for bending tests. Displacement and strain fields were measured for both types of tests using digital image correlation or video recordings of the tests. All experimental specimen data were successfully fitted to the model, and all constitutive parameter values were found to be correlated with anthropometric variables. Explicit entropy calculations indicate that microcracking is minimal for low strain and, initially, stress is nearly proportional to strain. After a certain point, significant microcracking occurs, and the relationship between stress and strain becomes invalid. Several significant associations between constitutive parameters and age have also been identified.

Some properties of the Sharma–Mittal statistical distribution

The statistical theory based on the two-parameter Sharma–Mittal functional is a generalization of the statistics of Gibbs, Renyi and Tsallis. In this paper, the formalism of statistical mechanics based on the Sharma–Mittal entropy functional is considered, and the theorem on the equidistribution of energy for classical statistical systems by degrees of freedom is proved. A generalized Maxwell distribution for the corresponding statistics is obtained and the characteristics of statistical systems described by the distribution are calculated: the average velocity modulus, the root-mean-square and the most probable velocities of gas particles. A generalized Sakura–Tetrode formula is also obtained.

Towards Quantum Computing Phase Diagrams of Gauge Theories with Thermal Pure Quantum States.

The phase diagram of strong interactions in nature at finite temperature and chemical potential remains largely theoretically unexplored due to inadequacy of Monte-Carlo-based computational techniques in overcoming a sign problem. Quantum computing offers a sign-problem-free approach, but evaluating thermal expectation values is generally resource intensive on quantum computers. To facilitate thermodynamic studies of gauge theories, we propose a generalization of the thermal-pure-quantum-state formulation of statistical mechanics applied to constrained gauge-theory dynamics, and numerically demonstrate that the phase diagram of a simple low-dimensional gauge theory is robustly determined using this approach, including mapping a chiral phase transition in the model at finite temperature and chemical potential. Quantum algorithms, resource requirements, and algorithmic and hardware error analysis are further discussed to motivate future implementations. Thermal pure quantum states, therefore, may present a suitable candidate for efficient thermal simulations of gauge theories in the era of quantum computing.

Finite Reservoirs Corrections to Hamiltonian Systems Statistics and Time Symmetry Breaking

We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. We analyze exactly solvable oscillator systems and perform simulations of relatively more complex ones. This indicates that the standard statistical mechanical formalism needs to be adjusted in the ever more investigated nano-scale science and technology. In particular, the hypothesis that heat reservoirs be considered infinite and be described by the classical ensembles is found to be critical when exponential quantities are considered since the large size limit may not coincide with the infinite size canonical result. Furthermore, process-dependent emergent irreversibility affects ensemble averages, effectively frustrating, on a statistical level, the time reversal invariance of Hamiltonian dynamics that are used to obtain numerous results.

The Mori–Zwanzig formulation of deep learning

We develop a new formulation of deep learning based on the Mori–Zwanzig (MZ) formalism of irreversible statistical mechanics. The new formulation is built upon the well-known duality between deep neural networks and discrete dynamical systems, and it allows us to directly propagate quantities of interest (conditional expectations and probability density functions) forward and backward through the network by means of exact linear operator equations. Such new equations can be used as a starting point to develop new effective parameterizations of deep neural networks and provide a new framework to study deep learning via operator-theoretic methods. The proposed MZ formulation of deep learning naturally introduces a new concept, i.e., the memory of the neural network, which plays a fundamental role in low-dimensional modeling and parameterization. By using the theory of contraction mappings, we develop sufficient conditions for the memory of the neural network to decay with the number of layers. This allows us to rigorously transform deep networks into shallow ones, e.g., by reducing the number of neurons per layer (using projection operators), or by reducing the total number of layers (using the decay property of the memory operator).

Lecture notes of the 15th international summer school on Fundamental Problems in Statistical Physics: Colloidal dispersions

In this brief contribution we review some basic concepts in the physics of soft matter focusing on colloidal dispersions. In the first part we discuss the phase behavior of hard colloidal particles and, building on a statistical mechanics formalism, we introduce the concept of effective interactions between colloids mediated by the presence of macromolecular additives in the dispersion. We then review some examples of effective interactions starting from the simplest entropy-driven mechanism observed in nature, i.e. the depletion effect, but also discussing more complex interactions as those mediated by a critical co-solute. In the second part, we introduce a novel type of colloids having particles “softness” as an extra control parameter. We focus on a particular type of soft colloid, called microgel, that has a polymeric nature since it is made of a cross-linked polymer network able to respond to external stimuli. Finally, to understand microgels responsiveness we discuss the thermodynamics of swelling of polymer networks described by the Flory-Rehner theory.

Proper normalization removes nonlocal behavior of canonical-based thermodynamic entropy expressions

Equilibrium, canonical based expressions of entropy are known to exhibit non-local effects that can complicate statistical mechanical formulations and applications. Here, we demonstrate that proper normalization of the underlying probability density functions can remove non-local effects in entropy. Our analysis is carried out through third order in density function expansions and lead to results consistent with macroscopic thermodynamics, as required. In addition, we provide complete expressions for smaller dimensional systems that may not conform to the thermodynamic limit.

Quantum dynamics using path integral coarse-graining.

The vibrational spectra of condensed and gas-phase systems are influenced by thequantum-mechanical behavior of light nuclei. Full-dimensional simulations of approximate quantum dynamics are possible thanks to the imaginary time path-integral (PI) formulation of quantum statistical mechanics, albeit at a high computational cost which increases sharply with decreasing temperature. By leveraging advances in machine-learned coarse-graining, we develop a PI method with the reduced computational cost of a classical simulation. We also propose a simple temperature elevation scheme to significantly attenuate the artifacts of standard PI approaches as well as eliminate the unfavorable temperature scaling of the computational cost. We illustrate the approach, by calculating vibrational spectra using standard models of water molecules and bulk water, demonstrating significant computational savings and dramatically improved accuracy compared to more expensive reference approaches. Our simple, efficient, and accurate method has prospects for routine calculations of vibrational spectra for a wide range of molecular systems - with an explicit treatment of the quantum nature of nuclei.

Braiding properties of worldline configurations in hardcore lattice bosons

In this manuscript, we study braiding properties of worldline configurations for a variety of ground-states of hardcore Bose–Hubbard models in two dimensions. Configurations are collections of particle paths and result from the path-integral formulation of statistical mechanics. For hard-core bosons, configurations can be seen as geometric braids and therefore can be assigned a certain topological structure, i.e. a way to classify braiding events among worldlines. By means of Monte Carlo calculations, we study superfluid phase and a variety of insulating phases and observe that ground-states of different quantum phases correspond to different probability distributions of braiding properties.

Molecular correlation function integrals for condensed systems with solid-liquid-liquid equilibria

Fluctuation Solution Theory (FST) is an exact statistical mechanical formulation for solution thermodynamic properties and their derivatives via integrals of direct correlation functions (DCFI) and total correlation functions (TCFI). This work tabulates DCFIs and TCFIs for a series of strongly nonideal systems involving condensed-phase equilibria (LLE, SLE and SLLE) over wide ranges of temperature, pressure, and composition. The values obtained can be used for FST modeling of such systems as well as to test the accuracy of other models such as equations of state.

Coarse-Graining of Imaginary Time Feynman Path Integrals: Inclusion of Intramolecular Interactions and Bottom-up Force-Matching.

Feynman's imaginary time path integral formalism of quantum statistical mechanics and the corresponding quantum-classical isomorphism provide a tangible way of incorporating nuclear quantum effect (NQE) in the simulation of condensed matter systems using well-developed classical simulation techniques. Our previous work has presented the many-body coarse-graining of path integral (CG-PI) theory that builds an isomorphism between the quantum partition function of N distinguishable particles and the classical partition function of 2N pseudoparticles. In this present work, we develop a generalized version of the many-body CG-PI theory that incorporates many-body interactions in the force field. Based on the new derivation, we provide a numerical CG-PI (n-CG-PI) modeling strategy parametrized from the underlying path integral molecular dynamics (PIMD) trajectories using force matching and Boltzmann inversion. The n-CG-PI models for two liquid systems are shown to capture well both the intramolecular and intermolecular structural correlations of the reference PIMD simulations. The generalized derivation of the many-body CG-PI theory and the n-CG-PI model presented in this work extend the scope of the CG-PI formalism by generalizing the previously limited theory to incorporate force fields of realistic molecular systems.

Analytical framework for non-equilibrium phase transition to Bose–Einstein condensate

The theoretical description of non-equilibrium Bose–Einstein condensate (BEC) is one of the main challenges in modern statistical physics and kinetics. The non-equilibrium nature of BEC makes it impossible to employ the well-established formalism of statistical mechanics. We develop a framework for the analytical description of a non-equilibrium phase transition to BEC that, in contrast to previously developed approaches, takes into account the infinite number of continuously distributed states. We consider the limit of fast thermalization and obtain an analytical expression for the full density matrix of a non-equilibrium ideal BEC which also covers the equilibrium case. For the particular cases of 2D and 3D, we investigate the non-equilibrium formation of BEC by finding the temperature dependence of the ground state occupation and second-order coherence function. We show that for a given pumping rate, the macroscopic occupation of the ground state and buildup of coherence may occur at different temperatures. Moreover, the buildup of coherence strongly depends on the pumping scheme. We also investigate the condensate linewidth and show that the Schawlow–Townes law holds for BEC in 3D and does not hold for BEC in 2D.

Nuclear quantum effects on the dynamics and glass behavior of a monatomic liquid with two liquid states.

We perform path integral molecular dynamics (PIMD) simulations of a monatomic liquid that exhibits a liquid-liquid phase transition and liquid-liquid critical point. PIMD simulations are performed using different values of Planck's constant h, allowing us to study the behavior of the liquid as nuclear quantum effects (NQE, i.e., atoms delocalization) are introduced, from the classical liquid (h = 0) to increasingly quantum liquids (h > 0). By combining the PIMD simulations with the ring-polymer molecular dynamics method, we also explore the dynamics of the classical and quantum liquids. We find that (i) the glass transition temperature of the low-density liquid (LDL) is anomalous, i.e., Tg LDL(P) decreases upon compression. Instead, (ii) the glass transition temperature of the high-density liquid (HDL) is normal, i.e., Tg HDL(P) increases upon compression. (iii) NQE shift both Tg LDL(P) and Tg HDL(P) toward lower temperatures, but NQE are more pronounced on HDL. We also study the glass behavior of the ring-polymer systems associated with the quantum liquids studied (via the path-integral formulation of statistical mechanics). There are two glass states in all the systems studied, low-density amorphous ice (LDA) and high-density amorphous ice (HDA), which are the glass counterparts of LDL and HDL. In all cases, the pressure-induced LDA-HDA transformation is sharp, reminiscent of a first-order phase transition. In the low-quantum regime, the LDA-HDA transformation is reversible, with identical LDA forms before compression and after decompression. However, in the high-quantum regime, the atoms become more delocalized in the final LDA than in the initial LDA, raising questions on the reversibility of the LDA-HDA transformation.

Reaction-path statistical mechanics of enzymatic kinetics.

We introduce a reaction-path statistical mechanics formalism based on the principle of large deviations to quantify the kinetics of single-molecule enzymatic reaction processes under the Michaelis-Menten mechanism, which exemplifies an out-of-equilibrium process in the living system. Our theoretical approach begins with the principle of equal a priori probabilities and defines the reaction path entropy to construct a new nonequilibrium ensemble as a collection of possible chemical reaction paths. As a result, we evaluate a variety of path-based partition functions and free energies by using the formalism of statistical mechanics. They allow us to calculate the timescales of a given enzymatic reaction, even in the absence of an explicit boundary condition that is necessary for the equilibrium ensemble. We also consider the large deviation theory under a closed-boundary condition of the fixed observation time to quantify the enzyme-substrate unbinding rates. The result demonstrates the presence of a phase-separation-like, bimodal behavior in unbinding events at a finite timescale, and the behavior vanishes as its rate function converges to a single phase in the long-time limit.

Counterfactual thermodynamics: Extracting work from a lack of macroscopic change

A sudden change in the macroscopic parameters of a system will cause it to depart from equilibrium. In this paper we study how a lack of change can also inform of such a departure, and allow for work extraction. Potential events that are unrealized can provide information pertaining the microstates of the system, essentially playing the role of a passive Maxwell demon, thereby allowing one to infer details about the microstate probability distribution. Here, we first qualitatively argue that this effect is present and consequential in almost every physical system, but is ignored in the standard formulation of equilibrium statistical mechanics. Then, as a case study, we quantitatively investigate the local and global thermodynamic properties of an ideal gas placed in a fragile container that nevertheless, does not burst. This non-event indicates a departure from equilibrium and allows work extraction from the system. It also leads to corrections to the heat capacity of the gas.

RADIAÇÃO TÉRMICA E A FÓRMULA DE PLANCK

THERMAL RADIATION AND PLANCK’S FORMULA. In this paper, theoretical and experimental research on blackbody thermal radiation is discussed and historically ordered. The main thermal radiation equations following Planck’s revolutionary idea are obtained using the formalism of statistical mechanics. These equations, already well known, are deduced in a very detailed and didactic way in order to facilitate the reader’s understanding. The consequent results of Planck’s formula for blackbody radiation are analyzed: the limits of the average energy for frequencies tending to zero and infinity, Stefan’s law and Wien’s displacement law.