Statistical Mechanical Considerations Research Articles

Static spherical perfect fluid stars with finite radius in general relativity: a review

In this article, we provide a pedagogical review of the Tolman-Oppenheimer-Volkoff (TOV) equation and its solutions which describe static, spherically symmetric gaseous stars in general relativity. Our discussion starts with a systematic derivation of the TOV equation from the Einstein field equations and the relativistic Euler equations. Next, we give a proof for the existence and uniqueness of solutions of the TOV equation describing a star of finite radius, assuming suitable conditions on the equation of state characterizing the gas. We also prove that the compactness of the gas contained inside a sphere centered at the origin satisfies the well-known Buchdahl bound, independent of the radius of the sphere. Further, we derive the equation of state for an ideal, classical monoatomic relativistic gas from statistical mechanics considerations and show that it satisfies our assumptions for the existence of a unique solution describing a finite radius star. Although none of the results discussed in this article are new, they are usually scattered in different articles and books in the literature; hence it is our hope that this article will provide a self-contained and useful introduction to the topic of relativistic stellar models.

Deep learning approach for predicting functional Z-DNA regions using omics data

Computational methods to predict Z-DNA regions are in high demand to understand the functional role of Z-DNA. The previous state-of-the-art method Z-Hunt is based on statistical mechanical and energy considerations about B- to Z-DNA transition using sequence information. Z-DNA CHiP-seq experiment results showed little overlap with Z-Hunt predictions implying that sequence information only is not sufficient to explain emergence of Z-DNA at different genomic locations. Adding epigenetic and other functional genomic mark-ups to DNA sequence level can help revealing the functional Z-DNA sites. Here we take advantage of the deep learning approach that can analyze and extract information from large volumes of molecular biology data. We developed a machine learning approach DeepZ that aggregates information from genome-wide maps of epigenetic markers, transcription factor and RNA polymerase binding sites, and chromosome accessibility maps. With the developed model we not only verify the experimental Z-DNA predictions, but also generate the whole-genome annotation, introducing new possible Z-DNA regions, which have not yet been found in experiments and can be of interest to the researchers from various fields.

Impact of domain anisotropy on the inverse cascade in geostrophic turbulent convection

The effect of domain anisotropy on the inverse cascade occurring within the geostrophic turbulence regime of rapidly rotating Rayleigh–Bénard convection is investigated. In periodic domains with square cross-section in the horizontal, a domain-filling dipole state is present. For rectangular periodic domains, a Kolmogorov-like flow parallel to the short side and consisting of a periodic array of alternating unidirectional jets with embedded vortices is observed, together with an underlying weak meandering transverse jet. Similar transitions occurring in weakly dissipative two-dimensional flows driven by externally imposed small-amplitude noise and in classical hydrostatic geostrophic turbulence are a consequence of inviscid conservation of energy and potential enstrophy, and can be understood using statistical mechanics considerations. Rotating Rayleigh–Bénard convection represents an important three-dimensional system with only one inviscid invariant which nonetheless exhibits large-scale structures driven by intrinsically generated fluctuations.

Geometric algebra and information geometry for quantum computational software

The art of quantum algorithm design is highly nontrivial. Grover’s search algorithm constitutes a masterpiece of quantum computational software. In this article, we use methods of geometric algebra (GA) and information geometry (IG) to enhance the algebraic efficiency and the geometrical significance of the digital and analog representations of Grover’s algorithm, respectively. Specifically, GA is used to describe the Grover iterate and the discretized iterative procedure that exploits quantum interference to amplify the probability amplitude of the target-state before measuring the query register. The transition from digital to analog descriptions occurs via Stone’s theorem which relates the (unitary) Grover iterate to a suitable (Hermitian) Hamiltonian that controls Schrodinger’s quantum mechanical evolution of a quantum state towards the target state. Once the discrete-to-continuos transition is completed, IG is used to interpret Grover’s iterative procedure as a geodesic path on the manifold of the parametric density operators of pure quantum states constructed from the continuous approximation of the parametric quantum output state in Grover’s algorithm. Finally, we discuss the dissipationless nature of quantum computing, recover the quadratic speedup relation, and identify the superfluity of the Walsh–Hadamard operation from an IG perspective with emphasis on statistical mechanical considerations.

A Phase Space Approach to Supercooled Liquids and a Universal Collapse of Their Viscosity

A broad fundamental understanding of the mechanisms underlying the phenomenology of supercooled liquids has remained elusive, despite decades of intense exploration. When supercooled beneath its characteristic melting temperature, a liquid sees a sharp rise in its viscosity over a narrow temperature range, eventually becoming frozen on laboratory timescales. Explaining this immense increase in viscosity is one of the principle goals of condensed matter physicists. To that end, numerous theoretical frameworks have been proposed which explain and reproduce the temperature dependence of the viscosity of supercooled liquids. Each of these frameworks appears only applicable to specific classes of glassformers and each possess a number of variable parameters. Here we describe a classical framework for explaining the dynamical behavior of supercooled liquids based on statistical mechanical considerations, and possessing only a single variable parameter. This parameter varies weakly from liquid to liquid. Furthermore, as predicted by this new classical theory and its earlier quantum counterpart, we find with the aid of a small dimensionless constant that varies in size from $\sim 0.05-0.12$, a universal (16 decade) collapse of the viscosity data as a function of temperature. The collapse appears in all known types of glass forming supercooled liquids (silicates, metallic alloys, organic systems, chalcogenide, sugars, and water).

Suggested methodology for direct and inverse characterization of marine sediments that contain mud

Mud consists mostly of particles of clay minerals: kaolinite, illite, and smectite. Particles are hexagonally shaped platelets and have widths of the order of microns. In contrast, sandy/silty sediments are composed of silicate particles that are spheroidal is shape and have sizes of the order of millimeters. One distinction is that the platelets commonly carry electrical charges that repel each other. Relevant physics underlying their acoustic properties is considerably different than that for sandy/silty sediments. Naturally occurring mud also contains sandy/silty content, biological matter, and gaseous bubbles. The present paper discusses various techniques by which one can estimate quantitative information about mud sediments. Much can be learned from detailed geochemical analysis and microscopic observation of core samples, but enough is known about mud's general properties so that indirect methods can be used. These can estimate the density and sound speed as functions of depth into the sediment. The sound speed in mud is better approximated by the Mallock-Wood equations than for sandy/silty sediments. Density gradients in conjunction with statistical mechanics considerations yield information on how much of the mud is clay. Measurement of compressional wave attenuation can give information about the forces between platelets. [Work supported by ONR.]

Effective binary theory of multi-component nucleation.

Classical theory of multi-component nucleation [O. Hirschfelder, J. Chem. Phys. 61, 2690 (1974)] belongs to the class of the so-called intractable problems: it requires computational time which is an exponential function of the number of components N. For a number of systems of practical interest with N > 10, the brute-force use of the classical theory becomes virtually impossible and one has to resort to an effective medium approach. We present an effective binary model which captures important physics of multi-component nucleation. The distinction between two effective species is based on the observation that while all N components contribute to the cluster thermodynamic properties, there is only a part of them which trigger the nucleation process. The proposed 2D-theory takes into account adsorption by means of the Gibbs dividing surface formalism and uses statistical mechanical considerations for the treatment of small clusters. Theoretical predictions for binary-, ternary-, and 14-component mixtures are compared with available experimental data and other models.

The low-density diffusion coefficient of soft-sphere fluids in nanopores: Accurate correlations from exact theory and criteria for applicability of the Knudsen model

Molecular transport in confined spaces is of central importance to many traditional and emerging applications, such as gas separation and storage, catalytic and non-catalytic fluid–solid reactions and nanofluidics. The classical method to model the influence of solid–fluid particles collisions in transport comes from the early works of Knudsen and Smoluchowski, in which van-der-Walls interactions are neglected. While such assumption may be adequate for wide channels or high temperatures, it has been shown to lead to significant overprediction of the low-density diffusion coefficient in narrow pores at low and moderate temperatures, when compared to that obtained through molecular dynamics simulations. However, while molecular dynamics offers a much more accurate route for the calculation of diffusivities, its implementation and use is time-consuming and the Knudsen formulation remains preferred, particularly for interpretation of experimental permeation data. On the other hand, the oscillator model, developed in recent years in this laboratory, provides a straightforward method to estimate the low-density diffusivity, taking into account the dispersive forces exerted by the wall through rigorous statistical–mechanical considerations. The computational demand of the oscillator model is several orders of magnitude lower than that of a typical molecular dynamics run, though much greater than that of the simple Knudsen formulation. In order to facilitate its use, we provide several simple correlations for the fast estimation of the oscillator model-based diffusion coefficient for LJ fluids in simple geometries, as a function of the pore size and the LJ fluid–solid interaction parameters. Moreover, the parameter values for which the Knudsen equation supplies a reasonable estimation of the diffusion coefficient are presented for various degrees of accuracy.

On Maxwell’s Relations of Thermodynamics for Polymeric Liquids away from Equilibrium

To describe complex systems deeply in the nonlinear regime, advanced formulations of nonequilibrium thermodynamics such as the extended irreversible thermodynamics (EIT), the matrix model, the generalized bracket formalism, and the GENERIC (=general equation for the nonequilibrium reversible−irreversible coupling) formalism consider generalized versions of thermodynamic potentials in terms of a few, well-defined position-dependent state variables (defining the system at a coarse-grained level). Straightforward statistical mechanics considerations then imply a set of equalities for its second derivatives with respect to the corresponding state variables, typically known as Maxwell’s relations. We provide here direct numerical estimates of these relations from detailed atomistic Monte Carlo (MC) simulations of an unentangled polymeric melt coarse-grained to the level of the chain conformation tensor, under both weak and strong flows. We also report results for the nonequilbrium (i.e., relative to the quiesc...

STATISTICAL MECHANICS OF COLLISIONLESS ORBITS. III. COMPARISON WITHN-BODY SIMULATIONS

We compare the DARKexp differential energy distribution, N(E) \propto \exp(\phi_0-E)-1, obtained from statistical mechanical considerations, to the results of N-body simulations of dark matter halos. We first demonstrate that if DARKexp halos had anisotropic velocity distributions similar to those of N-body simulated halos, their density and energy distributions could not be distinguished from those of isotropic DARKexp halos. We next carry out the comparison in two ways, using (1) the actual energy distribution extracted from simulations, and (2) N-body fitting formula for the density distribution as well as N(E) computed from the density using the isotropic Eddington formula. Both the methods independently agree that DARKexp N(E) with \phi_0\approx 4-5 is an excellent match to N-body N(E). Our results suggest (but do not prove) that statistical mechanical principles of maximum entropy can be used to explain the equilibrated final product of N-body simulations.

Binary nucleation beyond capillarity approximation

Large discrepancies between binary classical nucleation theory (BCNT) and experiments result from adsorption effects and inability of BCNT, based on the phenomenological capillarity approximation, to treat small clusters. We propose a model aimed at eliminating both of these deficiencies. Adsorption is taken into account within Gibbsian approximation. Binary clusters are treated by means of statistical-mechanical considerations: tracing out the molecular degrees of freedom of the more volatile component, we obtain a coarse-grained system described in terms of the single-component mean-field kinetic nucleation theory [V. I. Kalikmanov, J. Chem. Phys. 124, 124505 (2006)], allowing an adequate treatment of clusters of arbitrary size. The model opens a route toward studying binary nucleation in complex systems with nanosized critical clusters.

Generalized Kelvin equation and pseudospinodal in nucleation theory

The classical Kelvin equation, relating the size of the critical cluster to the supersaturation, is inadequate for very small, molecular-sized clusters emerging at deep quenches observed in recent nucleation experiments. Using statistical mechanical considerations, we propose a generalization of the Kelvin equation applicable up to the vicinity of the pseudospinodal, where the nucleation barrier is approximately k(B)T. The supersaturation at the pseudospinodal is expressed in terms of the second virial coefficient. It is shown that near the pseudospinodal the critical cluster size is close to the coordination number in the liquid phase. Comparisons with computer simulations are presented.

Vapor liquid equilibrium modeling of alkane systems with Equations of State: “Simplicity versus complexity”

Two types of Equations of State (EoS), which are characterized here as “simple” and “complex” EoS, are evaluated in this study. The “simple” type involves two versions of the Peng–Robinson (PR) EoS: the traditional one that utilizes the experimental critical properties and the acentric factor and the other, referred to as PR-fitted (PR-f), where these parameters are determined by fitting pure compound vapor pressure and saturated liquid volume data. As “complex” EoS in this study are characterized the EoS derived from statistical mechanics considerations and involve the Sanchez–Lacombe (SL) EoS and two versions of the Statistical Associating Fluid Theory (SAFT) EoS, the original and the Perturbed-Chain SAFT (PC-SAFT). The evaluation of these two types of EoS is carried out with respect to their performance in the prediction and correlation of vapor liquid equilibria in binary and multicomponent mixtures of methane or ethane with alkanes of various degree of asymmetry. It is concluded that for this kind of systems complexity offers no significant advantages over simplicity. Furthermore, the results obtained with the PR-f EoS, especially those for multicomponent systems that are encountered in practice, even with the use of zero binary interaction parameters, indicate that this EoS may become a powerful tool for reservoir fluid phase equilibria modeling.

Statistical mechanical considerations in the theory of the spread of the Hantavirus

Calculations in the theory of the spread of epidemics are described with particular focus on the estimation of motion parameters describing rodents that are the carriers of the Hantavirus epidemic. The data considered are of the “mark-recapture” kind, i.e., those collected by capturing, tagging and recapturing the animals in a prescribed finite region of space. The theoretical tool used is the Fokker–Planck equation, its characteristic quantities being the diffusion constant which describes the motion of the rodents, and the attractive potential which addresses their tendency to live near their burrows. The measurements are addressed through simple analytical calculations of the mean squared displacement of the animals relevant to the specific probing window in space corresponding to the trapping region. A Fourier prescription is provided to extract the home range of the animals from the observations. Applications of the theory to rodent movement in Panama and New Mexico are mentioned and several on-going generalizations of current models of Hantavirus epidemic spread are introduced.

A Statistical Mechanical View of the Determination of the Composition of Multi?Temperature Plasmas

It is important that plasma composition is calculated in a manner consistent with statistical mechanics, particularly since the Boltzmann equation is the basis from which transport coefficients and the fluid‐dynamic equations are derived. It is shown from statistical mechanical considerations based on the Boltzmann equation and the H‐theorem that it is (i) not possible for a plasma to have more than one temperature in equilibrium in the absence of external forces and gradients, and (ii) not possible to draw conclusions about the change in entropy of a plasma in the presence of external forces and gradients. Derivations of the two‐temperature Saha equation, and more generally calculations of the composition of a multi‐temperature plasma, that are based on entropy maximization are therefore invalid. A thermodynamic derivation of the composition of a multi‐temperature plasma that is consistent with the statistical mechanical results is presented. The derivation shows that the equilibrium composition of a plasma can be correctly calculated by minimization of the internal or free energy.

The concept of temperature in a nonextensive scenario

Abstract Tsallis’ non-extensive thermostatistics (Tsallis et al., Physica A 261 (1998) 534; Tsallis, in: Abe, Okamoto (Eds.), Nonextensive Statistical Mechanics and its Applications, Lecture Notes in Physics, Springer, Berlin, 2000; Tsallis, Braz. J. Phys. 29 (1991) 1; Plastino and Plastino, in: Ludena (Ed.), Condensed Matter Theories, Nova Science Publishers, New York, USA, 1996, p. 341; Plastino and Plastino, Braz. J. Phys. 29 (1999) 79) is by now recognized as a new paradigm for statistical mechanical considerations. In a nonextensive scenario, however, the concept of temperature should be handled with some care, as some difficulties ensue (Tsallis, 2000). In this work we discuss this problem and reach some useful conclusions.

Interface of General Relativity, Quantum Physics and Statistical Mechanics: Some Recent Developments

The arena normally used in black holes thermodynamics was recently generalized to incorporate a broad class of physically interesting situations. The key idea is to replace the notion of stationary event horizons by that of `isolated horizons.' Unlike event horizons, isolated horizons can be located in a space-time quasi-locally. Furthermore, they need not be Killing horizons. In particular, a space-time representing a black hole which is itself in equilibrium, but whose exterior contains radiation, admits an isolated horizon. In spite of this generality, the zeroth and first laws of black hole mechanics extend to isolated horizons. Furthermore, by carrying out a systematic, non-perturbative quantization, one can explore the quantum geometry of isolated horizons and account for their entropy from statistical mechanical considerations. After a general introduction to black hole thermodynamics as a whole, these recent developments are briefly summarized.

Interface of General Relativity, Quantum Physics and Statistical Mechanics: Some Recent Developments

AbstractThe arena normally used in black holes thermodynamics was recently generalized to incorporate a broad class of physically interesting situations. The key idea is to replace the notion of stationary event horizons by that of ‘isolated horizons.’ Unlike event horizons, isolated horizons can be located in a space‐time quasi‐locally. Furthermore, they need not be Killing horizons. In particular, a space‐time representing a black hole which is itself in equilibrium, but whose exterior contains radiation, admits an isolated horizon. In spite of this generality, the eroth and first laws of black hole mechanics extend to isolated horizons. Furthermore, by carrying out a systematic, non‐perturbative quantization, one can explore the quantum geometry of isolated horizons and account for their entropy from statistical mechanical considerations. After a general introduction to black hole thermodynamics as a whole, these recent developments are briefly summarized.

Quantum geometry of isolated horizons and black hole entropy

Using the earlier developed classical Hamiltonian framework as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polymer excitations of the bulk quantum geometry pierce the horizon endowing it with area. The intrinsic geometry of the horizon is then described by the quantum Chern-Simons theory of a U(1) connection on a punctured 2-sphere, the horizon. Subtle mathematical features of the quantum Chern-Simons theory turn out to be important for the existence of a coherent quantum theory of the horizon geometry. Heuristically, the intrinsic geometry is flat everywhere except at the punctures. The distributional curvature of the U(1) connection at the punctures gives rise to quantized deficit angles which account for the overall curvature. For macroscopic black holes, the logarithm of the number of these horizon microstates is proportional to the area, irrespective of the values of (non-gravitational) charges. Thus, the black hole entropy can be accounted for entirely by the quantum states of the horizon geometry. Our analysis is applicable to all non-rotating black holes, including the astrophysically interesting ones which are very far from extremality. Furthermore, cosmological horizons (to which statistical mechanical considerations are known to apply) are naturally incorporated. An effort has been made to make the paper self-contained by including short reviews of the background material.

Enzymes and the supramolecular organization of the living cell. Information transfer within supramolecular edifices and imprinting effects

Simple considerations of statistical mechanics show that the association of an enzyme with another protein or with an ‘inert’ surface results in a decrease of its information content and thus that it receives an ‘instruction’ from this protein or from the surface. As a consequence, the free energy stored in the enzyme increases, and this energy may be used to alter the intrinsic catalytic properties of the enzyme. This may imply, for instance, that an enzyme which is devoid of activity may have its activity enhanced when bound to another protein or to a membrane. A possible consequence of this communication between proteins is that, upon dissociation of the complex, one of these enzymes may transitorily retain an imprinting of the other protein and this imprinting may in turn alter the properties of the enzyme. Different enzyme systems may illustrate this view. A particular emphasis has been put on the study of a phosphoribulokinase-glyceraldehyde phosphate dehydrogenase complex from Chlamydomonas chloroplasts. Whereas the isolated oxidized phosphoribulokinase is almost completely inactive, it becomes active when bound to glyceraldehyde phosphate dehydrogenase. Moreover, upon dissociation of the complex, the phosphoribulokinase retains for a while an imprinting exerted by glyceraldehyde phosphate dehydrogenase. It then displays properties that are markedly different from those of the free stable enzyme. Thermodynamics allows us to calculate the amount of energy stored in this enzyme and used to facilitate substrate binding and catalysis. There is thus little doubt that information and instructions are transferred from protein to protein within enzyme complexes that result in a complete change of their biological function.