Equilibrium In Statistical Mechanics Research Articles

A Spectral Theorem for the Semigroup Generated by a Class of Infinitely Many Master Equations

In this article we investigate the spectral properties of the infinitesimal generator of an infinite system of master equations arising in the analysis of the approach to equilibrium in statistical mechanics. The system under consideration thus consists of infinitely many first-order differential equations governing the time evolution of probabilities susceptible of describing jumps between the eigenstates of a differential operator with a discrete point spectrum. The transition rates between eigenstates are chosen in such a way that the so-called detailed balance conditions are satisfied, so that for a large class of initial conditions the given system possesses a global solution which converges exponentially rapidly toward a time independent probability of Gibbs type. A particular feature and a challenge of the problem under consideration is that in the infinite-dimensional functional space where the initial-value problem is well posed, the infinitesimal generator is realized as a non normal and non dissipative compact operator, whose spectrum therefore does not exhibit a spectral gap around the zero eigenvalue.

On Boltzmann versus Gibbs and the Equilibrium in Statistical Mechanics

Charlotte Werndl and Roman Frigg discuss the relationship between the Boltzmannian and Gibbsian framework of statistical mechanics, addressing, in particular, the question when equilibrium values calculated in both frameworks agree. This note points out conceptual confusions that could arise from their discussion, concerning, in particular, the authors’ use of “Boltzmann equilibrium.” It also clarifies the status of the Khinchin condition for the equivalence of Boltzmannian and Gibbsian equilibrium predictions and shows that it follows, under the assumptions proposed by Werndl and Frigg, from standard arguments in probability theory.

Kinematic viscosity measurement of granular flows via low Reynolds number cylinder drag experiment

Confined, vibration-driven grain piles exhibit fluid-like properties, in particular, predictable, non-random flow patterns, hydrodynamic modal response to vibrational forcing, and a persistent, spatially uniform tendency toward local statistical mechanical equilibrium. This paper presents a technique that combines particle image velocimetry of vibration-driven grain flow over a submerged, instrumented cylinder and measurement of the flow-induced drag force on the cylinder to determine the grain flows effective kinematic viscosity. The fundamental basis of such measurements is provided by recent work showing that high-restitution grain piles subject to low-amplitude vibration are macroscopic dynamical analogs of liquid-state molecular hydrodynamic systems. Practically, the proposed viscometric method provides a key material property, the kinematic, or equivalently, dynamic viscosity, for use in computational fluid dynamic simulations of a variety of materials processing operations that utilize vibration-driven grain flows.

Interplay between DNA sequence and negative superhelicity drives R-loop structures

R-loops are abundant three-stranded nucleic-acid structures that form in cis during transcription. Experimental evidence suggests that R-loop formation is affected by DNA sequence and topology. However, the exact manner by which these factors interact to determine R-loop susceptibility is unclear. To investigate this, we developed a statistical mechanical equilibrium model of R-loop formation in superhelical DNA. In this model, the energy involved in forming an R-loop includes four terms-junctional and base-pairing energies and energies associated with superhelicity and with the torsional winding of the displaced DNA single strand around the RNA:DNA hybrid. This model shows that the significant energy barrier imposed by the formation of junctions can be overcome in two ways. First, base-pairing energy can favor RNA:DNA over DNA:DNA duplexes in favorable sequences. Second, R-loops, by absorbing negative superhelicity, partially or fully relax the rest of the DNA domain, thereby returning it to a lower energy state. In vitro transcription assays confirmed that R-loops cause plasmid relaxation and that negative superhelicity is required for R-loops to form, even in a favorable region. Single-molecule R-loop footprinting following in vitro transcription showed a strong agreement between theoretical predictions and experimental mapping of stable R-loop positions and further revealed the impact of DNA topology on the R-loop distribution landscape. Our results clarify the interplay between base sequence and DNA superhelicity in controlling R-loop stability. They also reveal R-loops as powerful and reversible topology sinks that cells may use to nonenzymatically relieve superhelical stress during transcription.

On the existence and blow-up of solutions for a mean field equation with variable intensities

We study an elliptic problem with exponential nonlinearities describing the statistical mechanics equilibrium of point vortices with variable intensities. For suitable values of the physical parameters we exclude the existence of blow-up points on the boundary, we prove a mass quantization property and we apply our analysis to the construction of minimax solutions.

Unnormalized probability: A different view of statistical mechanics

All teachers and students of physics have absorbed the doctrine that probability must be normalized. Nevertheless, there are problems for which the normalization factor only gets in the way. An important example of this counter-intuitive assertion is provided by the derivation of the thermodynamic entropy from the principles of statistical mechanics. Unnormalized probabilities provide a surprisingly effective teaching tool that can make it easier to explain to students the essential concept of entropy. The elimination of the normalization factor offers simpler equations for thermodynamic equilibrium in statistical mechanics, which then lead naturally to a new and simpler definition of the entropy in thermodynamics. Notably, this definition does not change the formal expression of the entropy based on composite systems that I have previously offered. My previous definition of entropy has been criticized by Dieks, based on what appears to be a misinterpretation. I believe that the new definition presented here has the advantage of greatly reducing the possibility of such a misunderstanding—either by students or by experts.

The approach towards equilibrium in Lanford’s theorem

This paper develops a philosophical investigation of the merits and faults of a theorem by Lanford (1975), Lanford (Asterisque 40, 117–137, 1976), Lanford (Physica 106A, 70–76, 1981) for the problem of the approach towards equilibrium in statistical mechanics. Lanford’s result shows that, under precise initial conditions, the Boltzmann equation can be rigorously derived from the Hamiltonian equations of motion for a hard spheres gas in the Boltzmann-Grad limit, thereby proving the existence of a unique solution of the Boltzmann equation, at least for a very short amount of time. We argue that, by establishing a statistical H-theorem, it offers a prospect to complete Boltzmann’s combinatorial argument, without running against the objections which plug other typicality-based approaches. However, we submit that, while recovering the irreversible approach towards equilibrium for positive times, it fails to predict a monotonic increase of entropy for negative times, and hence it yields the wrong retrodictions about the past evolution of a gas.

Entropy, Closures and Subgrid Modeling

Maximum entropy states or statistical mechanical equilibrium solutions have played an important role in the development of a fundamental understanding of turbulence and its role in geophysical flows. In modern general circulation models of the earth’s atmosphere and oceans most parameterizations of the subgrid-scale energy and enstrophy transfers are based on ad hoc methods or ideas developed from equilibrium statistical mechanics or entropy production hypotheses. In this paper we review recent developments in nonequilibrium statistical dynamical closure theory, its application to subgrid-scale modeling of eddy-eddy, eddy-mean field and eddy-topographic interactions and the relationship to minimum enstrophy, maximum entropy and entropy production arguments.

ASYMPTOTIC STATES IN COUPLED MAP LATTICES

Coupled Map Lattices (CML) are a kind of dynamical systems that appear naturally in some contexts, like the discretization of partial differential equations, and as a simple model of coupling between nonlinear systems. The coupling creates new and rich properties, that has been the object of intense investigation during the last decades. In this work we have two goals: first we give a nontechnical introduction to the theory of invariant measures and equilibrium in dynamics (with analogies with equilibrium in statistical mechanics) because we believe that sometimes a lot of interesting problems on the interface between physics and mathematics are not being developed simply due to the lack of a common language. Our second goal is to make a small contribution to the theory of equilibrium states for CML. More specifically, we show that a certain family of Coupled Map Lattices presents different asymptotic behaviors when some parameters (including coupling) are changed. The goal is to show that we start in a configuration with infinitely many different measures and, with a slight change in coupling, get an asymptotic state with only one measure describing the dynamics of most orbits coupling. Associating a symbolic dynamics with symbols +1, 0 and -1 to the system we describe a transition characterized by an asymptotic state composed only of symbols +1 or only of symbols -1. We rigorously prove our assertions and provide numerical experiments with two goals: first, as illustration of our rigorous results and second, to motivate some conjectures concerning the problem and some of its possible variations.

NOSÉ-HOOVER AND LANGEVIN THERMOSTATS DO NOT REPRODUCE THE NONEQUILIBRIUM BEHAVIOR OF LONG-RANGE HAMILTONIANS

We compare simulations performed using the Nosé-Hoover and the Langevin thermostats with the Hamiltonian dynamics of a long-range interacting system in contact with a reservoir. We find that while the statistical mechanics equilibrium properties of the system are recovered by all the different methods, the Nosé-Hoover and the Langevin nonequilibrium results differ from the Hamiltonian ones.

Hamiltonian Dynamics Reveals the Existence of Quasistationary States for Long-Range Systems in Contact with a Reservoir

We introduce a Hamiltonian dynamics for the description of long-range interacting systems in contact with a thermal bath (i.e., in the canonical ensemble). The dynamics confirms statistical mechanics equilibrium predictions for the Hamiltonian mean field model and the equilibrium ensemble equivalence. We find that long-lasting quasistationary states persist in the presence of the interaction with the environment. Our results indicate that quasistationary states are indeed reproducible in real physical experiments.

An Energy Interconversion Principle Applied in Reaction dynamics for the determination of equilibrium standard states

Chemical and other reaction theories involving thermodynamical equilibrium states utilize statistical mechanical equilibrium density distributions. Here, a definition of heat-work transformation termed thermo-mechanical coherence is first made, and it is conjectured that most molecular bonds have the above heat-work transformation property, which models a chemical bond as a “centrifugal heat engine”, where the internal energy state need not correspond to any of the standard equilibrium densities. Expressions are derived for the standard Gibbs free energy, enthalpy, and entropy where the bond coordinates need not conform to a non-degenerate Boltzmann state, since bond breakdown and formation are processes that have direction, whereas equilibrium distributions are derived when the Hamiltonian is of fixed form, which is not the case for chemical reactions using localized Hamiltonians. The empirically determined Gibbs free energy from a known molecular dynamics simulation of a dimer reaction $$ 2{\rm A} \rightleftharpoons {\rm A}_{\rm 2} $$ , accords rather well with the theoretical estimate. A relation connecting the rate of reaction with the equilibrium constant and other kinetic parameters is derived and could place the commonly observed linear relationship between the logarithms of the rate constant and equilibrium constant on a firmer theoretical footing. These relationships could include analogues of the Hammett correlations used extensively in physical organic chemistry, as well as others which are temperature dependent. One prediction of the principles developed here is that the equilibrium standard reaction free energy is more dependent on the height of the intermolecular potential than its depth, so that the sign of the ΔG θ can change for varying barrier height with fixed well depth, which may appear counter-intuitive. All the above developments can be tested directly in simulations and therefore provides a fertile ground for further research with significant implications on how standard states are determined in relation to the direction of chemical reaction.This work treats the molecular bond using standard thermodynamics as if it were a system, and it is anticipated that with the advent of single-molecule science and experiment, that might be one direction in which molecular statistical thermodynamics would develop.

Theoretical and computational models of biological ion channels

The goal of this review is to establish a broad and rigorous theoretical framework to describe ion permeation through biological channels. This framework is developed in the context of atomic models on the basis of the statistical mechanical projection-operator formalism of Mori and Zwanzig. The review is divided into two main parts. The first part introduces the fundamental concepts needed to construct a hierarchy of dynamical models at different level of approximation. In particular, the potential of mean force (PMF) as a configuration-dependent free energy is introduced, and its significance concerning equilibrium and non-equilibrium phenomena is discussed. In addition, fundamental aspects of membrane electrostatics, with a particular emphasis on the influence of the transmembrane potential, as well as important computational techniques for extracting essential information from all-atom molecular dynamics (MD) simulations are described and discussed. The first part of the review provides a theoretical formalism to 'translate' the information from the atomic structure into the familiar language of phenomenological models of ion permeation. The second part is aimed at reviewing and contrasting results obtained in recent computational studies of three very different channels: the gramicidin A (gA) channel, which is a narrow one-ion pore (at moderate concentration), the KcsA channel from Streptomyces lividans, which is a narrow multi-ion pore, and the outer membrane matrix porin F (OmpF) from Escherichia coli, which is a trimer of three beta-barrel subunits each forming wide aqueous multi-ion pores. Comparison with experiments demonstrates that current computational models are approaching semi-quantitative accuracy and are able to provide significant insight into the microscopic mechanisms of ion conduction and selectivity. We conclude that all-atom MD with explicit water molecules can represent important structural features of complex biological channels accurately, including such features as the location of ion-binding sites along the permeation pathway. We finally discuss the broader issue of the validity of ion permeation models and an outlook to the future.

Equilibrium and non-equilibrium effects in nucleus–nucleus collisions

Local thermal and chemical equilibration is studied for central A+A collisions at 10.7–160 AGeV in the Ultrarelativistic Quantum Molecular Dynamics model (UrQMD). The UrQMD model exhibits strong deviations from local equilibrium at the high density hadron–string phase formed during the early stage of the collision. Equilibration of the hadron–resonance matter is established in the central cell of volume V=125 fm 3 at later stages, t≥10 fm/ c, of the resulting quasi-isentropic expansion. The thermodynamical functions in the cell and their time evolution are presented. Deviations of the UrQMD quasi-equilibrium state from the statistical mechanics equilibrium are found. They increase with energy per baryon and lead to a strong enhancement of the pion number density as compared to statistical mechanics estimates at SPS energies.

Equilibrium spectra and implications for a two-field turbulence model

Analytic expressions are given for statistical mechanical equilibrium solutions of two-field turbulence model equations that are used in describing plasma drift waves and for passive scalar advection in a neutral fluid. These are compared with those previously proposed [Phys. Fluids B 1, 1331 (1989)], in particular regarding the role of the cross correlations between fields. Implications for two-point closure calculations are discussed.

Statistical mechanics approach to the structure determination of a crystal

The previously formulated new approach to the structure analysis of a crystal based on the profound analogy between the problem of determination of thermodynamic equilibrium in statistical mechanics and the optimization problem for a function of many variables [Khachaturyan, Semenovskaya & Vainshtein (1979). Soy. Phys. Crystallogr. 24, 519-524; (1981). Acta Cryst. A37, 742-754] is developed. In this approach, a crystal structure is determined by the equilibrium low-temperature state of a model non-ideal gas composed of the atoms within a crystal unit cell, the unit cell and the R factor being regarded as a vessel and an interatomic interaction Hamiltonian, respectively. In contrast to the above cited papers, the low-temperature equilibrium state is found by means of the Monte Carlo sampling scheme usually utilized in statistical mechanics applications. The main advantage of such a treatment is that the system reaches the equilibrium state avoiding all 'traps', metastable (local) minima of the free energy, and thus provides automatic determination of the crystal structure. The method is successfully tested on the known l-proline structure containing 32 non-hydrogen atoms per unit cell. The proposed statistical mechanics approach can also be applied with minor modifications to optimization problems in the multi-dimensional space of many variables.

Statistical dynamics of internal gravity waves-turbulence

Abstract Numerical simulations of internal gravity waves-turbulence are carried out for the inviscid, viscous and forced-dissipative two-dimensional primitive equations using the spectral method. Some of the results are compared with the predictions of the eddy damped quasi-normal Markovian (EDQNM) closure for internal waves of Carnevale and Frederiksen, generalized for periodic boundary conditions and possible random forcing and dissipation. The EDQNM reduces to the Boltzman equation of resonant interaction theory in the continuum space limit and as the forcing and dissipation vanish. However, the limit is singular in the sense that as well as conserving total energy, E, and total cross-correlation between the vorticity and buoyancy fields, C, an additional conservation law, viz. z-momentum, Pz , occurs in the limit. This means that the resonant interaction equilibrium (RIE) solution of the Boltzmann equation differs from the statistical mechanical equilibrium (SME) solution of the EDQNM closure. The sta...

Statistical mechanics of a finite difference approximation to the barotropic vorticity equation

AbstractA conventional finite‐difference representation of the barotropic vorticity equation is shown to possess a statistical mechanical equilibrium state, which is attained as a quasi‐equilibrium in the presence of weak dissipation. The state is isotropic only in suitably chosen wavenumber coordinates. The autocorrelation functions of the discrete complex Fourier amplitudes have integral time scales which are O(wave number)−1 at middle wavenumber, but which are infinite at the highest wavenumbers. This is demonstrated experimentally and confirmed by inspection of the interaction coefficients for the finite‐difference Jacobian. The results emphasize the dependence of predictability upon the choice of numerical method.

Decaying systems with degenerate Livsic matrix

The spectral analysis given by Wong for the resolvent of a non-self-adjoint operator with arbitrary multiplicity is utilized for the description of the time evolution of an unstable system. After studying the case for which the operator is independent of the resolvent variable z, the Wong analysis is extended to the physically interesting case for which the operator depends on z. The case of infinite multiplicity is treated, and it is found that the flow of probability through the generalized eigenstates is analogous to the approach to equilibrium in statistical mechanics.

Statistical Dynamics of Two-Dimensional Inviscid Flow on a Sphere

We derive the statistical mechanical equilibrium properties of two-dimensional flow on a sphere, described by the truncated inviscid nondivergent barotrapic model. It is found that probability distribution functions and expectation values are considerably different from those on a rotating beta-plane. The approach of numerical simulations to equilibrium is established by integrations for 208 days starting with observed meteorological global fields. It is found that the smaller scales of the observed initial field with a rhomboidal truncation wavenumber of 15 are indistinguishable from those of a realization of the equilibrium ensemble, while by days 70–80 this applies to all scales including the slowly changing zonal flow contributions. The effects of changing the resolution in the numerical simulations are examined and it is shown that the growth of differences between high-and low-resolution integrations may also be explained in terms of the nonuniforin relaxation of different scales to equilibrium. We find that differences between such simulations started with identical fields are under-estimated for the first few days, compared with using increased initial resolution in the high-resolution integration, while subsequently differences at the larger scales tend to be overestimated as the smaller scales equilibrate. For a given number of degrees of freedom, the equilibrium spectra are found to be relatively insensitive to the truncation scheme used and we propose the use of a more efficient parauelogrammic truncation scheme for numerical spectral models.