Ensemble Formalism Research Articles

Replica symmetry breaking, complexity and spin representation in the generalized random energy model

We study the random energy model with a hierarchical structure known as the generalized random energy model (GREM). In contrast to the original analysis by the microcanonical ensemble formalism, we investigate the GREM by the canonical ensemble formalism in conjunction with the replica method. In this analysis, spin-glass-order parameters are defined for the respective hierarchy level, and all possible patterns of replica symmetry breaking (RSB) are taken into account. As a result, we find that the higher step RSB ansatz is useful for describing spin-glass phases in this system. For investigating the nature of the higher step RSB, we generalize the notion of complexity developed for the one-step RSB to the higher step and demonstrate how the GREM is characterized by the generalized complexity. In addition, we propose a novel mean-field spin-glass model with a hierarchical structure, which is equivalent to the GREM at a certain limit. We also show that the same hierarchical structure can be implemented to other mean-field spin models than the GREM. Such models with hierarchy exhibit phase transitions of multiple steps in common.

Noncommutative effects in astrophysical objects: A survey

The main implications of noncommutativity over astrophysical objects are examined. Noncommutativity is introduced through a deformed dispersion relation E2 = p2c2 (1+λE)2 + m2c4 and the relevant thermodynamical quantities are calculated using the grand canonical ensemble formalism. These results are applied to simple physical models describing main-sequence stars, white-dwarfs and neutron stars. The stability of main-sequence stars and white dwarfs is discussed.

A localization technique for ensemble Kalman filters

AbstractEnsemble Kalman filter techniques are widely used to assimilate observations into dynamical models. The phase‐space dimension is typically much larger than the number of ensemble members, which leads to inaccurate results in the computed covariance matrices. These inaccuracies can lead, among other things, to spurious long‐range correlations, which can be eliminated by Schur‐product‐based localization techniques. In this article, we propose a new technique for implementing such localization techniques within the class of ensemble transform/square‐root Kalman filters. Our approach relies on a continuous embedding of the Kalman filter update for the ensemble members, i.e. we state an ordinary differential equation (ODE) with solutions that, over a unit time interval, are equivalent to the Kalman filter update. The ODE formulation forms a gradient system with the observations as a cost functional. Besides localization, the new ODE ensemble formulation should also find useful application in the context of nonlinear observation operators and observations that arrive continuously in time. Copyright © 2010 Royal Meteorological Society

The role of nonequilibrium thermo-mechanical statistics in modern technologies and industrial processes: an overview

The nowadays notable development of all the modern technology, fundamental for the progress and well being of world society, imposes a great deal of stress in the realm of basic Physics, more precisely on Thermo-Statistics. We do face situations in electronics and optoelectronics involving physical-chemical systems far-removed-from equilibrium, where ultrafast (in pico- and femto-second scale) and non-linear processes are present. Further, we need to be aware of the rapid unfolding of nano-technologies and use of low-dimensional systems (e.g., nanometric quantum wells and quantum dots in semiconductor heterostructures). All together this demands having an access to a Statistical Mechanics being efficient to deal with such requirements. It is worth noticing that the renowned Ryogo Kubo once stated that mechanics has been considered a theoretical endeavor. However, statistical mechanics exists for the sake of the real world, not for fictions. Further progress can only be hoped by close cooperation with experiment. Moreover, one needs to face the study of soft matter and fluids with complex structures (usually of the average self-affine fractal-like type). This is relevant for technological improvement in industries like, for example, that of polymers, petroleum, cosmetics, food, electronics and photonics (conducting polymers and glasses), in medical engineering, etc. It is then required to introduce a thermo-hydrodynamics going well beyond the classical (Onsagerian) one. Moreover, in the both type of situations above mentioned there often appear difficulties of description and objectivity (existence of so-called hidden constraints), which impair the proper application of the conventional ensemble approach used in the general, logically and physically sound, and well established Boltzmann-Gibbs statistics. A tentative to partially overcome such difficulties consists in resorting to non-conventional approaches. Here we briefly describe the construction of a Non-Equilibrium Statistical Ensemble Formalism (NESEF) that can deal, within a certain degree of success, with the situations above described. Several particular instances involving experimental observations and measurements in the area of semiconductor physics and in physics of fluids, which were analyzed in the context of the theory, are summarized. They comprise the cases of ultrafast optical spectroscopy; optical and transport processes in low-dimensional complex semiconductors; nonlinear transport in doped highly-polar semiconductors (of use in blue diodes) under moderate to high electric fields; nonlinear higher-order thermo-hydrodynamics in fluids under driven flow, in normal solutions and in complex situations as in solutions of polymers, micelles, DNA, and in microbatteries.

Statistical Ensemble Theory of Gompertz Growth Model

An ensemble formulation for the Gompertz growth function within the framework of statistical mechanics is presented, where the two growth parameters are assumed to be statistically distributed. The growth can be viewed as a self-referential process, which enables us to use the Bose-Einstein statistics picture. The analytical entropy expression pertain to the law can be obtained in terms of the growth velocity distribution as well as the Gompertz function itself for the whole process.

THERMO-STATISTICAL THEORY OF KINETIC AND RELAXATION PROCESSES

We describe, in a short overview, the construction of a Nonequilibrium Statistical Mechanics Ensemble Formalism, providing a thermo-statistical theory of kinetic and relaxation processes. Such construction has been approached along the recently past 20th century by a pleiad of distinguished scientists, a work that can be subsumed in a large systematization in the form of a physically sound, general and useful, theoretical framework. We briefly comment on the main questions associated to that construction. Among them are the relevant ones of choice of the basic variables, and of historicity and irreversibility. The derivation of a nonequilibrium grand-canonical statistical operator and a brief description of the all-important accompanying Nonlinear Quantum Kinetic Theory of relaxation processes are presented. The aspect of validation of the theory (comparison of theory and experiment) is reviewed in compact form, and its use is illustrated in a study of a nonequilibrium system of quantum oscillators embedded in a thermal bath and under the action of an external force, showing how a far-reaching generalization of Mori–Langevin equations arises.

Multiplicity fluctuations in the pion-fireball gas

Pion number fluctuations are considered in the system of pions and large mass fireballs decaying finally into pions. A formulation which gives an extension of the model of independent sources is suggested. The grand canonical and micro-canonical ensemble formulations of the pion-fireball gas are considered as particular examples.

Ensemble estimation of background‐error variances in a three‐dimensional variational data assimilation system for the global ocean

AbstractThis paper studies the sensitivity of global ocean analyses to two flow‐dependent formulations of the background‐error standard deviations (σb) for temperature and salinity in a three‐dimensional variational data assimilation (3D‐Var) system. The first formulation is based on an empirical parameterization of σb in terms of the vertical gradients of the background temperature and salinity fields, while the second formulation involves a more sophisticated approach that derives σb from the spread of an ensemble of background states. The ensembles are created by explicitly perturbing both the surface fluxes (wind stress, fresh water and heat) used to force the model and the observations (temperature and salinity profiles) used in the assimilation process. The two formulations are compared in two cycled 3D‐Var experiments for the period 1993‐‐2000. In both experiments, the observation‐error standard deviations (σo) are geographically dependent and estimated from a model‐data comparison prior to assimilation. An additional 3D‐Var experiment that employs the parametrized σb but a simpler σo formulation, and a control experiment involving no data assimilation, were also conducted and used for comparison.All 3D‐Var experiments produce a significant reduction in the mean and standard deviation of the temperature and salinity innovations compared to those of the control experiment. The largest differences between the two σb formulations occur in the upper 150 m, where the parametrized σb are notably larger than the ensemble σb. In this region, the innovation statistics are slightly better for the parametrized σb. Statistical consistency checks indicate that both schemes underestimate σb, the underestimation being stronger with the ensemble formulation. The error growth between cycles, however, is much reduced with the ensemble σb, suggesting that the analyses produced with the ensemble σb are in better balance than those produced with the parametrized σb. This claim is supported by independent data comparisons involving model fields not directly constrained by the assimilated temperature and salinity profiles. In particular, sea‐surface height (SSH) anomalies in the northwest Atlantic and zonal velocities in the equatorial Pacific are clearly better with the ensemble σb than with the parametrized σb. Results also show that while some aspects of those variables are improved with data assimilation (SSH anomalies and currents in the central and eastern Pacific), other aspects are degraded (SSH anomalies in the northwest Atlantic, currents in the western Pacific). Areas for improving the ensemble method and for making better use of the ensemble information are discussed. Copyright © 2009 Royal Meteorological Society

Positive-operator-valued-measure view of the ensemble approach to polarization optics

The statistical ensemble formalism of Kim et al [J. Opt. Soc. Am. A4, 433 (1987)] offers a realistic model for characterizing the effect of stochastic nonimage-forming optical media on the state of polarization of transmitted light. With suitable choice of the Jones ensemble, various Mueller transformations-some of which are hitherto unknown-are deduced. It is observed that the ensemble approach is formally identical to the positive-operator-valued measures (POVMs) on the quantum density matrix. This observation, in combination with the recent suggestion by Ahnert and Payne [Phys. Rev. A71, 012330-1 (2005)]-in the context of generalized quantum measurement on single photon polarization states-that linear optics elements can be employed in setting up all possible POVMs enables us to propose a way of realizing different types of Mueller devices.

Modeling of the Psychophysical Response Curves Using the Grand Canonical Ensemble in Statistical Physics

This paper aims to investigate the peripheral mechanism of taste perception by the use of the grand canonical ensemble in statistical physics. It allows a better understanding of this process and its interpretation at a microscopic level. The experimental part allowed us to obtain psychophysical curves relative to four nutritive sweeteners (sucrose, fructose, glucose, and maltitol). These curves represent the intensity of sweetness as a function of sugar concentration in water. A Sensory Measuring Unit for Recording Flux (SMURF) device is used to obtain intensity–time curves for each of the studied sweeteners. To model these curves we have established the expressions of some models using grand canonical ensemble formalism in statistical physics and applying some simplification approaches. The fitting of the psychophysical data with statistical models by numerical simulation demonstrates a good correlation between the models and the experimental curves. Some physicochemical parameters interfere in the expression of the established models. These parameters are classified in two categories: on one hand, the steric aspect, which is manifested by the density of taste receptor site, and the number of molecules per receptor site and on the other hand, an energetic aspect illustrated by the concentration at half saturation, which gives indirectly the sweet molecule-receptor site energy of interaction.

THEORY OF TRANSPORT PROCESSES AND THE METHOD OF THE NONEQUILIBRIUM STATISTICAL OPERATOR

The aim of this paper is to provide better understanding of a few approaches that have been proposed for treating nonequilibrium (time-dependent) processes in statistical mechanics with the emphasis on the interrelation between theories. The ensemble method, as it was formulated by Gibbs, has great generality and broad applicability to equilibrium statistical mechanics. Different macroscopic environmental constraints lead to different types of ensembles, with particular statistical characteristics. In the present work, the statistical theory of nonequilibrium processes which is based on nonequilibrium ensemble formalism is discussed. We also outline the reasoning leading to some other useful approaches to the description of the irreversible processes. The kinetic approach to dynamic many-body problems, which is important from the point of view of the fundamental theory of irreversibility, is alluded to. Appropriate references are made to papers dealing with similar problems arising in other fields. The emphasis is on the method of the nonequilibrium statistical operator (NSO) developed by Zubarev. The NSO method permits one to generalize the Gibbs ensemble method to the nonequilibrium case and to construct a nonequilibrium statistical operator which enables one to obtain the transport equations and calculate the transport coefficients in terms of correlation functions, and which, in the case of equilibrium, goes over to the Gibbs distribution. Although some space is devoted to the formal structure of the NSO method, the emphasis is on its utility. Applications to specific problems such as the generalized transport and kinetic equations, and a few examples of the relaxation and dissipative processes, which manifest the operational ability of the method, are considered.

Cell identification and verification of QPF ensembles using shape analysis techniques

This paper introduces a new verification technique designed for, but not limited to, quantitative precipitation forecasts. It is tested on, and examples are given for, an intercomparison of very short-period nowcasting schemes. One of these nowcasters is a Bayesian scheme that is used in an extensive ensemble formulation, and the verification scheme is uniquely capable of treating both the ensemble members and the mean forecast. The verification scheme uses Procrustes shape analysis methods that are well established in statistics but have not, to date, been applied to meteorological forecast assessment. The Procrustes methodology allows for a decomposition of the forecast error into any number of components such as location (displacement), shape, size, orientation and intensity. Each error component can be afforded a separate weighting such that a cost or value of the forecast can be calculated that accounts for different error types. For example, a forecaster who is concerned with the location of a storm would place greater emphasis on correct location in the forecast than other attributes. This ability to apply weights makes the system particularly suited to real-time verification applications where confidence in the performance of the forecast translates into improved dissemination to users. In addition, the decomposition of the error into parts enables diagnosis of the error sources that can lead to model adjustment and improvement.

Parametric equation of state of a weakly interacting Bose system

A uniform dilute Bose system with repulsive interactions is studied in the grand canonical ensemble formalism. We present a parametric equation of state that holds true from high temperatures down to below the transition temperature, thus providing a scheme for exploring the quantum-statistical nature of the Bose-Einstein condensation transition in interacting gases. As an application, the interaction-induced shift of the transition temperature is derived to be $\mathrm{\ensuremath{\Delta}}{T}_{c}∕{T}_{c}^{0}=2.83{n}^{1∕3}a$, where $n$ is the density and $a$ is the $S$-wave scattering length.

On the electronegativity nonlocality paradox

The electronegativity equalization principle states that, in its ground state, the electronegativity of every component in a system is the same. A paradox then arises: molecular fragments that are very far apart must still have the same electronegativity, which seems to contradict the common assumption that spatially separated molecular species can be described independently. Density-functional theory provides the tools needed to analyze this paradox at a fundamental level, and a resolution is found from the properties of the exact Hohenberg–Kohn functional. Specifically, there is no paradox because the electronegativity is not uniquely defined for separated systems. Instead, there is an “apparent electronegativity” that preserves locality. This may have implications for the treatment of charge-transfer excited states. A model for the energy as a function of the number of electrons is also presented. This model gives some insight into the utility of the grand canonical ensemble formulation (at nonzero temperature) and, unlike most previous models, this model recovers the appropriate behavior in the limits of infinitely separated and/or weakly interacting subsystems.

Novel Monte Carlo Molecular Simulation Scheme Using Identity-Altering Elementary Moves for the Calculation of Structure and Thermodynamic Properties of Polyolefin Blends

In this work, the semi-grand statistical ensemble formalism proposed by Pant and Theodorou (Macromolecules, 1995, 28, 7224) for polydisperse melts of linear chains is extended to mixtures of linear and branched chains in order to calculate stability of binary polymer blends based on Monte Carlo simulation. Furthermore, a new elementary Monte Carlo move is introduced that consists of chain identity-altering between branched and linear chain species and thus allows fluctuations in the blend composition. The new move, together with previously developed moves for long chain molecules, are shown to relax efficiently both the branched and the linear molecules in binary linear polyethylene−triarm polyethylene blends of various macromolecular sizes. Subsequently, they are used for the calculation of the microscopic structure and the thermodynamic properties of these systems. All blends examined are shown to be fully miscible.

Globular structures of a helix-coil copolymer: Self-consistent treatment

A self-consistent-field theory was developed in the grand canonical ensemble formulation to study transitions in a helix-coil multiblock globule. Helical and coil parts are treated as stiff rods and self-avoiding walks of variable lengths correspondingly. The resulting field theory takes, in addition to the conventional Zimm-Bragg, [J. Chem. Phys. 31, 526 (1959)] parameters, also three-dimensional interaction terms into account. The appropriate differential equations which determine the self-consistent fields were solved numerically with finite element method. Three different phase states are found: open chain, amorphous globule, and nematic liquid-crystalline (LC) globule. The LC-globule formation is driven by the interplay between the hydrophobic helical segment attraction and the anisotropic globule surface energy of an entropic nature. The full phase diagram of the helix-coil copolymer was calculated and thoroughly discussed. The suggested theory shows a clear interplay between secondary and tertiary structures in globular homopolypeptides.

Mapping a homopolymer onto a model fluid

We describe a linear homopolymer using a grand canonical ensemble formalism, a statistical representation that is very convenient for formal manipulations. We investigate the properties of a system where only next neighbor interactions and an external, confining, field are present and then show how a general pair interaction can be introduced perturbatively, making use of a Mayer expansion. Through a diagrammatic analysis, we shall show how constitutive equations derived for the polymeric system are equivalent to the Ornstein-Zernike and Percus-Yevick equations for a simple fluid and find the implications of such a mapping for the simple situation of Van der Waals mean field model for the fluid.

Extensive statistical mechanics based on nonadditive entropy: Canonical ensemble

The original canonical ensemble formalism for the nonextensive entropy thermostatistics is reconsidered. It is shown that the unambiguous connection of the statistical mechanics with the equilibrium thermodynamics is provided if the entropic parameter $1/(q-1)$ is an extensive variable of the state. Based on a particular example of the perfect gas, it is proved that the Tsallis thermostatistics meets all the requirements of equilibrium thermodynamics in the thermodynamic limit. In particular, the entropy of the system is extensive and the temperature is intensive. However, for finite systems both the Tsallis and Boltzmann-Gibbs entropies are nonextensive. The equivalence of the canonical and microcanonical ensembles of Tsallis thermostatistics in the thermodynamic limit is established. The issue associated with physical interpretation of the entropic variable is discussed in detail.

Statistical mean-field theory of finite quantum systems: canonical ensemble formulation

We develop a statistical mean-field theory of finite quantum systems in thermal equilibrium. Our formulation employs the canonical ensemble of statistical mechanics, and it enables us to analytically determine the occupation number distributions of interacting particles obeying Bose–Einstein or Fermi–Dirac statistics. We have also developed a numerical procedure that enables us to obtain a universal scaled occupation number distribution that, for a given total number of interacting particles in a finite system, makes it possible to determine the occupation number distribution for any temperature. The developed mean-field theory is applicable to a wide range of atomic, nuclear and condensed matter systems for which finite-size effects can play an important role. In particular, the present approach makes it possible to formulate a finite temperature mean-field theory for a specific ion in a dense plasma.

Nonlinear charge transport in III-N semiconductors: Mobility, diffusion, and a generalized Einstein relation

A theoretical study of nonlinear charge transport in polar semiconductors is presented. It is based on a nonequilibrium statistical ensemble formalism which provides a generalized Boltzmann-style nonlinear quantum kinetic theory. The mobility and the diffusion coefficients are obtained and, relating both, a Nernst-Townsend-Einstein relation is derived extended to the nonlinear regime (i.e., outside the Ohmic domain). Numerical calculations are performed considering the particular case of the strongly polar III nitrides, which have application in blue-emitting diodes.