Techniques Of Statistical Mechanics Research Articles

Statistical mechanics of lossy compression for nonmonotonic multilayer perceptrons

A lossy data compression scheme for uniformly biased Boolean messages is investigated via statistical mechanics techniques. We utilize a treelike committee machine (committee tree) and a treelike parity machine (parity tree) whose transfer functions are nonmonotonic. The scheme performance at the infinite code length limit is analyzed using the replica method. Both committee and parity treelike networks are shown to saturate the Shannon bound. The Almeida-Thouless stability of the replica symmetric solution is analyzed, and the tuning of the nonmonotonic transfer function is also discussed.

Projection operator techniques and Hilbert space averaging in the quantum theory of nonequilibrium systems

The development of efficient strategies for the treatment of the dynamics of relevant observables in complex quantum systems plays a decisive role in the theory of quantum relaxation and transport behavior. Here we discuss the most important tools that are based on the projection operator techniques of nonequilibrium statistical mechanics. For both the Nakajima-Zwanzig and the time-convolutionless projection operator technique we derive the equations of motion for a set of relevant observables and develop explicit expressions in second and fourth order of the corresponding perturbation expansions. We also discuss the Hilbert space average method which is based on the idea of a best guess of conditional quantum expectations determined by an average over a suitable region in the underlying Hilbert space, and relate this method to the projection operator technique.

Modeling protein synthesis from a physicist’s perspective: A toy model

Proteins are polymers of amino acids. These macromolecules are synthesized by intracellular machines called ribosomes. Although the experimental investigation of protein synthesis has been a traditional area of research in molecular cell biology, important quantitative models of protein synthesis have been reported in research journals devoted to statistical physics and related interdisciplinary topics. From the perspective of a physicist, protein synthesis is the classical transport of interacting ribosomes on a messenger RNA (mRNA) template that dictates the sequence of the amino acids on the protein. We discuss appropriate simplification of the models and methods. In particular, we develop and analyze a simple toy model using some elementary techniques of nonequilibrium statistical mechanics and predict the average rate of protein synthesis and the spatial organization of the ribosomes in the steady state.

Putative mechanisms of peroxybicarbonate formation

The mechanism for the formation of the oxidant peroxymonocarbonate from hydrogen peroxide and bicarbonate/carbon dioxide in aqueous solution is studied by electron structure- and statistical mechanics techniques. It is found that the reaction proceeds through a single transition state. The activation energy is 83.3 kJ/mol at the highest level of theory [CCSD(T)/aug-cc-pvdz//B3LYP/aug-cc-pvdz]. However, it is reduced by 17% when the counter ion ( viz. NH 4 + ) is incorporated in the calculations.

Thermal quantum electrodynamics of nonrelativistic charged fluids

The theory relevant to the study of matter in equilibrium with the radiation field is thermal quantum electrodynamics (TQED). We present a formulation of the theory, suitable for nonrelativistic fluids, based on a joint functional integral representation of matter and field variables. In this formalism cluster expansion techniques of classical statistical mechanics become operative. They provide an alternative to the usual Feynman diagrammatics in many-body problems, which is not perturbative with respect to the coupling constant. As an application we show that the effective Coulomb interaction between quantum charges is partially screened by thermalized photons at large distances. More precisely one observes an exact cancellation of the dipolar electric part of the interaction, so that the asymptotic particle density correlation is now determined by relativistic effects. It still has the r(-6) decay typical for quantum charges, but with an amplitude strongly reduced by a relativistic factor.

Non-Markovian generalization of the Lindblad theory of open quantum systems

A systematic approach to the non-Markovian quantum dynamics of open systems is given by the projection operator techniques of nonequilibrium statistical mechanics. Combining these methods with concepts from quantum information theory and from the theory of positive maps, we derive a class of correlated projection superoperators that take into account in an efficient way statistical correlations between the open system and its environment. The result is used to develop a generalization of the Lindblad theory to the regime of highly non-Markovian quantum processes in structured environments.

Traffic of interacting ribosomes: Effects of single-machine mechanochemistry on protein synthesis

Many ribosomes simultaneously move on the same messenger RNA (mRNA), each synthesizing separately a copy of the same protein. In contrast to the earlier models, here we develop a "unified" theoretical model that not only incorporates the mutual exclusions of the interacting ribosomes, but also describes explicitly the mechanochemistry of each of these macromolecular machines during protein synthesis. Using analytical and numerical techniques of nonequilibrium statistical mechanics, we analyze the rates of protein synthesis and the spatiotemporal organization of the ribosomes in this model. We also predict how these properties would change with the changes in the rates of the various chemomechanical processes in each ribosome. Finally, we illustrate the power of this model by making experimentally testable predictions on the rates of protein synthesis and the density profiles of the ribosomes on some mRNAs in E-coli.

Mechanisms and Free Energies of Enzymatic Reactions

Mechanisms and Free Energies of Enzymatic Reactions

Ionization Equilibria and Conformational Transitions in Polyprotic Molecules and Polyelectrolytes

The coupling between proton binding and conformational degrees of freedom in polyprotic molecules and polyelectrolytes is studied theoretically. Our approach combines the classical rotational isomeric state (RIS) model developed by Flory and the site binding (SB) model used to treat proton binding equilibria. The properties of the resulting SBRIS model, which treats conformational degrees of freedom and proton binding on equal footing, are studied with statistical mechanical techniques. Quantities of interest, such as titration curves, conformational probabilities, or macroscopic binding constants, are expressed as thermal averages and are evaluated by direct enumeration of states or by transfer matrix techniques. We further demonstrate that in the SBRIS model conformational degrees of freedom can be averaged out, leading to the contracted description within the SB model. In most cases, this contraction leads to higher order interactions, which may not be present at the SBRIS level (e.g., triplet interactions). Several examples are discussed to illustrate the concepts developed. The case of succinic acid exemplifies the situation in its simplest form. The model can further rationalize the very different titration behavior of poly(acrylic acid) (PAA) and poly(methacrylic acid) (PMAA). In particular, the characteristic "jump" in the titration curve of PMAA is described quantitatively and is interpreted in terms of a conformational transition.

Statistical mechanics of optimization problems

Here I will present an introduction to the results that have been recently obtained in constraint optimization of random problems using statistical mechanics techniques. After presenting the general results, in order to simplify the presentation I will describe in details the problems related to the coloring of a random graph.

NONEQUILIBRIUM PHASE TRANSITIONS IN MODEL FERROMAGNETS: A REVIEW

The thermodynamical behaviors of ferromagnetic systems in equilibrium are well studied. However, the ferromagnetic systems far from equilibrium became an interesting field of research in last few decades. Recent exploration of ferromagnetic systems in the presence of a steady magnetic field are also studied by using standard tools of equilibrium statistical physics. The ferromagnet in the presence of time-dependent magnetic field, shows various interesting phenomena. An usual response of a ferromagnet in the presence of a sinusoidally oscillating magnetic field is the hysteresis. Apart from this hysteretic response, the nonequilibrium dynamic phase transition is also a very interesting phenomenon. In this chapter, the nonequilibrium dynamic phase transitions of the model ferromagnetic systems in presence of time-dependent magnetic field are discussed. For this kind of nonequilibrium phase transition, one cannot employ the standard techniques of equilibrium statistical mechanics. The recent developments in this direction are mainly based on numerical simulation (Monte Carlo). The Monte Carlo simulation of kinetic Ising model, in presence of sinusoidally oscillating (in time but uniform over space) magnetic field, is extensively performed to study the nonequilibrium dynamic phase transition. The temperature variations of dynamic order parameter, dynamic specific heat, dynamic relaxation time etc. near the transition point are discussed. The appearance and behaviors of a dynamic length scale and a dynamic time scale near the transition point are also discussed. All these studies indicate that this proposed dynamic transition is a nonequilibrium thermodynamic phase transition. The disorder (quenched) induced zero temperature (athermal) dynamic transition is studied in random field Ising ferromagnet. The dynamic transition in the Heisenberg ferromagnet is also studied. The nature of this transition in the Heisenberg ferromagnet depends on the anisotropy and the polarisation of the applied time varying magnetic field. The anisotropic Heisenberg ferromagnet in the presence of elliptically polarised magnetic field shows multiple dynamic transitions. This multiple dynamic transitions in anisotropic Heisenberg ferromagnet are discussed here. Recent experimental evidences of dynamic transitions are also discussed very briefly.

Accelerated Monte Carlo for Optimal Estimation of Time Series

By casting stochastic optimal estimation of time series in path integral form, one can apply analytical and computational techniques of equilibrium statistical mechanics. In particular, one can use standard or accelerated Monte Carlo methods for smoothing, filtering and/or prediction. Here we demonstrate the applicability and efficiency of generalized (nonlocal) hybrid Monte Carlo and multigrid methods applied to optimal estimation, specifically smoothing. We test these methods on a stochastic diffusion dynamics in a bistable potential. This particular problem has been chosen to illustrate the speedup due to the nonlocal sampling technique, and because there is an available optimal solution which can be used to validate the solution via the hybrid Monte Carlo strategy. In addition to showing that the nonlocal hybrid Monte Carlo is statistically accurate, we demonstrate a significant speedup compared with other strategies, thus making it a practical alternative to smoothing/filtering and data assimilation on problems with state vectors of fairly large dimensions, as well as a large total number of time steps.

Statistical mechanics of spatial evolutionary games

We discuss the long-run behavior of stochastic dynamics of many interacting players in spatial evolutionary games. In particular, we investigate the effect of the number of players and the noise level on the stochastic stability of Nash equilibria. We discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy. We use concepts and techniques of statistical mechanics to study game-theoretic models. In order to obtain results in the case of the so-called potential games, we analyze the thermodynamic limit of the appropriate models of interacting particles.

Ability of static and statistical mechanics posturographic measures to distinguish between age and fall risk

Traditional posturographic analysis and four statistical mechanics techniques were applied to center-of-pressure (COP) trajectories of young, older “low-fall-risk” and older “high-fall-risk” individuals. Low-fall-risk older adults were active 3 days per week in a cardiac rehabilitation program, while high-fall-risk older adults were diagnosed with perilymph fistula. Subjects diagnosed with perilymph fistula must have experienced two of the following vestibular findings: constant disequilibrium, positional vertigo and/or a positive fistula test. Non-parametric statistical tests were used to determine whether the posturographic measures could detect differences between the young and older “low-fall-risk” groups (age comparison) and between the older “low-” and “high-risk” groups (risk of falling comparison). The statistical mechanics techniques were more sensitive than the traditional measures: detecting significant differences between the young and older “low-risk” groups, while none of the traditional measures were significantly different. In addition, interpretation of the statistical mechanics techniques may offer more insight into the nature of the process controlling the COP trajectories. However, the methods offered slightly different explanations. For instance, the Hurst rescaled range analysis suggests that the movement of the COP is governed solely by anti-persistent behavior, whereas the stabilogram diffusion analysis suggests a short-term persistence balanced by a long-term anti-persistence. These discrepancies highlight the need for a model that incorporates the biological systems responsible for maintaining balance and experimental methods to directly quantify their status and roles. Until such a model exists, however, the statistical mechanics techniques appear to have some advantages over traditional posturographic measures for studying balance control.

Non-equilibrium statistical mechanics of recurrent networks with realistic neurons

Experimental evidence suggests that spike timing might be used by neurons to process and store information. Unfortunately, the mathematical analysis of recurrent networks with spiking neurons is highly non trivial. Most analytical studies have therefore focused on rate-based models, whereas spiking models tend to be studied numerically. In order to bridge this gap, we propose an effective spiking neuron model which still allows for the application of non-equilibrium statistical mechanical techniques. The model is flexible and its parameters can be adjusted in order to match real data. We analyze the population dynamics in the simple case of constant excitatory synapses.

On the quantum density of states and partitioning an integer

This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For N bosons in a one-dimensional harmonic oscillator potential, it is well known that the asymptotic ( N→∞) density of states is identical to the Hardy–Ramanujan formula for the partitions p( n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for p s ( n), the latter being the number of partitions of n into a sum of sth powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain d s ( n) for distinct partitions. We find that the distinct square partitions d 2( n) show pronounced oscillations as a function of n about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdos–Lehner formula for restricted partitions for the s=1 case, we use the modified technique to obtain a new formula for distinct restricted partitions.

Statistical mechanical approach to competitive binding of metal ions to multi-center receptors

A microscopic site binding model to treat binding of several metal ions to multi-center receptors is proposed. The model introduces the appropriate parameterization in terms of microscopic complexation constants and metal-metal pair interaction energies. The model is solved with statistical mechanical techniques, including direct enumeration or transfer matrices. We obtain microscopic and macroscopic complexation constants, microstate probabilities, and binding isotherms for chain-like receptors, including the long-chain limit. Various examples to illustrate the usefulness of the model are given.

Voltammogram spikes interpreted as envelopes of spikes resulting from electrode crystals of various sizes: Application to the UPD of Cu on Au(111)

We study and explain shapes of voltammogram spikes, observed during underpotential deposition (UPD) on electrode surfaces, as averaged envelopes of mutually shifted spikes associated with first-order phase transitions that occur in crystalline domains of various sizes that are formed on the electrode surface. This concept, already used in our previous work for two-phase systems and symmetric voltammogram spike shapes, is here substantially generalized to systems with multiple-phase coexistence and asymmetric spike shapes, using the rigorous statistical mechanical techniques of Borgs and Kotecký. Rather than mere numerical plots, we extract explicit functions that accurately describe the spike shapes. For the sake of clarity, we present our analysis and apply our results to fit the voltammogram of the UPD of Cu on Au(111) in sulfuric acid medium. This voltammogram shows two distinct spikes with a broad foot region near the spike at higher potentials. As was done in earlier treatments, we explain each of the two spikes as a result of a first-order transition. Here, though, the spikes are obtained as envelopes of closely spaced spikes resulting from crystals of various sizes. In contrast to earlier studies, however, we also explain the foot region in the same way. The foot’s shape, despite its large width and small height, can be equally well obtained as an envelope of shifted crystal spikes that are broader and smaller than those giving rise to the two distinct spikes. We achieve very good agreement with experiment.

First-principles calculations on Li xNiO 2: phase stability and monoclinic distortion

The phase diagram of Li x NiO 2 (0< x<1) and the evolution of the monoclinic distortion as a function of the lithium content are calculated using a combination of first-principles energy methods and statistical–mechanics techniques. As a function of the temperature different ordered Li x NiO 2 structures appear in the phase diagram at x=0.25, 0.33, 0.4, 0.5 and 0.75. Noteworthy, a new and unsuspected phase, Li 0.4NiO 2, dominates the low-lithium region of the phase diagram. In agreement with experimental results, maxima in monoclinicity ( a m/ b m) are predicted to occur near Li 0.75NiO 2 and Li 0.4NiO 2. The coupling between the Li-vacancy ordering and the Jahn–Teller activity of Ni 3+ ions plays a crucial role in the stability of ordered Li x NiO 2 structures and is at the origin of the monoclinic distortion. As a result, the different electrochemical behavior of Li x NiO 2 versus Li x CoO 2 lies in the electronic nature of the involved transition metal cation.

Semiclassical calculation of thermal rate constants in full Cartesian space: The benchmark reaction D+H2→DH+H

Semiclassical (SC) initial-value representation (IVR) methods are used to calculate the thermal rate constant for the benchmark gas-phase reaction D+H2→DH+H. In addition to several technical improvements in the SC-IVR methodology, the most novel aspect of the present work is use of Cartesian coordinates in the full space (six degrees of freedom once the overall center-of-mass translation is removed) to carry out the calculation; i.e., we do not invoke the conservation of total angular momentum J to reduce the problem to fewer degrees of freedom and solve the problem separately for each value of J, as is customary in quantum mechanical treatments. With regard to the SC-IVR methodology, we first present a simple and straightforward derivation of the semiclassical coherent-state propagator of Herman and Kluk (HK). This is achieved by defining an interpolation operator between the Van Vleck propagators in coordinate and momentum representations in an a priori manner with the help of the modified Filinov filtering method. In light of this derivation, we examine the systematic and statistical errors of the HK propagator to fully understand the role of the coherent-state parameter γ. Second, the Boltzmannized flux operator that appears in the rate expression is generalized to a form that can be tuned continuously between the traditional half-split and Kubo forms. In particular, an intermediate form of the Boltzmannized flux operator is shown to have the desirable features of both the traditional forms; i.e., it is easy to evaluate via path integrals and at the same time it gives a numerically well-behaved flux correlation function at low temperatures. Finally, we demonstrate that the normalization integral required in evaluating the rate constant can be expressed in terms of simple constrained partition functions, which allows the use of well-established techniques of statistical mechanics.