Formulation Of Statistical Mechanics Research Articles

Folding Thermodynamics of a Beta-Hairpin Peptide: A Theoretical Study

Analyzing experimental data of thermodynamic properties of proteins reveals large discrepancies existing between data obtained using different experimental methods, for example, those obtained by spectroscopy methods and those by calorimetry. The interpretation is that some experimental methods probe local or site-specific properties of protein, others probe global properties. To develop a statistical mechanical model for site-specific properties of proteins we extend the Wako-Saito-Munoz-Eaton (WSME) model by introducing a set of site-dependent parameters, Ti, which has a mid-point temperature interpretation. This model has been applied to the GB1 C-terminal β-hairpin, which has also been investigated using Molecular Dynamics (MD) simulations. Our MD results support the novel modification to the WSME model by providing reasonable insight into the local thermodynamic behavior of the system. Furthermore, our model is equivalent to a statistical mechanical formulation of the foldon behavior, identified by Englander, et al in hydrogen-exchange (HX) experiments and other experimental techniques, e.g., absorption spectroscopy (Abs), circular dichroism (CD), small angle x-ray scattering (SAXA), etc. Using the simple β-hairpin model as an example, we hope to illustrate that our model is generally useful for the interpretation of site-dependent thermodynamic properties of proteins as well as for the study of protein folding.

Tsallis distributions and1/fnoise from nonlinear stochastic differential equations

Probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems. In this article we unite modeling of such distributions with the model of widespread 1/f noise. We propose a class of nonlinear stochastic differential equations giving both the q-exponential or q-Gaussian distributions of signal intensity, revealing long-range correlations and 1/f(β) behavior of the power spectral density. The superstatistical framework to get 1/f(β) noise with q-exponential and q-Gaussian distributions of the signal intensity is proposed, as well.

Statics and dynamics of inhomogeneous liquids via the internal-energy functional

We give a variational formulation of classical statistical mechanics where the one-body density and the local entropy distribution constitute the trial fields. Using Levy's constrained search method, it is shown that the grand potential is a functional of both distributions, that it is minimal in equilibrium, and that the minimizing fields are those at equilibrium. The functional splits into a sum of entropic, external energetic, and internal energetic contributions. Several common approximate Helmholtz free-energy density functionals, such as the Rosenfeld fundamental measure theory for hard-sphere mixtures, are transformed to internal-energy functionals. The variational derivatives of the internal-energy functional are used to generalize dynamical density-functional theory to include the dynamics of the microscopic entropy distribution, as is relevant for studying heat transport and thermal diffusion.

Macroscopic Time Evolution and MaxEnt Inference for Closed Systems with Hamiltonian Dynamics

MaxEnt inference algorithm and information theory are relevant for the time evolution of macroscopic systems considered as problem of incomplete information. Two different MaxEnt approaches are introduced in this work, both applied to prediction of time evolution for closed Hamiltonian systems. The first one is based on Liouville equation for the conditional probability distribution, introduced as a strict microscopic constraint on time evolution in phase space. The conditional probability distribution is defined for the set of microstates associated with the set of phase space paths determined by solutions of Hamilton's equations. The MaxEnt inference algorithm with Shannon's concept of the conditional information entropy is then applied to prediction, consistently with this strict microscopic constraint on time evolution in phase space. The second approach is based on the same concepts, with a difference that Liouville equation for the conditional probability distribution is introduced as a macroscopic constraint given by a phase space average. We consider the incomplete nature of our information about microscopic dynamics in a rational way that is consistent with Jaynes' formulation of predictive statistical mechanics. Maximization of the conditional information entropy subject to this macroscopic constraint leads to a loss of correlation between the initial phase space paths and final microstates. Information entropy is the theoretic upper bound on the conditional information entropy, with the upper bound attained only in case of the complete loss of correlation. In this alternative approach to prediction of macroscopic time evolution, maximization of the conditional information entropy is equivalent to the loss of statistical correlation. In accordance with Jaynes, irreversibility appears as a consequence of gradual loss of information about possible microstates of the system.

Irreversibility and entropy production in transport phenomena, II: Statistical–mechanical theory on steady states including thermal disturbance and energy supply

Some general aspects of nonlinear transport phenomena are discussed on the basis of two kinds of formulations obtained by extending Kubo’s perturbational scheme of the density matrix and Zubarev’s non-equilibrium statistical operator formulation. Both formulations are extended up to infinite order of an external force in compact forms and their relationship is clarified through a direct transformation. In order to make it possible to apply these formulations straightforwardly to thermal disturbance, its mechanical formulation is given (in a more convenient form than Luttinger’s formulation) by introducing the concept of a thermal field ET which corresponds to the temperature gradient and by defining its conjugate heat operator AH=∑jhjrj for a local internal energy hj of the thermal particle j. This yields a transparent derivation of the thermal conductivity κ of the Kubo form and the entropy production (dS/dt)irr=κET2/T. Mathematical aspects of the non-equilibrium density-matrix will also be discussed. In Paper I (M. Suzuki, Physica A 390 (2011)1904), the symmetry-separated von Neumann equation with relaxation terms extracting generated heat outside the system was introduced to describe the steady state of the system. In this formulation of the steady state, the internal energy 〈H0〉t is time-independent but the field energy 〈H1〉t(=−〈A〉t⋅F) decreases as time t increases. To overcome this problem, such a statistical–mechanical formulation is proposed here as includes energy supply to the system from outside by extending the symmetry-separated von Neumann equation given in Paper I. This yields a general theory based on the density-matrix formulation on a steady state with energy supply inside and heat extraction outside and consequently with both 〈H0〉t and 〈H1〉t constant. Furthermore, this steady state gives a positive entropy production.The present general formulation of the current yields a compact expression of the time derivative of entropy production, which yields the plausible justification of the principle of minimum entropy production in the steady state even for nonlinear responses.

Disjoining pressure of planar adsorbed films

Frumkin-Derjaguin theory of interfacial phase transitions and in particular the concept of the disjoining pressure of a planar adsorbed film is reviewed and then discussed in terms of statistical mechanical formulations of interfacial phase transitions beyond mean-field.

Hamiltonian formulation for the classical EM radiation-reaction problem: Application to the kinetic theory for relativistic collisionless plasmas

A notorious difficulty in the covariant dynamics of classical charged particles subject to non-local electromagnetic (EM) interactions arising in the EM radiation-reaction (RR) phenomena is due to the definition of the related non-local Lagrangian and Hamiltonian systems. The lack of a standard Lagrangian/Hamiltonian formulation in the customary asymptotic approximation for the RR equation may inhibit the construction of consistent kinetic and fluid theories. In this paper the issue is investigated in the framework of Special Relativity. It is shown that, for finite-size spherically-symmetric classical charged particles, non-perturbative Lagrangian and Hamiltonian formulations in standard form can be obtained, which describe particle dynamics in the presence of the exact EM RR self-force. As a remarkable consequence, based on axiomatic formulation of classical statistical mechanics, the covariant kinetic theory for systems of charged particles subject to the EM RR self-force is formulated in Hamiltonian form. A fundamental feature is that the non-local effects enter the kinetic equation only through the retarded particle 4-position, which permits the construction of the related non-local fluid equations. In particular, the moment equations obtained in this way do not contain higher-order moments, allowing as a consequence the adoption of standard closure conditions. A remarkable aspect of the theory concerns the short delay-time asymptotic expansions. Here it is shown that two possible expansions are permitted. Both can be implemented for the single-particle dynamics as well as for the corresponding kinetic and fluid treatments. In the last case, they are performed a posteriori on the relevant moment equations obtained after integration of the kinetic equation over the velocity space. Comparisons with literature are pointed out.

Fluctuation-Dissipation Relations in Minimal Models for Active Driving

Biological systems exhibit fluctuations driven by active (energy consuming) processes, such as molecular motors in the cytoskeleton and flagella propelling bacteria. It is tempting to treat these fluctuations in the same manner that thermal fluctuations are treated in equilibrium systems. Recent experiments have compared these inherent fluctuations with the response to small external perturbations. From these measurements, an effective temperature may be defined using the fluctuation-dissipation formalism of equilibrium statistical mechanics. In active systems, this effective temperature is frequency dependent (unlike the thermal case), and is higher than the ambient temperature. Another parameter that is useful for characterizing deviations from equilibrium of such fluctuations is the kurtosis of their distribution function; deviations of the kurtosis from its value for a Gaussian distribution indicate that the system may not be mapped onto an equilibrium ensemble.We address these issues theoretically by examining simple model systems of randomly kicked “particles”, with a variable number of kicking “motors”. Our generalized particles and motors represent different objects in different real systems, and we make specific choices when comparing to experimental observations. We exactly calculate the non-equilibrium fluctuation-dissipation relations and resulting effective temperatures. We show that their frequency dependence is model dependent and generally non-monotonous. We investigate the non-Gaussianity of the velocity distribution functions by a combination of numerical simulations and theoretical estimates. We show that the kurtosis has a non-monotonous dependence on the motor activity. We identify situations in which these two indicators of non-equilibrium behavior contradict. In particular, in the presence of multiple motors, the kurtosis of the distribution recovers its Gaussian value, while the effective temperature may still exhibit strong frequency dependence. We compare our results with two recent measurements of oscillations of the red-blood cell membrane, where these two indicators where independently measured.

Study of the Cooperativity of Calcium Binding in Calbindin D9k using 2D Replica-Exchange Umbrella Sampling

Department of Biochemistry and Molecular Biology, The University of Chicago, IL 60637 Argonne National Laboratory, Argonne, IL 60439 Calbindin D9k, a small protein possessing a pair of EF-hands that binds two calcium ions in a cooperative fashion, is chosen as a model system for studying the molecular mechanism of cooperativity.1 Binding is characterized in terms of a potential of mean force (PMF) as a function two variables: the distance r between the ion, and the binding pocket, and the root-mean-square deviation (RMSD) of the conformation of the EF-hand relative to its ion-bound structure. The PMF is calculated using a novel two-dimensional replica-exchange molecular dynamics (MD) umbrella sampling scheme, which is developed and implemented in the program CHARMM to increase the configuration space sampling. Using 2048 replicas on Blue Gene/P, the 2D-PMF/REMD calculation for the binding the second calcium ion converges within 800 ps, with an exchange probability of 30%. In contrast, standard umbrella sampling PMF/MD simulations without replica-exchange did not converge, even after 1.2 ns. The absolute binding free energy of a second ion to a singly occupied calbindin calculated from the 2D-PMF following the statistical mechanical formulation of noncovalent association2 is −9.4 kcal/mol, in excellent agreement with the experimental value3. The 2D-PMF/REMD simulations will be extended to provide important information about the molecular basis of calcium binding cooperativity. Footnotes 1 Marchand S. and Roux B. Protein Struct. Funct. Genet. 1998, 33, 265-284. 2 Woo H. and Roux B., PNAS, 2005, 19, 6825. 3 Linse S. et al. Biochemistry 1991, 30, 154.

Generalized statistical mechanics for superstatistical systems

Mesoscopic systems in a slowly fluctuating environment are often well described by superstatistical models. We develop a generalized statistical mechanics formalism for superstatistical systems, by mapping the superstatistical complex system onto a system of ordinary statistical mechanics with modified energy levels. We also briefly review recent examples of applications of the superstatistics concept for three very different subject areas, namely train delay statistics, turbulent tracer dynamics and cancer survival statistics.

Classical relativistic ideal gas in thermodynamic equilibrium in a uniformly accelerated reference frame

A classical (non-quantum-mechanical) relativistic ideal gas in thermodynamic equilibrium in a uniformly accelerated frame of reference is studied using Gibbs's microcanonical and grand canonical formulations of statistical mechanics. Using these methods explicit expressions for the particle, energy and entropy density distributions are obtained, which are found to be in agreement with the well-known results of the relativistic formulation of Boltzmann's kinetic theory. Explicit expressions for the total entropy, total energy and rest mass of the gas are obtained. The position of the center of mass of the gas in equilibrium is found. The non-relativistic and ultrarelativistic approximations are also considered. The phase space volume of the system is calculated explicitly in the ultrarelativistic approximation.

The brain's recognition dynamics: A statistical mechanical formulation

The brain's recognition dynamics: A statistical mechanical formulation

Polarization of a plasma in massive astrophysical objects

The macroscopic polarization of a plasma in massive astrophysical objects, which is created by gravitational and other inertial forces acting on a mass, is discussed. An additional polarization source, which appears due to the effect of Coulomb nonideality induced by the strong Coulomb interaction between charged particles, is considered. The solution of the problem can be obtained in the form of a final compact expression for a simplified case of a completely equilibrium (isothermal) star without the relativistic and magnetic-field effects. The variation formulation of statistical mechanics leads to the global equilibrium condition as an extremum of the thermodynamic potential of the star as a whole, as well as to two equivalent local forms of such an equilibrium condition in terms of the constancy of the sum of the potentials including the generalized electrochemical potential and the total balance of forces acting on each charged particle. The hypothetical consequences of the polarization of the plasma in the thermodynamics of massive astrophysical objects including the existence of a macroscopic charge localized at any interface deep in the star are discussed. The possible mechanisms of the suppression of hydrodynamic instability in massive astrophysical objects owing to the almost complete compensation (“screening”) of the effect of the gravitational field acting on each heavy charge particle by the electrostatic polarization field are analyzed.

Quantum path integral simulation of isotope effects in the melting temperature of ice Ih

The isotope effect in the melting temperature of ice Ih has been studied by free energy calculations within the path integral formulation of statistical mechanics. Free energy differences between isotopes are related to the dependence of their kinetic energy on the isotope mass. The water simulations were performed by using the q-TIP4P/F model, a point charge empirical potential that includes molecular flexibility and anharmonicity in the OH stretch of the water molecule. The reported melting temperature at ambient pressure of this model (T=251 K) increases by 6.5±0.5 and 8.2±0.5 K upon isotopic substitution of hydrogen by deuterium and tritium, respectively. These temperature shifts are larger than the experimental ones (3.8 and 4.5 K, respectively). In the classical limit, the melting temperature is nearly the same as that for tritiated ice. This unexpected behavior is rationalized by the coupling between intermolecular interactions and molecular flexibility. This coupling makes the kinetic energy of the OH stretching modes larger in the liquid than in the solid phase. However, the opposite behavior is found for intramolecular modes, which display larger kinetic energy in ice than in liquid water.

Ab initio statistical mechanics of surface adsorption and desorption. II. Nuclear quantum effects

We show how the path-integral formulation of quantum statistical mechanics can be used to construct practical ab initio techniques for computing the chemical potential of molecules adsorbed on surfaces, with full inclusion of quantum nuclear effects. The techniques we describe are based on the computation of the potential of mean force on a chosen molecule and generalize the techniques developed recently for classical nuclei. We present practical calculations based on density functional theory with a generalized-gradient exchange-correlation functional for the case of H(2)O on the MgO (001) surface at low coverage. We note that the very high vibrational frequencies of the H(2)O molecule would normally require very large numbers of time slices (beads) in path-integral calculations, but we show that this requirement can be dramatically reduced by employing the idea of thermodynamic integration with respect to the number of beads. The validity and correctness of our path-integral calculations on the H(2)O/MgO(001) system are demonstrated by supporting calculations on a set of simple model systems for which quantum contributions to the free energy are known exactly from analytic arguments.

Maximum entropy generation and [formula omitted]-exponential model

An increasing number of natural phenomena appear to deviate from standard statistical distributions. It has kindled interest in alternative formulation of statistical mechanics which should preserve most of the mathematical structures of the Boltzmann–Gibbs theory, while reproducing the phenomenology of the anomalous systems. The κ -theory has obtained a lot of results in the analysis of natural phenomena, but it was not extended to irreversible open systems. A modified κ -theory is proposed for its use in irreversible open systems.

Chemical potential of a Lennard Jones fluid

The aim of this paper is to present results of analytical calculation of chemical potential of a Lennard Jones (LJ) fluid performed in two ways: by using the thermodynamical formalism and the formalism of statistical mechanics. The integration range is divided into two regions. In the small distance region, which is r ? ? in the usual notation, the integration range had to be cut off in order to avoid the occurrence of divergences. In the large distance region, the calculation is technically simpler. The calculation reported here will be useful in all kinds of studies concerning phase equilibrium in a LJ fluid. Interesting kinds of such systems are the giant planets and the icy satellites in various planetary systems, but also the (so far) hypothetical quark stars.

Constrained Markovian Dynamics of Random Graphs

We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process, in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the `mobility' (the number of allowed moves for any given graph). As an application of the general theory we analyze the properties of degree-preserving Markov chains based on elementary edge switchings. We give an exact yet simple formula for the mobility in terms of the graph's adjacency matrix and its spectrum. This formula allows us to define acceptance probabilities for edge switchings, such that the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process). As a corollary we also derive a condition in terms of simple degree statistics, sufficient to guarantee that, in the limit where the number of nodes diverges, even for state-independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform. We test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology.

Reality of the Wigner Functions and Quantization

Here we use the extended phase space formulation of quantum statistical mechanics proposed in an earlier work to define an extended lagrangian for Wigner's functions (WFs). The extended action defined by this lagrangian is a function of ordinary phase space variables. The reality condition of WFs is employed to quantize the extended action. The energy quantization is obtained as a direct consequence of the quantized action. The technique is applied to find the energy states of harmonic oscillator, particle in the box, and hydrogen atom as the illustrative examples.

Generalized molecular chaos hypothesis and theHtheorem: Problem of constraints and amendment of nonextensive statistical mechanics

Quite unexpectedly, kinetic theory is found to specify the correct definition of average value to be employed in nonextensive statistical mechanics. It is shown that the normal average is consistent with the generalized Stosszahlansatz (i.e., molecular chaos hypothesis) and the associated H theorem, whereas the q average widely used in the relevant literature is not. In the course of the analysis, the distributions with finite cutoff factors are rigorously treated. Accordingly, the formulation of nonextensive statistical mechanics is amended based on the normal average. In addition, the Shore-Johnson theorem, which supports the use of the q average, is carefully re-examined and it is found that one of the axioms may not be appropriate for systems to be treated within the framework of nonextensive statistical mechanics.