Stratification as a general variance reduction method for Markov chain Monte Carlo.

Realization of sub 1 V polymeric EO modulators through systematic definition of material structure/function relationships

The Buckingham‐Darey flux law, an empirical equation that relates the volumetric flux density of water in an unsaturated soil to the gradient of the total potential, is derived from first principles by using the methods of statistical mechanics. The derivation given, a direct application of well‐known techniques in nonequilibrium statistical mechanics, proceeds through a detailed molecular description of two laboratory experiments for measuring the hydraulic conductivity tensor of a homogeneous unsaturated soil. In the first experiment the steady flow of water is induced by a gradient in the matric potential, while in the second, flow is induced by a gradient in the gravitational potential. In both cases the appropriate form of the Buckingham‐Darcy law is derived on the basis of a linear response approximation, and an expression for the hydraulic conductivity is given in terms of a time integral of the correlation function for the velocities of the water molecules in the soil. The problem of calculating the hydraulic conductivity of a soil thereby is reduced to quadrature and to the task of developing a molecular model of the velocity correlations among the water molecules. A recent successful model of this type is discussed briefly.

We have seen that macroscopic properties in a linear irreversible process are determined by the response function, the relaxation function, the complex admittance or the double-time correlation functions. This chapter briefly describes techniques for calculating these functions. Of course, there are many methods of calculation, each of which has its own merits and demerits and has particular key points to be considered. A simple example for the determination of the response function by using the kinetic theoretical method was given in .KeywordsSpectral FunctionPerturbational ExpansionExternal LineDyson EquationComplex Dielectric FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

<b>Fannes, M.; Verbeure, A.</b> (eds.), Mathematical Methods in Statistical Mechanics. Conference Proceedings, Leuven, 23–24 June 1988. Leuven, Leuven University Press 1989. III, 196 pp., Bef 600.‐. ISBN 90–6186–306‐6 (Leuven Notes in Mathematical and Theoretical Physics. Vol. 1. Ser. A: Mathematical Physics)

The structural and thermomechanical properties of zincblende ZnX (X = S, Se, Te) compounds have been investigated based on the moment method in statistical mechanics. Expressions for the lattice constant, atomic mean-square displacement (MSD), and elastic moduli (Young’s modulus, bulk modulus, and shear modulus) of the zincblende compounds were derived. The results show that the quantum-mechanical zero-point vibrations make the main contribution to the atomic MSDs at low temperature. At high temperature, due to the predominance of anharmonicity, the atomic MSDs increase rapidly with increasing temperature. The Young’s modulus and shear modulus of the zinc chalcogenides ZnX (X = S, Se, Te) were calculated. The data derived from this research can be seen as useful references for future experiments.

The pressure effects on thermo-mechanical properties of B2-type FeAl compound have been investigated based on the moment method in statistical mechanics. We derive analytical expressions of equation-of-state, isothermal bulk modulus and specific heats at constant volume and constant pressure of B2-type iron aluminide intermetallic compound. Numerical calculations for lattice parameter, isothermal bulk modulus, Young’s modulus, shear modulus and specific heat at constant pressure of B2-type Fe-40 at.% Al have been performed up to pressure of 10 GPa. Our research shows that the elastic moduli are linear proportional to pressure and the specific heat at constant pressure diminishes strongly at temperature below 300 K. The present statistical moment method results are compared with available experimental data as well as ab initio calculations when possible to verify the developed theory. This research proposes the potential of the moment method in the investigation of thermo-mechanical properties of materials under pressure.

The macroscopic differential equations of mass and momentum balance for water in a rigid unsaturated soil are derived from first principles by using the methods of statistical mechanics. The derivation begins with the development, at the molecular level, of expressions for the mass and linear momentum densities of water in a soil. The derivatives of these expressions with respect to the time then provide for local balance equations that are ensemble averaged and local volume averaged in order to produce the macroscopic balance equations. The resulting macroscopic equations agree with those derived for mass and linear momentum transport in a soil on the basis of the continuum theory of mixtures. The unified insight into mass transport processes afforded by a statistical mechanical theory as compared to one evolved from a solely macroscopic view is discussed.

The Green function method in statistical mechanics: Bonch-Bruevich and Tyablikov (North-Holland Publishing Co., Amsterdam, 1962. v-251 p. 27.50 fl.)

Molecular and atomic clusters attract considerable interest; this can be explained by their intermediate position between molecules, as such, and macroscopic particle ensembles, as well as their position in the hierarchy of systems described by methods of statistical mechanics. Gibbs statistical mechanics also remains valid for systems with a limited number of particles. For the rigorous derivation of an equilibrium distribution function, two conditions should be fulfilled: the size of a thermostat in equilibrium with a molecular system should be macroscopic and the energy of the interaction between the system and the thermostat should be infinitely small compared to the system internal energy. The thermodynamic limit for the number of particles N ∞ in the system in itself is not required [1]. The mechanism responsible for the formation of the collective behavior of particles, which, in the macroscopic limit, leads to phase transitions, is of fundamental interest. In small molecular systems, liquidcrystal transitions are of particular importance in application, since they occur at rather low temperatures, when molecular clusters, within a relatively wide range of external conditions, are stable with respect to decomposition. Strictly speaking, phase transitions accompanied by discontinuities in functional dependences of thermodynamic potentials are possible only in the limit of an infinite number of particles. Phase transitions represent the most complex problem in modern classical statistical equilibrium thermodynamics. In addition, there exists no theory capable of explaining, in the general form, the origin of phase transitions and linking the mechanism of particle interaction with the position of the boundaries separating different phase states in the phase diagram. According to present-day concepts of general statistical theory [2], phase transitions are related to the zeros of the grand statistical sum:

Methods in Statistical Mechanics

: Progress was made on the statistical mechanics of the Zakharov model which describes the interaction of an electromagnetic wave with a plasma. The model leads to a nonlinear set of partial differential equations which resemble in some ways equations arising in nonlinear field theories which have been studied by statistical mechanical methods. Some very exact results for this model have been obtained in one dimensional and computer simulations are being carried out on it. A new program has been launched involving both theoretical analysis and computer simulations on interacting quantum systems. Keywords: Plasma; Statistical Mechanics; Theory.

Book reviews

Monte Carlo methods in classical statistical mechanics

We review the history of DNA mechanics and its analysis. We evaluate several methods to analyze the structures of superhelical DNA molecules, each predicated on the assumption that DNA can be modeled with reasonable accuracy as an extended, linearly elastic polymer. Three main approaches are considered: mechanical equilibrium methods, which seek to compute minimum energy conformations of topologically constrained molecules; statistical mechanical methods, which seek to compute the Boltzmann distribution of equilibrium conformations that arise in a finite temperature environment; and dynamic methods, which seek to compute deterministic trajectories of the helix axis by solving equations of motion. When these methods include forces of self-contact, which prevent strand passage and preserve the topological constraint, each predicts plectonemically interwound structures. On the other hand, the extent to which these mechanical methods reliably predict energetic and thermodynamic properties of superhelical molecules is limited, in part because of their inability to account explicitly for interactions involving solvent. Monte Carlo methods predict the entropy associated with supercoiling to be negative, in conflict with a body of experimental evidence that finds it is large and positive, as would be the case if superhelical deformations significantly disrupt the ordering of ambient solvent molecules. This suggests that the large-scale conformational properties predicted by elastomechanical models are not the only ones determining the energetics and thermodynamics of supercoiling. Moreover, because all such models that preserve the topological constraint correctly predict plectonemic interwinding, despite these and other limitations, this constraint evidently dominates energetic and thermodynamic factors in determining supercoil geometry. Therefore, agreement between predicted structures and structures obtained experimentally, for example, by electron microscopy, does not in itself provide evidence for the correctness or completeness of any given model of DNA mechanics.

A theory developed by the author to describe macroscopic neocortical interactions demonstrates that empirical values of chemical and electrical parameters of synaptic interactions establish several minima of the path-integral Lagrangian as a function of excitatory and inhibitory columnar firings. The number of possible minima, their time scales of hysteresis and probable reverberations, and their nearest-neighbor columnar interactions are all consistent with well-established empirical rules of human short-term memory. Thus, aspects of conscious experience are derived from neuronal firing patterns, using modern methods of nonlinear nonequilibrium statistical mechanics to develop realistic explicit synaptic interactions.