Statistical Mechanical Functions Research Articles

Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, and Computational Views of Patterns in One-Dimensional Spin Systems.

We compare and contrast three different, but complementary views of “structure” and “pattern” in spatial processes. For definiteness and analytical clarity, we apply all three approaches to the simplest class of spatial processes: one-dimensional Ising spin systems with finite-range interactions. These noncritical systems are well-suited for this study since the change in structure as a function of system parameters is more subtle than that found in critical systems where, at a phase transition, many observables diverge, thereby making the detection of change in structure obvious. This survey demonstrates that the measures of pattern from information theory and computational mechanics differ from known thermodynamic and statistical mechanical functions. Moreover, they capture important structural features that are otherwise missed. In particular, a type of mutual information called the excess entropy—an information theoretic measure of memory—serves to detect ordered, low entropy density patterns. It is superior in several respects to other functions used to probe structure, such as magnetization and structure factors. -Machines—the main objects of computational mechanics—are seen to be the most direct approach to revealing the (group and semigroup) symmetries possessed by the spatial patterns and to estimating the minimum amount of memory required to reproduce the configuration ensemble, a quantity known as the statistical complexity. Finally, we argue that the information theoretic and computational mechanical analyses of spatial patterns capture the intrinsic computational capabilities embedded in spin systems—how they store, transmit, and manipulate configurational information to produce spatial structure.

A simple way of approximating the canonical partition functions in statistical mechanics

We propose a simple pedagogical way of introducing the Euler–MacLaurin summation formula in an undergraduate course on statistical mechanics. The reason is that the students may feel more comfortable and confident if they are able to deduce the main equations. To this end we put forward two alternative routes: the first one is the simplest and yields the first two terms of the expansion. The second one is somewhat more elaborate and takes into account all the correction terms. We apply both to the calculation of the simplest one-particle canonical partition functions for the translational, vibrational and rotational degrees of freedom. The more elaborate, systematic calculation of the correction terms is suitable for motivating the students to explore the possibility of using available computer algebra software that enable one to avoid long and tedious manipulation of algebraic equations.

統計力学における Green 関数

統計力学における Green 関数

Partition functions in statistical mechanics, symmetric functions, and group representations.

Partition functions for noninteracting particles are known to be symmetric functions. It is shown that powerful group-theoretical techniques can be used not only to derive these relationships, but also to significantly simplify calculation of the partition functions for particles that carry internal quantum numbers. The partition function is shown to be a sum of one or more group characters. The utility of character expansions in calculating the partition functions is explored. Several examples are given to illustrate these techniques.

Subgraph generating functions in chemistry— An example for perfect matchings on honeycomb fragments

The analogous role of subgraph generating functions in statistical mechanics and in quantum-chemical theory is noted. To illustrate the analogy, the example of perfect matchings on hexagonal animals is considered. Isomorphisms to two types of multiple-walk problems are noted to reveal a “hidden order”, with physical relevance. This is further used to accomplished an enumeration of the perfect matchings on a class of hexagonal-animal strips of with and length.

Correlation functions in statistical mechanics and astrophysics.

A general review of distribution functions reveals that the two- and three-point correlation functions familiar to galaxy studies may be identified with the total correlation functions relevant to the study of liquids. In this context a discussion is included of the Peebles-Groth three-point correlation function. A recently derived two-point correlation function in polynomial form is reviewed. Renormalization of this expression is found to bring it into very good agreement with the well-established ${r}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ form (where r is intergalaxy displacement and \ensuremath{\gamma} is a constant).

A variational approach for the free energy

To calculate thermodynamic functions in statistical mechanics the authors propose a new variational approach which has as particular cases the approaches of Bogoliubov, Oguchi and Tsallis-da Silva (1982), but introduces some improvements, mainly in the short-range order effects above the critical temperature. It requires a choice of a trial Hamiltonian that defines an approximate ensemble through which the thermodynamic functions may be calculated. To check the efficiency of their approach they apply it to the ferromagnetic Ising model by using the simplest possible trial Hamiltonian.

Inhomogeneous Coulomb fluids with image interactions between planar surfaces. I

Description of Coulomb particles near a surface is complicated by the effect of the surface on the ion–ion correlations, the electrostatic images (if the surface is a dielectric boundary), and the long range of the correlations in the lateral direction. We formulate the problem of the ion distribution for inhomogeneous Coulomb fluids with an arbitrary core potential and confined between two planar surfaces, with which the particles can interect via any short-ranged potential. All orders of image interactions are included into an effective position-dependent pair potential. The problem is solved by mapping the inhomogeneus three-dimensional system into a homogeneous two-dimensional one. The system is subdivided into M layers, and then shown to be isomorphic to an M-component fluid mixture in two dimensions. The mapping becomes exact as the number of layers (‘‘components’’) M→∞, and is accordingly an excellent approximation for finite, but large M. The correlations and other statistical mechanical functions can now be obtained with any conventional closure scheme. This forms the basis for a numerical method to solve the complete integral equations for the one- and two-particle distribution functions, using, e.g., the hypernetted chain (HNC) closure for the inhomogeneous pair correlations as the sole approximation. The associated numerical procedures are briefly described. The method could also be applied to study various problems for solute particles interacting via other pair potentials.

Modification of the Becker Technique for Obtaining Distribution Functions in Statistical Mechanics

The average values of the occupation numbers of the energy levels of a system in equilibrium are calculated by a modification of the Becker technique, in the Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics. No use is made of the Stirling approximation, or of Lagrange multipliers.

A Functional Calculus Approach to the Ursell-Mayer Functions

A more complete treatment of the Ursell-Mayer functions is given here by way of a functional calculus defined on the algebra of complex-valued functions on the exponential of a set. Through measure-theoretical considerations, we are able to establish relationships between these functions and certain linear operators which relate probability densities and correlation functions in statistical mechanics. Also we point out that these functions provide a way of representing states of an infinite system.

Becker Averaging Technique for Obtaining Distribution Functions in Statistical Mechanics

We discuss a general, basic approach for obtaining the distribution function in introductory statistical mechanics by averaging over the microcanonical ensemble, which is accomplished without the use of Stirling's approximation or Lagrange multipliers.

Many-Body Problems

W E Turner et al. New York: Benjamin 1969 pp vii + 265, price £2 The Institute of Physics and The Physical Society has decided to have republished selected articles, with a common theme, from Reports on Progress in Physics. The present volume on many-body problems is such a venture and contains reviews entitled 'Green functions in statistical mechanics' (Parry and Turner 1964), 'Theory and applications of the density matrix' (ter Haar 1961), 'The equation of state of dense systems' (Rowlinson 1965), 'The theory of irreversible processes' (Chester 1963), 'Quantum theory of many-body systems' (Hugenholtz 1965) and 'The fluctuation-dissipation theorem' (Kubo 1966).

Functional Integrals Representing Distribution Functions in Statistical Mechanics

We show how to obtain formal solutions of the chain of equations for distribution functions in classical statistical mechanics. These solutions are in the form of complex functional integrals. They are not unique, which fact is a fundamental property of the equations, and the different solutions are recognized by different integration paths in the complex function space. The different manners of integration correspond to different phases, of which some can be identified with the possible physical states. The treatment of the integrals in some cases is also discussed. They are closely related to generalizations of the molecular field approach to the problem. It is also shown that the functional integrals can be written as averages over an external field and that essentially the same form is valid in the quantum‐mechanical case.

STUDIES IN IRREVERSIBLE THERMODYNAMICS, V. STATISTICAL THERMODYNAMICS OF A SIMPLE STEADY-STATE MEMBRANE MODEL

We are currently interested in the question of the possible existence of a general thermodynanics of steady-state systems, neither restricted to near-equilibrium nor to local equilibrium. For this taski, it is very helpful to have available exact properties for some very simple models. We consider such a model in this paper. It is hoped that the treatment of this model, and others to be discussed subsequently, will stimulate some interest in the general question mentioned above, even though the moclels may not themselves lead to an answer. Another purpose of the present paper is to point out the relation of this work to that of Jaynes1 and Shannon and Fienberg2 on generalized (nonequilibrium) partition functions in statistical mechanics. A large number of related models, and methods of finding some of their steadystate properties, were discussed in the two previous papers of this series.' 4 The Model.-The system whose steady-state properties we shall investigate here is a membrane between two infinite baths, A and B, both at temperature T. Each bath is itself an equilibrium system, but in general the baths are not in equilibrium with each other. The steady-state system (mrembrane) consists of B independent and equivalent units (or channels) for transport of a single component from bath A to bath B. A unit can exist in four different states (Fig. 1). Figure 2 shows the allowed transitions between states and the assigned transition probabilities or rate constants. Thus, for example, if a unit is in state 0, the probability that it will undergo a transition to state 2 in a time interval dt is aBdt, where aB is a constant. The constants f (desorption) and K (hopping) are intrinsic properties of the system itself, but the a's (adsorption) also involve the baths: aA = IXA and aB =3 XB, where XA = e/A/kT (in bath A), etc. If we choose aA = aB, the steady state is an equilibrium state. But ordinarily we take aA 7? aB. We digress to remark that in a much more general treatment, the unit might be very large, even macroscopic, so that a unit becomes a "system" and a system becomes an "ensemble" (B -* 03). The states referred to above would then be energy eigenstates available to each "system" (as in the Pauli-van Hove master equation)i. The model outlined above is nodel 2 in pa-rt ITT,' with n = 2, except that, for simplicity, we shall here take the partition function for a moleecule aclsorbed on a site as q = 1 (ground state only; eniergy = 0). .Properties of the Mlodel. A stochastic argument leads immediately to

Some comments on the Hodgkin-Huxley equations

One of the important problems of electrophysiology is to provide a suitable model for the axon membrane. The relation of some of the Hodgkin-Huxley empirical equations to certain common functions in statistical mechanics is examined, and it is shown that the rate constants a n and a m can be considered as averaging functions. An illustrative abstract model is proposed, and it is shown that this leads directly to the empirical equations. The treatment suggests possible guidelines for the formulation of a useful physical theory.

THE QUANTUM THEORY OF FERROMAGNETISM

摘要 本文用双时格林函数方法计算了铁磁性金属在整个温度范围内的自发磁化强度和磁各向异性常数K1。计算中考虑了平均每个原子的d电子数非整数的事实和同一原子内d电子之间的交换作用。所得结果较普通Heisenberg模型的有所改进。K1随温度的变化性质可以得到定性的解释。 Abstract By taking account of the non-integer number of d electrons per atom and the intra- atomic interactions of electrons, a quantum theory of ferromagnetism is given. The spontaneous magnetization and the magnetic anisotropic constant K1 in the whole temperature range are calculated by using the method of double-time Green's functions in statistical mechanics. The results obtained are better than that of the simple Heisenberg's model. The properties of the temperature dependence of K1 may be explained qualitatively. 作者及机构信息 邝宇平, 易余萍 Authors and contacts KUANG YU-PING, YI YU-PING 参考文献 [1] 施引文献

Three New Partition Functions in Statistical Mechanics

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