Statistical mechanics lattice

In order to obtain a connection with statistical mechanics lattice systems, we discretize the position space. The position in time slice l is given by... 

If the classical pseudosystem Hamiltonian were independent of At, the classical pseudosystem would behave as a statistical mechanics lattice system with inverse temperature Ax/h. 

The analogy of the time-evolution operator in quantum mechanics on the one hand, and the transfer matrix and the Markov matrix in statistical mechanics on the other, allows the two fields to share numerous techniques. Specifically, a transfer matrix G of a statistical mechanical lattice system in d dimensions often can be interpreted as the evolution operator in discrete, imaginary time t of a quantum mechanical analog in d — 1 dimensions. That is, G exp(—tJf), where is the Hamiltonian of a system in d — 1 dimensions, the quantum mechanical analog of the statistical mechanical system. From this point of view, the computation of the partition function and of the ground-state energy are essentially the same problems finding... 

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

Density functional theory from statistical mechanics is a means to describe the thermodynamics of the solid phase with information about the fluid [17-19]. In density functional theory, one makes an ansatz about the structure of the solid, usually describing the particle positions by Gaussian distributions around their lattice sites. The free... 

Our discussion shows that the Ising model, lattice gas and binary alloy are related and present one and the same statistical mechanical problem. The solution to one provides, by means of the transcription tables, the solution to the others. Flistorically, however, they were developed independently before the analogy between the models was recognized. 

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

Chesnut D A and Salsburg Z W 1963 Monte Carlo procedure for statistical mechanical calculation in a grand canonical ensemble of lattice systems J. Chem. Phys. 38 2861-75... 

In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

We thus have that the time evolution of the one-dimensional PCA system is equivalent to the equilibrium statistical mechanics of a spin model on a triangular lattice ([domany84], [geor89]). ... 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

In the following we will review recent applications of the lattice-gas model to liquid-liquid interfaces. We will start by presenting the basics of the model and various ways of treating its statistical mechanics. Then we will present model calculations for interfacial properties and for electron- and ion-transfer reactions. It is one of the virtues of the lattice-gas model that it is sufficiently flexible to serve as a framework for practically all processes at these interfaces. 

In various forms, lattice-gas models permeate statistical mechanics. Consider a lattice in which each site has two states. If we interpret the states as full or empty , we have a lattice-gas model, and an obvious model for an intercalation compound. If the states are spin up and spin down , we have an Ising model for a magnetic system if the states are Atom A and Atom B , we have a model for a binary alloy. Many different approximation techniques have been derived for such models, and many lattices and interactions have been considered. 

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

Gibbs and DiMarzio [47] (GD) first developed a systematic statistical mechanical theory of glass formation in polymer fluids, based on experimental observations and on lattice model calculations by Meyer, Flory, Huggins, and... 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Recent developments in the theory of polymer solutions have been reviewed by Berry and Casassa (32), and by Casassa (71). Casassa, who has contributed very largely to these developments, has adopted a statistical mechanical approach using molecular distribution functions, as first outlined by Zimm (72), rather than using a lattice model like that used by Flory, Huggins, and many later workers. 

From the point of view of statistical mechanics there are many problems, such as strongly anharmonic lattices, to which the theory can be applied.14 It appears as a natural generalization of Landau s theory of quasi-particles in the case when dissipation can no longer be neglected. The most interesting feature is that equilibrium and nonequilibrium properties appear linked. The very definition of the strongly coupled anharmonic phonons depends on their lifetime. 

For sufficiently short chains it is possible to calculate C , u , and all other features of interest exactly. Such enumerations were initiated independently because of their application to the statistical mechanics of interacting systems on crystal lattices,10 and a variety of analytical and computational methods (including the use of digital computers) has been employed to extend the enumerations to as large a value of n as practicable. These exact results are then used to conjecture the pattern of asymptotic behavior,... 

The quantities K0 and Kt thus define the solution. As indicated in Appendix A, the result, Eqs. (5)-(9), is identical with the familiar statistical mechanical solution for the case of nearest-neighbor interactions, summarized for example, by Schwarz.2 We note the ease with which the results have been obtained here. The procedure could be extended to other cases, for example, a copolymer (i.e., a linear lattice with two types of sites) distributed in a prescribed manner and undergoing a transition to two other types of sites. For the finite chain, however, the use of nearest-neighbor conditional probabilities and detailed balancing will not yield the complete solution.3... 

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