Configuration integrals technique

In Chapter 2, we saw that the configuration integral is the key quantity to be calculated if one seeks to compute thermal properties of classical (confined) fluids. However, it is immediately apparent that this is a formidable task because it reejuires a calculation of Z, which turns out to involve a 3N-dimensional integration of a horrendously complex integrand, namely the Boltzmann factor exp [-C7 (r ) /k T] [ see Eq. (2.112)]. To evaluate Z we either need additional simplifjfing assumptions (such as, for example, mean-field approximations to be introduced in Chapter 4) or numerical approaches [such as, for instance, Monte Carlo computer simulations (see Chapters 5 and 6), or integral-equation techniques (see Chapter 7)]. 

The path integral technique was first proposed by Feynmann (Feynmann Hibbs, 1965). The purpose of this technique was to deal with questions in quantum mechanics. It has been applied to the study of the statistical mechanics of polymer systems (Kreed, 1972 Doi Edwards, 1986) and liquid crystalline polymers as well (Jahnig, 1981 Warner et al, 1985 Wang Warner, 1986). The path integrals relate the configurations of a polymer chain to the paths of a particle when the particle is undergoing Brownian or diffusive motion. 

Thermodynamic perturbation as well as multiconfiguration thermodynamic integration are based on the generation of ensembles of configurations. Both techniques rely on the implementation of exact equations, and there is, a priori, no reason to expect one to provide better results than the other. The difference in results from the two techniques for the same mutation originates in the difficulty of adequate sampling. 

The sensitivity of deuteron NMR to the molecular orientational order and to director field configurations turned out to be extremely useful in studies of liquid crystals confined into snbmicrometer pores. Moreover, the large surface-to-volume ratio of these composite systems render the interfacial and surface phenomena, induced by the liquid crystal-surface interactions, accessible even to an essentially integrative technique like NMR. Since the discovery of polymer dispersed liquid crystals (PDLCs) in 1986 [4], NMR of selectively deuterated liquid crystals was used to discriminate unambiguously among various director structures in cavities, resulting from an interplay between elastic forces, morphology and size of the cavity, and surface interactions. These structures include the escaped-radial, planar axial, planar-polar, and... 

In order to introduce the notation and some of the necessary concepts, as well as to motivate the introduction of the functional integral techniques, first some exact results from the configurational statistics of individual polymer chains are introduced. Functional integral techniques are then applied to these simpler problems before discussing the more difficult problems of polymer excluded volume and the description of polymers in bulk. 

The structure of a MEMS device typically involves a part that has been micromachined from a wafer integrated with other components to result in a three-dimensional configuration. A technique for integration of components with basic planar geometry is wafer bonding, usually achieved by pretreatment of the wafer to assure planarity and cleanliness, followed by a high temperature anneal with the surfaces to be bonded in firm contact. 

Given a reasonable starting configuration (Xi,k)°, integration of the above differential equation gives the trajectories of all atoms in the molecule. Of course the integration cannot be carried out analytically, but numerical integration techniques are cheaply and readily available [2]. 

Batch Distillation Chapter 4 is devoted to batch distillation. This is one of the most important and one of the most studied unit operations in batch industries. Separation is based on vapor-liquid equilibria. There are a number of configurations possible in conventional batch colmnn, namely, the constant reflux mode, the variable reflux mode, and the optimal reflux mode. There are a number of new configurations that have emerged in the literature for batch distillation. This chapter describes aU these operating modes and configurations. Various levels of models are available for different analysis. Different numerical integration techniques are needed to solve equations of these different models. Optimization and optimal control are well studied for this unit operation. 

It is difficult to analyze batch distillation without using computers due to the two reasons stated before (a) the process is time varying, and one has to resort to complex numerical integration techniques and different simulation models for obtaining the transients, and (b) this ever-changing process also provides flexibility in operating and configuring the column in numerous ways. Based on the current state of the... 

It is expected that in the limit of large n, (U will approach (U), that U will approach U, and that both E and Cy will converge to their correct values. But what are the uncertainties in the calculated values of E and Cy Because the Metropolis method is intrinsically based on the sampling of configurations from a probability distribution function, appropriate statistical error analysis methods can be applied. This fact alone is an improvement on most other numerical integration techniques, which typically lack such strict error bounds. 

Finite difference techniques are used to generate molecular dynamics trajectories with continuous potential models, which we will assume to be pairwise additive. The essential idea is that the integration is broken down into many small stages, each separated in time by a fixed time 6t. The total force on each particle in the configuration at a time t is calculated as the vector sum of its interactions with other particles. From the force we can determine the accelerations of the particles, which are then combined with the positions and velocities at a time t to calculate the positions and velocities at a time t + 6t. The force is assumed to be constant during the time step. The forces on the particles in their new positions are then determined, leading to new positions and velocities at time t - - 2St, and so on. 

The problem is that for diffusive systems the multidimensional configuration space is so vast that it can never be integrated by simulation techniques. This is immediately clear from the occurrence of N in Z. The number of integrand evaluations should vastly exceed N , which for N 1000 and one evaluation per picosecond on the futuristic ultrasupercomputer requires vastly longer then the age of the universe. 

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