Statistical theories mean-field theory

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

For general rules, a first-order statistical approximation for limiting densities Pi t —> oo) can be obtained by a method akin to the mean-field theory in statistical mechanics (more sophisticated approaches will be introduced in chapter 4). 

The LST is a finitely parameterized model of the action of a given CA rule, >, on probability measures on the space of configurations on an arbitrary lattice. In a very simple manner - which may be thought of as a generalization of the simple mean field theory (MFT) introduced in section 3.1.3. - the LST provides a sequence of approximations of the statistical features of evolving CA patterns. 

In what follows, we use simple mean-field theories to predict polymer phase diagrams and then use numerical simulations to study the kinetics of polymer crystallization behaviors and the morphologies of the resulting polymer crystals. More specifically, in the molecular driving forces for the crystallization of statistical copolymers, the distinction of comonomer sequences from monomer sequences can be represented by the absence (presence) of parallel attractions. We also devote considerable attention to the study of the free-energy landscape of single-chain homopolymer crystallites. For readers interested in the computational techniques that we used, we provide a detailed description in the Appendix. ... 

The only problem necessary for developing the condensation theory is to add to the above-mentioned equation of the state the equation defining the function x(r)- Unfortunately, it turns out that the exact equation for the joint correlation function, derived by means of basic equations of statistical physics, contains f/iree-particle correlation function x 3), which relates the correlations of the density fluctuations in three points of the reaction volume. The equation for this three-particle correlations contains four-particle correlation functions and so on, and so on [9], This situation is quite understandable, since the use of the joint correlation functions only for description of the fluctuation spectrum of a system is obviously not complete. At the same time, it is quite natural to take into account the density fluctuations in some approximate way, e.g., treating correlation functions in a spirit of the mean-field theory (i.e., assuming, in particular, that three-particle correlations could be expanded in two-particle ones). 

Helfand (25,26,27,28,29) has formulated a statistical thermodynamic model of the microphases similar to that of Meier. This treatment, however, requires no adjustable parameters. Using the so-called mean-field-theory approach, the necessary statistics of the molecules are embodied in the solutions of modified diffusion equations. The constraint at the boundary was achieved by a narrow interface approximation which is accomplished mathematically by applying reflection boundary conditions. 

Equation (40) may help to convert the examples presented in Figs. 7-9 to other cases of physical interest. Equation (40) makes it clear once more that the present mean field theory is too crude to describe the interplay between the configurational statistics of flexible polymers and surface enrichment in thin films ... 

Like all mean-field theories, SCF theories replace the detailed, configuration-dependent interaction potentials with a mean potential averaged over the distribution of molecular configurations. Unlike other mean-field theories, SCF theory explicitly calculates the mean field by accounting for the polymer chain statistics. This field, in turn, controls the distribution of polymer configurations Hence the term self-consistent. ... 

Hie pioneering work in this area was carried out by Seaton et al. (1989), who adapted a statistical mechanical approach originally known as mean field theory (Ball and Evans, 1989). At the time of their early work (Jessop et al., 1991) mean field theory was already known to become less accurate as the pore size was reduced, but even so it was claimed to offer a more realistic way of determining the pore size distribution than the classical methods based on the Kelvin equation. 

The variation of 5(7) near the N-I phase transition will be measured in this experiment and will be compared with the behavior predicted by Landau theory, " " which is a variant of the mean-field theory first introduced for magnetic order-disorder systems. In this theory, local variations in the environment of each molecule are ignored and interactions with neighbors are represented by an average. This type of theory for order-disorder phase transitions is a very useful approximate treatment that retains the essential features of the transition behavior. Its simplicity arises from the suppression of many complex details that make the statistical mechanical solution of 3-D order-disorder problems impossible to solve exactly. 

The mean-field theory does not work in good solvent because excluded volume interactions strongly affect chain statistics and reduce the probability of inter-chain contacts. In 0-solvents, the chain statistics and the probability of the inter-chain contacts-are almost unaffected by interac-tions and well approximated by the mean-field theory. 

However, one-dimensional confined fluids with purely repulsive interactions can be expected to be only of limited usefulness, especially if one is interested in phase transitions that cannot occur in any one-dimensional system. In treating confined fluids in such a broader context, a key theoretical tool is the one usually referred to as mean-field theory. This powerful theory, by which the key problem of statistical thermodynamics, namely the computation of a partition function, becomes tractable, is introduced in Chapter 4 where we focus primarily on lattice models of confined pure fluids and their binary mixtures. In this chapter the emphasis is on features rendering confined fluids unique among other fluidic systems. One example in this context is the solid-like response of a confined fluid to an applied shear strain despite the absence of any solid-like structure of the fluid phase. 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

The book by Hamley [65] is a good general resource for self-consistent mean field theory. This formalism is based on the assumptions that (a) every chain in the system obeys Gaussian statistics, (b) the fluid is incompressible, and (c) the interactions between different structural units are local so that they depend only on the chemical nature but not on the positions of the units along their respective chains. As a result, the equations describe an ensemble of ideal chains in an external field which, in turn, is determined self-consistently from the structural unit probability distributions. As illustrated by Matsen and Schick [66], solving the exact equations requires a significant amount of computational effort to determine the equilibrium... 

M.-H. Hao and H. A. Scheraga, /. Chem. Phys., 102, 1334 (1995). Statistical Thermodynamics of Protein Folding Comparison of Mean Field Theory with Monte Carlo Simulation. 

One of the first attempts to examine the consequences of an electric-field-induced cooperative response was by Hill/ This effort was the basis for the later work of Blumenthal, Changeux, and Lefever and subsequently the work of Hill and Chen/ Most of these efforts, however, were directed toward understanding electric-field-induced excitability in those classes of membranes commonly referred to as excitable membranes, for example as in nerve and muscle. They were not attempts to model the interaction of external time-varying electric fields with biological membranes. Most all of these formulations were based on some type of mean-field theory and the use of lattice statistics. More recently Grodsky " and Denner and Kaiser performed somewhat analogous calculations with reference to a specific dipole model of an excitable membrane. Both analyses used a type of mean field theory to generate the thermodynamic expressions used to describe the behavior of the systems. 

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