Thermodynamics, continued statistical

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle energy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soHdity of matter is an effect of scale. 

The thermodynamic and statistical mechanical analysis of first-order and/or continuous phase transitions demonstrated clearly that models based on the MEA are often inadequate. Lattice gas models (LGM), and approximations based on the quasiseries expansion techniques (e.g. the low-temperature series expansion (SE)) are more appropriate. MG simulations have also been developed as a powerful approach to unravel the role of lateral interactions in various phase formation processes. 

It is not the purpose of chemistry, but rather of statistical thermodynamics, to formulate a theory of the structure of water. Such a theory should be able to calculate the properties of water, especially with regard to their dependence on temperature. So far, no theory has been formulated whose equations do not contain adjustable parameters (up to eight in some theories). These include continuum and mixture theories. The continuum theory is based on the concept of a continuous change of the parameters of the water molecule with temperature. Recently, however, theories based on a model of a mixture have become more popular. It is assumed that liquid water is a mixture of structurally different species with various densities. With increasing temperature, there is a decrease in the number of low-density species, compensated by the usual thermal expansion of liquids, leading to the formation of the well-known maximum on the temperature dependence of the density of water (0.999973 g cm-3 at 3.98°C). 

In the examples described above the resulting probability distributions were discontinuous functions. However, it is not difficult to imagine cases in which the distributions become continuous in the limit of an infinite - or at least a very large - number of trials. Sucb is the case in the application of statistical arguments to problems in thermodynamics, as outlined in Section 10.5. 

The density of states is the central function in statistical thermodynamics, and provides the key link between the microscopic states of a system and its macroscopic, observable properties. In systems with continuous degrees of freedom, the correct treatment of this function is not as straightforward as in lattice systems - we, therefore, present a brief discussion of its subtleties later. The section closes with a short description of the microcanonical MC simulation method, which demonstrates the properties of continuum density of states functions. 

Recently there has emerged the beginning of a direct, operational link between quantum chemistry and statistical thermodynamic. The link is obtained by the ability to write E = V Vij—namely, to write the output of quantum-mechanical computations as the standard input for statistical computations, It seems very important that an operational link be found in order to connect the discrete description of matter (X-ray, nmr, quantum theory) with the continuous description of matter (boundary conditions, diffusion). The link, be it a transformation (probably not unitary) or other technique, should be such that the nonequilibrium concepts, the dissipative structure concepts, can be used not only as a language for everyday biologist, but also as a tool of quantitation value, with a direct, quantitative and operational link to the discrete description of matter. 

The most important assumptions for the applicability of thermodynamical statistics is the independence of the particles from one another and the absence of interchange effects between them. Boltzmann — as well as Bose- and Fermi-statistics consider individual particles without interaction. In the gaseous state, photons, electrons as well as molecules coexist. In applying these theories to condensed phases, the individual particle is to be considered, according to Schrodingerls, either in a continuous medium otherwise the interaction must be taken into account. 

The use of thermodynamically averaged solvent distributions replaces the discrete description with a continuum distribution (expressed as a distribution function). The discrete description of the system, introduced at the start of the procedure, is thus replaced in the final stage by a continuous distribution of statistical nature, from which the solvation energy may be computed. Molecular aspects of the solvation may be recovered at a further stage, especially for the calculation of properties, but a new, less extensive, average should again be applied. 

A number of statistical thermodynamic theories for the domain formation in block and graft copolymers have been formulated on the basis of this idea. The pioneering work in this area was done by Meier (43). In his original work, however, he assumed that the boundary between the two phases is sharp. Leary and Williams (43,44) were the first to recognize that the interphase must be diffuse and has finite thickness. Kawai and co-workers (31) treated the problem from the point of view of micelle formation. As the solvent evaporates from a block copolymer solution, a critical micelle concentration is reached. At this point, the domains are formed and are assumed to undergo no further change with continued solvent evaporation. Minimum free energies for an AB-type block copolymer were computed this way. 

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