## Statistical Mechanics of Small Systems

The objective of statistical mechanics is generally to develop predictive tools for computation of properties and local structure of fluids, solids and phase transitions from the knowledge of the nature of molecules comprising the systems as well as intra- and intermolecular interactions. The accuracy of the predictive tools developed through statistical mechanics will depend on two factors. The accuracy of molecular and intermolecular properties and parameters available for the material in mind and the accuracy of the statistical mechanical theory used for such calculations.

Statistical mechanical prediction of the behavior of matter in macroscopic scale, in the thermodynamic limit (V & Nand N/V=pN finite), is well developed and a variety of molecular-based theories and models are available for prediction of the behavior of macroscopic systems. There is also a wealth of data available for thermodynamic and transport properties of matter in macroscopic scale, which can be used for testing and comparison of molecular based theories of matter in macroscopic scale. In the case of nano (small) scale there is little or no such data available and the molecular theories of matter in nanoscale are in their infancy. With the recent advent of tools to observe, study and measure the behavior of matter in nanoscale it is expected that in a near future experimental nanoscale data will become available. In the present section we introduce an analytic statistical mechanical technique which has potential for application in nano systems. In the next chapter, we will introduce the computer simulation techniques, which have been proposed and used for molecular based study of matter in nanoscale.

Statistical mechanics of small systems is not a new subject. Many investigators have studied small systems consisting of one or more macromolecule, droplets, bubbles, clusters, etc. utilizing the techniques of statistical mechanics [4-8,25,26]. Computer simulation approaches, like Monte Carlo and molecular dynamics techniques, have been used extensively for such studies and they will be presented and reviewed in the next few chapters. However, a general analytic formalism for dealing with small systems and developing working equations for thermodynamic properties, without regard to their nature, through statistical mechanics is still lacking.

Recent nanotechnology advances, both bottom-up and top-down approaches, have made it possible to envision complex and advanced systems, processes, reactors, storage tanks, machines and other moving systems which include matter in all possible phases and phase transitions. There is a need to understand and develop analytic predictive models, for example, for the behavior of a matter confined in a fullerene, or flowing in a nanotube at various state conditions of pressures, temperatures and compositions. For such diverse circumstances and for application of such components in the design of nanomachinery, the development of analytic predictive approaches of properties of matter in nanoscale is necessary to build accurate computational techniques to model such nanomachinery.

It is a well-known fact that the behavior of matter confined in nano systems (inside a fullerene, in a nanotube, etc.) is a function of the environmental geometry, size and wall effects that are surrounding it. This has not been the case with macroscopic scale of matter where we have been able to develop universal correlations and equations of state to be applied in all possible applications regardless of the geometry of the confinement systems. For example the property-relations of water (its equation of state), is universally needed for many applications. Such applications include in the use of water as working fluid in thermal to mechanical energy conversion devices, in the use of water as the reagent or solvent in many chemical processes and in oceanographic, meteorological and geothermal studies. However, the same equations of state of water may not be applicable in small systems. Even if one develops such database for a particular nanosystem, the results may not be applicable in other nano system. This demonstrates the need for a methodology for the development of universal analytic techniques of thermodynamic property relations in nanoscale. One of the prime candidates for this endeavor seems to be the newly developed thermodynamics and statistical mechanics of nonextensive systems. In what follows we present the principle of thermodynamics and statistical mechanics of nonextensive systems and discuss its validity for small systems due to their nonextensive nature.

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