Basic Methods of Equilibrium Statistical Mechanics

In principle, the macroscopic (including thermodynamic) properties of matter ultimately derive from the underlying microscopic structures. Because the exact mechanics for a huge number of constituent particles is out of question, one is forced to seek statistical methods. The fundamental idea of statistical mechanics starts from the notion that an observed macroscopic property is the outcome of averaging over many underlying microscopic states. For a micro-canonical ensemble of an isolated system at equilibrium, we show how the entropy is obtained from information on the microstates, or, from the probabilities of finding the microstates. Once the entropy is given, the first order thermodynamic variables are obtained by taking derivatives of it with respect to their conjugate thermodynamic variables (as shown in Chap. 2). We then consider the microstates in canonical and grand ensembles of the system, which can exchange energy and matter with the surrounding kept at a constant temperature. From the probability of each microstate and the primary thermodynamic potentials for the ensembles, all the macroscopic properties are calculated. Statistical mechanics also allows us to obtain the information on the fluctuations of observed properties about the averages, which provides deeper understanding of the structures of matter. The standard ensemble theories of equilibrium statistical mechanics will be outlined in this chapter. In applying such methods to biological systems we face a shift of its old paradigm (of relating the macroscopic properties to the microscopic structures). Unlike ideal and simple interacting systems covered in typical statistical mechanics text books, biological systems are too complex to be explained directly in terms of the small molecules or other atoministic structures. Nevertheless, the structures and properties can be observed on nanoscales, thanks to various single-molecule experimental methods which are now available. Certain nanoscale subunits or even larger units, rather than small molecules, can emerge as the basic constituents and properties. Throughout this chapter we demonstrate the applicability of statistical mechanics for numerous mesoscopic biological models involving these subunits.