Thermodynamics of Small Systems

It is shown in this paper that the differential equations of macroscopic thermodynamics can be generalized in such a way that they apply as well to small (i.e., nonmacroscopic) systems. Conventional thermodynamic relations then follow from the present treatment as a limiting case (large system). As with macroscopic thermodynamics, there are two main classes of applications: (1) as an aid in analyzing, classifying, and correlating equilibrium experimental data on ``small systems'' such as (non-interacting) colloid particles, liquid droplets, crystallites, macromolecules, polymers, polyelectrolytes, nucleic acids, proteins, etc.; and (2) to verify, stimulate, and provide a framework for statistical mechanical analysis of models of finite (i.e., ``small'') systems. A well-known experimental and theoretical example (in which there are sizable effects of chain length) is the helix-coil transition in synthetic polypeptides and polynucleotides. Unlike macroscopic thermodynamics, thermodynamic functions are different for different environments (open, closed, isothermal, siobaric, etc.). Although it is possible to derive a single set of thermodynamic equations applicable to all environments, it proves useful to give a separate analysis for each environment. Several cases are discussed, and a few simple statistical mechanical models are used for purposes of illustration. The partition function for a ``completely open'' small system can be used without any special technique such as is required when this partition function is applied to a macroscopic system. Solvent effects are discussed and details are given in one case. The present method provides an invariant treatment of the spherical interface of a drop or bubble, independent of any choice of dividing surface. Usually, only mean values of fluctuating extensive variables appear in thermodynamic equations. This is justified in macroscopic thermodynamics because fluctuations are generally unimportant. The situation is different for small systems and we derive, in one case, a hierarchy of thermodynamic equations involving higher moments of the probability distribution of fluctuating extensive properties.