Describing phase coexistence in systems with small phases

Clusters of atoms can be studied in molecular beams and by computer simulation; 'liquid drops' provide elementary models for atomic nuclei and for the critical nuclei of nucleation theory. These clusters are often described in thermodynamic terms, but the behaviour of small clusters near a phase boundary is qualitatively different from the behaviour at a first order phase transition in idealized thermodynamics. In the idealized case the density and entropy show mathematically sharp discontinuities when the phase boundary is crossed. In large, but finite, systems, the phase boundaries become regions of state space wherein these properties vary rapidly but continuously. In small clusters with a large surface/volume ratio, however, the positive interfacial free energy makes it unlikely, even in states on phase boundaries, that a cluster will have a heterogeneous structure. What is actually seen in these states is a structure that fluctuates in time between homogeneous structures characteristic of the two sides of the phase boundary. That is, structural fluctuations are observed. Thermodynamics only predicts average properties; statistical mechanics is required to understand these fluctuations. Failure to distinguish thermodynamic properties and characterizations of fluctuations, particularly in the context of first order phase transitions, has led to suggestions that the classical rules for thermodynamic stability are violated in small systems and that classical thermodynamics provides an inconsistent description of these systems. Much of the confusion stems from taking statistical mechanical identifications of thermodynamic properties, explicitly developed for large systems, and applying them uncritically to small systems. There are no inconsistencies if thermodynamic properties are correctly identified and the distinction between thermodynamic properties and fluctuations is made clear.