I.C Phase Transitions

The most spectacular consequence of interactions among particles is the appearance of new phases of matter whose collective behavior bears little resemblance to that of a few particles. How do the particles then transform from one macroscopic state to a completely different one. From a formal perspective, all macroscopic properties can be deduced from the free energy or the partition function. Since phase transitions typically involve dramatic changes in various response functions they must correspond to singularities in the free energy. The canonical partition function for a finite collection of particles is always an analytical function. Hence phase transitions, and their associated non–analyticities, are only obtained for infinitely many particles, i.e. in the thermodynamic limit, N → ∞. The study of phase transitions is thus related to finding the origin of various singularities in the free energy and characterizing them. The classical example of a phase transition is the condensation of a gas into a liquid. Some important features of the liquid–gas condensation transition are: (1) In the temperature/pressure plane, (T, P ), the phase transition occurs along a line that terminates at a critical point (Tc, Pc). (2) In the volume/pressure plane, (P, v ≡ V/N), the transition appears as a coexistence interval, corresponding to a mixture of gas and liquids of densities ρg = 1/vg, and ρl = 1/vl, at temperatures T < Tc. (3) Due to the termination of the coexistence line, it is possible to go from the gas phase to the liquid phase continuously (without a phase transition) by going around the critical point. Thus there are no fundamental differences between liquid and gas phases. From a mathematical perspective, the free energy of the system is an analytical func­ tion in the (P, T ) plane, except for some form of branch cut along the phase boundary. Observations in the vicinity of the critical point further indicate that: (4) The difference between the densities of coexisting liquid and gas phases vanishes on approaching Tc, i.e. ρliquid ρgas, as T Tc − . → → (5) The pressure versus volume isotherms become progressively more flat on approaching TC from the high temperature side. This implies that the isothermal compressibility, κT = − ∂V/∂P |T /V , diverges as T → Tc + . (6) The fluid appears “milky” close to criticality. This phenomenon, known as critical opalescence, suggests collective fluctuations in the gas at long enough wavelengths to scatter visible light. These fluctuations must necessarily involve many particles, and a coarse graining procedure may thus be appropriate to their description.