The theory of phase transitions

The theory of phase transitions

The term phase transition is most commonly used to describe the transitions between the solid, liquid and gaseous states of matter. Gibbs phase rule tells us that if the solid and liquid states of a one-component substaance are in equilibrium only one thermodynamic variable is required to specify that systems thermodynamic state. In other words, if I am told that I have water and its vapour in equilibrium and I am told the pressure I can calculate the temperature. Consequently, if we draw a phase diagram showing what state the system will be in under different temperature and pressure conditions we find that the boundaries between the states are lines. Furthermore, when the system crosses one of these boundaries between different different states the values of thermodynamic variables change discontinuously. For example when a liquid is evaporated the volume of the system changes dramatically. As we have learnt elsewhere the values of all the thermodynamic variables can be calculated by taking suitable derivatives of the thermodynamic potential/logarithm of the partition function. When we cross a phase boundary and observe a discontinuous change in the value of a thermodynamic variable the thermodynamic potential must change in a manner that is not-analytic. In other words, the value of the thermodynamic potential cannot be calculated using the Taylor series: $$ F(x + \delta x ) = F(x) + \frac{\partial F}{\partial x} \delta x + \frac{1}{2!} \frac{\partial^2 F}{\partial x^2} (\delta x)^2 + \dots $$ if in moving from point $x$ to point $x+\delta x$ the system crosses a phase boundary. This observation about phase transitions proved difficult to reconcile with the predictions of statistical mechanics. In fact statistical mechanics predicts that finite sized systems do not undergo phase transitions. Phase transitions are only possible in systems containing infinite numbers of particles. Phase transitions only appear when we consider systems in the so-called thermodynamic limit.

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