Phase Transitions and Soft Modes

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Abstract
The stability of an equilibrium phase is described in thermodynamics in terms of certain convexity properties of the free energy. The phase is said to be unstable with respect to an equilibrium phase transition if the isothermal response of the system to a static external field is infinite. It is also known that associated with several second-order and some first-order transitions there are "soft collective modes" (for instance, in ferroelectric, antiferromagnetic, liquidgas, and structural phase transitions). In this paper we show that in systems where the order parameter can be treated as an ergodic variable, and which possess collective excitations, at least one of the collective modes must go soft when the system undergoes a second-order phase change (or more generally, at the stability limit of the system). Implications to discontinuous firstorder transitions are also briefly discussed. Examples of soft modes associated with phase changes are given.