Although the microscopic Hamiltonian contains all of the information needed to describe phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct reduced descriptions. Generally a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a useful starting point for theoretical models. The equations of motion of the macrovariables can be derived from the microscopic Hamiltonian, but in practice one often begins with a phenomenological description. The set of macrovariables is chosen to include the order parameter and all other slow variables to which it is coupled. Such slow variables are typically obtained from consideration of the conservation laws and broken symmetries of the system. The remaining degrees of freedom are assumed to vary on a much faster time scale and enter the phenomenological description as random thermal noise. The resulting coupled nonlinear stochastic differential equations for such a chosen “relevant” set of macrovariables are collectively referred to as the Langevin field theory description. In two of the simplest Langevin models, the order parameter ø is the only relevant macrovariable; in model A (introduced in this chapter) it is nonconserved and in model B (described in the next chapter) it is conserved. The labels A, B, etc. have an historical origin from the Langevin models of critical dynamics. The scheme is often referred to as the Hohenberg-Halperin classification scheme (Hohenberg and Halperin, 1977).

Langevin model A

For model A the Langevin description assumes that, on average, the time rate of change of the order parameter is proportional to the thermodynamic force that drives the phase transition.