Fluctuations in the phase-field model of solidification

We develop two analytical solutions for thermodynamic fluctuations that are present in the phase-field model of solidification. One solution deals with fluctuations in an isothermal single phase system. The other deals with fluctuations in a two-phase isothermal system having a planar diffuse interface. Explicit formulae are obtained in one, two and three dimensions. In the case of two-phase system, fluctuations of the phase field are somewhat suppressed in the interface region. These solutions provide targets for testing numerical solutions.