Equation of state of a classical anharmonic oscillator

Static properties of a particle moving in an anharmonic potential in equilibrium with a temperature bath and an external field are discussed in the framework of classical statistical mechanics. This model system represents the basic unit in current theories of structural phase transitions. Hierarchies of equations for the correlations (cumulants) and irreducible vertices are derived from the equilibrium condition. Approximate solutions are obtained from the hierarchies by truncation. Alternatively, one can write the equilibrium condition as differential equation which may be solved exactly, if appropriate initial conditions are known. Both methods have been worked out for a single- and a double-well potential. By truncation of the hierarchies one obtains as quantitatively correct result only a low-temperature expansion for the single-well potential.