Specific heat, susceptibility and frequency of an anharmonic oscillator: test the variational method

The quality of approximation of the Variational Method in Statistical Mechanics is tested in the case of a classical anharmonic oscillator with a potential of the x2 sup(n)-type and also of the (x2)+(x4)-type. Comparison between the variational and exact results for the specific heat, the electric susceptibility and the mean oscillation frequence, shows that the Variational Method leads to surprisingly good results in the whole domain of temperatures (from T=0 to T → infinite). In the particular case of the specific heat, this method leads to the exact result for a potential of the x2 sup(n)-type, and gives, for a potential of the (x2)+(x4)-type, the exact or almost exact asymptotic behaviours (T → 0 and T → infinite). (Author)