Optimal Gaussian approximation in the fluctuating field theory

We consider the problem of calculating the partition function given by the functional integral over an external field that fluctuates in space and in “time” $\tau\in[0,1/T]$ ($T$ is temperature). A method is presented for calculating such integrals with the help of the Gaussian approximation that takes into account dynamics and non-locality of the fluctuations. The method is based on the free energy minimum principle.