For an oscillator in a viscous medium under a gravitational filed and an external field, and iterative procedure of giving the stationary solution of the Fokker-Planck equation is presented, in which iterations give successive terms of the power series in the inverse powers of the viscosity η....

The solution of the Fokker-Planck equation with spatial coordinate-dependent moments is given in the form of the path integral, involving the conditional probability expressed in terms of the moments at the “postpoint”. It is easily seen that it satisfies the Fokker-Planck equation.

The integrated probability distribution functions of the effective field νi(h) on the site i satisfy a recurrence formula for the random Ising model on the linear chain or on the Cayley tree. Assuming that the values of the exchange integral are bounded, we prove for the system on the linear...

The solution of the Fokker-Planck equation, given in the form of the path integral, is estimated by the contribution from paths around the most probable path. A numerical procedure for this estimation is given and applied to the regression of an anharmonic oscillator under a random force, from...

We study the statistical mechanics of spin-one Ising models on the Bethe lattice assuming that the spins interact by dipolar and quadrupolar interactions. An exact calculation of the properties of the system is performed on the basis of the general formulation of Morita. An exact expression for...

The pair approximation of the cluster variation method is applied to the hole theory of classical fluids and the melting transition of the hard-sphere system is discussed. The exact free volume of the hard-sphere molecule is used to make up an effective ‘interaction’ assigned to a cluster...

It is shown that the thermodynamic quantities and spin correlation functions of the classical Heisenberg model on a linear chain are expressed in terms of the eigenvalue with the smallest absolute value and the corresponding eigenfunction of a homogeneous linear integral equation, where the...