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 solution of the Fokker-Planck equation with spatial coordinate-dependent and time-dependent moments in many dimensions is given in the form of the path integral.

We calculate the dimension of the set which is complementary to the complete devil's staircase of a family of piecewise linear mappings. We obtain universal values of fractal dimension 12 for one region of the staircase and 0 for the remaining region. The ways by which these values are...

A variational calculation is given of an approximation scheme, which is far easier to apply than the cluster variation method, in the higher approximations. The Curie temperature of the Ising model in the square approximation in the present scheme is obtained as 2.6253 and 4.7611, respectively,...

We give a one-dimensional mapping which is a simple example that the periodic orbits show an arithmetic furcation as a function of a parameter characterizing the mapping. The mapping is a piecewise linear function which consists of three parts, that is, a line with slope 1, a line with slope 0...

It is shown that the thermodynamic properties and the distribution functions of the Ising systems on the Cayley tree are generally obtained in terms of the solution of a recurrence formula, when the interaction is of finite range. For the Bethe lattice where the boundary effects are ignored, the...

The dimer problem on a two-dimensional lattice is reduced to a problem of random walks on the lattice, and then the latter problem is solved by the method which Vdovichenko developed in order to derive an exact expression for the partition function of the Ising model on a two-dimensional...

We study the random Ising model in the pair approximation of the cluster variation method. We show that the distribution function of the effective field is determined either by a reducibility condition of the distribution function of two sites to that of one site or by a stationarity condition...