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...

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...

Magnetic properties are numerically obtained for the random-bond Ising model of interactions + J (J 0) and - J with probabilities p(p ⩾/ 0.5) and 1 − p, on the Bethe lattice of coordination number 3 in zero and non-zero uniform external magnetic fields. It is confirmed that the phase...

The distribution functions and the free energy are expressed in terms of the effective fields for the regular and random Ising models of an arbitrary spin S on the generalized cactus tree. The same expressions apply to systems on the usual lattice in the “cactus approximation” in the cluster...

The present paper deals with the statistical mechanics of an asymmetrical Ising mounted on the Bethe lattice. An asymmetrical Ising model is defined by assuming the spin variable of the usual Ising Hamiltonian to take up the eigenvalues 1, − λ, where λ is an asymmetry parameter. It is shown...

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.