International Journal of Modern Physics B

A review of the classical and quantum thermodynamic properties of the Toda chain is provided, together with a survey of the techniques that have been used to work them out, i.e., the bilateral Laplace transform method for the classical system, the Bethe–Ansatz and the variational path-integral approach for the quantum one. For the classical Toda chain we also recall some of the main dynamical features, i.e. the integrability of the model and the soliton-like solutions of the equations of motion. In the quantum case a comparison between the Bethe–Ansatz and the variational method is made. In particular it is shown as the latter offers the possibility of also evaluating static correlation functions.

This paper reviews soliton theories for classical statistical mechanics of the massive-Thirring-sine-Gordon model. These theories attempt not only to reproduce the classical transfer matrix method result but also to provide thermodynamic information about exact nonlinear excitations of the model. Three theories are discussed. Firstly, the soliton–gas phenomenology which is successful in the Toda lattice is shown to be not even approximately correct for the sine-Gordon model. This breakdown is due to the incorrect counting of the number of degrees of freedom. Secondly, the soliton–phonon theory which reproduces the WKB result is also shown to be not exact. This non-exactness is ascribed to the fact that the nonlinear phonons are not exact elementary excitations in the model. Thirdly, a non-gas phenomenological soliton–breather theory is derived as a classical limit of quantum statistical mechanics of the model. The exactness of the last theory is demonstrated through an accurate numerical analysis of its basic equations.

The asymptotic behavior of random scalar fields with long tails is analyzed. The problem is illustrated by considering the distribution of rare minerals (e.g. gold) on Earth crust. The main assumptions of the model are the following: a) the concentration field is generated by a random number of independent sources obeying nonhomogeneous Poisson statistics; b) the contribution of a source to the concentration field is a random function selected from a certain probability density functional; c) the moments of the contributions of a source have long tails with respect to the distance from the source. An analytical description of the random process described by these assumptions is possible: the generating functional of the concentration field as well as the cumulants of the concentration can be computed exactly. We distinguish two different regimes. If the average distance between two points is large enough, the finiteness of the system determines the asymptotic behavior and the random process is neither Gaussian nor of the Lévy type. This behavior is due to the long tails: they result in a partial overlapping of the contributions of the different sources to the concentration field, the fluctuations are strong and the superior cumulants do not vanish; however, as the system is finite they are also finite. If the average distance between two sources tends to zero, by using an appropriate length scale the system may be assumed to be infinite; the enhancement of fluctuations due to the overlapping is much bigger than in the first case, the cumulants become infinite and the stochastic process is of the Lévy type. The new approach can be applied to a large class of natural phenomena such as the shape of the fitness landscape in evolutionary biology, the stochastic gravitational fluctuations in astrophysics or the behavior of Langevin force fields in random media.

The first order quantum correction is obtained for the characteristics of spontaneous radiation as a functional of a classical trajectory by means of quasi-classical trajectory–coherent states method for relativistic spinless particle in an arbitrary external electro-magnetic field. It is shown that the power of boson radiation coincides up to O (ħ²), ħ → 0 with the power of electron radiation averaged over spin states calculated by the TC-states of the Dirac equation. Certain specific examples are considered on the base of a general formula for the radiation power in the non-relativistic approximation, in particular, an expression for the radiation power in an axially symmetric magnetic field is obtained.

Dipole squeezing of the atom in the presence of another atom is investigated for the one-photon and two-photon transition mechanism with the initial atoms in the two-atom squeezed state and the field in a vacuum, a coherent, or a squeezed state. For a vacuum input, the degree of squeezing is shown to depend on the photon multiplicity 'm' and the superposition angles of the atoms, θ₁ and θ₂. One of the quadratures of the atomic polarization is found to exhibit permanent squeezing only for some nonzero values of θ₁, as well as θ₂. The effect of θ₂ on the dipole squeezing, however, is found to be negligible for the initial field in a coherent or a squeezed state. A comparison with the dipole squeezing of a single atom is also presented.

An effective wavefunction for the edge excitations in the Fractional quantum Hall effect can be found by dimensionally reducing the bulk wavefunction. Treated this way the Laughlin ν = 1/(2n + 1) wavefunction yields a Luttinger model ground state. We identify the edge-electron field with a Luttinger hyper-fermion operator, and the edge electron itself with a non-backscattering Bogoliubov quasi-particle. The edge-electron propagator may be calculated directly from the effective wavefunction using the properties of a one-dimensional one-component plasma, provided a prescription is adopted which is sensitive to the extra flux attached to the electrons.