E. Braun

IRREVERSIBLE BEHAVIOR OF A QUANTUM HARMONIC-OSCILLATOR COUPLED TO A HEAT BATH

In this work we analyze descriptions, in several languages, of a quantum mechanical harmonic oscillator interacting with a heat bath consisting of harmonic oscillators. The differences in the languages consist in selecting different variables in terms of which one will describe the dissipation. We consider for the force equation the cases in which the dissipation is proportional to the momentum and to the “velocity” operators. We also analyze the “acceleration” equation. The results are obtained in the rotating wave approximation (RWA), a case in which the momentum and “velocity” operators are not simply related.

Exact solution of a general two-dimensional Ising model: The partition function

The exact partition function per spin is obtained for a two-dimensional generalization of the Ising model. This general model consists of a unitary cell that is repeated throughout the system. The unitary cell is made up of spins in t columns and q rows with arbitrary energies between them. The exact result is given in terms of certain Ψ quantities (see section 2), which can be determined for particular cases.

The correlation function for a quantum oscillator in a low-temperature heat bath

The momentum autocorrelation function c(t) for a quantum oscillator coupled with harmonic forces to a heat bath of oscillators is calculated at low temperatures. It is found that c(t) contains two distinct terms: one, the zero-point contribution c0(t), is temperature independent, and the other, c1(t), does depend on temperature. We concentrate our attention on the low-temperature case. An expression for c1(t) is obtained, which is valid for arbitrary strenghts of the coupling and for arbitrary times. It is shown that c1(t) is governed by the low-frequency behaviour of F(λ) = A2(λ)ϱ(λ), whereϱ(λ) is the density of normal modes and A(λ) is the central-oscillator component of the λth normal mode; other details of the problem are irrelevant. It is found that c1(t) decays in time as an inverse-power law, with a relaxation time tq ≈ ħ/kT.

Reentrant phase transition in a particular (3 × 3) UC two-dimensional Ising model

Abstract Using the method presented in an earlier paper we obtain the exact partition function per spin, L ( T >), corresponding to the particular (3 × 3) unitary cell two-dimensional Ising model studied by Kitatani et al. In contrast with these authors that worked with numerical methods, we use algebraic procedures to obtain the analytic expression for L ( T ), and the exact equation for the critical temperature, T c , for other values of the interaction parameters. We find results for more extensive ranges of the values of the interaction parameters than obtained by Kitatani et al. We use these results to discuss physical properties of the systems. We exhibit an extended range of values for which double reentrancy is found.

Quantum random walk and transient phenomena

The quantum random walk (QRW) is a new microscopic model for diffusion in a one-dimensional lattice. In order to test its reliability we use the QRW mathematical model to solve the lattice equivalent of the well known perfect-absorbent shutter problem. The QRW is by construction a causal theory; therefore even though our solution lacks for times less than x/v the non-causal behavior, which is characteristic of an exact Moshinsky “diffraction in time” solution, our QRW-derived solution shows for times greater than x/v a damped-oscillatory behavior which resembles the causal Moshinsky solution. Therefore at least for the shutter problem, the oscillatory transient behavior is well approximated by the QRW theory. This gives a confidence test on the model.

Exact magnetic transitions in the “union jack” model

We present the exact expression for the partition function per spin in the thermodynamic limit, for the general “union jack” model. In effect, in this model we treat nearest and next nearest neighbors interactions. For the particular case in which the interaction parameters within the unit cell have the values e1 = e2, e3 = ae4, e5 = Ne3 and e6 = Ne4 with a = ±1 and N any real number, we present explicit phase diagrams and rigorous exact expressions for the loci of the transitions. These diagrams split up in several regions which we are able to characterize by their magnetic configurations. We do that by calculating the exact correlation functions between the spins in the unit cell.

The specific heat of a general two-dimensional Ising model

We analyze the specific heat for the class of models treated in a previous paper. It is shown that if a certain condition is fulfilled (eq. (3)), then the specific heat diverges logarithmically; This fact is established without recourse to the details of the interactions between the spins; thus, we show that the universality of the singularity holds for this class of models. The equations for the transition temperatures are presented for the models treated specifically in a previous paper.

Path integral method applied to the Brownian rotation of a symmetric top

The conditional probability density function in angular velocities space is obtained in an exact and closed expression for a symmetric top undergoing brownian motion. The distribution turns out to be non-gaussian. We obtain the distribution function by shifting the problem of solving stochastic differential equations to the problem of solving ordinary differential equations. This is done using the method of path integrals developed by Feynman and Hibbs for quantum mechanics.

HYDRODYNAMIC MOBILITIES IN A SUSPENSION WITH A FLUID WITH SPIN

Abstract The hydrodynamic interactions between any number of spheres immersed in a viscous fluid that has an internal structure have been studied. These effects are due to the internal angular momentum of the fluid molecules. The scheme proposed by Mazur and van Saarloos has been generalized, and general expressions for the translational, rotational and cross mobilities for the case under study have been obtained. The dependence on the distances of these mobilities is presented up to an order of R −7 and the spin dependences of their leading terms are explicitly given.

HYDRODYNAMIC INTERACTIONS IN A SUSPENSION WITH A STRUCTURED FLUID. I. GENERAL EXPRESSIONS FOR THE MOBILITIES

In this work we treat the hydrodynamic interactions between any number of spheres immersed in a viscous fluid that has internal structure. These effects are due to internal angular momentum of the fluid molecules. We generalize the scheme proposed by Mazur and van Saarloos, and obtain general expressions for the translational, rotational and cross mobilities for the case under treatment. The limit of a non-structured fluid agrees with earlier known results.