Peter Szepfalusy

Statistical properties of chaos demonstrated in a class of one‐dimensional maps

One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.

Structure of the perturbation series of the spin-1 Bose gas at low temperatures

The properties of Greens functions and various correlation functions of density and spin operators are considered in a homogeneous spin-1 Bose gas in different phases. The dielectric formalism is worked out and the partial coincidence of the one particle and collective spectra is pointed out below the temperature of Bose-Einstein condensation. As an application, the formalism is used to give two approximations for the propagators and the correlation functions and the spectra of excitations including shifts and widths due to the thermal cloud.

Kohn mode for trapped Bose gases within the dielectric formalism

The presence of undamped harmonic center of mass oscillations of a weakly interacting Bose gas in a harmonic trap is demonstrated within the dielectric formalism for a previously introduced finite temperature approximation including exchange. The consistency of the approximation with the Kohn theorem is thereby demonstrated. The Kohn modes are found explicitly, generalizing an earlier zero-temperature result found in the literature. It is shown how the Kohn mode disappears from the single-particle spectrum, while remaining in the density oscillation spectrum, when the temperature increases from below to above the condensation temperature.

Gradient corrections to the local-density approximation for trapped superfluid Fermi gases

Two species superfluid Fermi gas is investigated on the BCS side up to the Feshbach resonance. Using the Greenss function technique gradient corrections are calculated to the generalized Thomas-Fermi theory including Cooper pairing. Their relative magnitude is found to be measured by the small parameter (d/R{sub TF}){sup 4}, where d is the oscillator length of the trap potential and R{sub TF} is the radial extension of the density n in the Thomas-Fermi approximation. In particular, at the Feshbach resonance the universal corrections to the local density approximation are calculated and a universal prefactor {kappa}{sub W}=7/27 is derived for the von Weizsaecker-type correction {kappa}{sub W}(({h_bar}/2{pi}){sup 2}/2m)({nabla}{sup 2}n{sup 1/2}/n{sup 1/2}).

Static properties and spin dynamics of the ferromagnetic spin-1 Bose gas in a magnetic field

The properties of spin-1 Bose gases with ferromagnetic interactions in the presence of a nonzero magnetic field are studied. The equation of state and thermodynamic quantities are worked out with the help of a mean-field approximation. The phase diagram besides Bose-Einstein condensation contains a first-order transition where two values of the magnetization coexist. The dynamics is investigated with the help of the random phase approximation. The soft mode corresponding to the critical point of the magnetic phase transition is found to behave like in conventional theory.

Quasiparticle excitations and dynamical structure function of trapped Bose condensates in the WKB approximation

The Bogoliubov equations of the quasi-particle excitations in a weakly interacting trapped Bose-condensate are solved in the WKB approximation in an isotropic harmonic trap, determining the discrete quasi-particle energies and wave functions by torus (Bohr-Sommerfeld) quantization of the integrable classical quasi-particle dynamics. The results are used to calculate the position and strengths of the peaks in the dynamic structure function which can be observed by off-resonance inelastic light-scattering.

Renormalization group analysis of relaxational dynamics in systems with many-component order-parameter II - Scaling fields and scaling variables

A general discussion of scaling fields and scaling variables in the dynamic renormalization group is given using path probability formalism. It is shown that scaling variables are the derivatives of the action with respect to scaling fields. The general ideas are illustrated on the multicomponent relaxational model in the large-n limit, where scaling fields and scaling variables are calculated explicitly and flow lines, crossover and universality are discussed. Critical points of higher order are also included in the investigation.

Clustering of Fermi particles with arbitrary spin

A single l-shell model is investigated for a system of fermions of spin s and an attractive s-wave, spin channel independent, interaction. The spectra and eigenvectors are determined exactly for different l,s values and particle numbers N. As a generalization of Cooper pairing it is shown that when N=mu(2s+1), mu=1,2,..., 2l+1, the ground state consists of clusters of (2s+1) particles. The relevance of the results for more general situations including the homogeneous system is briefly discussed.

Dielectric formalism and damping of collective modes in trapped Bose-Einstein condensed gases

We present the general dielectric formalism for Bose-Einstein condensed systems in an external potential at finite temperatures. On the basis of a model arising within this framework as a first approximation in an intermediate-temperature region for the large condensate we calculate the damping of low-energy excitations in the collisionless regime.

Critical Dynamics near a Hard Mode Instability

Scaling hypothesis and a renormalization group procedure are formulated in the vicinity of the bifurcation point, where the behaviour is governed by inhomogeneous fluctuations. The working of the general ideas is illustrated in a model system in which the number of components of the complex order parameter field goes to infinity.