Some exact results in kinetic theory

It is shown that for fluids composed of particles interacting with pairwise-additive, spherically symmetric forces, the exact linearized transport equation admits mass, momentum, and kinetic energy as homogeneous solutions and that the kinetic part of the bulk viscosity is identically zero.     

Some exact results on the superpotential from Calabi-Yau compactifications

We prove that certain superpotential couplings in compactified string theories are given exactly by the values calculated at sigma-model tree level. In particular, the 27³ coupling in (2,2) compactifications is a coupling of this sort. We explicitly check our results by examining the couplings in an exactly soluble model proposed by Gepner as a particular point in the moduli space of (2,2) compactifications on Y_{4; 5}. We thus determine the exact values of the normalized 27³ couplings for any value of the radius of Y_{4; 5}.     

Some exact results for the three-layer zamolodchikov model

In this talk, we present our recent results on the three-layer Zamolodchikov model. We discuss solutions to the Bethe ansatz equations following from functional relations. We consider two regimes I and II that differ by the signs of the spherical sides (a₁, a₂, a₃) → (?a₁, ?a₂, ?a₃). Also, we accept the two-line hypothesis for regime I and the one-line hypothesis for regime II. In the thermodynamic limit, we derive integral equations for distribution densities and solve them exactly. Using this solution, we calculate the partition function for the three-layer Zamolodchikov model and check the compatibility of this result with functional relations. We also discuss the reasons for the discrepancy with Baxter’s result of 1986.     

Average dynamics of noisy maps

The average trajectories and fluctuations around them resulting from noisy, non-linear maps are analyzed. After describing their scaling properties they are discussed using a deterministic average equation of motion. A bifurcation gap is found and the existence of exceptional attractors for special initial conditions is predicted.     

Coherent backscattering by polydisperse discrete random media: exact T-matrix results

The numerically exact superposition T-matrix method is used to compute, for the first time to our knowledge, electromagnetic scattering by finite spherical volumes composed of polydisperse mixtures of spherical particles with different size parameters or different refractive indices. The backscattering patterns calculated in the far-field zone of the polydisperse multiparticle volumes reveal unequivocally the classical manifestations of the effect of weak localization of electromagnetic waves in discrete random media, thereby corroborating the universal interference nature of coherent backscattering. The polarization opposition effect is shown to be the least robust manifestation of weak localization fading away with increasing particle size parameter.     

Slaving principle for stochastic differential equations with additive and multiplicative noise and for discrete noisy maps

We first treat multidimensional nonlinear noisy maps. We assume that the variables can be split into two classes of variablesu ands so that the linearized equations would give rise to growth or decay foru ands, respectively. We show how the slaved variabless can be explicitly expressed by the order parametersu by making use of the fully nonlinear equations. By taking the limit of vanishing time steps and using a Wiener process and the Îto calculus we derive the corresponding formulas for stochastic differential equations (including multiplicative noise). In this way a high-dimensional problem can be reduced to a problem of much lower dimensions described again by stochastic equations of theÎto type. A similar procedure holds for theStratonovich calculus.     

Critical fluctuations of noisy period-doubling maps

We extend the theory of quasipotentials in dynamical systems by calculating, within a broad class of period-doubling maps, an exact potential for the critical fluctuations of pitchfork bifurcations in the weak noise limit. These far-from-equilibrium fluctuations are described by finite-size mean field theory, placing their static properties in the same universality class as the Ising model on a complete graph. We demonstrate that the effective system size of noisy period-doubling bifurcations exhibits universal scaling behavior along period-doubling routes to chaos.     

Some exact results for the anderson model of an impurity

Summary General formulae for the valence, charge and spin susceptibility, and specific heat at low temperature and zero magnetic field are obtained for the Anderson model of an impurity with the use of the Betheansatz method, following the work of Tsvelick and Wiegmann.

---

Riassunto Si ottengono formule generali per la valenza, la carica e suscettibilità di spin e per il calore specifico a temperatura bassa e campo magnetico zero per il modello di Anderson di un'impurità con l'uso del metodo dell'ansatz di Bethe seguendo il lavoro di Tsvelick e Wiegmann.

---

Резюме Получаются общие формулы для валентности, заряда, спиновой восприимчивости и удельной теплоемкости при низкой температуре и нулевом магнитном поле для модели Андерсона для описания примеси с использованием подхода Бете.

     

Some exact results for the Kondo lattice with infinite exchange interaction

Using an exact equivalence between the Kondo lattice with infinite J and the Hubbard model with infinite U, we show that the ground state of the Kondo lattice is non-magnetic for concentrations of conduction electrons close to 1, but there are still some magnetic regions even for J → ∞.     

Stochastic analysis of a nonequilibrium phase transition: Some exact results

A system involving all-or-none transitions away from equilibrium is considered. Under the assumption of spatially homogeneous fluctuations an integral representation of the solution of the master equation is derived, which permits an exact evaluation of the variance in the thermodynamic limit. A systematic perturbative solution of the master equation is also developed. Both approaches yield “classical” exponents describing the divergence of the second-order variance as the instability point is approached on either side. Finally, at the instability point the second-order variance is shown to diverge as the $\text{3}\text{2}$ power of the volume.     

Some exact results for one-dimensional two-band SU(2) bosons

A one-dimensional model for two-band SU(2) bosons with isospin exchange interaction is solved by means of the nested Bethe-ansatz method. The features of the ground state and low-lying excitation state are discussed explicitly by numerical and analytical method. The thermodynamics of the system is analyzed by means of the thermodynamic Bethe-ansatz method, and some physical quantities, such as magnetization, specific heat, etc. are obtained explicitly in some special cases.Received: 14 March 2004, Published online: 12 August 2004PACS: 03.65.-w Quantum mechanics - 72.15.Nj Collective modes (e.g., in one-dimensional conductors) - 03.65.Ge Solutions of wave equations: bound states     

Computing the invariant density of bistable noisy maps

A numerical method is presented allowing the computation of the invariant density of a time-discrete bi- or multistable map perturbed by weak noise. It permits the examination of noise-induced transitions between different stable states in the limit of weak but not amplitude-limited noise. The method is tested by comparing the results with computer experiments. For this purpose a new one-parameter family of bistable maps is introduced. It turns out that the numerics are in good agreement with the experiments. The results suggest the conjecture that in the limit of weak but transition-inducing noise the probability of being in one basin of attraction approaches one. A simple example which can be solved in closed form and which illustrates these findings is discussed.     

Quasipotentials for simple noisy maps with complicated dynamics

The theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems, leading to exponential estimates of invariant probabilities and mean first exit times. In the present article we develop the mathematical foundation of this theory for discrete-time systems, following and extending the work of Freidlin and Wentzell, and Kifer. We discuss strategies for calculating and estimating quasipotentials and show their application to one-dimensionalS-unimodal maps. The method proves to be especially suited for describing the noise scaling behavior of invariant probabilities, e.g., for the map occurring as the limit of the Feigenbaum period-doubling sequence. We show that the method allows statements about the scaling behavior in the case of localized noise, too, which does not originally lie within the scope of the quasipotential formalism.     

Locally exact modifications of discrete gradient schemes

Locally exact integrators preserve linearization of the original system at every point. We construct energy-preserving locally exact discrete gradient schemes for arbitrary multidimensional canonical Hamiltonian systems by modifying classical discrete gradient schemes. Modifications of this kind are found for any discrete gradient.     

Average trajectories and fluctuations from noisy,nonlinear maps

The average trajectories and fluctuations around them resulting from an ensemble of noisy, nonlinear maps are analyzed. The bifurcation diagram for the average value obtained from the computer simulation of noisy maps ensemble is discussed first. Then a deterministic average equation of motion describing in an approximate way the time evolution of the average value and of the variance is analyzed numerically. This equation predicts the existence of the bifurcation gap and of the exceptional attractors for special initial points. The scaling properties of the average value and of the variance are obtained with the help of this equation.On leave from Institute for Theoretical Physics, Warsaw University, 00-681 Warsaw, Hoza 69, Poland.