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arXiv1995

Semiclassical Quantization of Billiards with Mixed Boundary Conditions

The semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived. The parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards. It is also used to check in detail some recently proposed parametric spectral statistics. The methods are illustrated in the analysis of the spectrum of the Sinai billiard and its parametric dependence on the boundary condition on the dispersing arc.

chao-dyn↗

arXiv1995

Quantum state diffusion, localization and computation

Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wave packets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle, we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested.

quant-ph↗

arXiv1994

The Dynamics of Vortex Structures and States of Current in Plasma-Like Fluids and the Electrical Explosion of Conductors: 3. Comparison with Experiment

The present paper is a concluding part of the series [N.B. Volkov and A.M. Iskoldsky: 1; 2; [1], [2]]. Here on the basis of the results of the above papers the experiments on the electric explosion in conductors have been discussed and the mechanism of "hot" plasma spots formation which was repeatedly observed in the experiments on a plasma focus, z-pinches, vacuum sparks and electric explosion of microwires has been proposed. To realize this mechanism, the formation of sausage-type instabilities is not necessary. Moreover, the possibility of hot spots formation in a liquid metal (in the model under discussion, the plasma-like medium is assumed incompressible and the local kinetic transport coefficients - constant) has been shown. The latter is of great interest for understanding the mechanism of initiation and operation of vacuum arc cathode spots and also for the explanation to the phenomenon of explosive electron emission which is widely employed in applied physics.

chao-dyn↗

arXiv1995

The Level Splitting Distribution in Chaos-assisted Tunneling

A compound tunneling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of phase space was recently introduced to explain peculiar features of tunneling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunneling. We study its consequences for the distribution of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunneling is the only operative mechanism. %The validity of this form can then in %principle be checked either numerically or even experimentally. We have checked that the analytical form we obtain agrees with splitting distributions calculated numerically for a model system in which chaos-assisted tunneling is known to be the dominant mechanism. The distribution depends on two parameters: The first gives the scale of the splittings and is related to the magnitude of the classically forbidden processes, the second gives a measure of the efficiency of possible barriers to classical transport which may exist in the chaotic region. If these are weak, this latter parameter is irrelevant; otherwise it sets an energy scale at which the splitting distribution crosses over from one type of behavior to another. The detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos-assisted tunneling.

chao-dyn↗

arXiv1996

Generalized Coherent States and Spin $S\geq 1$ Systems

Generalized Coherent States (GCS) are constructed (and discussed) in order to study quasiclassical behaviour of quantum spin models of the Heisenberg type. Several such models are taken to their semiclassical limits, whose form depends on the spin value as well as the Hamiltonian symmetry. In the continuum approximation, SU(2)/U(1) GCS when applied give rise to the well-known Landau-Lifshitz classical phenomenology. For arbitrary spin values one obtains a lattice of coupled nonlinear oscillators. Corresponding classical continuum models are described as well.

chao-dyn↗

arXiv1996

Secular determinants of random unitary matrices

We consider the characteristic polynomials of random unitary matrices $U$ drawn from various circular ensembles. In particular, the statistics of the coefficients of these polynomials are studied. The variances of these ``secular coefficients'' are given explicitly for arbitrary dimension and continued analytically to arbitrary values of the level repulsion exponent $β$. The latter secular coefficients are related to the traces of powers of $U$ by Newton's well-known formulae. While the traces tend to have Gaussian distributions and to be statistically independent among one another in the limit as the matrix dimension grows large, the secular coefficients exhibit strong mutual correlations due to Newton's mixing of traces to coefficients. These results might become relevant for current efforts at combining semiclassics and random-matrix theory in quantum treatments of classically chaotic dynamics.

chao-dyn↗

arXiv1996

Exact statistics of complex zeros for Gaussian random polynomials with real coefficients

k-point correlations of complex zeros for Gaussian ensembles of Random Polynomials of order N with Real Coefficients (GRPRC) are calculated exactly, following an approach of Hannay for the case of Gaussian Random Polynomials with Complex Coefficients (GRPCC). It is shown that in the thermodynamic limit $N \to \infty$ of Gaussian random holomorphic functions all the statistics converge to the their GRPCC counterparts as one moves off the real axis, while close to the real axis the two cases are essentially different. Special emphasis is given to 1 and 2 point correlation functions in various regimes.

chao-dyn↗

arXiv1996

Parametric statistics of zeros of Husimi representations of quantum chaotic eigenstates and random polynomials

Local parametric statistics of zeros of Husimi representations of quantum eigenstates are introduced. It is conjectured that for a classically fully chaotic systems one should use the model of parametric statistics of complex roots of Gaussian random polynomials which is exactly solvable as demonstrated below. For example, the velocities (derivatives of zeros of Husimi function with respect to an external parameter) are predicted to obey a universal (non-Maxwellian) distribution ${d P(v)}/{dv^2} = 2/(πσ^2)(1 + |v|^2/σ^2)^{-3},$ where $σ^2$ is the mean square velocity. The conjecture is demonstrated numerically in a generic chaotic system with two degrees of freedom. Dynamical formulation of the ``zero-flow'' in terms of an integrable many-body dynamical system is given as well.

chao-dyn↗

arXiv1995

Thermal Fluctuations in Quantized Chaotic Systems

We consider a quantum system with $N$ degrees of freedom which is classically chaotic. When $N$ is large, and both $\hbar$ and the quantum energy uncertainty $ΔE$ are small, quantum chaos theory can be used to demonstrate the following results: (1) given a generic observable $A$, the infinite time average $\overline A$ of the quantum expectation value $ $ is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of $A$; (2) the time variations of $ - \overline A$ are too small to represent thermal fluctuations; (3) however, the time variations of $ - ^2$ can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when $N$ is small.

chao-dyn↗

arXiv1996

Predictability in the large: an extension of the concept of Lyapunov exponent

We investigate the predictability problem in dynamical systems with many degrees of freedom and a wide spectrum of temporal scales. In particular, we study the case of $3D$ turbulence at high Reynolds numbers by introducing a finite-size Lyapunov exponent which measures the growth rate of finite-size perturbations. For sufficiently small perturbations this quantity coincides with the usual Lyapunov exponent. When the perturbation is still small compared to large-scale fluctuations, but large compared to fluctuations at the smallest dynamically active scales, the finite-size Lyapunov exponent is inversely proportional to the square of the perturbation size. Our results are supported by numerical experiments on shell models. We find that intermittency corrections do not change the scaling law of predictability. We also discuss the relation between finite-size Lyapunov exponent and information entropy.

chao-dyn↗

arXiv1996

The role of tank-treading motions in the transverse migration of a spheroidal vesicle in a shear flow

The behavior of a spheroidal vesicle, in a plane shear flow bounded from one side by a wall, is analysed when the distance from the wall is much larger than the spheroid radius. It is found that tank treading motions produce a transverse drift away from the wall, proportional to the spheroid eccentricity and the inverse square of the distance from the wall. This drift is independent of inertia, and is completely determined by the characteristics of the vesicle membrane. The relative strength of the contribution to drift from tank-treading motions and from the presence of inertial corrections, is discussed.

chao-dyn↗

arXiv1996

Quantum chaos, random matrix theory, and statistical mechanics in two dimensions - a unified approach

We present a theory where the statistical mechanics for dilute ideal gases can be derived from random matrix approach. We show the connection of this approach with Srednicki approach which connects Berry conjecture with statistical mechanics. We further establish a link between Berry conjecture and random matrix theory, thus providing a unified edifice for quantum chaos, random matrix theory, and statistical mechanics. In the course of arguing for these connections, we observe sum rules associated with the outstanding counting problem in the theory of braid groups. We are able to show that the presented approach leads to the second law of thermodynamics.

chao-dyn↗

arXiv1998

Symmetries of the stochastic Burgers equation

All Lie symmetries of the Burgers equation driven by an external random force are found. Besides the generalized Galilean transformations, this equation is also invariant under the time reparametrizations. It is shown that the Gaussian distribution of a pumping force is not invariant under the symmetries and breaks them down leading to the nontrivial vacuum (instanton). Integration over the volume of the symmetry groups provides the description of fluctuations around the instanton and leads to an exactly solvable quantum mechanical problem.

hep-th↗

arXiv1996

Symmetry Decomposition of Potentials with Channels

We discuss the symmetry decomposition of the average density of states for the two dimensional potential $V=x^2y^2$ and its three dimensional generalisation $V=x^2y^2+y^2z^2+z^2x^2$. In both problems, the energetically accessible phase space is non-compact due to the existence of infinite channels along the axes. It is known that in two dimensions the phase space volume is infinite in these channels thus yielding non-standard forms for the average density of states. Here we show that the channels also result in the symmetry decomposition having a much stronger effect than in potentials without channels, leading to terms which are essentially leading order. We verify these results numerically and also observe a peculiar numerical effect which we associate with the channels. In three dimensions, the volume of phase space is finite and the symmetry decomposition follows more closely that for generic potentials --- however there are still non-generic effects related to some of the group elements.

chao-dyn↗

arXiv1997

Comment on Energy Level Statistics in the Mixed Regime

We comment on the recent paper by Abul-Magd (J.Phys.A: Math.Gen. 29 (1996) 1) concerning the energy level statistics in the mixed regime, i.e. such having the mixed classical dynamics where regular and chaotic regions coexist in the phase space. We point out that his basic assumption on the additive property of the level-repulsion function $r(S)$ (conditional probability density) in the sense of dividing it linearly into the regular and chaotic part in proportion to the classical fractional phase space volumes $ρ_1$ and $ρ_2=q$ is not justified, since among other things, it relies on the type of Berry's ergodic assumption, which however is right only in a homogeneous ensemble of ergodic systems, but not in the neighbourhood of an integrable system. Thus his resulting distribution cannot be regarded as a theoretically well founded object. We point out that the semiclassical limiting energy level spacing distribution must be of Berry-Robnik (1984) type, and explain what transitional behaviour of the Brody-type (with fractional power-law energy level repulsion) we observe in the near semiclassical regime where effective $\hbar$ is not yet small enough. Thus we refer to the derivation, arguments and conclusions in our paper (Prosen and Robnik, J.Phys.A: Math.Gen. 26 (1994) 8059), and explain again the behaviour in this double transition region.

chao-dyn↗

arXiv1996

Supersymmetric quantum mechanics based on higher excited states

We generalize the formalism and the techniques of the supersymmetric (susy) quantum mechanics to the cases where the superpotential is generated/defined by higher excited eigenstates. The generalization is technically almost straightforward but physically quite nontrivial since it yields an infinity of new classes of susy-partner potentials, whose spectra are exactly identical except for the lowest m+1 states, if the superpotential is defined in terms of the (m+1)-st eigenfunction, with m=0 reserved for the ground state. It is shown that in case of the infinite 1-dim potential well nothing new emerges (the partner potential is still of Pöschl-Teller type I, for all m), whilst in case of the 1-dim harmonic oscillator we get a new class of infinitely many partner potentials: for each m the partner potential is expressed as the sum of the quadratic harmonic potential plus rational function, defined as the derivative of the ratio of two consecutive Hermite polynomials. These partner potentials of course have m singularities exactly at the locations of the nodes of the generating (m+1)-st wavefunction. The susy formalism applies everywhere between the singularities. A systematic application of the formalism to other potentials with known spectra would yield an infinitely rich class of "solvable" potentials, in terms of their partner potentials. If the potentials are shape invariant they can be solved at least partially and new types of analytically obtainable spectra are expected. PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Sq

chao-dyn↗

arXiv1996

WKB expansion for the angular momentum and the Kepler problem: from the torus quantization to the exact one

We calculate the WKB series for the angular momentum and the non--relativistic 3-dim Kepler problem. This is the first semiclassical treatment of the angular momentum for terms beyond the leading WKB approximation. We explain why the torus quantization (the leading WKB term) of the full problem is exact, even if the individual torus quantization of the angular momentum and of the radial Kepler problem separately is not exact. PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Sq

chao-dyn↗

arXiv1996

Time-reversal symmetry and random polynomials

We analyze the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation. Moreover, a class of time-reverse invariant quantum systems is shown, for which spectra display fluctuations characteristic of orthogonal ensemble, while eigenvectors confer to predictions of unitary ensemble.

chao-dyn↗