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Chaos in dynamical systems has best been understood in terms of Hamiltonian systems. A primary method of diagnosis of chaos in these systems is the Lyapunov exponent. According to general relativity, space-time is itself a dynamical system. When the evolution of a model universe is expressed in the ADM form it can be described as a Hamiltonian system. Among the various model cosmologies, the Mixmaster or Bianchi IX cosmology has been extensively studied as a candidate to exhibit chaos. However, the Lyapunov exponents in this system have shown contradictory properties, including a seeming dependence on the coordinates used to describe space-time. Such dependencies, if true, would be surprising as the time coordinate of space-time is unrelated to the parameterization of phase space. Further, this sort of dependence would relegate chaos to a “bad” coordinate choice rather than a dynamic property of the system. The problem with the Lyapunov exponent lies in the ambiguities remaining in the ADM action integral. The current interpretation involves an arbitrary Lagrange multiplier—thought to be necessary for the coordinate invariance of space-time. An arbitrary multiplier turns out to be unnecessary for coordinate invariance, and in addition destroys the symplectic structure of phase space. In reality, the geometry selects the parameterization of phase space, and any change in the parameter results in a changed Hamiltonian system. It must be emphasized that the fixing of the phase space parameter does NOT impose a coordinate choice on space-time. The parameter is selected by the symplectic structure of phase space and full coordinate invariance of space-time is left intact. Once the demands of both geometries, space-time and phase space, have been satisfied, the Lyapunov exponent becomes independent of the coordinate imposed on space-time. Additionally, the correction of the phase space structure leads to a Hamiltonian that is more general, in that it describes a gravitational system with a cosmological constant, than is currently the case.

For ideal Bose atoms in an isotropic harmonic trap, we consider thermodynamic variables obtained from microcanonical, canonical, and grand canonical ensembles, each with certain variables specified and other variables fluctuating. For the first two of these ensembles, we derive recursion relations that link partition functions for different dimensions. We discuss fluctuations in general, and obtain expressions for variances of the atom number N, the chemical potential \ensuremath{\mu}, and the temperature T for small, Gaussian fluctuations in the grand canonical ensemble. Then from our recursion relations and others given elsewhere, we obtain probability distributions for N, for ground-state occupation number ${n}_{0}$, for \ensuremath{\mu}, and for T. Below the critical temperature, the shape of the distributions for N, ${n}_{0},$ and \ensuremath{\mu} are definitely not Gaussian for the grand canonical ensemble. For given temperature and small N, we find that the chemical potential values pertaining to the three ensembles differ. We compare the specific-heat function ${C}_{N}(T)$ for the three ensembles and propose to use the minimum of ${\mathrm{dC}}_{N}/dT$ to define the critical temperature to facilitate comparisons with similar configurations of interacting atoms.

Non-relativistic cellular automata can model non-relativistic hydrodynamical flows. In this article we show that if the hopping occurs on a space-time lattice which is generated by discrete subgroups of the Poincare group and if the collision rules embody the relativistic conservation laws, we can modelrelativistic flows. The simplest version of the relativistic model is formally isomorphic with the non-relativistic Hardy, de Pazzis and Pomeau (HPP) lattice model, provided we reinterpret the various quantities that appear there. This observation explains the non-Galilean invariant results of HPP.

The quantum mechanics of some maps have been studied as means to understand the implications of “chaos” in quantum systems. Maps exhibiting highly complex classical dynamics can also be quantized. Linear maps that are classically unstable are simple systems that can be exactly solved, and here we do the same for their quantum version, namely find the eigenfunctions of the unitary operator that propagates states in one discrete time step. Here we discuss maps in a two-dimensional (non-compact) phase space where the instability is a hyperbolic fixed point that is either ordinary or “reflecting.” The essential technique uses a basis that separates the time evolution of the dynamical variables. This also provides us a convenient basis to study the transition in the eigenfunctions when the classical map's fixed point changes from a stable elliptic case to a hyperbolic one.

The general solution of the one flavour integral equation for string fragmentation is presented and is approximated by a finite sum. The N pion phase space distribution is calculated for the emission points; thence the hadronic fragmentation function, the number density, and the energy density vs. rapidity are obtained. We discuss the effects of the left-right asymmetry embedded into the integral equation, and present numerical results based on the Lund fragmentation function.

“ … our proposed way is contrary to the advice given by many mathematicians, that one should concentrate in order to be able to achieve something.

It is shown that the free motion of any three-dimensional rigid body colliding elastically between two parallel, flat walls is equivalent to a three-dimensional billiard system. Depending upon the inertial parameters of the problem, the billiard system may possess a potential energy field and a non-Euclidean configuration space. The corresponding curvilinear motion of the billiard ball does not necessarily lead to a decrease of the stable periodic orbits found in the analogous rectilinear system.

We demonstrate that the free motion of any two-dimensional rigid body colliding elastically with two parallel, flat walls is equivalent to a billiard system. Using this equivalence, we analyze the integrable and chaotic properties of this new class of billiards. This provides a demonstration that coin tossing, the prototypical example of an independent random process, is a completely chaotic (Bernoulli) problem. The related question of which billiard geometries can be represented as rigid body systems is examined., 16 pages, LaTex

Periodic points of a classical map may belong to a fractal set. It is shown here on a simple model that upon quantising this classical map the influence of the classical periodic points upon the quantal results is the same whether the periodic point belongs to a fractal set, or not. A simple intuitive explanation is given.

We quantise and study several versions of finite multibaker maps. Classically these are exactly solvable K-systems with known exponential decay to global equilibrium. This is an attempt to construct simple models of relaxation in quantum systems. The effect of symmetries and localization on quantum transport is discussed., Comment: 32 pages. LaTex file. 9 figures, not included. For figures send mail to first author at 'arul@max.physics.sunysb.edu'

There are two opposing desires in choosing a model in theoretical physics. On the one hand it should contain all the important physics under consideration; on the other hand it should be simple enough to pursue the calculations easily and exhibit the results in an unequivocal manner.

We discuss within the framework of an analytical model central collisions between large nuclei at an intermediate energy. The model assumes three stages: the ignition stage in which thermalized nucleons are created within the reaction volume; the expansion phase in which the hot nuclear matter expands and cools; and finally, the evaporation stage in which the system disintegrates through a surface evaporation. The model provides a qualitative insight into the connections between the main physical processes throughout the collision event.

By means of the Weyl transform we propose to associate phase-spaceoperators, and not functions with the dynamical operators of quantum-mechanics. This way one achieves a consistent and one-to-one mapping of the quantum-mechanical operator calculus unto a phase-space operator calculus in which only variables associated with the canonical variables of position and momenta appear. Contrary to the usual Weyl transform technique, here no composition law is needed for the Weyl transforms. As a consequence the calculus involving Weyl transforms is greatly simplified, which we demonstrate at two examples.

It is shown how to generalize the semiclassical treatment of dense Fermi gases in the presence of spin-dependent forces. One constructs the Wigner transform of the singlet density matrix in that representation in which the position operator and thez component of the spin operator are diagonal. This way one obtains a matrix in the spin indices which is a function of the semiclassicalp andq variables. Rules are then given how to calculate with this matrix, theμ-matrix. Next, this matrix is explicitly evaluated for independent Fermions in a spherically symmetrical potential, subject to spin-orbit forces. Finally a brief discussion is given about the nature of this matrix and possible uses are proposed.

It has been shown that strange and non-strange quarks generally possess different hadronization rates in a quark-nucleon phase transition. Since the main process driving the rehadronization is the expansion and cooling of the fireball of quarks, this complicated phase transition is intimately connected with relativistic hydrodynamics. Here the consistent hydro+thermodynamical description of the transition is presented.

An inference procedure is required to assign a density operator D to a system when the only available information is the set of expectation values a i of some observables A i , commuting or not. To do this, we consider as a single “supersystem” the Gibbsian ensemble consisting of N replicas of the system, on which thought experiments compatible with the data a i can be performed. In this supersystem, we identify the expectation values a i with mean values, and implement the principle that equal ignorance should be represented by equiprobability. New features are brought in through the non-commutativity of the observables A i , and through the operator nature of the quantum states. These generate difficulties, both conceptual and technical, which we discuss and resolve. The density operator D of the system considered is obtained from that of the supersystem by taking the partial trace over the N − 1 other systems. The rather involved calculations are accomplished by field theoretical techniques. They provide a generalized canonical distribution, with deviations which vanish as N → ∞, the same result as would be obtained by maximizing von Neumann's entropy with the constraints 〈 A i 〉 = a i . We succeed thus to justify the quantal maximum entropy principle and to construct simultaneously von Neumann's entropy, starting from the idea of equiprobability in the supersystem.

The thermodynamics is studied of a quark plasma in equilibrium with a hadronic phase assuming strangeness conservation. It turns out that the presence of strangeness conservation does not lead to a discontinuity in the chemical potential μ s associated with the strange particles but to a coexistence region in the phase diagram. The discontinuity computed before is in fact the difference in the values of μ s at the boundaries of the coexistence region keeping T fixed.

We show that for a wave ψ in the form of an Airy function the probability density ‖ψ‖2 propagates in free space without distortion and with constant acceleration. This ’’Airy packet’’ corresponds classically to a family of orbits represented by a parabola in phase space; under the classical motion this parabola translates rigidly, and the fact that no other curve has this property shows that the Airy packet is unique in propagating without change of form. The acceleration of the packet (which does not violate Ehrenfest’s theorem) is related to the curvature of the caustic (envelope) of the family of world lines in spacetime. When a spatially uniform force F (t) acts the Airy packet continues to preserve its integrity. We exhibit the solution of Schrodinger’s equation for general F (t) and discuss the motion for some special forms of F (t).

Consider an atomic nucleus, or any closed system of Fermions under the influence of a saturating short-range interaction. If the range is sufficiently short, one finds that the thickness of the surface layer is not determined by the range of this force, but by a new,quantum mechanical characteristic length, L=(ħ2/8mF)1/3, whereF is the restoring force generated at the boundary by the self-consistent potential acting on the most energetic particles. Moreover, in equilibrium, thisforce adjusts itself in such a manner that k F L ∼ 1 wherek F is the Fermi wave number. As a consequence, one is now able to compute the binding energy, and thus the volume and surface energy coefficients in an explicit analytical fashion as a function of the coupling constants of the interaction, which in the present case is a modified Skyrme interaction. In addition, speculations are made on the existence ofsurface isomers. These are nuclei at several hundred MeV excitation with essentially the same central density, but with an increased surface region.

An approximation is constructed to the phase‐space, or Wigner, distribution function for a three‐dimensional, dense Fermi gas in a spherically symmetric potential well. Near the surface separating the classically forbidden region from the classically allowed region, quantum oscillations occur. The oscillations are expressed in terms of a universal function.

The study theoretically and numerically the role of the fluctuations of eigenvalue spectra /..mu../sub n/ in a particular analytical continuation process applied to the (generalized) zeta function Z(s) = ..sigma../sub n/..mu../sub n//sup -s/ for s large and positive. A particularly interesting example is the spectrum of the Laplacian on a triangular domain which tessellates a compact surface of constant negative curvature (of genus two). The authors indeed find that the fluctuations restrict the abscissa of convergence, and also affect the rate of convergence. This then initiates a new approach to the exploration of spectral fluctuations through the convergence of analytical continuation processes.

In this paper Cerenkov radiation is treated using a generalization of Dirac's representation of the four potentials. This way the case of moving media can also be discussed.

We generalize Helmholtz's theorem and apply it to four-dimensional vector fields and tensor fields. For vector fields the generalization is straightforward. Antisymmetric tensor fields of rank two exhibit a beautiful symmetry between the irrotational part of the tensor and the dual of the solenoidal component. The physical applications show that in Maxwell's theory the irrotational part of the four-potential field has no physical meaning and the Lorentz condition makes it identically zero. In Dirac's new electrodynamics an algebraic condition is imposed on the four-potential. Hence in this theory the irrotational part is not zero, and the algebraic condition establishes a relation between the sources and vortices of the four-potential field. If we apply the resolution to the electromagnetic field tensor we can see that the free charges are responsible for the sources, and the magnetic poles, if they exist, for the vortices, provided we use the customary association between the components of the electromagnetic field tensor and the components of the electric and magnetic fields.

This present paper is an attempt to describe the operation of a laser by a method patterned closely after the classical theory of dispersion. There, the electromagnetic field is treated as a classical system interacting with a collection of classical harmonic oscillators which are at rest. In the present case the radiation field is still treated classically, but, in accordance with the correspondence principle, the oscillators are now virtual oscillators associated with the up-and-down transitions in effective two-state atoms which are described quantum mechanically. The atoms carrying these virtual oscillators are not at rest but move with different velocities. We show how the Lorentz averaging can be performed for such systems, and derive a closed set of equations linking the average electric field E, with the average polarization density P„ and the average population inversion density Mp associated with atoms moving with the velocity v. Next, we investigate the conditions for the existence of a steady state if pumping is present, and if the electric field is represented by a standing wave, or a travelling wave of a single frequency (single-mode case). It turns out that, notwithstanding the nonlinear nature of the equations, the steady-state conditions give a simple complex dispersion relation; moreover, the real and imaginary part of this dispersion relation are equivalent to the usual heuristic expressions which specify the operating frequency in terms of the index of refraction and cavity length, and which balance the gain against the losses. The single-mode case is analysed in detail without any smallness assumptions for the resultant intensity. The small-intensity case gives the usual results exhibiting the tuning dip. For high intensities the relative depth of the dip tends to zero. At the end of the paper we discuss proposed extensions and additional applications.

In this note we plan to show how the evenness and oddness of the spatial dimensions influence the general properties of wave propagation. The solution of the inhomogeneous wave equation will be given by a multiple integral; one of the integrations is to be performed in a complex time-plane. The singularities of the integrand in this plane have an alternating character depending on the evenness or oddness of the spatial dimensions. If the space is odd-dimensional, the singularities are poles; if it is even-dimensional, the singularities are branch points. This, then, will alter profoundly the properties of the solution.

The chemical equilibrium is studied for the reactionA+B⇄C, assuming that, initially, the particlesB form a lattice and the particlesA are statistically distributed on interstices. A mass action law is derived which defines the numbersnA, nB, nC of particlesA, B,C in the chemical equilibrium assuming the initial distribution to be known. It predicts a considerably larger numbernC of fused particlesC compared to the mass action law for the gaseous phase. The result holds for an ordinary as well as for a nuclear lattice. Its possible relevance for the production of proton-rich isotopes in the universe is discussed.

We consider quantized motion of neutral atoms cooled below the recoil limit in a quadrupole magnetostatic trap. Because of Majorana transitions to untrapped levels near the point of zero field at the trap center, all quantum levels have a nonzero decay rate. The Schrodinger equation associated with the potential gμB · S (S is the total atomic spin) takes the form of coupled equations in r when the spinor components are expanded in spherical harmonics. We integrate the multichannel problem numerically to obtain asymptotic phase shifts, resonance energies, and widths. For S = 1/2, the lowest levels have widths somewhat less than their spacing. Thus the trap quantum-level structure might possibly be observable if the atoms are sufficiently cold, namely, in the 0.1-μK regime for most atoms and attainable trap field gradients. The width decreases rapidly with increasing MJ, the angular momentum about the symmetry axis. Spectroscopic linewidths of a few hertz are possible if there is enough population in the lowest levels with a few MJ quanta. The decay rate of the lowest levels, however, is probably too rapid for studying Bose–Einstein condensation in such a trap.