arXiv subjects
Publications and source records attributed to Salman Habib.
We present a new method for the computation of Lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of M or MM_trans where M is the tangent map. This method uses a minimal set of variables, does not require renormalization or reorthogonalization, can be used to efficiently compute partial Lyapunov spectra, and does not break down when the Lyapunov spectrum is degenerate.
We study the classical thermodynamics of a 1+1-dimensional double-well sinh-Gordon theory. Remarkably, the Schrodinger-like equation resulting from the transfer integral method is quasi-exactly solvable at several temperatures. This allows exact calculation of the partition function and some correlation functions above and below the short-range order (``kink'') transition, in striking agreement with high resolution Langevin simulations. Interesting connections with the Landau-Ginzburg and double sine-Gordon models are also established.
Dynamical chaos has recently been shown to exist in the Gaussian approximation in quantum mechanics and in the self-consistent mean field approach to studying the dynamics of quantum fields. In this study, we first show that any variational approximation to the dynamics of a quantum system based on the Dirac action principle leads to a classical Hamiltonian dynamics for the variational parameters. Since this Hamiltonian is generically nonlinear and nonintegrable, the dynamics thus generated can be chaotic, in distinction to the exact quantum evolution. We then restrict attention to a system of two biquadratically coupled quantum oscillators and study two variational schemes, the leading order large N (four canonical variables) and Hartree (six canonical variables) approximations. The chaos seen in the approximate dynamics is an artifact of the approximations: this is demonstrated by the fact that its onset occurs on the same characteristic time scale as the breakdown of the approximations when compared to numerical solutions of the time-dependent Schrodinger equation.
The time evolution of O(N) symmetric lambda Phi^4 scalar field theory is studied in the large N limit. In this limit the mean field and two-point correlation function evolve together as a self-consistent closed Hamiltonian system, characterized by a Gaussian density matrix. The static part of the effective Hamiltonian defines the True Effective Potential U_eff for configurations far from thermal equilibrium. Numerically solving the time evolution equations for energy densities corresponding to a quench in the unstable spinodal region, we find results quite different from what might be inferred from the equilibrium free energy ``effective'' potential F. Typical time evolutions show effectively irreversible energy flow from the coherent mean fields to the quantum fluctuating modes, due to the creation of massless Goldstone bosons near threshold. The plasma frequency and collisionless damping rate of the mean fields are calculated in terms of the particle number density by a linear response analysis and compared with the numerical results. Dephasing of the fluctuations leads also to the growth of an effective entropy and the transition from quantum to classical behavior of the ensemble. In addition to casting some light on fundamental issues of nonequilibrium quantum statistical mechanics, the general framework presented in this work may be applied to a study of the dynamics of second order phase transitions in a wide variety of Landau-Ginsburg systems described by a scalar order parameter.
Winding number transitions in the two dimensional softly broken O(3) nonlinear sigma model are studied at finite energy and temperature. New periodic instanton solutions which dominate the semiclassical transition amplitudes are found analytically at low energies, and numerically for all energies up to the sphaleron scale. The Euclidean period beta of these finite energy instantons increases with energy, contrary to the behavior found in the abelian Higgs model or simple one dimensional systems. This results in a sharp crossover from instanton dominated tunneling to sphaleron dominated thermal activation at a certain critical temperature. Since this behavior is traceable to the soft breaking of conformal invariance by the mass term in the sigma model, semiclassical winding number transition amplitudes in the electroweak theory in 3+1 dimensions should exhibit a similar sharp crossover. We argue that this is indeed the case in the standard model for M_H < 4 M_W.
We critically examine the recent claim (gr-qc/9603008) of a ``new effect'' of gravitationally induced quantum mechanical phases in neutrino oscillations. A straightforward exercise in the Schwarzschild coordinates appropriate to a spherically symmetric non-rotating star shows that, although there is a general relativistic effect of the star's gravity on neutrino oscillations, it is not of the form claimed, and is too small to be measured.
The time evolution of a closed system of mean fields and fluctuations is Hamiltonian, with the canonical variables parameterizing the general time-dependent Gaussian density matrix of the system. Yet, the evolution manifests both quantum decoherence and apparent irreversibility of energy flow from the coherent mean fields to fluctuating quantum modes. Using scalar QED as an example we show how this collisionless damping and decoherence may be understood as the result of {\em dephasing} of the rapidly varying fluctuations and particle production in the time varying mean field.
It has long been recognized that the dynamics of linear quantum systems is classical in the Wigner representation. Yet many conceptually important linear problems are typically analyzed using such generally applicable techniques as influence functionals and Bogoliubov transformations. In this Letter we point out that the classical equations of motion provide a simpler and more intuitive formalism for linear quantum systems. We examine the important problem of Brownian motion in the independent oscillator model, and show that the quantum dynamics is described directly and completely by a c-number Langevin equation. We are also able to apply recent insights into quantum Brownian motion to show that the classical Fokker-Planck equation is always local in time, regardless of the spectral density of the environment.
We study the thermodynamics of kinks in the Phi^6 model using a Langevin code implemented on a massively parallel computer. This code can be used to study first order dynamical phase transitions which exhibit multiple length and time scales. The classical statistical mechanics of a 1+1-dimensional field theory reduces to a time-independent quantum problem in one dimension via the transfer integral method. Exact solutions of the Schrodinger equation exist for the Phi^6 potential (unlike the case for Phi^4) and can be used to check results from the simulations. The Phi^6 model is also much richer than the Phi^4 model in terms of the variety of coherent structures and possible phases accompanying a phase transition. Specifically, we have calculated (in a one dimensional model) such quantities as the probability density function (PDF) and field-field correlation functions. These quantities help us understand the contribution to the specific heat from coherent structures such as domain walls (kinks) and other transformation structures as opposed to the contribution from lattice vibrations. We have calibrated our results against known exact solutions for limiting cases with very high accuracy. Having understood this problem, we are now extending our Langevin code to higher dimensions.
Some recent investigations of the thermal equilibrium properties of kinks in a $1+1$-dimensional, classical $Φ^4$ field theory are reviewed. The distribution function, kink density, correlation function, and certain thermodynamic quantities were studied both theoretically and via large scale simulations. A simple double Gaussian variational approach within the transfer operator formalism was shown to give good results in the intermediate temperature range where the dilute gas theory is known to fail.
We consider winding number transitions in the two dimensional O(3) non-linear sigma model, modified by a suitable conformal symmetry breaking term. We discuss the general properties of the relevant instanton solutions which dominate the transition amplitudes at finite energy, and find the solutions numerically. The Euclidean period of the solution increases with energy, contrary to the behavior found in the abelian Higgs model or simple one dimensional systems. This indicates that there is a sharp crossover from instanton dominated tunneling to sphaleron dominated thermal activation at a certain critical temperature in this model. We argue that the electroweak theory in four dimensions should exhibit a similar behavior.
We consider the dynamical system consisting of a quantum degree of freedom $A$ interacting with $N$ quantum oscillators described by the Lagrangian \bq L = {1\over 2}\dot{A}^2 + \sum_{i=1}^{N} \left\{{1\over 2}\dot{x}_i^2 - {1\over 2}( m^2 + e^2 A^2)x_i^2 \right\}. \eq In the limit $N \rightarrow \infty$, with $e^2 N$ fixed, the quantum fluctuations in $A$ are of order $1/N$. In this limit, the $x$ oscillators behave as harmonic oscillators with a time dependent mass determined by the solution of a semiclassical equation for the expectation value $\VEV{A(t)}$. This system can be described, when $\VEV{x(t)}= 0$, by a classical Hamiltonian for the variables $G(t) = \VEV{x^2(t)}$, $\dot{G}(t)$, $A_c(t) = \VEV{A(t)}$, and $\dot{A_c}(t)$. The dynamics of this latter system turns out to be chaotic. We propose to study the nature of this large-$N$ limit by considering both the exact quantum system as well as by studying an expansion in powers of $1/N$ for the equations of motion using the closed time path formalism of quantum dynamics.
A recently developed method for the calculation of Lyapunov exponents of dynamical systems is described. The method is applicable whenever the linearized dynamics is Hamiltonian. By utilizing the exponential representation of symplectic matrices, this approach avoids the renormalization and reorthogonalization procedures necessary in usual techniques. It is also easily extendible to damped systems. The method is illustrated by considering two examples of physical interest: a model system that describes the beam halo in charged particle beams and the driven van der Pol oscillator.
We consider a simple quantum system subjected to a classical random force. Under certain conditions it is shown that the noise-averaged Wigner function of the system follows an integro-differential stochastic Liouville equation. In the simple case of polynomial noise-couplings this equation reduces to a generalized Fokker-Planck form. With nonlinear noise injection new ``quantum diffusion'' terms arise that have no counterpart in the classical case. Two special examples that are not of a Fokker-Planck form are discussed: the first with a localized noise source and the other with a spatially modulated noise source.
ABBREVIATED ABSTRACT: This paper summarises an investigation of the effects of weak friction and noise in time-independent, nonintegrable potentials which admit both regular and stochastic orbits. The aim is to understand the qualitative effects of internal and external irregularities associated, e.g., with discreteness effects or couplings to an external environment, which stars in any real galaxy must experience. The two principal conclusions are: (1) These irregularities can be important on time scales much shorter than the natural relaxation time scale t_R associated with the friction and noise. For stochastic orbits friction and noise induce an average exponential divergence from the unperturbed Hamiltonian trajectory at a rate set by the value of the local Lyapunov exponent. Even weak noise can make a pointwise interpretation of orbits suspect already on time scales much shorter than t_R. (2) The friction and noise can also have significant effects on the statistical properties of ensembles of stochastic orbits, these also occurring on time scales much shorter than t_R. Potential implications for galactic dynamics are discussed, including the problem of shadowing.
Results are reported from a numerical investigation of orbits in a truncated Toda potential which is perturbed by weak friction and noise. Two significant conclusions are shown to emerge: (1) Despite other nontrivial behaviour, configuration, velocity, and energy space moments associated with these perturbations exhibit a simple scaling in the amplitude of the friction and noise. (2) Even very weak friction and noise can induce an extrinsic diffusion through cantori on a time scale much shorter than that associated with intrinsic diffusion in the unperturbed system.
The Lyapunov exponents of a chaotic system quantify the exponential divergence of initially nearby trajectories. For Hamiltonian systems the exponents are related to the eigenvalues of a symplectic matrix. We make use of this fact to develop a new method for the calculation of Lyapunov exponents of such systems. Our approach avoids the renormalization and reorthogonalization of usual techniques. It is also easily extendible to damped systems. We apply our method to two examples of physical interest: a model system that describes the beam halo in charged particle beams and the driven van der Pol oscillator.
LAGEOS is an accurately-tracked, dense spherical satellite covered with 426 retroreflectors. The tracking accuracy is such as to yield a medium term (years to decades) inertial reference frame determined via relatively inexpensive observations. This frame is used as an adjunct to the more difficult and data intensive VLBI absolute frame measurements. There is a substantial secular precession of the satellite's line of nodes consistent with the classical, Newtonian precession due to the non-sphericity of the earth. Ciufolini has suggested the launch of an identical satellite (LAGEOS-3) into an orbit supplementary to that of LAGEOS-1: LAGEOS-3 would then experience an equal and opposite classical precession to that of LAGEOS-1. Besides providing a more accurate real-time measurement of the earth's length of day and polar wobble, this paired-satellite experiment would provide the first direct measurement of the general relativistic frame-dragging effect. Of the five dominant error sources in this experiment, the largest one involves surface forces on the satellite, and their consequent impact on the orbital nodal precession. The surface forces are a function of the spin dynamics of the satellite. Consequently, we undertake here a theoretical effort to model the spin ndynamics of LAGEOS. In this paper we present our preliminary results.