Project Results      

In FY 98/99/00 our total allocation has been insufficient to run many of our large codes in true production mode. In FY 98 we focused on transfering most of our codes from the CM-5 to the T3E and obtaining reasonable performance. We succeeded in porting almost of all our older CM-5 codes to the T3E. New code development with a view towards future production capability was also carried out especially in the area of Master equation solvers. [For more details on the codes, see Code Descriptions.] In FY 99 we focused mainly on dynamics of nonlinear coherent structures, initial value problems in the early Universe, and dynamics of open quantum systems. In FY00, we carried out quantum control simulations and careful studies on kink nucleation and dynamics.

Due to resource limitations, we have so far concentrated on running problems that do not require a very large investment of computer time but are nevertheless important to our research program. This has already resulted in several publications, four of them in Physical Review Letters (and one more in submission).

We studied 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, which was found to be in striking agreement with high resolution Langevin simulations (carried out at NERSC) [See Figure ]. Interesting connections with the Landau-Ginzburg and double sine-Gordon models were also established.

One of the main results of this work was the validation, via comparison with nontrivial exact results, of the high accuracy of the Langevin simulations. This allows the use of the numerically determined probability distribution function to directly compute several thermodynamic quantities thus providing an alternative to conventional methods based on fluctuations.

Reference(s): Statistical Mechanics of Double sinh-Gordon Kinks, S. Habib, A. Khare, and A. Saxena, Physica D 123, 341 (1998) cond-mat/9808008; Exact Thermodynamics of the Double sinh-Gordon Theory in 1+1-Dimensions, A. Khare, S. Habib, and A. Saxena, Phys. Rev. Lett. 79, 3797 (1997) cond-mat/9707222

We investigated the nucleation, annihilation, and dynamics of kinks in a classical 1+1 dimensional Landau-Ginzburg field theory at finite temperature. From large scale Langevin simulations, we established that the nucleation rate is proportional to the square of the equilibrium density of kinks. We identified two annihilation time scales: one due to kink-antikink pair recombination after nucleation, the other from non-recombinant annihilation. We introduced a mesoscopic model of diffusing kinks based on ``paired'' and ``survivor'' kinks/antikinks. Analytical predictions for the dynamical time scales, as well as the corresponding length scales, were in good agreement with the simulations. This work settled a controversy (both analytically and numerically) regarding the kink nucleation rate and also provided a new framework on which to build a macrscopic rate theory for this system. The identification and physical importance of multiple length and time scales is also a new feature.

Stochastic evolutions of classical field theories have recently become popular in the study of problems such as determination of the rates of topological transitions and the statistical mechanics of nonlinear coherent structures. To obtain high precision results from numerical calculations, a careful accounting of spacetime discreteness effects is essential, as well as the development of schemes to systematically improve convergence to the continuum. Using the field theory above as the application arena, we presented such an analysis for a 1+1-dimensional Langevin system, including a simple scheme to improve the order of spatial lattice errors. Analytical predictions and results from high resolution numerical solutions were found to be in excellent agreement.

Reference(s): Dynamics of Kinks: Nucleation, Diffusion and Annihilation, S. Habib and G. Lythe, Phys. Rev. Lett. 84, 1070 (2000) cond-mat/9911228"; Controlling One-Dimensional Langevin Dynamics on the Lattice, L.M.A. Bettencourt, S. Habib, and G. Lythe, Phys. Rev. D 60, 105039 (1999) hep-lat/9903007

We have applied our earlier work to the more general problem of reaction=diffusion in one dimension by mapping the kink problem onto the A+A -> 0 problem of interest to chemical physicists. We have found several new results here, including corrections to previous work regarding the proper treatment of paired nucleation.

Reference(s): Reaction-Diffusion in One Dimension: Paired and Unpaired Nucleation, Salman Habib, Katja Lindenberg, Grant Lythe, Carmen Molina-Paris, Phys. Rev. E (to be submitted)

We carried out an analytical and numerical study of the motion of an isolated vortex, defined as a point singularity of a complex scalar field obeying a nonlinear stochastic Schrodinger (Gross-Pitaevski) equation. It turns out that the vortex does not execute a simple random walk and the probability distribution of vortex flights has non-Gaussian (exponential) tails. We quantified these effects and explained their origin as due to the convective transport of the vortex by the random flow field of thermal excitations. These results present a dynamical picture of the vortex both as (1) a passive scalar advected to random background superflow and (2) as a heavy Brownian particle. We will next study multiple-vortex effects, the possibility of modifying the phonon spectrum to correct for quantum effects, and extension to higher dimensions.

Reference(s): Thermal Vortex Motion in a Two-Dimensional Condensate, R. Sasik, L.M.A. Bettencourt, and S. Habib, Phys. Rev. B 62, 1238 (2000) cond-mat/9907501

Gauge theories of the strong and electroweak interactions are characterized by a multiple vacuum structure. Tunneling transitions between different vacua are responsible for physically interesting effects, such as baryon number violation in the electroweak theory. At finite temperatures thermal activation over the potential barrier separating the multiple vacua can take place, in addition to quantum tunneling. The static classical solution whose energy is equal to the top of this barrier between neighboring vacua is called the sphaleron. At sufficiently high temperatures (but not so high that symmetry restoration occurs), transitions between different winding number sectors are dominated not by quantum tunneling but by classical thermal activation, with a rate controlled by the energy of the sphaleron. As the temperature is lowered, the topological transition rate gets contributions both from quantum tunneling and thermal activation. The rate can now be controlled by classical solutions of the Euclidean field equations with period equal to the inverse temperature. These solutions are called periodic instantons. While their properties have been previously studied in simpler models, very little is known about the situation in the electroweak theory.

The difficulty in finding these solutions in a very nonlinear regime of the theory is such that analytic techniques have very limited utility and one must of necessity resort to numerical methods. To make the problem tractable, we employed a reduction of the four dimensional SU(2)-Yang-Mills-Higgs theory (i.e. the bosonic sector of the of the standard electroweak theory with theta_W=0) to two dimensions by assuming spherical symmetry. Our numerical exploration is now almost complete. The main results are as follows: (1) bifurcations expected from some of our earlier work were found numerically, (2) the associated complex solutions were also found, (3) a perturbative calculation for the periodic instanton action and energy at high temperatures was in good agreement with the numerical calculations. The rich structure of bifurcating periodic instanton solutions promises interesting physical consequences for B and L violating transitions in the electroweak theory. These ramifications are now under investigation.

Reference(s): Periodic Instantons in SU(2) Yang-Mills-Higgs Theory, G. Bonini, S. Habib, E. Mottola, C. Rebbi, R. Singleton, and P. Tinyakov, Phys. Lett. B 474, 113 (2000) hep-ph/9905243

We worked out a general formulation of the time-dependent initial value problem for a quantum scalar field of arbitrary mass and curvature coupling in a FRW cosmological model. We introduced an adiabatic number basis which has the virtue that the divergent parts of the quantum expectation value of the energy-momentum tensor are isolated in the vacuum piece of , and may be removed using adiabatic subtraction. The resulting renormalized is conserved, independent of the cutoff, and has a physically transparent, quasiclassical form in terms of the average number of created adiabatic `particles'. By analyzing the evolution of the adiabatic particle number in de Sitter spacetime we exhibited the time structure of the particle creation process, which can be understood in terms of the time at which different momentum scales enter the horizon. A numerical scheme to compute as a function of time with arbitrary adiabatic initial states (not necessarily de Sitter invariant) was introduced. We found that for minimally coupled, massless fields, at late times the renormalized goes asymptotically to the de Sitter invariant state previously found by Allen and Folacci, and not to the zero mass limit of the Bunch-Davies vacuum. If the mass m and the curvature coupling xi differ from zero, but satisfy m^2+xi R=0, the energy density and pressure of the scalar field grow linearly in cosmic time demonstrating that, at least in this case, quantum backreaction effects become significant and cannot be neglected in de Sitter spacetime. Ramifications of this work include applications to inflation, the problem of the cosmological constant, and semiclassical quantum gravity.

Reference(s): Energy-Momentum Tensor of Particles Created in an Expanding Universe, S. Habib, C. Molina-Paris, and E. Mottola, Phys. Rev. D 61, 024010 (1999) gr-qc/9906120

We extended our work from the previous year and proved a theorem regarding the existence of attractor states in de Sitter space. The existence of this theorem was first suggested by numerical simulations carried out by us at NERSC. We proved quite generally that the expectation value of the energy-momentum tensor has a fixed point attractor behavior at late times, which depends only on the mass and coupling to gravity, for any (initial) fourth order adiabatic state that is infrared finite.

Reference(s): Attractor States and Infrared Scaling in de Sitter Space, Paul R. Anderson, Wayne Eaker, Salman Habib, Carmen Molina-Paris, and Emil Mottola, Phys. Rev. D (in press) gr-qc/0005102

Quantum dynamics of nonlinear systems has associated with it several interesting issues such as the status of the correspondence principle, the signatures of classical chaos, the existence of various relaxation mechanisms, and effects due to quantum decoherence. Studying these problems in the full field theoretic setting can only be done via approximations. However, much may be learnt by first concentrating on the case of single particle quantum mechanics. We have recently presented evidence that decoherence can produce a smooth quantum-to-classical transition in nonlinear dynamical systems. High-resolution tracking of quantum and classical evolutions was used to reveal differences in expectation values of corresponding observables in the two evolutions. Solutions of master equations were then used to demonstrate that decoherence destroys quantum interference in Wigner distributions and washes out fine structure in classical distributions in such a way as to bring the two closer together [See Figure ]. Consequently, the correspondence between quantum and classical expectation values is also re-established. Tools employed were very high-resolution simulations of the time-dependent Schrodinger, quantum and classical Liouville, and master equations implemented on massively parallel computers (the CM-5 at Los Alamos and the T3E at NERSC).

Reference(s): Decoherence, Chaos, and the Correspondence Principle, S. Habib, K. Shizume, and W.H. Zurek, Phys. Rev. Lett. 80, 4361 (1998) quant-ph/9803042

We formulated the conditions under which the dynamics of a continuously measured quantum system becomes indistinguishable from that of the corresponding classical system. In particular, we demonstrated that even in a classically chaotic system the quantum state vector conditioned by the measurement remains localized and, under these conditions, follows a trajectory characterized by the classical Lyapunov exponent. We have a host of new results in the very interesting quantum/classical intermediate dynamical regime, these are now being written up. Some of these results can be tested in quantum optics experiments.

Reference(s): Continuous Quantum Measurement and the Emergence of Classical Chaos, T. Bhattacharya, S. Habib, and K. Jacobs, Phys. Rev. Lett. (submitted) quant-ph/9906092

A longer paper on our work in FY99 is in preparation. We have also almost completed a new paper which aims at establishing a classification of dynamical systems based on violations of the quantum-classical correspondence. A central result of this work is the realization that quantum decoherence can succeed in producing (almost) positive Wigner distributions, yet the dynamics that connect two such positive, and hence classically interpretable distributions, need not be classical.

There is now considerable interest in the computation of transport coefficients in quantum field theories by attempting to exploit possible connections with the underlying Lyapunov exponents of the classical lattice theory. We have found a new method for the computation of Lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of M or MM^T 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. Since our method is based on exact differential equations for the Lyapunov exponents, global invariances of the Lyapunov spectrum can be preserved. Finally, since the structure of the coupled differential equations is of a special form, they may also turn out to be useful for analytic studies of evolution in tangent space.

Simulations on the T3E were used to study long time convergence properties of the Lyapunov exponents. These simulations were also used to establish the property of exact preservation of invariances in the Lyapunov spectra.

Reference(s): Lyapunov Exponents without Rescaling and Reorthogonalization, G. Rangarajan, S. Habib, and R.D. Ryne, Phys. Rev. Lett. 80, 3747 (1998) chao-dyn/9803017

An extended version of our work from the previous year is now available. We have extended our method to discrete maps and provided full proofs of the previously reported results.

Reference(s): Computation of Lyapunov Spectrum for Continuous-Time Dynamical Systems and Discrete Maps, T.M. Janaki, G. Rangarajan, S. Habib, and R.D. Ryne, Phys. Rev. E 60, 6614 (1999)

We wrote a new code for the simulation of a measurement-based stochastic master equation for single-atom cavity quantum electrodynamics (QED). For integrating such an equation this code is the first of its kind. In order to keep the integration accurate and stable, it was necessary to use a semi-implicit Milstein integration algorithm and a timestep on the order of one thousandth the largest term in the equation. Unlike most stochastic integration scenarios, we had to achieve strong convergence in order to produce accurate individual trajectories (as opposed to accurate reproduction of statistical moments). The purpose of the code is to simulate the conditional evolution of an open quantum system, under conditions of continuous but partial observation. A rigorous mathematical basis for deriving the equations of motion has only recently been established, and experiments in quantum optics are just beginning to be able to investigate this type of physical system. Our intention is to use this code for the development of schemes for quantum feedback control of open quantum systems, which will provide crucial guidance for initial experimental work in this area. During the past year we have validated the code by comparison with predictions based on analytic simplifications of the full equations of motion.

Due to resource limitations we could not carry out large scale simulations in FY 99. However, a new and more efficient version of the code is now under development and should be in production mode by late Fall 1999. The corresponding experiments should be online during Winter 1999.

A new and completely rewritten code for this problem was developed. We have F90/MPI and C++/MPI versions which are much more efficient than the previous code. However, production runs were not carried out at NERSC due to resource limitations (this work was carried out at the ACL).

Salman Habib / LANL / habib@lanl.gov / revised August 00