Parallel algorithm with spectral convergence for nonlinear integro-differential equations

We discuss a numerical algorithm for solving nonlinear integro-differential equations, and illustrate our findings for the particular case of Volterra type equations. The algorithm combines a perturbation approach meant to render a linearized version of the problem and a spectral method where unknown functions are expanded in terms of Chebyshev polynomials (El-gendi's method). This approach is shown to be suitable for the calculation of two-point Green functions required in next to leading order studies of time-dependent quantum field theory.Comment: 15 pages, 9 figure

Hydrodynamic transport functions from quantum kinetic theory

Starting from the quantum kinetic field theory [E. Calzetta and B. L. Hu, Phys. Rev. D37, 2878 (1988)] constructed from the closed-time-path (CTP), two-particle-irreducible (2PI) effective action we show how to compute from first principles the shear and bulk viscosity functions in the hydrodynamic-thermodynamic regime. For a real scalar field with $\lambda \Phi ^{4}$ self-interaction we need to include 4 loop graphs in the equation of motion. This work provides a microscopic field-theoretical basis to the ``effective kinetic theory'' proposed by Jeon and Yaffe [S. Jeon and L. G. Yaffe, Phys. Rev. D53, 5799 (1996)], while our result for the bulk viscosity reproduces their expression derived from linear response theory and the imaginary-time formalism of thermal field theory. Though unavoidably involved in calculations of this sort, we feel that the approach using fundamental quantum kinetic field theory is conceptually clearer and methodically simpler than the effective kinetic theory approach, as the success of the latter requires clever rendition of diagrammatic resummations which is neither straightforward nor failsafe. Moreover, the method based on the CTP-2PI effective action illustrated here for a scalar field can be formulated entirely in terms of functional integral quantization, which makes it an appealing method for a first-principles calculation of transport functions of a thermal non-abelian gauge theory, e.g., QCD quark-gluon plasma produced from heavy ion collisions.Comment: 25 pages revtex, 11 postscript figures. Final version accepted for publicatio

Time evolution of the chiral phase transition during a spherical expansion

We examine the non-equilibrium time evolution of the hadronic plasma produced in a relativistic heavy ion collision, assuming a spherical expansion into the vacuum. We study the $O(4)$ linear sigma model to leading order in a large-$N$ expansion. Starting at a temperature above the phase transition, the system expands and cools, finally settling into the broken symmetry vacuum state. We consider the proper time evolution of the effective pion mass, the order parameter $\langle \sigma \rangle$, and the particle number distribution. We examine several different initial conditions and look for instabilities (exponentially growing long wavelength modes) which can lead to the formation of disoriented chiral condensates (DCCs). We find that instabilities exist for proper times which are less than 3 fm/c. We also show that an experimental signature of domain growth is an increase in the low momentum spectrum of outgoing pions when compared to an expansion in thermal equilibrium. In comparison to particle production during a longitudinal expansion, we find that in a spherical expansion the system reaches the ``out'' regime much faster and more particles get produced. However the size of the unstable region, which is related to the domain size of DCCs, is not enhanced.Comment: REVTex, 20 pages, 8 postscript figures embedded with eps

Numerical Approximations Using Chebyshev Polynomial Expansions

We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N (El-gendi's method). The solutions are exact at these points, apart from round-off computer errors and the convergence of other numerical methods used in connection to solving the linear system of equations. Applications to initial value problems in time-dependent quantum field theory, and second order boundary value problems in fluid dynamics are presented.Comment: minor wording changes, some typos have been eliminate

Neutrino collective excitations in the Standard Model at high temperature

Neutrino collective excitations are studied in the Standard Model at high temperatures below the symmetry breaking scale. Two parameters determine the properties of the collective excitations: a mass scale $m_\nu=gT/4$ which determines the \emph{chirally symmetric} gaps in the spectrum and $\Delta=M^2_W(T)/2m_\nu T$. The spectrum consists of left handed negative helicity quasiparticles, left handed positive helicity quasiholes and their respective antiparticles. For $\Delta < \Delta_c = 1.275...$ there are two gapped quasiparticle branches and one gapless and two gapped quasihole branches, all but the higher gapped quasiparticle branches terminate at end points. For $\Delta_c < \Delta < \pi/2$ the quasiparticle spectrum features a pitchfork bifurcation and for $\Delta >\pi/2$ the collective modes are gapless quasiparticles with dispersion relation below the light cone for $k\ll m_\nu$ approaching the free field limit for $k\gg m_\nu$ with a rapid crossover between the soft non-perturbative to the hard perturbative regimes for $k\sim m_\nu$.The \emph{decay} of the vector bosons leads to a \emph{width} of the collective excitations of order $g^2$ which is explicitly obtained in the limits $k =0$ and $k\gg m_\nu \Delta$. At high temperature this damping rate is shown to be competitive with or larger than the collisional damping rate of order $G^2_F$ for a wide range of neutrino energy.Comment: 32 pages 16 figs. Discussion on screening corrections. Results unchanged to appear in Phys. Rev.

The Role of Nonequilibrium Dynamical Screening in Carrier Thermalization

We investigate the role played by nonequilibrium dynamical screening in the thermalization of carriers in a simplified two-component two-band model of a semiconductor. The main feature of our approach is the theoretically sound treatment of collisions. We abandon Fermi's Golden rule in favor of a nonequilibrium field theoretic formalism as the former is applicable only in the long-time regime. We also introduce the concept of nonequilibrium dynamical screening. The dephasing of excitonic quantum beats as a result of carrier-carrier scattering is brought out. At low densities it is found that the dephasing times due to carrier-carrier scattering is in picoseconds and not femtoseconds, in agreement with experiments. The polarization dephasing rates are computed as a function of the excited carrier density and it is found that the dephasing rate for carrier-carrier scattering is proportional to the carrier density at ultralow densities. The scaling relation is sublinear at higher densities, which enables a comparison with experiment.Comment: Revised version with additional refs. 12 pages, figs. available upon request; Submitted to Phys. Rev.

Renormalization of initial conditions and the trans-Planckian problem of inflation

Understanding how a field theory propagates the information contained in a given initial state is essential for quantifying the sensitivity of the cosmic microwave background to physics above the Hubble scale during inflation. Here we examine the renormalization of a scalar theory with nontrivial initial conditions in the simpler setting of flat space. The renormalization of the bulk theory proceeds exactly as for the standard vacuum state. However, the short distance features of the initial conditions can introduce new divergences which are confined to the surface on which the initial conditions are imposed. We show how the addition of boundary counterterms removes these divergences and induces a renormalization group flow in the space of initial conditions.Comment: 22 pages, 4 eps figures, uses RevTe

Non-Equilibrium Quantum Fields in the Large N Expansion

An effective action technique for the time evolution of a closed system consisting of one or more mean fields interacting with their quantum fluctuations is presented. By marrying large $N$ expansion methods to the Schwinger-Keldysh closed time path (CTP) formulation of the quantum effective action, causality of the resulting equations of motion is ensured and a systematic, energy conserving and gauge invariant expansion about the quasi-classical mean field(s) in powers of $1/N$ developed. The general method is exposed in two specific examples, $O(N)$ symmetric scalar \l\F^4 theory and Quantum Electrodynamics (QED) with $N$ fermion fields. The \l\F^4 case is well suited to the numerical study of the real time dynamics of phase transitions characterized by a scalar order parameter. In QED the technique may be used to study the quantum non-equilibrium effects of pair creation in strong electric fields and the scattering and transport processes in a relativistic $e^+e^-$ plasma. A simple renormalization scheme that makes practical the numerical solution of the equations of motion of these and other field theories is described.Comment: 43 pages, LA-UR-94-783 (PRD, in press), uuencoded PostScrip

Quantum kinetics and thermalization in a particle bath model

We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and Non-Markovian quantum kinetics. There are two different cases that are studied in detail: i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and ii) a stable renormalized `particle' state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.Comment: 39 pages, RevTex, 14 figures (13 eps figures

High-field noise in metallic diffusive conductors

We analyze high-field current fluctuations in degenerate conductors by mapping the electronic Fermi-liquid correlations at equilibrium to their semiclassical non-equilibrium form. Our resulting Boltzmann description is applicable to diffusive mesoscopic wires. We derive a non-equilibrium connection between thermal fluctuations of the current and resistive dissipation. In the weak-field limit this is the canonical fluctuation- dissipation theorem. Away from equilibrium, the connection enables explicit calculation of the excess ``hot-electron'' contribution to the thermal spectrum. We show that excess thermal noise is strongly inhibited by Pauli exclusion. This behaviour is generic to the semiclassical metallic regime.Comment: 13 pp, one fig. Companion paper to cond-mat/9911251. Final version, to appear in J. Phys.: Cond. Ma