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The dynamics of symmetry breaking during out of equilibrium phase transitions is a topic of great importance in many disciplines, from condensed matter to particle physics and early Universe cosmology with definite experimental impact. In these notes we provide a summary of the relevant aspects of the dynamics of symmetry breaking in many different fields with emphasis on the experimental realizations. In condensed matter we address the dynamics of phase ordering, the emergence of condensates, coarsening and dynamical scaling. In QCD the possibility of disoriented chiral condensates of pions emerging during a strongly out of equilibrium phase transition is discussed. We elaborate on the dynamics of phase ordering in phase transitions in the Early Universe, in particular the emergence of condensates and scaling in FRW cosmologies. We mention some experimental efforts in different fields that study this wide ranging phenomena and offer a quantitative theoretical description both at the phenomenological level in condensed matter, introducing the scaling hypothesis as well as at a microscopic level in quantum field theories. The emergence of semiclassical condensates and a dynamical length scale is shown in detail, in quantum field theory this length scale is constrained by causality. The large N limit provides a natural bridge to compare the solutions in different settings and to establish similarities and differences., Comment: 35 pages, Revtex, 10 figures. Invited by the National Academy of Sciences of India, to appear in ANSI-2000

We derive quantum kinetic equations from a quantum field theory implementing a diagrammatic perturbative expansion improved by a resummation via the dynamical renormalization group. The method begins by obtaining the equation of motion of the distribution function in perturbation theory. The solution of this equation of motion reveals secular terms that grow in time, the dynamical renormalization group resums these secular terms in real time and leads directly to the quantum kinetic equation. We used this method to study the relaxation in a cool gas of pions and sigma mesons in the O(4) chiral linear sigma model. We obtain in relaxation time approximation the pion and sigma meson relaxation rates. We also find that in large momentum limit emission and absorption of massless pions result in threshold infrared divergence in sigma meson relaxation rate and lead to a crossover behavior in relaxation. We then study the relaxation of charged quasiparticles in scalar electrodynamics (SQED). While longitudinal, Debye screened photons lead to purely exponential relaxation, transverse photons, only dynamically screened by Landau damping lead to anomalous relaxation, thus leading to a crossover between two different relaxational regimes. We emphasize that infrared divergent damping rates are indicative of non-exponential relaxation and the dynamical renormalization group reveals the correct relaxation directly in real time. Finally we also show that this method provides a natural framework to interpret and resolve the issue of pinch singularities out of equilibrium and establish a direct correspondence between pinch singularities and secular terms. We argue that this method is particularly well suited to study quantum kinetics and transport in gauge theories., Comment: RevTeX, 40 pages, 4 eps figures, published version

We study the generation of a non-equilibrium plasma in scalar QED with N charged scalar fields through spinodal instabilities in the case of a super cooled second order phase transition and parametric amplification when the order parameter oscillates with large amplitude around the minimum of the potential.The focus is to study the non-equilibrium electromagnetic properties of the plasma, such as photon production, electric and magnetic screening and conductivity. A novel kinetic equation is introduced to compute photon production far away from equilibrium in the large N limit and lowest order in the electromagnetic coupling.During the early stages of the dynamics the photon density grows exponentially and asymptotically the amplitude and frequency distribution becomes \sim alpha m^2/[lambda^2 \omega^3] with lambda the scalar self-coupling and m the scalar mass.In the case of a phase transition,electric and magnetic fields are correlated on distances xi(t) \sim sqrt{t} during the early stages and the power spectrum is peaked at low momentum. Magnetic and Debye screening masses are defined out of equilibrium. The magnetic mass vanishes out of equilibrium in this abelian models. The Debye mass turns to be m^2_{Deb} \sim alpha m^2/lambda for a supercooled phase transition while in the case of an oscillating order parameter an interpolating time dependent Debye mass grows as alpha sqrt{m t}/lambda due to a non-linear resonance at low momentum in the charged particle distribution. The build-up of the transverse electric conductivity is studied during the formation of the non-equilibrium plasma. Its long wavelength limit reaches a value sigma_{k\approx 0} \sim alpha m/lambda at the end of the stage of linear instabilities. Its asymptotic form in terms of the non-equilibrium particle distribution functions is derived., Comment: LaTex, 57 pages, 11 .ps figures, improved and completed version to appear in Phys. Rev. D

The real time non-equilibrium evolution of condensates in field theory requires an initial value problem specifying an initial quantum state or density matrix. Arbitrary specifications of the initial quantum state (pure or mixed) results in initial time singularities which are not removed by the usual renormalization counterterms. We study the initial time singularities in the linearized equation of motion for the scalar condensate in a renormalizable Yukawa theory in 3+1 dimensions. In this renormalizable theory the initial time singularities are enhanced. We present a consistent method for removing these initial time singularities by specifying initial states where the distribution of high energy quanta is determined by the initial conditions and the interaction effects. This is done through a Bogoliubov transformation which is consistently obtained in a perturbative expansion.The usual renormalization counterterms and the proper choice of the Bogoliubov coefficients lead to a singularity free evolution equation. We establish the relationship between the evolution equations in the linearized approximation and linear response theory. It is found that only a very specific form of the external source for linear response leads to a real time evolution equation which is singularity free. We focus on the evolution of spatially inhomogeneous scalar condensates by implementing the initial state preparation via a Bogoliubov transformation up to one-loop. As a concrete application, the evolution equation for an inhomogenous condensate is solved analytically and the results are carefully analyzed. Symmetry breaking by initial quantum states is discussed., Comment: LaTex, 26 pages, 2 .ps figures

We study the viscosity corrections to the growth rate of nucleating bubbles in a first order phase transition in scalar field theory. We obtain the non-equilibrium equation of motion of the coordinate that describes small departures from the critical bubble and extract the growth rate consistently in weak coupling and in the thin wall limit. Viscosity effects arise from the interaction of this coordinate with the stable quantum and thermal fluctuations around a critical bubble. In the case of 1+1 dimensions we provide an estimate for the growth rate that depends on the details of the free energy functional. In 3+1 dimensions we recognize robust features that are a direct consequence of the thin wall approximation and give the leading viscosity corrections.These are long-wavelength hydrodynamic fluctuations that describe surface waves, quasi-Goldstone modes which are related to ripples on interfaces in phase ordered Ising-like systems. We discuss the applicability of our results to describe the growth rate of hadron bubbles in a quark-hadron first order transition., Comment: 40 pages, 4 figures, revtex, minor changes, to be published in Phys. Rev. D

These lectures address the dynamics of phase ordering out of equilibrium in condensed matter and in quantum field theory in cosmological settings, emphasizing their similarities and differences. In condensed matter we describe the phenomenological approach based on the Time Dependent Ginzburg-Landau (TDGL) description. After a general discussion of the main experimental and theoretical features of phase ordering kinetics and the description of linear (spinodal) instabilities we introduce the scaling hypothesis and show how a dynamical correlation length emerges in the large N limit in condensed matter systems. The large N approximation is a powerful tool in quantum field theory that allows the study of non-perturbative phenomena in a consistent manner. We study the exact solution to the dynamics after a quench in this limit in Minkowski space time and in radiation dominated Friedman-Robertson-Walker Cosmology. There are some remarkable similarities between these very different settings such as the emergence of a scaling regime and of a dynamical correlation length at late times that describe the formation and growth of ordered regions. In quantum field theory and cosmology this length scale is constrained by causality and its growth in time is also associated with coarsening and the onset of a condensate. We provide a density matrix interpretation of the formation of defects and the classicalization of quantum fluctuations., Comment: 23 pages (revtex), 7 eps figures. Lectures delivered at the NATO Advanced Study Institute: Topological Defects and the Non-Equilibrium Dynamics of Symmetry Breaking Phase Transitions

We study the transport coefficients, damping rates and mean free paths of soft fermion collective excitations in a hot fermion-gauge-scalar plasma with the goal of understanding the main physical mechanisms that determine transport of chirality in scenarios of non-local electroweak baryogenesis. The focus is on identifying the different transport coefficients for the different branches of soft collective excitations of the fermion spectrum. These branches correspond to collective excitations with opposite ratios of chirality to helicity and different dispersion relations. By combining results from the hard thermal loop (HTL) resummation program with a novel mechanism of fermion damping through heavy scalar decay, we obtain a robust description of the different damping rates and mean free paths for the soft collective excitations to leading order in HTL and lowest order in the Yukawa coupling. The space-time evolution of wave packets of collective excitations unambiguously reveals the respective mean free paths. We find that whereas both the gauge and scalar contribution to the damping rates are different for the different branches, the difference of mean free paths for both branches is mainly determined by the decay of the heavy scalar into a hard fermion and a soft collective excitation. We argue that these mechanisms are robust and are therefore relevant for non-local scenarios of baryogenesis either in the Standard Model or extensions thereof., Comment: REVTeX, 19 pages, 4 eps figures, published version

We study the kinetics of the distribution function for charged particles of hard momentum in scalar QED. The goal is to understand the effects of infrared divergences associated with the exchange of quasistatic magnetic photons in the relaxation of the distribution function. We begin by obtaining a kinetic transport equation for the distribution function for hard charged scalars in a perturbative expansion that includes hard thermal loop resummation. Solving this transport equation, the infrared divergences arising from absorption and emission of soft quasi-static magnetic photons are manifest in logarithmic secular terms. We then implement the dynamical renormalization group resummation of these secular terms in the relaxation time approximation. The distribution function (in the linearized regime) is found to approach equilibrium as $\delta n_k(t) =\delta n_k(t_o) e^{-2\alpha T (t-t_o) \ln[(t-t_o)\bar{\mu}]}$, with $\bar{\mu}\approx \omega_p$ the plasma frequency and $\alpha =e^2/4\pi$. This anomalous relaxation is recognized to be the square of the relaxation of the single particle propagator, providing a generalization of the usual relation between the damping and the interaction rate. The renormalization group approach to kinetics reveals clearly the time scale $t_{rel} \approx (\alpha T \ln[1/\alpha])^{-1}$ arising from infrared physics and hinges upon the separation of scales $t_{rel} >>\omega_p^{-1}$., Comment: 16 pages, no figures

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)

We analyze the non-equilibrium dynamics of the O(N) Phi^4 model in the large N limit and for states of large energy density. The dynamics is dramatically different when the energy density is above the top of the tree level potential V_0 than when it is below it.When the energy density is below V_0, we find that non-perturbative particle production through spinodal instabilities provides a dynamical mechanism for the Maxwell construction. The asymptotic values of the order parameter only depend on the initial energy density and all values between the minima of the tree level potential are available, the asymptotic dynamical `effective potential' is flat between the minima. When the energy density is larger than V_0, the evolution samples ergodically the broken symmetry states, as a consequence of non-perturbative particle production via parametric amplification. Furthermore, we examine the quantum dynamics of phase ordering into the broken symmetry phase and find novel scaling behavior of the correlation function. There is a crossover in the dynamical correlation length at a time scale t_s \sim \ln(1/lambda). For t < t_s the dynamical correlation length \xi(t) \propto \sqrt{t} and the evolution is dominated by spinodal instabilities, whereas for t>t_s the evolution is non-linear and dominated by the onset of non-equilibrium Bose-Einstein condensation of long-wavelength Goldstone bosons.In this regime a true scaling solution emerges with a non- perturbative anomalous scaling length dimension z=1/2 and a dynamical correlation length \xi(t) \propto (t-t_s). The equal time correlation function in this scaling regime vanishes for r>2(t-t_s) by causality. For t > t_s the equal time correlation function falls of as 1/r. A semiclassical but stochastic description emerges for time scales t > t_s., Comment: Minor improvements, to appear in Phys. Rev. D. Latex file, 48 pages, 12 .ps figures

In this article we study the dynamics of fermions in a fermion-scalar plasma. We begin by obtaining the effective in-medium Dirac equation in real time which is fully renormalized and causal and leads to the initial value problem. For a heavy scalar we find the novel result that the decay of the scalar into fermion pairs in the medium leads to damping of the fermionic excitations and their in-medium propagation as quasiparticles. That is, the fermions acquire a width due to the decay of the heavier scalar in the medium. We find the damping rate to lowest order in the Yukawa coupling for arbitrary values of scalar and fermion masses, temperature and fermion momentum. An all-order expression for the damping rate in terms of the exact quasiparticle wave functions is established. A kinetic Boltzmann approach to the relaxation of the fermionic distribution function confirms the damping of fermionic excitations as a consequence of the induced decay of heavy scalars in the medium. A linearization of the Boltzmann equation near equilibrium clearly displays the relationship between the damping rate of fermionic mean fields and the fermion interaction rate to lowest order in the Yukawa coupling directly in real time., Comment: REVTEX, 16 pages, 3 eps figures included, published version. To be published in Phys. Rev. D

The real time evolution of field condensates is solved for small and large field amplitudes in scalar theories.For small amplitudes,the quantum equations of motion for the condensate can be linearized and solved by Laplace transform. The late time evolution turns to be determined by the singularities in the complex plane (one-particle poles, two- and multi- particle cuts, Landau cuts for non-zero initial temperature). In hot scalar electrodynamics, we solve the real time evolution of field condensates with soft length scales \sim k^{-1}>(eT)^{-1}. Transverse gauge invariant condensates relax as 1/t^2 to amplitudes determined by the quasiparticle poles. We rederive the HTL action using the non-equilibrium field theory techniques.In the nonlinear regime (for large initial energy densities) we analyze the dynamics of dissipation and relaxation in scalar theory after linear unstabilities are shut-off by the quantum back-reaction. A new time scale emerges that separates the linear from the non-linear regimes. This scale is non-perturbative in the coupling and initial amplitude. A combination of numerical analysis and the implementation of a dynamical renormalization group resummation via multi-time scale analysis reveals the presence of unstable bands in the nonlinear regime. These are associated with power law growth of quantum fluctuations, that result in power law relaxation and dissipation with {\bf non-universal and non-perturbative dynamical anomalous exponents., Comment: Latex file, 10 pages, 3 .eps figures. Lecture delivered at TFT98, the Vth International Workshop on Thermal Field Theories and Their Applications, Ratisbona (Regensburg), Germany, 10-14 August 1998

We introduce the method of dynamical renormalization group to study relaxation and damping out of equilibrium directly in real time and applied it to the study of infrared divergences in scalar QED. This method allows a consistent resummation of infrared effects associated with the exchange of quasistatic transverse photons and leads to anomalous logarithmic relaxation of the form $e^{-\alpha T t \ln[t/t_0]}$ which prevents a quasiparticle interpretation of charged collective excitations at finite temperature. The hard thermal loop resummation program is incorporated consistently into the dynamical renormalization group yielding a picture of relaxation and damping phenomena in a plasma in real time that trascends the conceptual limitations of the quasiparticle picture and other type of resummation schemes. We derive a simple criterion for establishing the validity of the quasiparticle picture to lowest order., Comment: REVTEX, 33 pages, 2 figures,improved discussion,more examples,same physics to appear in phys. rev. d

We study the dynamics of relaxation and thermalization in an exactly solvable model with the goal of understanding the effects of off-shell processes. The focus is to compare the exact evolution of the distribution function with different approximations to the relaxational dynamics: Boltzmann, non-Markovian and Markovian quantum kinetics. The time evolution of the distribution function is evaluated exactly using two methods: time evolution of an initially prepared density matrix and by solving the Heisenberg equations of motion. There are two different cases that are studied in detail: i) no stable particle states below threshold of the bath and a quasiparticle resonance above it and ii) a stable discrete exact `particle' state below threshold. For the case of quasiparticles in the continuum (resonances) the exact quasiparticle distribution asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes. In the case ii), the distribution of particles does not thermalize with the bath. We study the kinetics of thermalization and relaxation by deriving 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 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 of the model. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important., Comment: 49 pages, LaTex, 17 figures (16 eps figures)

We establish a self-consistent variational framework that allows us to study numerically the non-equilibrium evolution of non-perturbative inhomogeneous field configurations including quantum backreaction effects. After discussing the practical merits and disadvantages of different approaches we provide a closed set of local and renormalizable update equations that determine the dynamical evolution of inhomogeneous condensates and can be implemented numerically. These incorporate self-consistently the backreaction of quantum fluctuations and particle production. This program requires the solution of a self-consistent inhomogeneous problem to provide initial Cauchy data for the inhomogeneous condensates and Green's functions. We provide a simple solvable ansatz for such an initial value problem for the Sine-Gordon and phi^4 quantum field theories in one spatial dimension. We compare exact known results of the Sine Gordon model to this simple ansatz. We also study the linear sigma model in the large N limit in three spatial dimensions as a microscopic model for pion production in ultrarelativistic collisions. We provide a solvable self- consistent ansatz for the initial value problem with cylindrical symmetry. For this case we also obtain a closed set of local and renormalized update equations that can be numerically implemented. A novel phenomenon of spinodal instabilities and pion production arises as a result of a Klein paradox for large amplitude inhomogeneous condensate configurations., Comment: Latex file, 44 pages, 3 .eps figures. To appear in Phys. Rev. D. Improved presentation and references added

The energy and time scales during the inflationary stage of the universe calls for an out of equilibrium quantum field treatment. Moreover, the high energy densities involved make necessary the use of non-perturbative approaches as large N and Hartree methods. We start these lectures by introducing such non-perturbative out of equilibrium methods in cosmological universes. We discuss the renormalization procedure and the choice of initial conditions. We then study the nonlinear dynamics of quantum fields in matter and radiation dominated FRW and de Sitter universes. For a variety of initial conditions, we compute the evolution of the inflaton,its quantum fluctuations and the equation of state. We investigate the explosive particle production due to spinodal unstabilities and parametric amplification in FRW and de Sitter universes with and without symmetry breaking.We find that the particle production is sensitive to the expansion of the universe.For symmetry breaking scenarios, we determine generic late time fields behavior for FRW and deSitter cosmologies.We find that quantum fluctuations damp in FRW as the square of the scale factor while the order parameter approaches a minimum of the potential at the same rate.We con- sider an O(N) inflaton model coupled self-consistently to gravity in the semi- classical approximation for a `new inflation' scenario. We find that spinodal instabilities drive the growth of non-perturbatively large quantum fluctuations which shut off inflation. These fluctuations assemble with the inflaton zero mode yielding a new effective field that actually rolls down behaving classi- cally. We compute the amplitude and index for scalar density and tensor perturbations. In all models of this type the spinodal instabilities produce a `red' spectrum of primordial scalar density perturbations., Comment: Latex file, 76 pages, 26 figures in .eps files. To appear in the Proceedings of the VIth. Erice Chalonge School on Astrofundamental Physics, N. S\'anchez and A. Zichichi eds., Kluwer, 1998

We analyze the dynamics of dissipation and relaxation in the unbroken and broken symmetry phases of scalar theory in the nonlinear regime for large initial energy densities, and after linear unstabilities (parametric or spinodal) are shut-off by the quantum backreaction. A new time scale emerges that separates the linear from the non-linear regimes. This scale is non- perturbative in the coupling and initial amplitude. The non-perturbative evolution is studied in the large the N limit for the O(N) vector model. A combination of numerical analysis and the multitime scale analysis reveals the presence of unstable bands in the nonlinear regime. These are associated with power law growth of quantum fluctuations, that result in power law relaxation and dissipation with non-universal and non-perturbative dynamical anomalous exponents.We find substantial particle production during this non-linear evolu- tion which is of the same order as that in the linear regime. The expectation value of the scalar field vanishes asymptotically transferring all of the initial energy into produced particles via the non-linear resonances in the unbroken symmetry phase.The effective mass for the quantum modes tends asympto- tically to a constant plus oscillating O(1/t) terms. This slow decay causes the power behaviour in the modes which become free for t=infty. We derive a simple expression for the equation of state for the fluid of produced particles that interpolates between radiation-type and dust-type equations according to the initial value of the order parameter for unbroken symmetry. For broken symmetry the produced particles are massless Goldstone bosons with a radiation-type equation of state. We find the onset of a novel form of dynamical Bose condensation in the collisionless regime in the absence of thermalization., Comment: ReV Tex, 43 pages, 37 .ps figures

We study the non-equilibrium dynamics of domain walls in real time for $\phi^4$ and Sine Gordon models in 1+1 dimensions in the dilute regime. The equation of motion for the collective coordinate is obtained by integrating out the meson excitations around the domain wall to one-loop order. The real-time non-equilibrium relaxation is studied analytically and numerically to this order. The constant friction coefficient vanishes but there is dynamical friction and relaxation caused by off-shell non-Markovian effects. The validity of a Markovian description is studied in detail. The proper Langevin equation is obtained to this order, the noise is Gaussian and additive but colored. We analyze the classical and hard thermal loop contributions to the self-energy and noise kernels and show that at temperatures larger than the meson mass the hard contributions are negligible and the finite temperature contribution to the dynamics is governed by the classical soft modes of the meson bath. The long time relaxational dynamics is completely dominated by classical Landau damping resulting in that the corresponding time scales are not set by the temperature but by the meson mass. The noise correlation function and the dissipative kernel obey a generalized form of the Fluctuation-Dissipation relation., Comment: 39 pages, LaTex, 9 figures (3 EPS; 6 GIF), minor changes

We consider an O(N) model coupled self-consistently to gravity in the semiclassical approximation, where the field is subject to `new inflation' type initial conditions. We study the dynamics self-consistently and non-perturbatively with non-equilibrium field theory methods in the large N limit. We find that spinodal instabilities drive the growth of non-perturbatively large quantum fluctuations which shut off the inflationary growth of the scale factor. We find that a very specific combination of these large fluctuations plus the inflaton zero mode assemble into a new effective field. This new field behaves classically and it is the object which actually rolls down. We show how this reinterpretation saves the standard picture of how metric perturbations are generated during inflation and that the spinodal growth of fluctuations dominates the time dependence of the Bardeen variable for superhorizon modes during inflation. We compute the amplitude and index for the spectrum of scalar density and tensor perturbations and argue that in all models of this type the spinodal instabilities are responsible for a `red' spectrum of primordial scalar density perturbations. A criterion for the validity of these models is provided and contact with the reconstruction program is established validating some of the results within a non-perturbative framework. The decoherence aspects and the quantum to classical transition through inflation are studied in detail by following the full evolution of the density matrix and relating the classicality of cosmological perturbations to that of long-wavelength matter fluctuations., Comment: 39 pages, 12 epsf figures

We study the non-equilibrium dynamics of solitons in model Hamiltonians for Peierls dimerized quasi-one dimensional conducting polymers and commensurate charge density wave systems. The real time equation of motion for the collective coordinate of the soliton and the associated Langevin equation is found in a consistent adiabatic expansion in terms of the ratio of the optical phonon or phason frequency to the soliton mass. The equation of motion for the soliton collective coordinate allows to obtain the frequency dependent soliton conductivity. In lowest order we find that although the coefficient of static friction vanishes, there is dynamical dissipation represented by a non-Markovian dissipative kernel associated with two-phonon processes. The correlation function of the noise in the quantum Langevin equation and the dissipative kernel are related by a generalized quantum fluctuation dissipation relation. To lowest adiabatic order we find that the noise is gaussian, additive and colored. We numerically solve the equations of motion in lowest adiabatic order and compare to the Markovian approximation which is shown to fail both in the $\phi^4$ and the Sine Gordon models even at large temperatures., Comment: 44 pages, LaTex, 8 GIF figures, to be published in Phys. Rev. B

The physics of the inflationary universe requires the study of the out of equilibrium evolution of quantum fields in curved spacetime. We present the evolution for both the geometry and the matter (described by the quantum inflaton field) by means of the non-perturbative large N limit combined with semi-classical gravitational dynamics including the back-reaction of quantum fluctuations self-consistently for a new inflation scenario. We provide a criterion for the validity of the classical approximation and a full analysis of the case in which spinodal quantum fluctuations drive the evolution of the scale factor. Under carefully determined conditions, we show that the full field equations may be well approximated by those of a single composite field which obeys the classical equation of motion in all cases. The de Sitter stage is found to be followed by a matter dominated phase. We compute the spectrum of scalar density perturbations and argue that the spinodal instabilities are responsible for a `red' spectrum with more power at longer wavelengths. A criterion for the validity of these models is provided and contact with the reconstruction program is established., Comment: 26 pages, 5 epsf figures, to appear in the Proceedings of SEWM'97

We study a simplified version of the Standard Electroweak Model and introduce the concept of the physical gauge invariant effective potential in terms of matrix elements of the Hamiltonian in physical states. This procedure allows an unambiguous identification of the symmetry breaking order parameter and the resulting effective potential as the energy in a constrained state. We explicitly compute the physical effective potential at one loop order and improve it using the RG. This construction allows us to extract a reliable, gauge invariant bound on the Higgs mass by unambiguously obtaining the scale at which new physics should emerge to preclude vacuum instability. Comparison is made with popular gauge fixing procedures and an ``error'' estimate is provided between the Landau gauge fixed and the gauge invariant results., Comment: 23 pages, 2 figures, REVTEX

We study production of photons via the non-equilibrium relaxation of a Disoriented Chiral Condensate with the chiral order parameter having a large initial amplitude along the \pi^0 direction. Assuming the validity of the low energy coupling of the neutral pion to photons via the U_A(1) anomalous vertex, we find that for large initial amplitudes along the \pi^0 direction, photon production is enhanced by parametric amplification. These processes are non-perturbative with a large contribution during the non-equilibrium stages of the evolution and result in a distinct distribution of the produced photons and a polarization asymmetry. For initial amplitudes of the \pi^0 component of the order parameter between 200-400 MeV, corresponding to energy densities between 1-12 GeV/fm^3 we find a peak in the photon distribution at energies between \approx 300 -600 MeV. We also find polarization asymmetries typically between 5-10%. We discuss the potential experimental impact of these results as well as the problems associated with its detection., Comment: 36 pages, 14 figures, uses revtex

We study the non-linear dynamics of quantum fields in matter and radiation dominated universes, using the non-equilibrium field theory approach combined with the non-perturbative Hartree and the large N approximations. We examine the phenomenon of explosive particle production due to spinodal instabilities and parametric amplification in expanding universes with and without symmetry breaking. For a variety of initial conditions, we compute the evolution of the inflaton, its quantum fluctuations, and the equation of state. We find explosive growth of quantum fluctuations, although particle production is somewhat sensitive to the expansion of the universe. In the large N limit for symmetry breaking scenarios, we determine generic late time solutions for any flat Friedman-Robertson-Walker cosmology. We also present a complete and numerically implementable renormalization scheme for the equation of motion and the energy momentum tensor in flat FRW cosmologies. In this scheme the renormalization constants are independent of time and of the initial conditions., Comment: 43 pages, 12 figures, uses epsf

We study photoproduction during the non-equilibrium stages of the formation of chiral condensates within the ``quench'' scenario of the chiral phase transition. The dynamics is modeled with a gauged linear sigma model. A novel quantum kinetic approach to the description of photoproduction far off equilibrium is developed. We find that non-equilibrium spinodal instabilities of long wavelength pion fluctuations are responsible for an enhanced photoproduction rate for energies $\leq 80$ MeV at order $\alpha$. These non-equilibrium effects lead to a larger contribution than the typical processes in the medium, including that of the anomalous neutral pion decay $\pi^0 \rightarrow 2 \gamma$ (which is of order $\alpha^2$). We follow the evolution of the dynamics throughout the phase transition, which in this scenario occurs on a time scale of about $2.5-3$ fm/c and integrate the photon yield through its evolution. The spectrum of photons produced throughout the phase transition is a non- equilibrium one. For thermal initial conditions at the time of the quench it interpolates between a thermal distribution about 6% above the initial temperature (at the time of the quench) for low energy $\leq 80$ MeV photons, and a high energy tail in thermal equilibrium at the initial temperature, with a smooth crossover at 100 MeV. The rate displays a peak at $\sim 35$ MeV which receives a larger enhancement the closer the initial temperature at the time of the quench is to the critical temperature. It is found that the enhancement of photoproduction at low energies is not an artifact caused by the initial distribution of the photons, but is due to the pionic instabilities. We suggest that these strong out of equilibrium effects may provide experimental signatures for the formation and relaxation of DCC's in heavy ion collisions., Comment: 33 pages, 11 figures, uses revtex and epsfig

At the end of the inflationary stage of the early universe, profuse particle production leads to the reheating of the universe. Such explosive particle production is due to parametric amplification of quantum fluctuations for the unbroken symmetry case (appropriate for chaotic inflation), or spinodal instabilities in the broken symmetry phase(which is the case in new inflation). The self-consistent methods presented in these lectures are the only approaches, so far, that lead to reliable quantitative results on the reheating mechanism in the inflationary universe. They fully use the field theoretical Schwinger-Keldysh out of equilibrium method in a non-perturbative way. These approaches take into account the non-linear interaction between the quantum modes and exactly conserve energy (covariantly). Simplified analysis that do not include the full backreaction and do not conserve energy, result in unbound particle production and lead to quantitatively erroneous results. For spontaneously broken theories the issue of whether the symmetry may be restored or not by the quantum fluctuations is analyzed. The precise criterion for symmetry restoration is presented. The field dynamics is symmetric when the energy density in the initial state is larger than the top of the tree level potential. When the initial energy density is below the top of the tree level potential, the symmetry is broken., Comment: Latex file, 59 pages and 30 figures in .ps files. Two figure captions replaced by the appropriate ones. Analytic calculation of Floquet indices in Hartree corrected. To appear in the Proceedings of the 5th. Erice Chalonge School on Astrofundamental Physics, N. S\'anchez and A. Zichichi eds., World Scientific, 1997

We study the non-linear dynamics of an inflationary phase transition in a quartically self coupled inflaton model within the framework of a de Sitter background. Large N and Hartree non-perturbative approximations combined with non-equilibrium field theory methods are used to study the self-consistent time evolution including backreaction effects. We find that when the system cools down from an initial temperature T_i > T_c to below T_c with the initial value of the zero mode of the inflaton phi(0) << m lambda^{-1/4}, the dynamics is determined by the growth of long-wavelength quantum fluctuations. For phi(0) >> m lambda^{-1/4} the dynamics is determined by the evolution of the classical zero mode. In the regime where spinodal quantum fluctuations give the most important contribution to the non-equilibrium dynamics, we find that they modify the equation of state providing a graceful exit from the inflationary stage. Inflation ends through this new mechanism at a time scale t_s >= [H/m^2]ln[lambda^{-1}] which for H >= m and very weak coupling allows over one hundred e-folds during the de Sitter phase. Spatially correlated domains grow to be of horizon-size and quantum fluctuations ``freeze-out'' for times t> t_s., Comment: 33 pages plus 13 figures, RevTex, replaced with published version (minor comments added)

The nonlinear time evolution of the quantum fields is studied in the O(N) model for large N in a radiation dominated FRW universe, with a view towards the phenomenon of explosive particle production due to either spinodal instabilities or parametric amplification, i.e. preheating. Quantum backreaction effects due to the produced particles are included consistently within the large N approximation. We find that preheating persists when the expansion is included, although the amount of particle production is reduced compared to the values found in Minkowski space. We also see that the behavior of the evolving zero mode is very different from that in Minkowski space, though the late time behavior in all cases is determined by a sum rule that implies the existence of Goldstone bosons in the final state., Comment: 6 pages, RevTex, 2 epsf figures

Reply to the comment by L. Kofman, A. Linde and A.A. Starobinsky (hep-ph/9608341) to our article ``Analytic and Numerical Study of Preheating Dynamics'' (hep-ph/9608205)., Comment: 5 pages, revtex

Recent progress in the preheating phenomena for inflationary cosmology is reviewed. We first discuss estimates of the preheating time scale and particle production at the early stages of parametric amplification within the Mathieu and Lam'e approximations and we analyze their precision and limitations. The necessity of self-consistent calculations including the non-linearity of the field theory equations in an energy conserving scheme is stressed. The large N calculations including the field back-reaction are reviewed. For spontaneously broken theories the issue of symmetry restoration is analyzed. A discussion of the possibility and criterion for symmetry restoration is presented., Comment: Latex file, expanded (+50%) and improved version, eight pages and thirteen .ps figures. To appear in the Proceedings of the Paris Euronetwork Meeting `String Gravity'

We analyze the phenomenon of preheating,i.e. explosive particle production due to parametric amplification of quantum fluctuations in the unbroken case, or spinodal instabilities in the broken phase, using the Minkowski space $O(N)$ vector model in the large $N$ limit to study the non-perturbative issues involved. We give analytic results for weak couplings and times short compared to the time at which the fluctuations become of the same order as the tree level,as well as numerical results including the full backreaction.In the case where the symmetry is unbroken, the analytic results agree spectacularly well with the numerical ones in their common domain of validity. In the broken symmetry case, slow roll initial conditions from the unstable minimum at the origin, give rise to a new and unexpected phenomenon: the dynamical relaxation of the vacuum energy.That is, particles are abundantly produced at the expense of the quantum vacuum energy while the zero mode comes back to almost its initial value.In both cases we obtain analytically and numerically the equation of state which turns to be written in terms of an effective polytropic index that interpolates between vacuum and radiation-like domination. We find that simplified analysis based on harmonic behavior of the zero mode, giving rise to a Mathieu equation forthe non-zero modes miss important physics. Furthermore, analysis that do not include the full backreaction do not conserve energy, resulting in unbound particle production. Our results do not support the recent claim of symmetry restoration by non-equilibrium fluctuations.Finally estimates of the reheating temperature are given,as well as a discussion of the inconsistency of a kinetic approach to thermalization when a non-perturbatively large number of particles is created., Comment: Latex file, 52 pages and 24 figures in .ps files. Minor changes. To appear in Physical Review D, 15 December 1996

We study the decay of scalar fields, in particular the inflaton, into lighter scalars in a De Sitter spacetime background. After providing a practical definition of the rate, we focus on the case of an inflaton interacting with a massless scalar field either minimally or conformally coupled to the curvature. The evolution equation for the expectation value of the inflaton is obtained to one loop order in perturbation theory and the decay rate is recognized from the solution. We find the remarkable result that this decay rate displays an equilibrium Bose-enhancement factor with an effective temperature given by the Hawking temperature $H\slash{2 \pi}$, where $H$ is the Hubble constant. This contribution is interpreted as the ``stimulated emission'' of bosons in a thermal bath at the Hawking temperature. In the context of new inflation scenarios, we show that inflaton decay into conformally coupled massless fields slows down the rolling of the expectation value. Decay into Goldstone bosons is also studied. Contact with stochastic inflation is established by deriving the Langevin equation for the coarse-grained expectation value of the inflaton field to one-loop order in this model. We find that the noise is gaussian and correlated (colored) and its correlations are related to the dissipative (``decay'') kernel via a generalized fluctuation-dissipation relation., Comment: Replaced with version published in Phys. Rev. D

We propose a gauge invariant formulation of the effective potential in terms of a gauge invariant order parameter, for the Abelian Higgs model. The one-loop contribution at zero and finite temperature is computed explicitly, and the leading terms in the high temperature expansion are obtained. The result is contrasted to the effective potential obtained in several covariant gauge-fixing schemes, and the gauge invariant quantities that can be reliably extracted from these are identified. It is pointed out that the gauge invariant effective potential in the one-loop approximation is complex for {\em all values} of the order parameter between the maximum and the minimum of the tree level potential, both at zero and non-zero temperature. The imaginary part is related to long-wavelength instabilities towards phase separation. We study the real-time dynamics of initial states in the spinodal region, and relate the imaginary part of the effective potential to the growth rate of equal-time gauge invariant correlation functions in these states. We conjecture that the spinodal instabilities may play a role in non-equilibrium processes {\em inside} the nucleating bubbles if the transition is first order., Comment: 27 pages revtex 3.0, no figures; one reference added

A new approach to the dynamics of relaxation and kinetics of thermalization in a scalar field theory is presented that incorporates the relevant time scales through the resummation of hard thermal loops. An alternative derivation of the kinetic equations for the ``quasiparticle'' distribution functions is obtained that allows a clear understanding of the different ``coarse graining'' approximations usually involved in a kinetic description. This method leads to a systematic perturbative expansion to obtain the kinetic equations including hard-thermal loop resummation and to an improvement including renormalization, off-shell effects and contributions that change chemical equilibrium on short time scales. As a byproduct of these methods we establish the relation between the relaxation time scale in the linearized equation of motion of the quasiparticles and the thermalization time scale of the quasiparticle distribution function in the ``relaxation time approximation''. Hard thermal loop resummation dramatically modifies the scattering rate for long wavelength modes as compared to the usual (semi) classical estimate. Relaxation and kinetics are studied both in the unbroken and broken symmetry phases of the theory. The broken symmetry phase also provides the setting to obtain the contribution to the kinetic equations from processes that involve decay of a heavy scalar into light scalar particles in the medium., Comment: 28 pages, revtex 3.0, two figures available upon request

Using the O(4) linear $\sigma$ model, we address the topic of non-equilibrium relaxation of an inhomogeneous initial configuration due to quantum and thermal fluctuations. The space-time evolution of an inhomogeneous fluctuation of the condensate in the isoscalar channel decaying via the emission of pions in the medium is studied within the context of disoriented chiral condensates. We use out of equilibrium closed time path methods in field theory combined with the amplitude expansion. We give explicit expressions for the asymptotic space-time evolution of an initial inhomogeneous configuration including the contribution of thresholds at zero and non-zero temperature. At non-zero temperature we find new relaxational processes due to thermal cuts that have no counterpart in the homogeneous case. Within the one-loop approximation, we find that the space time evolution of such inhomogeneous configuration out of equilibrium is effectively described in terms of a rapidity dependent temperature $T(\vartheta)=T/\cosh[\vartheta]$ as well as a rapidity dependent decay rate $\Gamma(\vartheta, T(\vartheta))$. This rate is to be interpreted as the production minus absorption rate of pions in the medium and approaches the zero temperature value at large rapidities. An initial configuration localized on a bounded region spreads and decays in spherical waves with slower relaxational dynamics at large rapidity., Comment: 25 pages Revtex 3.0, two figures available upon request

The reheating stage in post-inflationary cosmologies is reanalyzed. New techniques from non-equilibrium quantum field theory allow a consistent derivation of the equation of motion including the non-linearity of the dynamics. These offer a rationale for the elementary theory of reheating based on single particle decay which is seen to be valid only in the linear regime of coherent oscillations of the scalar field. A new non-perturbative mechanism of induced amplification of quantum fluctuations is introduced and studied in detail, both analytically and numerically. This is a non-linear mechanism that is typically a far more efficient way of transfering energy out of the zero mode and into production of lighter particles than single particle decay. Thermalization is discussed and we estimate the reheating temperature to be of the order of the inflaton mass, thus providing a potential solution to the Polonyi and moduli problems., Comment: To appear in the Proceedings of the School on String Gravity and Physics at the Planck Scale, Erice, Sicily, 8-19 September, Editor: N. Sanchez, Revtex, 39 pages, 9 figures available upon request

We consider the case of a scalar field, the inflaton, coupled to both lighter scalars and fermions, and the study the relaxation of the inflaton via particle production in both the linear and non-linear regimes. This has an immediate application to the reheating problem in inflationary universe models. The linear regime analysis offers a rationale for the standard approach to the reheating problem, but we make a distinction between relaxation and thermalization. We find that particle production when the inflaton starts in the {\it non-linear} region is typically a far more efficient way of transfering energy out of the inflaton zero mode and into the quanta of the lighter scalar than single particle decay. For the non-linear regime we take into account self-consistently the evolution of the expectation value of the inflaton field coupled to the evolution of the quantum fluctuations. An exhaustive numerical analysis reveals that the distribution of produced particles is far from thermal, and the effect of open channels. In the fermionic case, Pauli blocking begins to hinder the transfer of energy into the fermion modes very early on in the evolution of the inflaton. We examine the implications of our results to the question of how to calculate the reheating temperature of the universe after inflation., Comment: Revtex 3.0; 34 pages

We consider the non-equilibrium evolution of the inflaton field coupled to both lighter scalars and fermions. The dissipational dynamics of this field is studied and found to be quite different than that believed in inflationary models. In particular, the damping time scale for the expectation value of the zero momentum mode of the inflaton can be much shorter than that given by the single particle decay rate when the inflaton amplitudes are large, as in chaotic inflation scenarios. We find that the reheating temperature may depart considerably from the usual estimates., Comment: 9 pages, revtex 3.0, 3 figures available upon request

We solve the equations of motion for a scalar field with domain wall boundary conditions in a Friedmann-Robertson-Walker (FRW) spacetime. We find (in agreement with Basu and Vilenkin) that no domain wall solutions exist in de Sitter spacetime for h = H/m >= 1/2, where H is the Hubble parameter and m is the scalar mass. In the general FRW case we develop a systematic perturbative expansion in h to arrive at an approximate solution to the field equations. We calculate the energy momentum tensor of the domain wall configuration, and show that the energy density can become negative at the core of the defect for some values of the non-minimal coupling parameter xi. We develop a translationally invariant theory for fluctuations of the wall, obtain the effective Lagrangian for these fluctuations, and quantize them using the Bunch-Davies vacuum in the de Sitter case. Unlike previous analyses, we find that the fluctuations act as zero-mass (as opposed to tachyonic) modes. This allows us to calculate the distortion and the normal-normal correlators for the surface. The normal-normal correlator decreases logarithmically with the distance between points for large times and distances, indicating that the interface becomes rougher than in Minkowski spacetime., Comment: 23 pages, LaTeX, 7 figures using epsf.tex. Now auto-generates PS

In this brief review we introduce the methods of quantum field theory out of equilibrium and study the non-equilibrium aspects of phase transitions. Specifically we critically study the picture of the ``slow-roll'' phase transition in the new inflationary models, we show that the instabilities that are the hallmark of the phase transition, that is the formation of correlated domains, dramatically change this picture. We analyze in detail the dynamics of phase separation in strongly supercooled phase transitions in Minkowski space. We argue that this is typically the situation in weakly coupled scalar theories. The effective evolution equations for the expectation value and the fluctuations of an inflaton field in a FRW cosmology are derived both in the loop expansion and in a self-consistent non-perturbative scheme. Finally we use these non-equilibrium techniques and concepts to study the influence of quantum and thermal fluctuations on the dynamics of a proposed mechanism for the formation of disoriented chiral condensates during a rapid phase transition out of equilibrium. This last topic may prove to be experimentally relevant at present accelerator energies. To appear in the Proceedings of the `2nd. Journ\'ee Cosmologie', Observatoire de Paris, 2-4, June 1994. H J de Vega and N. S\'anchez, Editors, World Scientific., Comment: 72 pages, LaTex, Figures availables from the authors upon request

The response and nonconserved dynamics of a two-phase interface in the presence of a temperature gradient oriented normally to the interface are considered. Two types of boundary conditions on the order parameter are considered, and the structure of the effective free energy and the Langevin equation for the collective coordinate specifying the interface position are analyzed., Comment: 15 pages, Revtex 3.0, 5 figures available upon request

The non-equilibrium dynamics of the first stage of the reheating process, that is dissipation via particle production is studied in scalar field theories in the unbroken and in the broken symmetry phase. We begin with a perturbative study to one loop and show explicitly that the mechanism of dissipation via particle production cannot be explained with a simple derivative term in the equation of motion. The dissipative contribution is non-local and there does not exist a local (Markovian) limit at zero temperature. Furthermore, we show that both an amplitude as well as a one-loop calculation present instabilities, requiring a non-perturbative resummation. Within the same approximations, we study an O(2) linear sigma model that allows to study dissipation by Goldstone bosons. We find infrared divergences that require non-perturbative resummation in order to understand the long-time dynamics. We obtain a perturbative Langevin equation that exhibits a generalized fluctuation-dissipation relation, with non-Markovian kernels and colored noise. We then study a Hartree approximation and clearly exhibit dissipative effects related to the thresholds to particle production. The asymptotic dynamics depends on the coupling and initial conditions but does not seem to lead to exponential relaxation. The effect of dissipation by Goldstone bosons is studied non-perturbatively in the large N limit in an O(N) theory. Dissipation produced by Goldstone bosons dramatically changes the picture of the phase transition. We find the remarkable result that for ``slow-roll'' initial conditions (with the expectation value of the field initially near the origin) the final value of the expectation value of the scalar field is very close to its initial value. We find that the minima of the effective action depend on the initial conditions. We provide extensive numerical analysis of the dynamics., Comment: 57 pages, 41 figures available upon request by mail, LATEX, PITT-94-07

We address the problem of how to incorporate quantum effects into the calculation of finite-temperature decay rates for a metastable state of a quantum field theory. To do this, we consider the Gross-Neveu model with an explicit chiral symmetry breaking term, which allows for a metastable state. This theory can be shown to have a "critical bubble" which is a solution to the *exact* equations of motions (i.e. to all orders in perturbation theory, including all higher derivative, quantum and thermal corrections). This configuration mediates the thermal activation of the metastable vacuum to the true ground state, with a decay rate $\Gamma \propto \exp(-F_c/T)$, where $F_c$ is the free energy of the critical bubble. We then compare this exact calculation to various approximations that have been used in previous work. We find that these approximations all *overestimate* the activation rate. Furthermore, we study the effect of finite baryon number upon the bubble profile and the activation barriers. We find that beyond a critical baryon number the activation barriers disappear altogether., Comment: 20 pages, LaTeX, 9 figures using epsf.tex. Now auto-generates PS

We present a study of the nucleation mechanism that allows the decay of the metastable phase (trans-cisoid) to the stable phase (cis-transoid) in quasi one-dimensional non-degenerate polymers within the continuum electron-phonon model. The electron-phonon configurations that lead to the decay, i.e. the critical droplets (or transition state), are identified as polarons of the metastable phase. We obtain an estimate for the decay rate via thermal activation within a range of parameters consistent with experimental values for the gap of the cis-configuration. It is pointed out that, upon doping, the activation barriers of the excited states are quite smaller and the decay rate is greatly enhanced. Typical activation energies for electron or hole polarons are $\approx 0.1$ eV and the typical size for a critical droplet (polaron) is about $20 \AA$. Decay via quantum nucleation is also studied and it is found that the crossover temperature between quantum nucleation and thermal activation is of order $T_c \leq 40 ^oK$. Metastable configurations of non-degenerate polymers may provide examples for mesoscopic quantum tunneling., Comment: REVTEX 3.0, 28 PAGES, 3 FIGURES AVAILABLE UPON REQUEST, PITT 94-05

We address the issue of whether a region of disordered chiral condensate (DCC), in which the chiral condensate has components along the pion directions, can form. We consider a system going through the chiral phase transition either via a quench, or via relaxation of the high temperature phase to the low temperature one within a given time scale (of order $\sim 1 \rm{fm/c}$). We use a density matrix based formalism that takes both thermal and quantum fluctuations into account non-perturbatively to argue that if the $O(4)$ linear sigma model is the correct way to model the situation in QCD, then it is very unlikely at least in the Hartree approximation, that a large ($> 10\ \rm{fm}$) DCC region will form. Typical sizes of such regions are $\sim 1 -2 \ \rm{fm}$ and the density of pions in such regions is at most of order $\sim 0.2 / \rm{fm}^3$. We end with some speculations on how large DCC regions may be formed., Comment: 21 pages LATEX, 12 figures available upon request via regular mail, PITT-94-01

We derive the effective equations for the out of equilibrium time evolution of the order parameter and the fluctuations of a scalar field theory in spatially flat FRW cosmologies.The calculation is performed both to one-loop and in a non-perturbative, self-consistent Hartree approximation.The method consists of evolving an initial functional thermal density matrix in time and is suitable for studying phase transitions out of equilibrium. The renormalization aspects are studied in detail and we find that the counterterms depend on the initial state. We investigate the high temperature expansion and show that it breaks down at long times. We also obtain the time evolution of the initial Boltzmann distribution functions, and argue that to one-loop order or in the Hartree approximation, the time evolved state is a ``squeezed'' state. We illustrate the departure from thermal equilibrium by numerically studying the case of a free massive scalar field in de Sitter and radiation dominated cosmologies. It is found that a suitably defined non-equilibrium entropy per mode increases linearly with comoving time in a de Sitter cosmology, whereas it is {\it not} a monotonically increasing function in the radiation dominated case., Comment: 29 pages, revtex 3.0, 11 figures available upon request, PITT-93-6; LPTHE-93-52; CMU-HEP-93-21

The dynamics of typical phase transitions is studied out of equilibrium in weakly coupled inflaton-type scalar field theories in Minkowski space. The shortcomings of the effective potential and equilibrium descriptions are pointed out. A case of a rapid supercooling from $T_i>T_c$ to $T_f \ll T_c$ is considered. The equations of motion up to one-loop for the order parameter are obtained and integrated for the case of ``slow rollover initial conditions''. It is shown that the instabilities responsible for the process of phase separation introduce dramatic corrections to the evolution. Domain formation and growth (spinodal decomposition) is studied in a non-perturbative self-consistent approximation. For very weakly coupled theories domains grow for a long time, their final size is several times the zero temperature correlation length. For strongly coupled theories the final size of the domains is comparable to the zero temperature correlation length and the transition proceeds faster. We also obtain the evolution equations for the order parameter and the fluctuations to one-loop order and in a non-perturbative Hartree approximation in spatially flat FRW cosmologies. The renormalization, and leading behavior of the high temperature limit are analyzed., Comment: 26 pages;Latex;4 figures not included, available upon request; PITT-93-9