Search

The scaling properties of a phase-ordering system with a conserved order parameter are studied. The theory developed leads to scaling functions satisfying certain general properties including the Tomita sum rule. The theory also gives good agreement with numerical results for the order parameter scaling function in three dimensions. The values of the associated nonequilibrium decay exponents are given by the known lower bounds., Comment: 15 pages, 6 figures

The perturbation theory expansion presented earlier to describe the phase-ordering kinetics in the case of a nonconserved scalar order parameter is generalized to the case of the $n$-vector model. At lowest order in this expansion, as in the scalar case, one obtains the theory due to Ohta, Jasnow and Kawasaki (OJK). The second-order corrections for the nonequilibrium exponents are worked out explicitly in $d$ dimensions and as a function of the number of components $n$ of the order parameter. In the formulation developed here the corrections to the OJK results are found to go to zero in the large $n$ and $d$ limits. Indeed, the large-$d$ convergence is exponential., 20 pages, no figures

A consistent perturbation theory expansion is presented for phase-ordering kinetics in the case of a nonconserved scalar order parameter. At zeroth order in this expansion one obtains the theory due to Ohta, Jasnow and Kawasaki (OJK). At the next nontrivial order in the expansion, worked out in d dimensions, one has small corrections to the OJK result for the nonequilibrium exponent $\lambda$ and the introduction of a new exponent $\nu$ governing the algebraic component of the decay of the order parameter scaling function at large scaled distances., Comment: 26 pages, LaTex

There appears to be a longtime, very slowly evolving state in dense simple fluids which, for high enough density, approaches a glassy nonergodic state. The nature of the nonergodic state can be characterized by the associated static equilibrium state. In particular, systems driven by Smoluchowski or Newtonian dynamics share the same static equilibrium and nonergodic states. That these systems share the same nonergodic states is a highly nontrivial statement and requires establishing a number of results. In the high-density regime one finds that an equilibrating system decays via a three-step process identified in mode-coupling theory (MCT). For densities greater than a critical density one has time-power-law decay with exponents a and b. There are sets of linear fluctuation dissipation relations (FDRs) which connect the cumulants of these two fields. The form of the FDRs is the same for both Smoluchowski or Newtonian dynamics. While we show this universality of nonergodic states within perturbation theory, we expect it to be true more generally. The nature of the approach to the nonergodic state has been suggested by MCT. It has been a point of contention that MCT is a phenomenological theory and not a systematic theory with prospects for improvement. Recently a systematic theory has been developed. It naturally allows one to calculate self-consistently density cumulants in a perturbation expansion in a pseudo-potential. At leading order one obtains a kinetic kernel quadratic in the density. This is a "one-loop" theory like MCT. At this one-loop level one finds vertex corrections which depend on the three-point equilibrium cumulants. Here we assume these vertex-corrections can be ignored and focus on the higher-order loops. We show that one can sum up all of the loop contributions. The higher-order loops do not change the nonergodic state parameters substantially., Comment: 34 pages

The theory of phase ordering kinetics for the O(2) model using the gaussian auxiliary field approach is reexamined from two points of view. The effects of fluctuations about the ordering field are included and we organize the theory such that the auxiliary field correlation function is analytic in the short-scaled distance (x) expansion. These two points are connected and we find in the refined theory that the divergence at the origin in the defect-defect correlation function $\tilde{g}(x)$ obtained in the original theory is removed. Modifications to the order-parameter autocorrelation exponent $\lambda$ are computed., Comment: 29 pages, REVTeX, to be published in Phys. Rev. E. Minor grammatical/syntax changes from the original

It is well known that mode coupling theory (MCT) leads to a two step power-law time decay in dense simple fluids. We show that much of the mathematical machinery used in the MCT analysis can be taken over to the analysis of the systematic theory developed in the Fundamental Theory of Statistical Particle Dynamics (arXiv:0905.4904). We show how the power-law exponents can be computed in the second-order approximation where we treat hard-sphere fluids with statics described by the Percus-Yevick solution., 25 pages, 12 figures

We review recent reformulations of the mode coupling theory (MCT) of the liquid-glass transition which incorporate the notion of metastability. These models provide an improved theoretical picture on the observed slowing down of dynamics compared to the conventional interpretation, since the temper-tature dependence of the models turns out to be smooth without any evidence for the special transition temperature conventionally assumed in MCT.

The element of metastability is incorporated in the fluctuating nonlinear hydrodynamic description of the mode coupling theory (MCT) of the liquid-glass transition. This is achieved through the introduction of the defect density variable $n$ into the set of slow variables with the mass density $\rho$ and the momentum density ${\bf g}$. As a first approximation, we consider the case where motions associated with $n$ are much slower than those associated with $\rho$. Self-consistently, assuming one is near a critical surface in the MCT sense, we find that the observed slowing down of the dynamics corresponds to a certain limit of a very shallow metastable well and a weak coupling between $\rho$ and $n$. The metastability parameters as well as the exponents describing the observed sequence of time relaxations are given as smooth functions of the temperature without any evidence for a special temperature. We then investigate the case where the defect dynamics is included. We find that the slowing down of the dynamics corresponds to the system arranging itself such that the kinetic coefficient $\gamma_v$ governing the diffusion of the defects approaches from above a small temperature-dependent value $\gamma^c_v$., Comment: 38 pages, 14 figures (6 figs. are included as a uuencoded tar- compressed file. The rest is available upon request.), RevTEX3.0+epsf

The theory of growth kinetics developed previously is extended to the asymmetric case of off-critical quenches for systems with a conserved scalar order parameter. In this instance the new parameter $M$, the average global value of the order parameter, enters the theory. For $M=0$ one has critical quenches, while for sufficiently large $M$ one approaches the coexistence curve. For all $M$ the theory supports a scaling solution for the order parameter correlation function with the Lifshitz-Slyozov-Wagner growth law $L \sim t^{1/3}$. The theoretically determined scaling function depends only on the spatial dimensionality $d$ and the parameter $M$, and is determined explicitly here in two and three dimensions. Near the coexistence curve oscillations in the scaling function are suppressed. The structure factor displays Porod's law $Q^{-(d+1)}$ behaviour at large scaled wavenumbers $Q$, and $Q^{4}$ behaviour at small scaled wavenumbers, for all $M$. The peak in the structure factor widens as $M$ increases and develops a significant tail for quenches near the coexistence curve. This is in qualitative agreement with simulations., Comment: 30 pages, REVTeX 3.0, 11 figures available upon request

We study the role of the Jacobian arising from a constraint enforcing the nonlinear relation: ${\bf g}=\rho{\bf V}$, where $\rho,\: {\bf g}$ and ${\bf V}$ are the mass density, the momentum density and the local velocity field, respectively, in the field theoretic formulation of the nonlinear fluctuating hydrodynamics of simple fluids. By investigating the Jacobian directly and by developing a field theoretic formulation without the constraint, we find that no changes in dynamics result as compared to the previous formulation developed by Das and Mazenko (DM). In particular, the cutoff mechanism discovered by DM is shown to be a consequence of the $1/\rho$ nonlinearity in the problem not of the constraint. The consequences of this result for the static properties of the system is also discussed., Comment: 17 pages (5 figs. upon request), ReVTEX 3.0

We use the recently introduced theory for the kinetics of systems of classical particles to investigate systems driven by Smoluchowski dynamics. We investigate the existence of ergodic-nonergodic (ENE) transitions near the liquid-glass transition. We develop a self-consistent perturbation theory in terms of an effective two-body potential. We work to second order in this potential. At second order we have an explicit relationship between the static structure factor and the effective potential. We choose the static structure factor in the case of hard spheres to be given by the solution of the Percus-Yevick approximation for hard spheres. Then using the analytically determined ENE equation for the ergodicity function we find an ENE transition for packing fraction, eta, greater than a critical value eta*=0.76 which is physically unaccessible. The existence of a linear fluctuation-dissipation theorem in the problem is shown and used to great advantage., 51 pages, 6 figures

We show how kinetic theory, the statistics of classical particles obeying Newtonian dynamics, can be formulated as a field theory. The field theory can be organized to produce a self-consistent perturbation theory expansion in an effective interaction potential. The need for a self-consistent approach is suggested by our interest in investigating ergodic-nonergodic transitions in dense fluids. The formal structure we develop has been implemented in detail for the simpler case of Smoluchowski dynamics. One aspect of the approach is the identification of a core problem spanned by the variables \rho the number density and B a response density. In this paper we set up the perturbation theory expansion with explicit development at zeroth and first order. We also determine all of the cumulants in the noninteracting limit among the core variables \rho and B., Comment: 45 pages

We continue our study of the statistical properties of particles in equilibrium obeying Smoluchowski dynamics. We show that the system is governed by a kinetic equation of the memory function form and that the memory function is given by one of the self-energies available via perturbation theory as introduced in previous work. We determine the memory function explicitly to second-order in an expansion in a pseudo-potential. The method we use allows for a straightforward computation of corrections via a formal expansion and we therefore view it as an improvement over the conventional mode-coupling theory (MCT) formalism where it is not clear how to make systematic corrections. In addition, the formalism we have introduced is flexible enough to allow for a wide array of different approximation schemes, including density expansions. The convergence criteria for our formal series are not worked out here, but the second order equation that we derive is promising in the sense that it leads to analytic and numerical results consistent with expectations from computer simulations of the hard sphere system in addition to replicating the desired features from conventional MCT (e.g., a two-step decay). These particular solutions will be discussed in forthcoming work., Comment: 35 pages

We introduce a fundamental theory for the kinetics of systems of classical particles. The theory represents a unification of kinetic theory, Brownian motion and field theory. It is self-consistent and is the dynamic generalization of the functional theory of static equilibrium fluids. This gives one a powerful tool for investigating the existence of ergodic-nonergodic transitions near the liquid-glass transition., 47 pages, 3 appendices

The random diffusion model is a continuum model for a conserved scalar density field driven by diffusive dynamics where the bare diffusion coefficient is density dependent. We generalize the model from one with a sharp wavenumber cutoff to one with a more natural large-wavenumber cutoff. We investigate whether the features seen previously -- namely a slowing down of the system and the development of a prepeak in the dynamic structure factor at a wavenumber below the first structure peak -- survive in this model. A method for extracting information about a hidden prepeak in experimental data is presented., Comment: 13 pages, 8 figures

We investigate the ordering process of an unstable system governed by a nonconserved scalar order parameter using a theoretical approach developed previously. The two-time order-parameter correlation function is shown to obey asymptotic dynamical scaling. The temporal evolution of the autocorrelation function exhibits power-law decay with a nonequilibrium exponent. We find a relation between this exponent and a nonlinear eigenvalue controlling the equal-time structure factor. This relation, as well as the predicted values of the exponent, is compared with direct numerical simulations of a cell-dynamical-system (CDS) model of the ordering process.

The full set of fluctuating nonlinear hydrodynamic equations for normal fluids is derived from the conventional Langevin equations extended to include multiplicative noise. The equations describing the set of conserved variables (the mass densityρ, the momentum densityg, the energy densityɛ) agree with those found by Morozov for a case of a driving free energy which is a local function of the hydrodynamic variables. We show here that if the standard form of the hydrodynamic equations is to hold in the absence of noise, then the driving free energy must be a local function ofg ande, but it may have to be a nonlocal function of the mass density.

Recent work on the glass transition from the liquid side is reviewed. Particular emphasis is placed on the so-called mode coupling theory.

The theory of domain growth developed previously to treat a nonconserved-order-parameter (NCOP) system is extended to treat the conserved-order-parameter (COP) case (spinodal decomposition). The theory here, as for the NCOP case, leads to universal scaling behavior for the order-parameter structure factor, which depends only on the spatial dimensionality of the system. The short-distance ordering in the system is found to be identical to that found for the NCOP case indicating that the structure near the interface is independent of the driving dynamics. Porod's law, signifying sharp interfaces, is of exactly the same form as for the NCOP case. There are significant differences between the two cases, however. In the COP case the growth mechanism works through a coupling between the ordering component and a diffusing component. This coupling increases the growth law, L(t), from the rather slow surface-diffusion form, ${\mathit{t}}^{1/4}$, to the classical Lifshitz-Slyzov-Wagner form, ${\mathit{t}}^{1/3}$. In the NCOP case, the ordering component is strongly decoupled from any fluctuating component. While the structure factors for the two cases are the same for small values of the scaled lengths (x=R/L\ensuremath{\ll}1) they differ significantly over the rest of the range of x. In the COP case the theoretical expression for the scaled structure factor F(x) agrees well with the best available simulational results. A striking feature of the theory in the NCOP case was the existence of a nonlinear eigenvalue problem associated with the determination of the scaling function F(x). In the COP case one has two such eigenvalues. The additional eigenvalue can be associated with the coefficient ${\mathit{x}}^{2}$ in the expansion of F(x) in powers of x. The nonzero value of this coefficient renders invalid the symmetry [1-F(x)]=-[1-F(-x)] found in the NCOP case.

Despite its appeal, real and simulated glass forming systems do not undergo an ergodic-nonergodic (ENE) transition. We reconsider whether the fluctuating nonlinear hydrodynamics (FNH) model for this system, introduced by us in 1986, supports an ENE transition. Using nonperturbative arguments, with no reference to the hydrodynamic regime, we show that the FNH model does not support an ENE transition. Our results support the findings in the original paper. Assertions in the literature questioning the validity of the original work are shown to be in error.

Etude de la nucleation de la phase stable dans un modele de Ginzburg-Landau dependant du temps, avec une structure de double puits de potentiel asymetrique. On considere que le systeme est au depart dans la phase metastable et qu'il evolue (on le fait evoluer) vers l'equilibre stable par le bruit thermique ou par les fluctuations aleatoires pilotant le systeme, dans les conditions initiales. Presentation des resultats en fonction du champ magnetique h, qui decompose une paire de minima degeneres, et de la force du bruit ou de la dispersion dans les valeurs initiales du champ h . Sur le domaine des valeurs des parametres suffisamment grand considere, on trouve que la vitesse de nucleation peut etre bien precisement decrite par une forme activee. L'energie d'activation est une fonction beaucoup plus forte de h pour des valeurs intermediaires de h, que ne le predit la theorie classique ; et cette energie est une fonction faible de la force (intensite) du bruit.

We study here the random diffusion model. This is a continuum model for a conserved scalar density field $\phi$ driven by diffusive dynamics. The interesting feature of the dynamics is that the {\it bare} diffusion coefficient $D$ is density dependent. In the simplest case $D=\bar{D}+D_{1}\delta \phi $ where $\bar{D}$ is the constant average diffusion constant. In the case where the driving effective Hamiltonian is quadratic the model can be treated using perturbation theory in terms of the single nonlinear coupling $D_{1}$. We develop perturbation theory to fourth order in $D_{1}$. The are two ways of analyzing this perturbation theory. In one approach, developed by Kawasaki, at one-loop order one finds mode coupling theory with an ergodic-nonergodic transition. An alternative more direct interpretation at one-loop order leads to a slowing down as the nonlinear coupling increases. Eventually one hits a critical coupling where the time decay becomes algebraic. Near this critical coupling a weak peak develops at a wavenumber well above the peak at $q=0$ associated with the conservation law. The width of this peak in Fourier space decreases with time and can be identified with a characteristic kinetic length which grows with a power law in time. For stronger coupling the system becomes metastable and then unstable. At two-loop order it is shown that the ergodic-nonergodic transition is not supported. It is demonstrated that the {\it critical} properties of the direct approach survive going to higher order in perturbation theory., Comment: 71 pages, 27 figures

We study the motion of vortices in the conserved and non-conserved phase-ordering models. We give an analytical method for computing the speed and position distribution functions for pairs of annihilating point vortices based on heuristic scaling arguments. In the non-conserved case this method produces a speed distribution function consistent with previous analytic results. As two special examples, we simulate the conserved and non-conserved O(2) model in two dimensional space numerically. The numerical results for the non-conserved case are consistent with the theoretical predictions. The speed distribution of the vortices in the conserved case is measured for the first time. Our theory produces a distribution function with the correct large speed tail but does not accurately describe the numerical data at small speeds. The position distribution functions for both models are measured for the first time and we find good agreement with our analytic results. We are also able to extend this method to models with a scalar order parameter., 21 pages, 9 figures

We introduce a model for describing the defected growth of striped patterns. This model, while roughly related to the Swift-Hohenberg model, generates a quite different mixture of defects during phase ordering. We find two characteristic lengths in the system: the scaling length L(t), and the average width of the domain walls. The growth law exponent is larger than the value of 1/2 found in typical point defect systems., 7 pages, 14 figures

The calculation of the point vortex velocity probability distribution function (vvpdf) is extended to a larger class of systems beyond the nonconserved TDGL model treated earlier. The range is extended to include certain anisotropic models and the conserved order parameter case. The vvpdf still satisfies scaling with large velocity tails as for the nonconserved isotropic case. It is shown that the average vortex speed can be self-consistently expressed in terms of correlation functions associated with a Gaussian auxiliary field. In the conserved order parameter case the average vortex speed decays as $t^{-1}$ compared to the $t^{-1/2}$ decay for the nonconserved case., 27 pages, Added treatment of conserved order parameter case

The vortex velocity distribution function for a 2-dimensional coarsening non-conserved O(2) time-dependent Ginzburg-Landau model is determined numerically and compared to theoretical predictions. In agreement with these predictions the distribution function scales with the average vortex speed which is inversely proportional to t^x, where t is the time after the quench and x is near to 1/2. We find the entire curve, including a large speed algebraic tail, in good agreement with the theory., 4 pages, 5 figures

The growth of striped order resulting from a quench of the two-dimensional Swift-Hohenberg model is studied in the regime of a small control parameter and quenches to zero temperature. We introduce an algorithm for finding and identifying the disordering defects (dislocations, disclinations and grain boundaries) at a given time. We can track their trajectories separately. We find that the coarsening of the defects and lowering of the effective free energy in the system are governed by a growth law $L(t)\approx t^{x}$ with an exponent x near 1/3. We obtain scaling for the correlations of the nematic order parameter with the same growth law. The scaling for the order parameter structure factor is governed, as found by others, by a growth law with an exponent smaller than x and near to 1/4. By comparing two systems with different sizes, we clarify the finite size effect. We find that the system has a very low density of disclinations compared to that for dislocations and fraction of points in grain boundaries. We also measure the speed distributions of the defects at different times and find that they all have power-law tails and the average speed decreases as a power law., Comment: 13 pages, 19 figures, accepted by Physical Review E