arXiv subjects

Gene F. Mazenko

Publications and source records attributed to Gene F. Mazenko.

At least 19 recordsLinked to original sources

arXiv2013

Universal Longtime Dynamics in Dense Simple Fluids

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.

cond-mat.stat-mech↗

arXiv2013

Newtonian Kinetic Theory and the Ergodic-Nonergodic Transition

In a recent work we have discussed 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. In the present work we use this development for investigating ergodic-nonergodic (ENE) transitions in dense fluids. The theory is developed in terms of a core problem spanned by the variables $ρ$, the number density, and $B$, a response density. We set up the perturbation theory expansion for studying the self-consistent model which gives rise to a ENE transition. Our main result is that the low-frequency dynamics near the ENE transition is the same for Smoluchowski and Newtonian dynamics. This is true despite the fact that term by term in a density expansion the results for the two dynamics are fundamentally different.

cond-mat.stat-mech↗

arXiv2013

Power-law Decay and the Ergodic-Nonergodic Transition in Simple Fluids

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.

cond-mat.stat-mech↗

arXiv2012

Kinetic Equations Governing Smoluchowski Dynamics in Equilibrium

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.

cond-mat.stat-mech↗

arXiv2011

Field Theoretic Formulation of Kinetic theory: I. Basic Development

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 ρthe 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 ρand B.

cond-mat.stat-mech↗

arXiv2011

Smoluchowski dynamics and the ergodic-nonergodic transition

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.

cond-mat.soft↗

arXiv2010

Fundamental Theory of Statistical Particle Dynamics

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.

cond-mat.stat-mech↗

arXiv2010

Random Diffusion Model with Structure Corrections

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.

cond-mat.soft↗

arXiv2008

Does Fluctuating Nonlinear Hydrodynamics Support an Ergodic-Nonergodic Transition?

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.

cond-mat.soft↗

arXiv2007

Random Diffusion Model

We study here the random diffusion model. This is a continuum model for a conserved scalar density field $ϕ$ 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 $D=\bar{D}+D_{1}δϕ$ 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.

cond-mat.soft↗

arXiv2006

Growth of Order in An Anisotropic Swift-Hohenberg Model

We have studied the ordering kinetics of a two-dimensional anisotropic Swift-Hohenberg (SH) model numerically. The defect structure for this model is simpler than for the isotropic SH model. One finds only dislocations in the aligned ordering striped system. The motion of these point defects is strongly influenced by the anisotropic nature of the system. We developed accurate numerical methods for following the trajectories of dislocations. This allows us to carry out a detailed statistical analysis of the dynamics of the dislocations. The average speeds for the motion of the dislocations in the two orthogonal directions obey power laws in time with different amplitudes but the same exponents. The position and velocity distribution functions are only weakly anisotropic.

cond-mat.soft↗

arXiv2004

Long time simulations of granular hydrodynamics : instabilities and attractors

Using a hydrodyamic model of granular flows, we present very long time simulations of a granular fluid in two dimensions without gravity and with periodic boundary conditions in a square domain. Depending upon the values of the viscosity, thermal conductivity and dissipation, we find for intermediate times a metastable clustering state. For longer times the system is attracted to either a shear band or a vortex state. Our results are in general agreement with molecular dynamics simulations.

cond-mat.soft↗

arXiv2004

The Vortex Kinetics of Conserved and Non-conserved O(n) Models

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.

cond-mat.soft↗

arXiv2004

Vortex Velocity Probability Distributions in Phase Ordering Kinetics

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.

cond-mat.soft↗

arXiv2003

Response Functions in Phase Ordering Kinetics

We discuss the behavior of response functions in phase ordering kinetics within the perturbation theory approach developed earlier. At zeroth order the results agree with previous gaussian theory calculations. At second order the nonequilibrium exponents λand λ_{R} are changed but remain equal.

cond-mat.soft↗

arXiv2003

A Model for Striped Growth

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.

cond-mat.soft↗

arXiv2003

Vortex Dynamics in a Coarsening Two Dimensional XY Model

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.

cond-mat.soft↗

arXiv2003

Granular clustering in a hydrodynamic simulation

We examine the hydrodynamics of a granular gas using numerical simulation. We demonstrate the appearance of shearing and clustering instabilities predicted by linear stability analysis, and show that their appearance is directly related to the inelasticity of collisions in the material. We discuss the rate at which these instabilities arise and the manner in which clusters grow and merge.

cond-mat.soft↗