Mode-coupling theory of the glass transition for colloidal liquids in slit geometry

ABSTRACT

We provide a detailed derivation of the mode-coupling equations for a colloidal liquid confined by two parallel smooth walls. We introduce irreducible memory kernels for the different relaxation channels thereby extending the projection operator technique to colloidal liquids in slit geometry. Investigating both the collective dynamics as well as the tagged-particle motion, we prove that the mode-coupling functional assumes the same form as in the Newtonian case corroborating the universality of the glass-transition singularity with respect to the microscopic dynamics.     

Violation of causality in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">T</Emphasis>) gravity

In the standard formulation, the f(T) field equations are not invariant under local Lorentz transformations, and thus the theory does not inherit the causal structure of special relativity. Actually, even locally violation of causality can occur in this formulation of f(T) gravity. A locally Lorentz covariant f(T) gravity theory has been devised recently, and this local causality problem seems to have been overcome. The non-locality question, however, is left open. If gravitation is to be described by this covariant f(T) gravity theory there are a number of issues that ought to be examined in its context, including the question as to whether its field equations allow homogeneous Gödel-type solutions, which necessarily leads to violation of causality on non-local scale. Here, to look into the potentialities and difficulties of the covariant f(T) theories, we examine whether they admit Gödel-type solutions. We take a combination of a perfect fluid with electromagnetic plus a scalar field as source, and determine a general Gödel-type solution, which contains special solutions in which the essential parameter of Gödel-type geometries, \(m^2\), defines any class of homogeneous Gödel-type geometries. We show that solutions of the trigonometric and linear classes (\(m^2 < 0\) and \(m=0\)) are permitted only for the combined matter sources with an electromagnetic field matter component. We extended to the context of covariant f(T) gravity a theorem which ensures that any perfect-fluid homogeneous Gödel-type solution defines the same set of Gödel tetrads \(h_A^{~\mu }\) up to a Lorentz transformation. We also showed that the single massless scalar field generates Gödel-type solution with no closed time-like curves. Even though the covariant f(T) gravity restores Lorentz covariance of the field equations and the local validity of the causality principle, the bare existence of the Gödel-type solutions makes apparent that the covariant formulation of f(T) gravity does not preclude non-local violation of causality in the form of closed time-like curves.     

A statistical-mechanical theory of slow dynamics near the glass transition

A statistical-mechanical theory of slow dynamics near the glass transition in two kinds of glass-forming systems, (M) molecular systems and (S) suspensions of colloids, is presented from a unified point of view based on the Tokuyama-Mori projection operator method. The exact diffusion equations for the coherent- and the incoherent-intermediate scattering functions are first derived, whose memory functions are convolutionless in time and contain the correlation effects due to the hydrodynamic interactions in (S). The analytic expressions of the memory functions are then calculated within the mode-coupling theory (MCT) approximation and are shown to coincide with the conventional ones obtained by MCT. Alternative mode-coupling equations are thus obtained in (M) and (S) separately. Self-diffusion is also discussed. Alternative equations for the mean-square displacement and the non-Gaussian parameter are also derived within MCT approximation. All results in both the systems are compared with those obtained by MCT.     

High-order mode-coupling theory for the colloidal glass transition

A theoretical approach is developed to derive a hierarchy of mode-coupling equations for the dynamics of concentrated colloidal suspensions, which improves the prediction of the colloidal glass transition. Our derivation is based on a matrix formalism for stochastic dynamics and the resulting recursive expressions for irreducible memory functions. The 1st order truncation of the generalized mode-coupling closure recovers mode-coupling theory, whereas its 2nd and 3rd order truncations provide corrections. The predictions of the transition volume fraction and Debye-Waller parameter for the hard-sphere colloidal system improve with the increasing mode-coupling order and compare favorably with experimental measurements.     

Gödel type metrics in three dimensions

We show that the Gödel type metrics in three dimensions with arbitrary two dimensional background space satisfy the Einstein-perfect fluid field equations. We also show that there exists only one first order partial differential equation satisfied by the components of fluid’s velocity vector field. We then show that the same metrics solve the field equations of the topologically massive gravity where the two dimensional background geometry is a space of constant negative Gaussian curvature. We discuss the possibility that the Gödel type metrics to solve the Ricci and Cotton flow equations. When the vector field u ^μ is a Killing vector field, we came to the conclusion that the stationary Gödel type metrics solve the field equations of the most possible gravitational field equations where the interaction lagrangian is an arbitrary function of the electromagnetic field and the curvature tensors.     

Various velocity correlations functions in a Lorentz gas - simulation and mode coupling theory

We present computer simulation results for several types of velocity correlation function in the two dimensional, overlapping Lorentz gas. Only the normal velocity autocorrelation function, whose integral gives the diffusion constant, shows obvious anomalous behaviour at the percolation transition. The other functions are fairly well approximated by the Lorentz-Boltzmann equation, even for densities at which the travelling particle is trapped. We do, however, at a sub-percolation density, examine the long time behaviour of the autocorrelation function corresponding to the second rank, irreducible tensor of the velocity, and find an algebraic decay with an exponent of 3.0 ± 0.1, consistent with the theoretically expected value of 3. With these observations in mind we re-examine the mode coupling theory of Götze, Leutheusser and Yip (Phys. Rev. A 23 (1981) 2634,) replacing their one (frequency dependent) relaxation time approximation to a kinetic operator by a two (frequency dependent) relaxation time model. We find that this leads to a significantly better estimate of the diffusions constant at low density. Furthermore the theory correctly predicts no striking anomalous behaviour in the types of velocity correlation function that are unrelated to diffusion as the percolation threshold is crossed.     

Bond-induced ergodicity breakdown in reactive mixtures

We have studied by inelastic x-ray scattering, at wave vectors 1 nm < or =q< or =15 nm(-1), the high-frequency dynamics of epoxy-amine mixtures as the monomers irreversibly polymerize. We find that chemical bonding, while inducing molecular ordering on a mesoscopic length scale, also efficiently realizes the mechanism of dynamical arrest described by the mode-coupling theory, as manifested by a cusp singularity in the behavior of the nonergodicity factor as a function of the number of chemical bonds. These results confront positively the mode-coupling theory with a new control parameter.     

The Decay of Massive Scalar Field in Non-Static Gödel Type Universe with Viscous Fluid and Heat Flow

In this study, we have investigated the dynamics of non-static Gödel type rotating universe with massive scalar field, viscous fluid and heat flow in the presence of cosmological constant. For various cosmic matter forms, the behavior of the cosmological constant (Λ), shear (η) and bulk (ξ) viscosity coefficients and other kinematic quantities have studied in the early universe. We have showed the decay of massive scalar field in the non-static rotating Gödel type universe and we have obtained constant scalar field with and without source density. Also, we have investigated the effects of massive scalar field on the matter density and pressure. From solutions of the field equations, we have a cosmological model with non-zero expansion, shear, heat flux and rotation. Also some physical and geometrical aspects of the model discussed.     

Noisy nonlinear coupled equations—Some new insights

We discuss the use of coupled nonlinear stochastic differential equations to model the dynamics of complex systems, and present some analytical insights into their critical behaviour. These concern in particular the role of infrared divergences which show up in a self-consistent resummation of perturbation theory (mode-coupling approximation), and their effects on critical exponents obtained in earlier work.     

Mode-coupling theory for purely diffusive systems

A mode-coupling formalism is developed for multicomponent systems of particles performing diffusive motion in a uniform host medium. The mode-coupling equations are derived from a set of nonlinear fluctuating diffusion equations by expanding the concentration-dependent diffusion constants about their equilibrium values. From the mode-coupling equations the dominant long time behavior of current-current and super-Burnett correlation functions is derived. As specific applications I consider the long time behaviors of these correlation functions for collective and tracer diffusion in a one-component lattice gas with particle-conserving stochastic dynamics. The results agree with those from exactly solvable models and computer simulations.     

Mode-coupling approach to the dynamics of simple liquids

Summary A new derivation of the mode-coupling theory of liquid dynamics is presented which is applicable to both quantum and classical systems. A clear distinction between the basic underlying hyphotheses and the more technical approximations has been attempted. Asymptotic corrections to the dynamics of modes due to mode-coupling effects and selfconsistent mode-coupling theories emerge from the original idea as two markedly different branches of application. Paper presented at the workshop ?Highlights on Simple Liquids?, held in Turin at ISI on 1–3 May, 1989.     

Renormalized field theory of critical dynamics

We formulate a Gell'Mann-Low-type renormalization group approach to the critical dynamics of stochastic models described by Langevin or Fokker-Planck equations including mode-coupling terms.Dynamical correlation and response functions are expressed in terms of path integrals, which are investigated by well-known methods of renormalized perturbation theory.Dynamical scaling laws and relations between static and dynamic critical exponents are derived. The leading temperature-dependence of correlation and response functions is obtained from the Kadanoff-Wilson short-distance expansion. We also consider corrections to dynamic scaling which are due to a finite lattice constant.     

The orientational glass instability and localization in (KCN) x (KBr)1−x and (KCN) x (NaCN)1−x

Substitutional disorder in mixed crystals with orientational and translational degrees of freedom leads to the appearance of random strain fields. The strains act as static scattering centers for dynamic modes. Near a ferroelastic phase transition where the crystal is very soft, internal friction processes dominate the dynamic restoring forces. The resulting nonergodic instability marks the onset of the orientational glass state, which is characterized by a freezing-in of orientational and translational modes without long range order. The method is based on the study of dynamic equations of mode-coupling type and was originally developed by Götze for the Anderson localization.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

Impurity spin dynamics in the classical Kondo system Cu: Fe. NMR relaxation of impurity neighbour nuclei

The correlation time τ_i of the impurity electronic spin in the Kondo system Cu:Fe has been measured over the temperature range 4.2 K?T?300 K (T_K=27.6 K) by means of NMR relaxation of impurity neighbour nuclei. τ_i being in the sub-picosecond region varies with temperatures as predicted by the model calculation of Götze and Schlottmann.     

Fermi field and Dirac oscillator in a Som–Raychaudhuri space-time

We investigate the relativistic dynamics of a Dirac field in the Som–Raychaudhuri space-time, which is described by a Gödel-type metric and a stationary cylindrical symmetric solution of Einstein field equations for a charged dust distribution in rigid rotation. In order to analyze the effect of various physical parameters of this space-time, we solve the Dirac equation in the Som–Raychaudhuri space-time and obtain the energy levels and eigenfunctions of the Dirac operator by using the Nikiforov–Uvarov method. We also examine the behaviour of the Dirac oscillator in the Som–Raychaudhuri space-time, in particular, the effect of its frequency and the vorticity parameter.     

Stability of closed timelike curves in the Gödel universe

We study, in some detail, the linear stability of closed timelike curves in the Gödel universe. We show that these curves are stable. We present a simple extension (deformation) of the Gödel metric that contains a class of closed timelike curves similar to the ones associated to the original metric. This extension correspond to the addition of matter whose energy-momentum tensor is analyzed. We find the conditions to have matter that satisfies the usual energy conditions. We study the stability of closed timelike curves in the presence of usual matter as well as in the presence of exotic matter (matter that does satisfy the above mentioned conditions). We find that the closed timelike curves in the Gödel universe with or without the inclusion of regular or exotic matter are stable under linear perturbations. We also find a sort of structural stability.     

On heat diffusion mode of structural phase transition by two-fluid model

With the help of the two-fluid model developed by Götze and Michel for phonons it is shown for a simple model Hamiltonian that in the low temperature phase the optical soft mode becomes isothermal, the heat diffusion mode is dominant near the transition temperatureT _c and the quasiparticle interaction is of great importance in determining the thermodynamic quantities nearT _c. Green function techniques are applied to describe the two-fluid model functions in a microscopic way. The simplest approximations are discussed for the model equations describing nonequilibrium phenomena of the soft optical phonon mode in the low temperature phase. The quasiparticle interaction operator can be related to the interaction operator between quasiparticles and the condensed mode. This relation enables one to understand the behaviour of the thermodynamic quantities near the transition temperature on a microscopic way. The first order displacive phase transition is also discussed.     

Excited states from range-separated density-functional perturbation theory

We explore the possibility of calculating electronic excited states by using perturbation theory along a range-separated adiabatic connection. Starting from the energies of a partially interacting Hamiltonian, a first-order correction is defined with two variants of perturbation theory: a straightforward perturbation theory and an extension of the Görling–Levy one that has the advantage of keeping the ground-state density constant at each order in the perturbation. Only the first, simpler, variant is tested here on the helium and beryllium atoms and on the hydrogen molecule. The first-order correction within this perturbation theory improves significantly the total ground- and excited-state energies of the different systems. However, the excitation energies mostly deteriorate with respect to the zeroth-order ones, which may be explained by the fact that the ionisation energy is no longer correct for all interaction strengths. The second (Görling–Levy) variant of the perturbation theory should improve these results but has not been tested yet along the range-separated adiabatic connection.