Time-dependent Density Correlation Function Research Articles

Slow dynamics and strong finite-size effects in many-body localization with random and quasiperiodic potentials

We investigate charge relaxation in disordered and quasi-periodic quantum-wires of spin-less fermions ($t{-}V$-model) at different inhomogeneity strength $W$ in the localized and nearly-localized regime. Our observable is the time-dependent density correlation function, $\Phi(x,t)$, at infinite temperature. We find that disordered and quasi-periodic models behave qualitatively similar: Although even at longest observation times the width $\Delta x(t)$ of $\Phi(x,t)$ does not exceed significantly the non-interacting localization length, $\xi_0$, strong finite-size effects are encountered. Our findings appear difficult to reconcile with the rare-region physics (Griffiths effects) that often is invoked as an explanation for the slow dynamics observed by us and earlier computational studies. As a relatively reliable indicator for the boundary towards the many-body localized (MBL) regime even under these conditions, we consider the exponent function $\beta(t) {=} d\ln \Delta x(t) / d\ln t$. Motivated by our numerical data for $\beta$, we discuss a scenario in which the MBL-phase splits into two subphases: in MBL$_\text{A}$ $\Delta x(t)$ diverges slower than any power, while it converges towards a finite value in MBL$_\text{B}$. Within the scenario the transition between MBL$_\text{A}$ and the ergodic phase is characterized by a length scale, $\xi$, that exhibits an essential singularity $\ln \xi \sim 1/|W-W_\text{c}|$. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.

Dynamics of interacting Brownian particles: A diagrammatic formulation

We present a diagrammatic formulation of a theory for the time dependence of density fluctuations in equilibrium systems of interacting Brownian particles. To facilitate derivation of the diagrammatic expansion, we introduce a basis that consists of orthogonalized many-particle density fluctuations. We obtain an exact hierarchy of equations of motion for time-dependent correlations of orthogonalized density fluctuations. To simplify this hierarchy we neglect contributions to the vertices from higher-order cluster expansion terms. An iterative solution of the resulting equations can be represented by diagrams with three- and four-leg vertices. We analyze the structure of the diagrammatic series for the time-dependent density correlation function and obtain a diagrammatic interpretation of reducible and irreducible memory functions. The one-loop self-consistent approximation for the latter function coincides with mode-coupling approximation for Brownian systems that was derived previously using a projection operator approach.

Temperature dependence of spatially heterogeneous dynamics in a model of viscous silica

Molecular dynamics simulations are performed to study spatially heterogeneous dynamics in a model of viscous silica above and below the critical temperature of the mode coupling theory, T(MCT) . Specifically, we follow the evolution of the dynamic heterogeneity as the temperature dependence of the transport coefficients shows a crossover from non-Arrhenius to Arrhenius behavior when the melt is cooled. It is demonstrated that, on intermediate time scales, a small fraction of oxygen and silicon atoms are more mobile than expected from a Gaussian approximation. These highly mobile particles form transient clusters larger than that resulting from random statistics, indicating that the dynamics are spatially heterogeneous. An analysis of the clusters reveals that the mean cluster size is maximum at times intermediate between ballistic and diffusive motion, and the maximum size increases with decreasing temperature. In particular, the growth of the clusters continues into the crossover to Arrhenius behavior for the transport coefficients. These findings imply that the structural relaxation in silica cannot be understood as a statistical bond breaking process. Although the mean cluster sizes for silica are at the lower end of the spectrum of values reported in the literature for other model liquids, we find that spatially heterogeneous dynamics in models of strong and fragile glass formers are similar on a qualitative level. However, unlike the results for fragile liquids, we show that correlated particle motion along quasi-one-dimensional, stringlike paths is of little importance for the structural relaxation in this model of silica, suggesting that stringlike motion is suppressed by the presence of a network structure. To study transient clusters of localized particles, we calculate a generalized susceptibility corresponding to the self-part of a four-point time dependent density correlation function. We find that this generalized susceptibility is maximum on the time scale of the structural relaxation, where a strong increase of the peak height indicates a growing length of spatial correlations between localized particles upon cooling. Characterizing the local structural environments of the most mobile and the most immobile particles, respectively, we show that high particle mobility is facilitated by, but not limited to, the vicinity of defects in the network structure.

Spatially heterogeneous dynamics investigated via a time-dependent four-point density correlation function

Relaxation in supercooled liquids above their glass transition and below the onset temperature of “slow” dynamics involves the correlated motion of neighboring particles. This correlated motion results in the appearance of spatially heterogeneous dynamics or “dynamical heterogeneity.” Traditional two-point time-dependent density correlation functions, while providing information about the transient “caging” of particles on cooling, are unable to provide sufficiently detailed information about correlated motion and dynamical heterogeneity. Here, we study a four-point, time-dependent density correlation function g4(r,t) and corresponding “structure factor” S4(q,t) which measure the spatial correlations between the local liquid density at two points in space, each at two different times, and so are sensitive to dynamical heterogeneity. We study g4(r,t) and S4(q,t) via molecular dynamics simulations of a binary Lennard-Jones mixture approaching the mode coupling temperature from above. We find that the correlations between particles measured by g4(r,t) and S4(q,t) become increasingly pronounced on cooling. The corresponding dynamical correlation length ξ4(t) extracted from the small-q behavior of S4(q,t) provides an estimate of the range of correlated particle motion. We find that ξ4(t) has a maximum as a function of time t, and that the value of the maximum of ξ4(t) increases steadily from less than one particle diameter to a value exceeding nine particle diameters in the temperature range approaching the mode coupling temperature from above. At the maximum, ξ4(t) and the α relaxation time τα are related by a power law. We also examine the individual contributions to g4(r,t), S4(q,t), and ξ4(t), as well as the corresponding order parameter Q(t) and generalized susceptibility χ4(t), arising from the self and distinct contributions to Q(t). These contributions elucidate key differences between domains of localized and delocalized particles.

Growing correlation length on cooling below the onset of caging in a simulated glass-forming liquid.

We present a calculation of a fourth-order, time-dependent density correlation function that measures higher-order spatiotemporal correlations of the density of a liquid. From molecular dynamics simulations of a glass-forming Lennard-Jones liquid, we find that the characteristic length scale of this function has a maximum as a function of time which increases steadily beyond the characteristic length of the static pair correlation function g(r) in the temperature range approaching the mode coupling temperature from above. This length scale provides a measure of the spatially heterogeneous nature of the dynamics of the liquid in the alpha-relaxation regime.

Generalized Fokker-Planck Kinetic Model

We study some consequences of a nonlinear kinetic model that has the structure of a constant-collision-time, Fokker-Planck equation. The collision operator, however, satisfies momentum and energy conservation in addition to particle conservation. One can find the relaxation spectrum for the nonlinear as well as linear problem. It consists of an infinite sequence of decays with the spectrum of the quantum harmonic oscillator. It is also possible to solve exactly the initial-value problem for the linearized kinetic equation. The Green's function for the complete operator obeys a degenerate integral equation containing the Green's function of the usual Fokker-Planck equation. We find the exact time-dependent density correlation function and thus the inelastic scattering cross section that determines the neutron scattering and the Brillouin scattering of light from the system.

Application of a Kinetic Model to Time-Dependent Density Correlations in Fluids

The kinetic model of Bhatnagar, Gross, and Krook is used to study the double Fourier transform of the time-dependent density correlation function $G(r, t)$. The results are appropriate to a dilute fluid for arbitrary ratio of wavelength to mean free path. The results of the model calculations are compared to those derived from the linearized hydrodynamic equations. Since neutron and light-scattering experiments can be analyzed in terms of $G(r, t)$, this comparison indicates that the hydrodynamic description should be applicable for momentum transfers less than ($\frac{\ensuremath{\hbar}}{2\ensuremath{\lambda}}$), where $\ensuremath{\lambda}$ is the collision mean free path in the fluid.