Statistical and Biological Physics

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Crossover in the Slow Decay of Dynamic Correlations in the Lorentz Model

Felix Höfling and Thomas Franosch

Spatial heterogeneities often give rise to intriguing slow dynamics in complex materials, manifested for example by broad frequency-dependent relaxation processes in colloidal gels which form stress-sustaining networks close to the sol-gel transition. A further prominent example are sodium silicates, where the formation of a space-filling network of channels allows for slow diffusion of sodium ions in an arrested host matrix. A minimal model that encompasses spatial disorder and slow dynamics is provided by the Lorentz model, i.e., classical point particles explore without mutual interaction a d-dimensional space in the presence of a frozen array of randomly distributed (possibly overlapping) hard spherical obstacles.

The dynamics of the Lorentz model exhibits a series of intriguing properties. First it is known that at a certain critical density the system exhibits a localization transition and transport ceases to be diffusive. Close to the critical density a dynamic scaling hypothesis holds, and the critical behavior is determined by divergent length and time scales.

Second, a low-density expansion revealed that the velocity autocorrelation functions should exhibit an algebraic long-time anomaly rather than an exponential decay expected from the Lorentz-Boltzmann equation. A direct observation of the tail in computer simulations has been attempted before, however, the amplitude was in disaccord with theoretical prediction. Furthermore the exponent of the algebraic decay appeared to vary with density.

In extensive Molecular Dynamics simulations we have demonstrated that the algebraic tail is indeed universal, however, due to the presence of the underlying percolation transition, the onset of the tail depends sensitively on the density of the scatterers. We have rationalized the growing amplitude as a manifestation of the divergent crossover time scale. In addition, we have calculated the (super-)Burnett coefficient and reported for the first time a linear increase indicative of the breakdown of Fickian diffusion due to frozen heterogeneities.