Session R47: Coalescence, Fragmentation, Mixing and Anomalous Diffusion

Precipitation Accumulations, Intensities and Durations

Precipitation accumulations (the integrated precipitation amount over the course of an event) exhibit large fluctuations, yielding distributions with a long power law regime, followed by a sharp decrease at a characteristic large-event cutoff scale. This cutoff scale limits the largest events experienced in a particular region. First-passage processes for fluctuations of moisture relative to a temperature-dependent threshold for onset of precipitation yield quantitative prototypes for this behavior including expressions for the cutoff in terms of the interplay between moisture loss by precipitation and moisture convergence fluctuations. The behavior of accumulations under coarse graining (e.g., daily precipitation) is important as these quantities are routinely used for characterization and assessment, for example of climate change effects. Earlier work has sometimes fit Gamma distributions to daily intensities, with no physical explanation. In this work we show that daily intensity distributions can be explained through these stochastic prototypes, with the traditional Gamma distributions as approximations. These results help to understand relations in US accumulations from hourly observations and daily rainfall.
  

Universal scaling laws for the activation of pre-existing natural fractures

The activation of sparsely distributed weak planes (fractures) in low permeability rocks by injecting a pressurized fluid creates flow paths that facilitate extraction of geothermal energy from hot dry rocks and gas from shale. Using a large-cell renormalization-group approach, we show that the process of forming percolating paths of activated fractures can be viewed as a critical phenomenon similar to the dilute n-vector model under the following assumptions: 1) a fracture is activated, following Mohr’s criterion, when the fluid pressure reaches a critical value that depends on its orientation with respect to the in-situ stress field; 2) the viscous pressure drop is negligible compared to the variability in the critical pressures; and 3) the lengths of the natural fractures are all of the same order of magnitude. Similar to the n-vector model, a crossover occurs as the linear dimension of the cluster of activated fractures becomes comparable to the correlation length of the natural fractures. The crossover exponent is found to be equal to 1 and the critical exponent ν is equal to the critical exponent of regular percolation. Based on the results, scaling laws relating the cluster radius with the transient injection pressure are provided.   

Limits on Inferring the Past

Nature is irreversible. Any state of knowledge, regardless of how exact, will invariably and universally deteriorate into a entropy maximizing probability distribution. Our loss of information, forward in time, can be quantified by the entropy generation rate. Here we address the converse question of how well one can reconstruct a past state given exact information concerning the present. To this end, we invert Langevin dynamics to estimate the original starting point of a number of particles, given their exact positions at a later time. We define "reconstruction entropy" - a measure of how many good candidates there are for a past state, given the present state. We then evaluate reconstruction entropy for Langevin dynamics exactly for simple cases, explore general properties of reconstruction entropy, and make several conjectures about the behavior of Langevin reconstruction entropy as a function of time. Finally, we obtain a formula connecting reconstruction entropy, thermodynamic entropy, and system parameters.   

Large deviation analysis of rapid onset of rain showers

Rainfall from ice-free cumulus clouds requires collisions
of large numbers of microscopic droplets to create every raindrop.
The onset of rain showers can be surprisingly rapid, much faster than the mean time
required for a single collision. Large-deviation theory is used to explain
this observation.

The presentation is based upon a recent paper: Phys. Rev. Lett., 116, 018501 (2016).   

Statistical Modeling for Desiccation Cracking Based on Shape-Dependent Fragmentation Process

We investigate statistical properties of desiccation crack patterns of a thin layer of paste experimentally and theoretically. As an experimental result, we have discovered two characteristic properties associated with the fragment distributions. One is that the fragment size distribution varies with time but can be collapsed into a time-invariant distribution by scaling with its mean, and another is that the aspect ratio distribution also converges with a time-invariant distribution. In order to explain these statistical properties, we have constructed a statistical model based on an elastic theory that describes the dynamics of the paste driven by the shrinkage owing the desiccation. We have confirmed that the statistical model can reproduce the statistical properties observed in the experiments, and have revealed through its analytical calculation that the statistical properties arise from a characteristic scaling relation between a critical stress needed to crack a fragment and the shape of the fragment.   

The Mpemba index and anomalous relaxation

The Mpemba effect is a counter-intuitive relaxation phenomenon, in which a system prepared at a hot temperature cools faster than an identical system starting from a cold temperature when both are quenched to an even colder bath. Such non-monotonic relaxations were observed in various systems, including water, magnetic alloys, and driven granular gases. We analyze the Mpemba effect for Markovian dynamics and show that under proper conditions, exponentially faster relaxations are possible. These special relaxations, which we named ``the strong Mpemba effect'', can be classified by a topological index. Using the parity of this index, we study the occurrence of the strong Mpemba effect for a large class of thermal quench processes and show that it happens with non-zero probability even in the thermodynamical limit. Thus, we expect that the strong Mpemba effect can be observed experimentally in a wide variety of systems. We analyze the different types of Mpemba relaxations in the mean field anti-ferromagnet Ising model and show surprisingly rich Mpemba phase diagram.   

Viscoelastic subdiffusion in random Gaussian potentials

Subdiffusion in a fluctuating environment is archetypal for viscoelastic cytosol of living cells, where a spatial disorder also naturally emerges. We model it by a generalized Langevin equation dynamics with long-range memory in stationary random Gaussian potentials featured by decaying spatial correlations. It is shown that for a relatively small potential energy disorder in units of thermal energy (several k_BT) viscoelastic subdiffusion in the ensemble sense easily overcomes the potential disorder. Asymptotically it is not distinguishable from the unobstructed subdiffusion. However, diffusion on the level of single-trajectory averages still exhibits transiently a characteristic scatter featuring weak ergodicity breaking. With an increase of the disorder strength to 5-10 k_BT, a very long transient regime of logarithmic or Sinai-like diffusion emerges. This nominally ultraslow Sinai diffusion can, however, be transiently even faster than the free-space viscoelastic subdiffusion, in the absolute terms, on the ensemble level. On the level of single-trajectories, such a strongly obstructed viscoelastic subdiffusion is always slower and exhibits a strong scatter in single-trajectory averages.   

A Multispecies Exclusion Process Inspired by Intraflagellar Transport: Effects of Fusion and Fission

Inspired by the “fusion and fission” of Intraflagellar transport (IFT) trains, we have developed a multispecies exclusion model where length-conserving probabilistic fusion and fission of the hard rods are allowed. A monodisperse population of rods of length L=1 enter the system, but their lengths keep fluctuating because of fusion and fission, as they move in a step-by-step manner towards the distal end obeying mutual exclusion. Fusion of two neighboring hard rods of lengths L1 and L2 results in a single rod of longer length L=L1+L2 provided L is less than, or equal to, N, which is the maximum allowed length for a rod inside the system. Similarly, length-conserving fission of a rod results in two shorter daughter rods. The density- and the flux-profiles of the rods are in excellent agreement with computer simulations and leads to a ‘transition zone’ in the fusion dominated regime. The insights gained from this simple model, emerging from the interplay of fusion and fission, are likely to have important implications for IFT and for other similar transport phenomena in long cell protrusions.   

Weak Galilean invariance as a selection principle for stochastic coarse-grained diffusive models

Galilean invariance states that the equations of motion of closed systems do not change under Galilei transformations to different inertial frames. However, real world systems typically violate it, as they are described by coarse-grained models integrating complex microscopic interactions indistinguishably as friction and stochastic forces. This leaves no alternative principle to assess a priori the physical consistency of a given stochastic model. Here, we use the Kac-Zwanzig model of Brownian motion to clarify how Galilean invariance is broken during the coarse graining procedure to derive stochastic equations. This analysis yields a set of rules characterizing systems in different inertial frames, called "weak Galilean invariance". Several stochastic processes, generating normal and anomalous diffusion, are shown to be invariant in these terms, except the continuous-time random walk whose correct invariant description is discussed. These results are particularly relevant for the modelling of biological systems as they provide a theoretical principle to select stochastic models of complex dynamics prior to their validation against experimental data.   

Cluster Size Distributions of Single Particle Jumps in Amorphous SiO2

We study the aging dynamics of the strong glass former SiO₂
via molecular dynamics simulations. Our system consists of
38416 Si and 76832 O atoms, is well equilibrated at 5000 K,
then instantly quenched to a temperature below the glass transition,
and then aged for up to 16 ns. At the low temperature,
we use single particle trajectories to identify single particle
jump events when a particle escapes its cage formed by its neighbors.
To study how these jump events are correlated in space
and time, we find clusters of jumping particles. We will present
cluster size distributions for three low temperatures (2500 K,
2750 K, 3000 K),for several aging times, and for various cluster definitions:
clusters of simultaneously jumping particles,
space-time clusters (jump events which are neighbors in
space and time), and clusters of irreversible
and reversible (to the previous cage site returning)
jumping particles.   

Random Searches in Non-Euclidean Spaces

Many animals display characteristic foraging patterns in their behavior when searching for food. Previous studies on foraging have shown that, in many cases, animals follow a Levy flight pattern with a power law distribution of step sizes that might be tuned for optimal search efficiency. While all of biology is constrained to live in Euclidean geometry, natural search processes may take place in effectively more complex spaces with a network topology such as networks of caves or other ecological niches. Motivated by the recent equivalency that has been shown to exist between complex scale-free networks and hyperbolic geometries, we consider the question of optimal foraging in the case when searching occurs in a curved geometry. We study the search process in an appropriate projection of the hyperbolic space and make use of the equivalency to infer connections between optimal Levy flight searching in hyperbolic space and searching on a scale-free network.
  

Exact Results for the Nonergodicity of Generalized Lévy Walks

We study the generalized Lévy walk introduced by Shlesinger et. al. [1], which was introduced to explain Richardson's t³-law for turbulent diffusion of passive scalars. This model, which can describe subdiffusive, diffusive, or superdiffusive motion, is considered as very physical because of the continuous nature of its trajectories. It applies to many processes in physics and biology, where velocity and duration or distance of travel are coupled. In extension of our results for integrated Brownian motion [2], we are able to obtain exact results for the ensemble- and the time-averaged squared displacement, and for the ergodicity breaking parameter in the full parameter space of this model. In certain regions of the latter we obtain surprising results such as the divergence of the mean-squared displacements, at variance with the t³-law, the divergence of the ergodicity breaking parameter despite a finite mean-squared displacement, and also subdiffusion which appears superdiffusive when one only considers time averages.
[1] M. Shlesinger, B. West and J. Klafter, Phys. Rev. Lett. 58, 1100 (1987)
[2] T. Albers and G. Radons, Phys. Rev. Lett. 113, 184101 (2014)   

A path from fractional Schrödinger to design and discovery of novel quantum materials

Transport phenomena in multi-scale classical systems, such as disordered media, porous materials, and turbulent fluids, are characterized by multiple spatial and temporal scales, nonlocality, fractional geometry, and non-Gaussian statistics. Transport in multi-scale classical materials is described by the fractional diffusion equation, while its quantum analog, fractional Schr\"{o}dinger equation, governs the dynamics of quantum materials. The fundamental processes of multi-scale quantum materials are carried out on a local fractional space-time metric. We show that the minimization of action on fractional space-time metric with a subsequent evaluation of the Feynman path integral leads to a self-consistent derivation of the fractional Schr\"{o}dinger equation, which is valid for any order of fractional space-time. We apply the derived fractional Schr\"{o}dinger equation to multi-scale quantum materials and show that they can be effectively modeled by a system of cold atoms in a multi-frequency optical potential. Specifically we demonstrate that the tunneling matrix element in fractional quantum materials embedded in a single frequency optical potential exactly matches the corresponding matrix element in multi-frequency optical potential.

  

Anomalous Diffusion on a Growing Domain

The ubiquity of subdiffusive transport in physical and biological systems has led to intensive efforts to provide robust theoretical models for this phenomena. Additionally many physical and biological phenomena occur on domains which evolve with time. We have derived a master equation, using a continuous time random walk, for particles diffusing on a domain that grows with time. Our method supports systems undergoing either standard diffusion and subdiffusion. This allows us to construct models that represent physical and biological systems which incorporate both diffusion and a domain that is growing. The resulting equations feature fractional derivatives. The implementation of the master equation is illustrated with a simple model of subdiffusing proteins in a growing membrane.