Janett Prehl

Tsallis Relative Entropy and Anomalous Diffusion

In this paper we utilize the Tsallis relative entropy, a generalization of the Kullback–Leibler entropy in the frame work of non-extensive thermodynamics to analyze the properties of anomalous diffusion processes. Anomalous (super-) diffusive behavior can be described by fractional diffusion equations, where the second order space derivative is extended to fractional order α ∈ (1, 2). They represent a bridging regime, where for α = 2 one obtains the diffusion equation and for α = 1 the (half) wave equation is given. These fractional diffusion equations are solved by so-called stable distributions, which exhibit heavy tails and skewness. In contrast to the Shannon or Tsallis entropy of these distributions, the Kullback and Tsallis relative entropy, relative to the pure diffusion case, induce a natural ordering of the stable distributions consistent with the ordering implied by the pure diffusion and wave limits.

Time Evolution of Relative Entropies for Anomalous Diffusion

The entropy production paradox for anomalous diffusion processes describes a phenomenon where one-parameter families of dynamical equations, falling between the diffusion and wave equations, have entropy production rates (Shannon, Tsallis or Renyi) that increase toward the wave equation limit unexpectedly. Moreover, also surprisingly, the entropy does not order the bridging regime between diffusion and waves at all. However, it has been found that relative entropies, with an appropriately chosen reference distribution, do. Relative entropies, thus, provide a physically sensible way of setting which process is “nearer” to pure diffusion than another, placing pure wave propagation, desirably, “furthest” from pure diffusion. We examine here the time behavior of the relative entropies under the evolution dynamics of the underlying one-parameter family of dynamical equations based on space-fractional derivatives.

An embarrassingly parallel algorithm for random walk simulations on random fractal structures

Abstract Anomalous diffusion is often simulated by random walks on random fractal structures. As existing simulation methods either lack a high degree of parallelism or impose restrictions on the choice of fractal structures, a new approach is proposed here. We present a parallel algorithm for simulating random walks on fractal structures that is suitable for a wide variety of hardware architectures. The degree of parallelism of the algorithm equals the number of random walkers, which is achieved by its communication-avoiding design. In contrast to other approaches, the random fractal structure is not pre-computed at whole. Instead, only the surrounding of each random walker is calculated by the parallel threads while the random walker moves around on the fractal structure.

Symmetric Fractional Diffusion and Entropy Production

The discovery of the entropy production paradox (Hoffmann et al., 1998) raised basic questions about the nature of irreversibility in the regime between diffusion and waves. First studied in the form of spatial movements of moments of H functions, pseudo propagation is the pre-limit propagation-like movements of skewed probability density function (PDFs) in the domain between the wave and diffusion equations that goes over to classical partial differential equation propagation of characteristics in the wave limit. Many of the strange properties that occur in this extraordinary regime were thought to be connected in some manner to this form of proto-movement. This paper eliminates pseudo propagation by employing a similar evolution equation that imposes spatial unimodal symmetry on evolving PDFs. Contrary to initial expectations, familiar peculiarities emerge despite the imposed symmetry, but they have a distinct character.

Desiccation of a clay film: Cracking versus peeling

We report a simulation study on competition between cracking and peeling, in a layer of clay on desiccation and how this is affected by the rate of drying, as well as the roughness of the substrate. The system is based on a simple 2-dimensional spring model. A vertical section through the layer with finite thickness is represented by a rectangular array of nodes connected by linear springs on a square lattice. The effect of reduction of the natural length of the springs, which mimics the drying is studied. Varying the strength of adhesion between sample and substrate and the rate of penetration of the drying front produces an interesting phase diagram, showing cross-over from peeling to cracking behavior. Changes in the number and width of cracks on varying the layer thickness is observed to reproduce experimental reports.

Random walks of oriented particles on fractals

Random walks of point particles on fractals exhibit subdiffusive behavior, where the anomalous diffusion exponent is smaller than one, and the corresponding random walk dimension is larger than two. This is due to the limited space available in fractal structures. Here, we endow the particles with an orientation and analyze their dynamics on fractal structures. In particular, we focus on the dynamical consequences of the interactions between the local surrounding fractal structure and the particle orientation, which are modeled using an appropriate move class. These interactions can lead to particles becoming temporarily or permanently stuck in parts of the structure. A surprising finding is that the random walk dimension is not affected by the orientation while the diffusion constant shows a variety of interesting and surprising features.

A dual power-law distribution for the stellar initial mass function

We introduce a new dual power law (DPL) probability distribution function for the mass distribution of stellar and substellar objects at birth, otherwise known as the initial mass function (IMF). The model contains both deterministic and stochastic elements, and provides a unified framework within which to view the formation of brown dwarfs and stars resulting from an accretion process that starts from extremely low mass seeds. It does not depend upon a top down scenario of collapsing (Jeans) masses or an initial lognormal or otherwise IMF-like distribution of seed masses. Like the modified lognormal power law (MLP) distribution, the DPL distribution has a power law at the high mass end, as a result of exponential growth of mass coupled with equally likely stopping of accretion at any time interval. Unlike the MLP, a power law decay also appears at the low mass end of the IMF. This feature is closely connected to the accretion stopping probability rising from an initially low value up to a high value. This might be associated with physical effects of ejections sometimes (i.e., rarely) stopping accretion at early times followed by outflow driven accretion stopping at later times, with the transition happening at a critical time (therefore mass). Comparing the DPL to empirical data, the critical mass is close to the substellar mass limit, suggesting that the onset of nuclear fusion plays an important role in the subsequent accretion history of a young stellar object.

Modeling Reaction Kinetics of Twin Polymerization via Differential Scanning Calorimetry

Abstract We present a kinetic model for the reaction mechanism of acid-catalyzed twin polymerization. Our model characterizes the reaction mechanism not by the reactants, intermediate structures, and products, but via reaction-relevant moieties. We apply our model for three different derivatives of 2,2’-Spirobi[4H-1,3,2-benzodioxasiline] and determine activation energies, reaction enthalpies, and reaction rate constants for the reaction steps in our mechanism. We compare our findings to previously reported values obtained from density functional theory calculations. Furthermore, with this approach we are also able to follow the time development of the concentrations of the reaction-relevant moieties.