Properties of the Interstellar Medium and the Propagation of Cosmic Rays in the Galaxy

The problem of the origin of cosmic rays in the shocks produced by supernova explosions at energies below the so called 'knee' (at ~3*10$^6$ GeV) in the energy spectrum is addressed, with special attention to the propagation of the particles through the inhomogenious interstellar medium and the need to explain recent anisotropy results, [1]. It is shown that the fractal character of the matter density and magnetic field distribution leads to the likelihood of a substantial increase of spatial fluctuations in the cosmic ray energy spectra. While the spatial distribution of cosmic rays in the vicinity of their sources (eg. inside the Galactic disk) does not depend much on the character of propagation and is largely determined by the distribution of their sources, the distribution at large distances from the Galactic disk depends strongly on the character of the propagation. In particular, the fractal character of the ISM leads to what is known as 'anomalous diffusion' and such diffusion helps us to understand the formation of Cosmic Ray Halo. Anomalous diffusion allows an explanation of the recent important result from the Chacaltaya extensive air shower experiment [1], viz. a Galactic Plane Enhancement of cosmic ray intensity in the Outer Galaxy, which is otherwise absent for the case of the so-called 'normal' diffusion. All these effects are for just one reason: anomalous diffusion emphasizes the role of local phenomena in the formation of cosmic ray characteristics in our Galaxy and elsewhere.Comment: 18 pages, 5 figures, accepted by Astropartoicle Physic

Diffusion maps for changing data

Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph associated with it changes depending on some set of parameters. We analyze this type of data in terms of the diffusion distance and the corresponding diffusion map. As the data changes over the parameter space, the low dimensional embedding changes as well. We give a way to go between these embeddings, and furthermore, map them all into a common space, allowing one to track the evolution of X in its intrinsic geometry. A global diffusion distance is also defined, which gives a measure of the global behavior of the data over the parameter space. Approximation theorems in terms of randomly sampled data are presented, as are potential applications.Comment: 38 pages. 9 figures. To appear in Applied and Computational Harmonic Analysis. v2: Several minor changes beyond just typos. v3: Minor typo corrected, added DO

Random Deposition Model with a Constant Capture Length

We introduce a sequential model for the deposition and aggregation of particles in the submonolayer regime. Once a particle has been randomly deposited on the substrate, it sticks to the closest atom or island within a distance \ell, otherwise it sticks to the deposition site. We study this model both numerically and analytically in one dimension. A clear comprehension of its statistical properties is provided, thanks to capture equations and to the analysis of the island-island distance distribution.Comment: 14 pages, minor corrections. Accepted for publication in Progress of Theoretical Physic

Development of Morphogen Gradient: The Role of Dimension and Discreteness

The fundamental processes of biological development are governed by multiple signaling molecules that create non-uniform concentration profiles known as morphogen gradients. It is widely believed that the establishment of morphogen gradients is a result of complex processes that involve diffusion and degradation of locally produced signaling molecules. We developed a multi-dimensional discrete-state stochastic approach for investigating the corresponding reaction-diffusion models. It provided a full analytical description for stationary profiles and for important dynamic properties such as local accumulation times, variances and mean first-passage times. The role of discreteness in developing of morphogen gradients is analyzed by comparing with available continuum descriptions. It is found that the continuum models prediction about multiple time scales near the source region in two-dimensional and three-dimensional systems is not supported in our analysis. Using ideas that view the degradation process as an effective potential, the effect of dimensionality on establishment of morphogen gradients is also discussed. In addition, we investigated how these reaction-diffusion processes are modified with changing the size of the source region

Optimal diffusion in ecological dynamics with Allee effect in a metapopulation

How diffusion impacts on ecological dynamics under the Allee effect and spatial constraints? That is the question we address. Employing a microscopic minimal model in a metapopulation (without imposing nonlinear birth and death rates) we evince --- both numerically and analitically --- the emergence of an optimal diffusion that maximises the survival probability. Even though, at first such result seems counter-intuitive, it has empirical support from recent experiments with engineered bacteria. Moreover, we show that this optimal diffusion disappears for loose spatial constraints.Comment: 16 pages; 6 figure

Many-body effects in tracer particle diffusion with applications for single-protein dynamics on DNA

30% of the DNA in E. coli bacteria is covered by proteins. Such high degree of crowding affect the dynamics of generic biological processes (e.g. gene regulation, DNA repair, protein diffusion etc.) in ways that are not yet fully understood. In this paper, we theoretically address the diffusion constant of a tracer particle in a one dimensional system surrounded by impenetrable crowder particles. While the tracer particle always stays on the lattice, crowder particles may unbind to a surrounding bulk and rebind at another or the same location. In this scenario we determine how the long time diffusion constant ${\cal D}$ (after many unbinding events) depends on (i) the unbinding rate of crowder particles $k_{\rm off}$, and (ii) crowder particle line density $\rho$, from simulations (Gillespie algorithm) and analytical calculations. For small $k_{\rm off}$, we find ${\cal D}\sim k_{\rm off}/\rho^2$ when crowder particles are immobile on the line, and ${\cal D}\sim \sqrt{D k_{\rm off}}/\rho$ when they are diffusing; $D$ is the free particle diffusion constant. For large $k_{\rm off}$, we find agreement with mean-field results which do not depend on $k_{\rm off}$. From literature values of $k_{\rm off}$ and $D$, we show that the small $k_{\rm off}$-limit is relevant for in vivo protein diffusion on a crowded DNA. Our results applies to single-molecule tracking experiments.Comment: 10 pages, 8 figure

Geometry and violent events in turbulent pair dispersion

The statistics of Lagrangian pair dispersion in a homogeneous isotropic flow is investigated by means of direct numerical simulations. The focus is on deviations from Richardson eddy-diffusivity model and in particular on the strong fluctuations experienced by tracers. Evidence is obtained that the distribution of distances attains an almost self-similar regime characterized by a very weak intermittency. The timescale of convergence to this behavior is found to be given by the kinetic energy dissipation time measured at the scale of the initial separation. Conversely the velocity differences between tracers are displaying a strongly anomalous behavior whose scaling properties are very close to that of Lagrangian structure functions. These violent fluctuations are interpreted geometrically and are shown to be responsible for a long-term memory of the initial separation. Despite this strong intermittency, it is found that the mixed moment defined by the ratio between the cube of the longitudinal velocity difference and the distance attains a statistically stationary regime on very short timescales. These results are brought together to address the question of violent events in the distribution of distances. It is found that distances much larger than the average are reached by pairs that have always separated faster since the initial time. They contribute a stretched exponential behavior in the tail of the inter-tracer distance probability distribution. The tail approaches a pure exponential at large times, contradicting Richardson diffusive approach. At the same time, the distance distribution displays a time-dependent power-law behavior at very small values, which is interpreted in terms of fractal geometry. It is argued and demonstrated numerically that the exponent converges to one at large time, again in conflict with Richardson's distribution.Comment: 21 page