Forcing anomalous scaling on demographic fluctuations

We discuss the conditions under which a population of anomalously diffusing individuals can be characterized by demographic fluctuations that are anomalously scaling themselves. Two examples are provided in the case of individuals migrating by Gaussian diffusion, and by a sequence of L\'evy flights.Comment: 5 pages 2 figure

Anomalous diffusion and stretched exponentials in heterogeneous glass-forming liquids: Low-temperature behavior

We propose a model of a heterogeneous glass forming liquid and compute the low-temperature behavior of a tagged molecule moving within it. This model exhibits stretched-exponential decay of the wavenumber-dependent, self intermediate scattering function in the limit of long times. At temperatures close to the glass transition, where the heterogeneities are much larger in extent than the molecular spacing, the time dependence of the scattering function crosses over from stretched-exponential decay with an index $b=1/2$ at large wave numbers to normal, diffusive behavior with $b = 1$ at small wavenumbers. There is a clear separation between early-stage, cage-breaking $\beta$ relaxation and late-stage $\alpha$ relaxation. The spatial representation of the scattering function exhibits an anomalously broad exponential (non-Gaussian) tail for sufficiently large values of the molecular displacement at all finite times.Comment: 9 pages, 6 figure

Relativistic Weierstrass random walks

The Weierstrass random walk is a paradigmatic Markov chain giving rise to a L\'evy-type superdiffusive behavior. It is well known that Special Relativity prevents the arbitrarily high velocities necessary to establish a superdiffusive behavior in any process occurring in Minkowski spacetime, implying, in particular, that any relativistic Markov chain describing spacetime phenomena must be essentially Gaussian. Here, we introduce a simple relativistic extension of the Weierstrass random walk and show that there must exist a transition time $t_c$ delimiting two qualitative distinct dynamical regimes: the (non-relativistic) superdiffusive L\'evy flights, for $t < t_c$, and the usual (relativistic) Gaussian diffusion, for $t>t_c$. Implications of this crossover between different diffusion regimes are discussed for some explicit examples. The study of such an explicit and simple Markov chain can shed some light on several results obtained in much more involved contexts.Comment: 5 pages, final version to appear in PR

Theory of Single File Diffusion in a Force Field

The dynamics of hard-core interacting Brownian particles in an external potential field is studied in one dimension. Using the Jepsen line we find a very general and simple formula relating the motion of the tagged center particle, with the classical, time dependent single particle reflection ${\cal R}$ and transmission ${\cal T}$ coefficients. Our formula describes rich physical behaviors both in equilibrium and the approach to equilibrium of this many body problem.Comment: 4 Phys. Rev. page

Migration and proliferation dichotomy in tumor cell invasion

We propose a two-component reaction-transport model for the migration-proliferation dichotomy in the spreading of tumor cells. By using a continuous time random walk (CTRW) we formulate a system of the balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration. The transport process is formulated in terms of the CTRW with an arbitrary waiting time distribution law. Proliferation is modeled by a standard logistic growth. We apply hyperbolic scaling and Hamilton-Jacobi formalism to determine the overall rate of tumor cell invasion. In particular, we take into account both normal diffusion and anomalous transport (subdiffusion) in order to show that the standard diffusion approximation for migration leads to overestimation of the overall cancer spreading rate.Comment: 9 page

Rates of convergence of nonextensive statistical distributions to Levy distributions in full and half spaces

The Levy-type distributions are derived using the principle of maximum Tsallis nonextensive entropy both in the full and half spaces. The rates of convergence to the exact Levy stable distributions are determined by taking the N-fold convolutions of these distributions. The marked difference between the problems in the full and half spaces is elucidated analytically. It is found that the rates of convergence depend on the ranges of the Levy indices. An important result emerging from the present analysis is deduced if interpreted in terms of random walks, implying the dependence of the asymptotic long-time behaviors of the walks on the ranges of the Levy indices if N is identified with the total time of the walks.Comment: 20 page

Another analytic view about quantifying social forces

Montroll had considered a Verhulst evolution approach for introducing a notion he called "social force", to describe a jump in some economic output when a new technology or product outcompetes a previous one. In fact, Montroll's adaptation of Verhulst equation is more like an economic field description than a "social force". The empirical Verhulst logistic function and the Gompertz double exponential law are used here in order to present an alternative view, within a similar mechanistic physics framework. As an example, a "social force" modifying the rate in the number of temples constructed by a religious movement, the Antoinist community, between 1910 and 1940 in Belgium is found and quantified. Practically, two temple inauguration regimes are seen to exist over different time spans, separated by a gap attributed to a specific "constraint", a taxation system, but allowing for a different, smooth, evolution rather than a jump. The impulse force duration is also emphasized as being better taken into account within the Gompertz framework. Moreover, a "social force" can be as here, attributed to a change in the limited need/capacity of some population, coupled to some external field, in either Verhulst or Gompertz equation, rather than resulting from already existing but competing goods as imagined by Montroll.Comment: 4 figures, 29 refs., 15 pages; prepared for Advances in Complex System

Subordinated Langevin Equations for Anomalous Diffusion in External Potentials - Biasing and Decoupled Forces

The role of external forces in systems exhibiting anomalous diffusion is discussed on the basis of the describing Langevin equations. Since there exist different possibilities to include the effect of an external field the concept of {\it biasing} and {\it decoupled} external fields is introduced. Complementary to the recently established Langevin equations for anomalous diffusion in a time-dependent external force-field [{\it Magdziarz et al., Phys. Rev. Lett. {\bf 101}, 210601 (2008)}] the Langevin formulation of anomalous diffusion in a decoupled time-dependent force-field is derived

Exit times in non-Markovian drifting continuous-time random walk processes

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.Comment: 9 pages, 3 color plots, two-column revtex 4; new Appendix and references adde

Multifractals of Normalized First Passage Time in Sierpinski Gasket

The multifractal behavior of the normalized first passage time is investigated on the two dimensional Sierpinski gasket with both absorbing and reflecting barriers. The normalized first passage time for Sinai model and the logistic model to arrive at the absorbing barrier after starting from an arbitrary site, especially obtained by the calculation via the Monte Carlo simulation, is discussed numerically. The generalized dimension and the spectrum are also estimated from the distribution of the normalized first passage time, and compared with the results on the finitely square lattice.Comment: 10 pages, Latex, with 3 figures and 1 table. to be published in J. Phys. Soc. Jpn. Vol.67(1998