481 research outputs found
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 at
large wave numbers to normal, diffusive behavior with at small
wavenumbers. There is a clear separation between early-stage, cage-breaking
relaxation and late-stage 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
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 and transmission 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
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
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 delimiting two qualitative distinct dynamical
regimes: the (non-relativistic) superdiffusive L\'evy flights, for ,
and the usual (relativistic) Gaussian diffusion, for . 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
Cluster growth in far-from-equilibrium particle models with diffusion, detachment, reattachment and deposition
Monolayer cluster growth in far-from-equilibrium systems is investigated by
applying simulation and analytic techniques to minimal hard core particle
(exclusion) models. The first model (I), for post-deposition coarsening
dynamics, contains mechanisms of diffusion, attachment, and slow activated
detachment (at rate epsilon<<1) of particles on a line. Simulation shows three
successive regimes of cluster growth: fast attachment of isolated particles;
detachment allowing further (epsilon t)^(1/3) coarsening of average cluster
size; and t^(-1/2) approach to a saturation size going like epsilon^(-1/2).
Model II generalizes the first one in having an additional mechanism of
particle deposition into cluster gaps, suppressed for the smallest gaps. This
model exhibits early rapid filling, leading to slowing deposition due to the
increasing scarcity of deposition sites, and then continued power law (epsilon
t)^(1/2) cluster size coarsening through the redistribution allowed by slow
detachment. The basic (epsilon t)^(1/3) domain growth laws and epsilon^(-1/2)
saturation in model I are explained by a simple scaling picture. A second,
fuller approach is presented which employs a mapping of cluster configurations
to a column picture and an approximate factorization of the cluster
configuration probability within the resulting master equation. This allows
quantitative results for the saturation of model I in excellent agreement with
the simulation results. For model II, it provides a one-variable scaling
function solution for the coarsening probability distribution, and in
particular quantitative agreement with the cluster length scaling and its
amplitude.Comment: Accepted in Phys. Rev. E; 9 pages with figure
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
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
Dimers on two-dimensional lattices
We consider close-packed dimers, or perfect matchings, on two-dimensional
regular lattices. We review known results and derive new expressions for the
free energy, entropy, and the molecular freedom of dimers for a number of
lattices including the simple-quartic (4^4), honeycomb (6^3), triangular (3^6),
kagome (3.6.3.6), 3-12 (3.12^2) and its dual [3.12^2], and 4-8 (4.8^2) and its
dual Union Jack [4.8^2] Archimedean tilings. The occurrence and nature of phase
transitions are also analyzed and discussed.Comment: Typos corrections in Eqs. (28), (32) and (43
Anomalous Drude Model
A generalization of the Drude model is studied. On the one hand, the free
motion of the particles is allowed to be sub- or superdiffusive; on the other
hand, the distribution of the time delay between collisions is allowed to have
a long tail and even a non-vanishing first moment. The collision averaged
motion is either regular diffusive or L\'evy-flight like. The anomalous
diffusion coefficients show complex scaling laws. The conductivity can be
calculated in the diffusive regime. The model is of interest for the
phenomenological study of electronic transport in quasicrystals.Comment: 4 pages, latex, 2 figures, to be published in Physical Review Letter
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
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