59,876 research outputs found
Semi-Markov models and motion in heterogeneous media
In this paper we study continuous time random walks (CTRWs) such that the
holding time in each state has a distribution depending on the state itself.
For such processes, we provide integro-differential (backward and forward)
equations of Volterra type, exhibiting a position dependent convolution kernel.
Particular attention is devoted to the case where the holding times have a
power-law decaying density, whose exponent depends on the state itself, which
leads to variable order fractional equations. A suitable limit yields a
variable order fractional heat equation, which models anomalous diffusions in
heterogeneous media
Multicritical continuous random trees
We introduce generalizations of Aldous' Brownian Continuous Random Tree as
scaling limits for multicritical models of discrete trees. These discrete
models involve trees with fine-tuned vertex-dependent weights ensuring a k-th
root singularity in their generating function. The scaling limit involves
continuous trees with branching points of order up to k+1. We derive explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with fractional
derivative couplings. We prove universality by rederiving the same results
within a purely continuous axiomatic approach based on the resolution of a set
of consistency relations for the multi-point correlations. The average profile
is shown to obey a fractional differential equation whose solution involves
hypergeometric functions and matches the integral formula of the discrete
approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps
Fluid limit of the continuous-time random walk with general LĂ©vy jump distribution functions.
The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions Ï(t) and general jump distribution functions η(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times, and algebraic decaying jump distributions, corresponding to LĂ©vy stable processes, the fluid limit leads to the fractional diffusion equation of order α in space and order ÎČ in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general LĂ©vy stochastic processes in the LĂ©vy-Khintchine representation for the jump distribution function and obtain an integro-differential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated LĂ©vy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as Ïc ⌠λ âα/ÎČ where 1/λ is the truncation length scale. The asymptotic behavior of the propagator (Greenâs function) of the truncated fractional equation exhibits a transition from algebraic decay for t > Ïc
On the fluid limit of the continuous-time random walk with general LĂ©vy jump distribution functions.
The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions Ï(t) and general jump distribution functions η(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times, and algebraic decaying jump distributions, corresponding to LĂ©vy stable processes, the fluid limit leads to the fractional diffusion equation of order α in space and order ÎČ in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general LĂ©vy stochastic processes in the LĂ©vy-Khintchine representation for the jump distribution function and obtain an integro-differential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated LĂ©vy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as Ïc ⌠λ âα/ÎČ where 1/λ is the truncation length scale. The asymptotic behavior of the propagator (Greenâs function) of the truncated fractional equation exhibits a transition from algebraic decay for t << Ïc to stretched Gaussian decay for t >> Ïc
On the fluid limit of the continuous-time random walk with general LĂ©vy jump distribution functions
The continuous time random walk (CTRW) is a natural generalization of the Brownian random
walk that allows the incorporation of waiting time distributions Ï(t) and general jump distribution functions η(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential
decaying waiting times and Gaussian jump distribution functions the fluid limit leads to
the diffusion equation. On the other hand, for algebraic decaying waiting times, and
algebraic decaying jump distributions, corresponding to LĂ©vy stable processes, the
fluid limit leads to the fractional diffusion equation of order α in space and order ÎČ in time. However,
these are two special cases of a wider class of models. Here we consider the CTRW for the most
general LĂ©vy stochastic processes in the LĂ©vy-Khintchine representation for the jump distribution
function and obtain an integro-differential equation describing the dynamics in the fluid limit. The
resulting equation contains as special cases the regular and the fractional diffusion equations. As an
application we consider the case of CTRWs with exponentially truncated LĂ©vy jump distribution
functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional
derivatives which describes the interplay between memory, long jumps, and truncation effects
in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to
subdiffusion with the crossover time scaling as Ïc ⌠λ âα/ÎČ where 1/λ is the truncation length scale.
The asymptotic behavior of the propagator (Greenâs function) of the truncated fractional equation
exhibits a transition from algebraic decay for t > Ï
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