9,118 research outputs found
Coarse-graining complex dynamics: Continuous Time Random Walks vs. Record Dynamics
Continuous Time Random Walks (CTRW) are widely used to coarse-grain the
evolution of systems jumping from a metastable sub-set of their configuration
space, or trap, to another via rare intermittent events. The multi-scaled
behavior typical of complex dynamics is provided by a fat-tailed distribution
of the waiting time between consecutive jumps. We first argue that CTRW are
inadequate to describe macroscopic relaxation processes for three reasons:
macroscopic variables are not self-averaging, memory effects require an
all-knowing observer,and different mechanisms whereby the jumps affect
macroscopic variables all produce identical long time relaxation behaviors.
Hence, CTRW shed no light on the link between microscopic and macroscopic
dynamics. We then highlight how a more recent approach, Record Dynamics (RD)
provides a viable alternative, based on a very different set of physical ideas:
while CTRW make use of a renewal process involving identical traps of infinite
size, RD embodies a dynamical entrenchment into a hierarchy of traps which are
finite in size and possess different degrees of meta-stability. We show in
particular how RD produces the stretched exponential, power-law and logarithmic
relaxation behaviors ubiquitous in complex dynamics, together with the
sub-diffusive time dependence of the Mean Square Displacement characteristic of
single particles moving in a complex environment.Comment: 6 pages. To appear in EP
Convergence rate of numerical solutions to SFDEs with jumps
In this paper, we are interested in numerical solutions of stochastic functional differential equations with jumps. Under a global Lipschitz condition, we show that the pth-moment convergence of Euler–Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1/p for any p ≥ 2. This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1/2 for any p ≥ 2. It is therefore best to use the mean-square convergence for stochastic functional differential equations with jumps. Moreover, under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1/2, provided that local Lipschitz constants, valid on balls of radius j, do not grow faster than log j
Modelling background charge rearrangements near single-electron transistors as a Poisson process
Background charge rearrangements in metallic single-electron transistors are
modelled in two-level tunnelling systems as a Poisson process with a scale
parameter as only variable. The model explains the recent observation of
asymmetric Coulomb blockade peak spacing distributions in metallic
single-electron transistors. From the scale parameter we estimate the average
size of the tunnelling systems, their density of states, and the height of
their energy barrier. We conclude that the observed background charge
rearrangements predominantly take place in the substrate of the single-electron
transistor.Comment: 7 pages, 2 eps figures, used epl.cls macro include
Spreading Processes over Socio-Technical Networks with Phase-Type Transmissions
Most theoretical tools available for the analysis of spreading processes over
networks assume exponentially distributed transmission and recovery times. In
practice, the empirical distribution of transmission times for many real
spreading processes, such as the spread of web content through the Internet,
are far from exponential. To bridge this gap between theory and practice, we
propose a methodology to model and analyze spreading processes with arbitrary
transmission times using phase-type distributions. Phase-type distributions are
a family of distributions that is dense in the set of positive-valued
distributions and can be used to approximate any given distributions. To
illustrate our methodology, we focus on a popular model of spreading over
networks: the susceptible-infected-susceptible (SIS) networked model. In the
standard version of this model, individuals informed about a piece of
information transmit this piece to its neighbors at an exponential rate. In
this paper, we extend this model to the case of transmission rates following a
phase-type distribution. Using this extended model, we analyze the dynamics of
the spread based on a vectorial representations of phase-type distributions. We
illustrate our results by analyzing spreading processes over networks with
transmission and recovery rates following a Weibull distribution
Useful martingales for stochastic storage processes with L\'{e}vy-type input
In this paper we generalize the martingale of Kella and Whitt to the setting
of L\'{e}vy-type processes and show that the (local) martingales obtained are
in fact square integrable martingales which upon dividing by the time index
converge to zero a.s. and in . The reflected L\'{e}vy-type process is
considered as an example.Comment: 15 pages. arXiv admin note: substantial text overlap with
arXiv:1112.475
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