10 research outputs found
Periodic Homogenization for Inertial Particles
We study the problem of homogenization for inertial particles moving in a
periodic velocity field, and subject to molecular diffusion. We show that,
under appropriate assumptions on the velocity field, the large scale, long time
behavior of the inertial particles is governed by an effective diffusion
equation for the position variable alone. To achieve this we use a formal
multiple scale expansion in the scale parameter. This expansion relies on the
hypo-ellipticity of the underlying diffusion. An expression for the diffusivity
tensor is found and various of its properties studied. In particular, an
expansion in terms of the non-dimensional particle relaxation time (the
Stokes number) is shown to co-incide with the known result for passive
(non-inertial) tracers in the singular limit . This requires the
solution of a singular perturbation problem, achieved by means of a formal
multiple scales expansion in Incompressible and potential fields are
studied, as well as fields which are neither, and theoretical findings are
supported by numerical simulations.Comment: 31 pages, 7 figures, accepted for publication in Physica D. Typos
corrected. One reference adde
Homogenization of Parabolic Equations with a Continuum of Space and Time Scales
This paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available. Namely, we consider divergence form linear parabolic operators in with -coefficients. It appears that the inverse operator maps the unit ball of into a space of functions which at small (time and space) scales are close in norm to a functional space of dimension . It follows that once one has solved these equations at least times it is possible to homogenize them both in space and in time, reducing the number of operation counts necessary to obtain further solutions. In practice we show under a Cordes-type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in (instead of with Euclidean coordinates). If the medium is time-independent, then it is sufficient to solve times the associated elliptic equation in order to homogenize the parabolic equation
Langevin dynamics with space-time periodic nonequilibrium forcing
We present results on the ballistic and diffusive behavior of the Langevin
dynamics in a periodic potential that is driven away from equilibrium by a
space-time periodic driving force, extending some of the results obtained by
Collet and Martinez. In the hyperbolic scaling, a nontrivial average velocity
can be observed even if the external forcing vanishes in average. More
surprisingly, an average velocity in the direction opposite to the forcing may
develop at the linear response level -- a phenomenon called negative mobility.
The diffusive limit of the non-equilibrium Langevin dynamics is also studied
using the general methodology of central limit theorems for additive
functionals of Markov processes. To apply this methodology, which is based on
the study of appropriate Poisson equations, we extend recent results on
pointwise estimates of the resolvent of the generator associated with the
Langevin dynamics. Our theoretical results are illustrated by numerical
simulations of a two-dimensional system
Temporal homogenization of linear ODEs, with applications to parametric super-resonance and energy harvest
We consider the temporal homogenization of linear ODEs of the form ẋ =Ax+ϵP(t)x+f(t), where P(t) is periodic and ϵ is small. Using a 2-scale expansion approach, we obtain the long-time approximation x(t)≈exp(At) (Ω(t)+∫^t_0exp(−Aτ)f(τ)dτ), where Ω solves the cell problem Ω=ϵBΩ+ϵF(t) with an effective matrix B and an explicitly-known F(t). We provide necessary and sufficient conditions for the accuracy of the approximation (over a O(ϵ^(−1)) time-scale), and show how B can be computed (at a cost independent of ϵ). As a direct application, we investigate the possibility of using RLC circuits to harvest the energy contained in small scale oscillations of ambient electromagnetic fields (such as Schumann resonances). Although a RLC circuit parametrically coupled to the field may achieve such energy extraction via parametric resonance, its resistance R needs to be smaller than a threshold κ proportional to the fluctuations of the field, thereby limiting practical applications. We show that if n RLC circuits are appropriately coupled via mutual capacitances or inductances, then energy extraction can be achieved when the resistance of each circuit is smaller than nκ. Hence, if the resistance of each circuit has a non-zero fixed value, energy extraction can be made possible through the coupling of a sufficiently large number n of circuits (n≈1000 for the first mode of Schumann resonances and contemporary values of capacitances, inductances and resistances). The theory is also applied to the control of the oscillation amplitude of a (damped) oscillator
Homogenization in a periodic and time-dependent potential
This paper contains a study of the long time behavior of a diffusion process in a periodic potential. The first goal is to determine a suitable rescaling of time and space so that the diffusion process converges to some homogeneous limit. The issue of interest is to characterize the effective evolution equation. The main result is that in some cases large drifts must be removed in order to get a diffusive asymptotic behavior. This is applied to the homogenization of parabolic differential equations