3,062 research outputs found
Statistics of the first passage time of Brownian motion conditioned by maximum value or area
We derive the moments of the first passage time for Brownian motion
conditioned by either the maximum value or the area swept out by the motion.
These quantities are the natural counterparts to the moments of the maximum
value and area of Brownian excursions of fixed duration, which we also derive
for completeness within the same mathematical framework. Various applications
are indicated.Comment: 29 pages, 4 figures include
The first-passage area for drifted Brownian motion and the moments of the Airy distribution
An exact expression for the distribution of the area swept out by a drifted
Brownian motion till its first-passage time is derived. A study of the
asymptotic behaviour confirms earlier conjectures and clarifies their range of
validity. The analysis also leads to a simple closed-form solution for the
moments of the Airy distribution.Comment: 13 page
Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
We calculate analytically the probability density of the time
at which a continuous-time Brownian motion (with and without drift) attains its
maximum before passing through the origin for the first time. We also compute
the joint probability density of the maximum and . In the
driftless case, we find that has power-law tails: for large and for small . In
presence of a drift towards the origin, decays exponentially for large
. The results from numerical simulations are in excellent agreement with
our analytical predictions.Comment: 13 pages, 5 figures. Published in Journal of Statistical Mechanics:
Theory and Experiment (J. Stat. Mech. (2007) P10008,
doi:10.1088/1742-5468/2007/10/P10008
Area distribution and the average shape of a L\'evy bridge
We consider a one dimensional L\'evy bridge x_B of length n and index 0 <
\alpha < 2, i.e. a L\'evy random walk constrained to start and end at the
origin after n time steps, x_B(0) = x_B(n)=0. We compute the distribution
P_B(A,n) of the area A = \sum_{m=1}^n x_B(m) under such a L\'evy bridge and
show that, for large n, it has the scaling form P_B(A,n) \sim n^{-1-1/\alpha}
F_\alpha(A/n^{1+1/\alpha}), with the asymptotic behavior F_\alpha(Y) \sim
Y^{-2(1+\alpha)} for large Y. For \alpha=1, we obtain an explicit expression of
F_1(Y) in terms of elementary functions. We also compute the average profile <
\tilde x_B (m) > at time m of a L\'evy bridge with fixed area A. For large n
and large m and A, one finds the scaling form = n^{1/\alpha}
H_\alpha({m}/{n},{A}/{n^{1+1/\alpha}}), where at variance with Brownian bridge,
H_\alpha(X,Y) is a non trivial function of the rescaled time m/n and rescaled
area Y = A/n^{1+1/\alpha}. Our analytical results are verified by numerical
simulations.Comment: 21 pages, 4 Figure
On the Inelastic Collapse of a Ball Bouncing on a Randomly Vibrating Platform
We study analytically the dynamics of a ball bouncing inelastically on a
randomly vibrating platform, as a simple toy model of inelastic collapse. Of
principal interest are the distributions of the number of flights n_f till the
collapse and the total time \tau_c elapsed before the collapse. In the strictly
elastic case, both distributions have power law tails characterised by
exponents which are universal, i.e., independent of the details of the platform
noise distribution. In the inelastic case, both distributions have exponential
tails: P(n_f) ~ exp[-\theta_1 n_f] and P(\tau_c) ~ exp[-\theta_2 \tau_c]. The
decay exponents \theta_1 and \theta_2 depend continuously on the coefficient of
restitution and are nonuniversal; however as one approches the elastic limit,
they vanish in a universal manner that we compute exactly. An explicit
expression for \theta_1 is provided for a particular case of the platform noise
distribution.Comment: 32 page
Precise Asymptotics for a Random Walker's Maximum
We consider a discrete time random walk in one dimension. At each time step
the walker jumps by a random distance, independent from step to step, drawn
from an arbitrary symmetric density function. We show that the expected
positive maximum E[M_n] of the walk up to n steps behaves asymptotically for
large n as, E[M_n]/\sigma=\sqrt{2n/\pi}+ \gamma +O(n^{-1/2}), where \sigma^2 is
the variance of the step lengths. While the leading \sqrt{n} behavior is
universal and easy to derive, the leading correction term turns out to be a
nontrivial constant \gamma. For the special case of uniform distribution over
[-1,1], Coffmann et. al. recently computed \gamma=-0.516068...by exactly
enumerating a lengthy double series. Here we present a closed exact formula for
\gamma valid for arbitrary symmetric distributions. We also demonstrate how
\gamma appears in the thermodynamic limit as the leading behavior of the
difference variable E[M_n]-E[|x_n|] where x_n is the position of the walker
after n steps. An application of these results to the equilibrium
thermodynamics of a Rouse polymer chain is pointed out. We also generalize our
results to L\'evy walks.Comment: new references added, typos corrected, published versio
On the time to reach maximum for a variety of constrained Brownian motions
Published: J. Phys. A: Math. Theor. 41, 365005 (2008).International audienceWe derive P(M,t_m), the joint probability density of the maximum M and the time t_m at which this maximum is achieved for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over M, the marginal density P(t_m) is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of P(t_m) in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path integral technique; however, an alternative approach is also outlined which is founded on certain "agreement formulae" that are encountered more generally in probabilistic studies of Brownian motion processes
Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution
We introduce an alternative definition of the relative height h^\kappa(x) of
a one-dimensional fluctuating interface indexed by a continuously varying real
paramater 0 \leq \kappa \leq 1. It interpolates between the height relative to
the initial value (i.e. in x=0) when \kappa = 0 and the height relative to the
spatially averaged height for \kappa = 1. We compute exactly the distribution
P^\kappa(h_m,L) of the maximum h_m of these relative heights for systems of
finite size L and periodic boundary conditions. One finds that it takes the
scaling form P^\kappa(h_m,L) = L^{-1/2} f^\kappa (h_m L^{-1/2}) where the
scaling function f^\kappa(x) interpolates between the Rayleigh distribution for
\kappa=0 and the Airy distribution for \kappa=1, the latter being the
probability distribution of the area under a Brownian excursion over the unit
interval. For arbitrary \kappa, one finds that it is related to, albeit
different from, the distribution of the area restricted to the interval [0,
\kappa] under a Brownian excursion over the unit interval.Comment: 25 pages, 4 figure
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Ultrasound-triggered disruption and self-healing of reversibly cross-linked hydrogels for drug delivery and enhanced chemotherapy
Foldable photoelectronics and muscle-like transducers require highly stretchable and transparent electrical conductors. Some conducting oxides are transparent, but not stretchable. Carbon nanotube films, graphene sheets and metal-nanowire meshes can be both stretchable and transparent, but their electrical resistances increase steeply with strain <100%. Here we present highly stretchable and transparent Au nanomesh electrodes on elastomers made by grain boundary lithography. The change in sheet resistance of Au nanomeshes is modest with a one-time strain of ~160% (from ~21 Ω per square to ~67 Ω per square), or after 1,000 cycles at a strain of 50%. The good stretchability lies in two aspects: the stretched nanomesh undergoes instability and deflects out-of-plane, while the substrate stabilizes the rupture of Au wires, forming distributed slits. Larger ratio of mesh-size to wire-width also leads to better stretchability. The highly stretchable and transparent Au nanomesh electrodes are promising for applications in foldable photoelectronics and muscle-like transducers.Engineering and Applied Science
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