3,347 research outputs found
Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
We consider the one-dimensional target search process that involves an
immobile target located at the origin and searchers performing independent
Brownian motions starting at the initial positions all on the positive half space. The process stops when the target is
first found by one of the searchers. We compute the probability distribution of
the maximum distance visited by the searchers till the stopping time and
show that it has a power law tail: for large . Thus all moments of up to the order
are finite, while the higher moments diverge. The prefactor increases
with faster than exponentially. Our solution gives the exit probability of
a set of particles from a box through the left boundary.
Incidentally, it also provides an exact solution of the Laplace's equation in
an -dimensional hypercube with some prescribed boundary conditions. The
analytical results are in excellent agreement with Monte Carlo simulations.Comment: 18 pages, 9 figure
The digital library DigilibLT
The aim of this paper is to describe the main strengths of the digital library DigilibLT: an open work environment that serves first of all as a reference point for research ends and secondly as an interdisciplinary developmental platform for young students; it is a hive of ideas that contributed to the creation of new projects, such as TBL and Vertερε, that both share the winning choice of using the universal language XML-TEI. This project proves that the successful union between information technology and humanistic studies generates great scientific and didactic advantages in relation to the sharing of ideas, collaboration and communication
High-precision simulation of the height distribution for the KPZ equation
The one-point distribution of the height for the continuum
Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping
to the directed polymer in a random potential at high temperature. Using an
importance sampling approach, the distribution is obtained over a large range
of values, down to a probability density as small as 10^{-1000} in the tails.
Both short and long times are investigated and compared with recent analytical
predictions for the large-deviation forms of the probability of rare
fluctuations. At short times the agreement with the analytical expression is
spectacular. We observe that the far left and right tails, with exponents 5/2
and 3/2 respectively, are preserved until large time. We present some evidence
for the predicted non-trivial crossover in the left tail from the 5/2 tail
exponent to the cubic tail of Tracy-Widom, although the details of the full
scaling form remains beyond reach.Comment: 6 pages, 5 figure
Maximum Distance Between the Leader and the Laggard for Three Brownian Walkers
We consider three independent Brownian walkers moving on a line. The process
terminates when the left-most walker (the `Leader') meets either of the other
two walkers. For arbitrary values of the diffusion constants D_1 (the Leader),
D_2 and D_3 of the three walkers, we compute the probability distribution
P(m|y_2,y_3) of the maximum distance m between the Leader and the current
right-most particle (the `Laggard') during the process, where y_2 and y_3 are
the initial distances between the leader and the other two walkers. The result
has, for large m, the form P(m|y_2,y_3) \sim A(y_2,y_3) m^{-\delta}, where
\delta = (2\pi-\theta)/(\pi-\theta) and \theta =
cos^{-1}(D_1/\sqrt{(D_1+D_2)(D_1+D_3)}. The amplitude A(y_2,y_3) is also
determined exactly
Local time of a system of Brownian particles on the line with step-like initial condition
We consider a system of non-interacting Brownian particles on a line with a
step-like initial condition and investigate the behavior of the local time at
the origin at large times. We compute the mean and the variance of the local
time and show that the memory effects are governed by the Fano factor
associated to the initial condition. For the uniform initial condition, we show
that the probability distribution of the local time admits large deviation form
and compute the corresponding large deviation functions for the annealed and
quenched averaging schemes. The two resulting large deviation functions are
very different. Our analytical results are supported by extensive numerical
simulations
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