Speaker
Gleb Oshanin
(Physique Theorique de la Matiere Condensee, Universite Pierre & Marie Curie, Paris)
Description
In many problems in physics, biophysics or chemistry one is
often interested to know how fast a randomly moving
”particle” (a ligand, an ion, a protein, a reactant, a
phage, etc) may find a particle of another species, a binding
or a ”specific” site, an entrance/exit to a bounded domain, a
receptor, a target, a source of odor, any ob ject in space,
etc. A commonly used characteristic is the first moment of
the first passage time distribution - the mean first passage
time (MFPT), which in many cases can be evaluated or, at
least, estimated analytically. The question is, however, how
the MFPT is representative of an actual behavior. In the
first, central part of this talk we overview some recent
results [C. Mejia-Monasterio, G. Oshanin and G. Schehr, J.
Stat. Mech. 2011 P06022 (2011)] for unbiased (centrally
biased) Brownian search for an immobile target in bounded
(or infinite) spherical domains. We use here a special novel
”diagnostic” of the first passage events: in our ”imaginary”
experiment, instead of considering a single searcher, we
launch simultaneously from the same point in space two
absolutely identical non-interacting ones and calculate the
distribution function P (ω) of a random variable ω = t1/(t1
+ t2), where t1 and t2 are the (unordered) times of the first
passage to the target for the searcher 1 and searcher 2,
respectively. We show that P (ω) exhibits quite a
non-trivial and sometimes a counterintuitive behaviour so
that its very shape (a broad bell-shaped form with a maximum
at ω = 1/2 or an M-like form with a minimum at ω = 1/2 and
maxima close to 0 and 1) depends, remarkably, on the size of
the system, the location of the starting point relative to
the target and/or on the magnitude of the bias, if any. The
shape of P (ω) appears also to be sensitive to the form of
the domain: we demonstrate that for an infinite (finite) wedge
or a 3D cone, or a triangle with adsorbing boundaries, the
distribution P (ω) may have a bell-shaped or an M-like forms
depending whether the value of the opening angle is less
than or exceeds some critical value. Our results thus
indicate that, despite the fact that the single-searcher
first passage time distribution has moments of arbitrary
order, the sample-to-sample fluctuations are very significant
and the mean first passage time is not, in fact, a reliable
measure of the first passage events.
Further on, we will consider, within the framework of a
model originally proposed by de Gennes [J. Stat. Phys. 12
463 (1975)], dynamics of a boundary separating the helix and
coil phases in a partially melted heteropolymer bearing a
random alphabet. For this model, a random variable ω is the
alphabet-dependent probability that a randomly moving
boundary will first hit the left extremity of the polymer (so
that the chain will return to its native helix state)
without having ever hit the right extremity, while t1 and t2
are the resistances of the two respective intervals to the
left and to the right of the starting point. Hence, P (ω) is
the probability distribution (averaged over all possible
alphabets) of such a hitting probability. We show [G.
Oshanin and S. Redner, EPL 85, 10008 (2009)] that also in
this situation P(ω) exhibits a transition from a bell-shaped
(for sufficiently short polymers) form to an M-like shape (for
sufficiently long polymers). We therefore conclude that the
evolution of a partially melted long random heteropolymer is
controlled by the arrangement of monomers along the chain so
that each heteropolymer realization has a unique kinetics
and unique final fate that is not representative of the
average behavior of an ensemble of such polymers.
Finally, we will show that an analogous shape-reversal
transition occurs in Black-Scholes model of stock options
evolution. Considering two identical Black-Scholes
stochastic equations, which produce two identical
(uncorrelated or correlated) either European- or Asian-style
options t1 and t2,
we will demonstrate that the distribution function of the
relative weight ω of either of the options in a portofolio
composed of two such options undergoes at a certain moment
of time a transition from a unimodal form with a maximum at
ω = 1/2 to a bimodal form with a minimum at ω = 1/2, which
reflects the symmetry breaking between two identical options
[G. Oshanin and G. Schehr, Quantitative Finance, to appear;
arXiv:1005.1760v2].
Primary author
Gleb Oshanin
(Physique Theorique de la Matiere Condensee, Universite Pierre & Marie Curie, Paris)
