268 research outputs found

### Finding passwords by random walks: How long does it take?

We compare an efficiency of a deterministic "lawnmower" and random search
strategies for finding a prescribed sequence of letters (a password) of length
M in which all letters are taken from the same Q-ary alphabet. We show that at
best a random search takes two times longer than a "lawnmower" search.Comment: To appear in J. Phys. A, special issue on "Random Search Problem:
Trends and Perspectives", eds.: MEG da Luz, E Raposo, GM Viswanathan and A
Grosber

### Proportionate vs disproportionate distribution of wealth of two individuals in a tempered Paretian ensemble

We study the distribution P(\omega) of the random variable \omega = x_1/(x_1
+ x_2), where x_1 and x_2 are the wealths of two individuals selected at random
from the same tempered Paretian ensemble characterized by the distribution
\Psi(x) \sim \phi(x)/x^{1 + \alpha}, where \alpha > 0 is the Pareto index and
$\phi(x)$ is the cut-off function. We consider two forms of \phi(x): a bounded
function \phi(x) = 1 for L \leq x \leq H, and zero otherwise, and a smooth
exponential function \phi(x) = \exp(-L/x - x/H). In both cases \Psi(x) has
moments of arbitrary order.
We show that, for \alpha > 1, P(\omega) always has a unimodal form and is
peaked at \omega = 1/2, so that most probably x_1 \approx x_2. For 0 < \alpha <
1 we observe a more complicated behavior which depends on the value of \delta =
L/H. In particular, for \delta < \delta_c - a certain threshold value -
P(\omega) has a three-modal (for a bounded \phi(x)) and a bimodal M-shape (for
an exponential \phi(x)) form which signifies that in such ensembles the wealths
x_1 and x_2 are disproportionately different.Comment: 9 pages, 8 figures, to appear in Physica

### Single-Species Reactions on a Random Catalytic Chain

We present an exact solution for a catalytically-activated annihilation A + A
\to 0 reaction taking place on a one-dimensional chain in which some segments
(placed at random, with mean concentration p) possess special, catalytic
properties. Annihilation reaction takes place, as soon as any two A particles
land from the reservoir onto two vacant sites at the extremities of the
catalytic segment, or when any A particle lands onto a vacant site on a
catalytic segment while the site at the other extremity of this segment is
already occupied by another A particle. We find that the disorder-average
pressure $P^{(quen)}$ per site of such a chain is given by $P^{(quen)} =
P^{(lan)} + \beta^{-1} F$, where $P^{(lan)} = \beta^{-1} \ln(1+z)$ is the
Langmuir adsorption pressure, (z being the activity and \beta^{-1} - the
temperature), while $\beta^{-1} F$ is the reaction-induced contribution, which
can be expressed, under appropriate change of notations, as the Lyapunov
exponent for the product of 2 \times 2 random matrices, obtained exactly by
Derrida and Hilhorst (J. Phys. A {\bf 16}, 2641 (1983)). Explicit asymptotic
formulae for the particle mean density and the compressibility are also
presented.Comment: AMSTeX, 17 pages, 1 figure, submitted to J. Phys.

### Binary Reactive Adsorbate on a Random Catalytic Substrate

We study the equilibrium properties of a model for a binary mixture of
catalytically-reactive monomers adsorbed on a two-dimensional substrate
decorated by randomly placed catalytic bonds. The interacting $A$ and $B$
monomer species undergo continuous exchanges with particle reservoirs and react
($A + B \to \emptyset$) as soon as a pair of unlike particles appears on sites
connected by a catalytic bond.
For the case of annealed disorder in the placement of the catalytic bonds
this model can be mapped onto a classical spin model with spin values $S =
-1,0,+1$, with effective couplings dependent on the temperature and on the mean
density $q$ of catalytic bonds. This allows us to exploit the mean-field theory
developed for the latter to determine the phase diagram as a function of $q$ in
the (symmetric) case in which the chemical potentials of the particle
reservoirs, as well as the $A-A$ and $B-B$ interactions are equal.Comment: 12 pages, 4 figure

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