37 research outputs found
Two-species diffusion-annihilation process on the fully-connected lattice: probability distributions and extreme value statistics
We study the two-species diffusion-annihilation process, \O,
on the fully-connected lattice. Probability distributions for the number of
particles and the reaction time are obtained for a finite-size system using a
master equation approach. Mean values and variances are deduced from generating
functions. When the reaction is far from complete, i.e., for a large number of
particles of each species, mean-field theory is exact and the fluctuations are
Gaussian. In the scaling limit the reaction time displays extreme-value
statistics in the vicinity of the absorbing states. A generalized Gumbel
distribution is obtained for unequal initial densities, . For
equal or almost equal initial densities, , the fluctuations
of the reaction time near the absorbing state are governed by a probability
density involving derivatives of , the Jacobi theta function.Comment: 30 pages, 18 figures. Continuation of arXiv:1711.01248. published
versio
On a random walk with memory and its relation to Markovian processes
We study a one-dimensional random walk with memory in which the step lengths
to the left and to the right evolve at each step in order to reduce the
wandering of the walker. The feedback is quite efficient and lead to a
non-diffusive walk. The time evolution of the displacement is given by an
equivalent Markovian dynamical process. The probability density for the
position of the walker is the same at any time as for a random walk with
shrinking steps, although the two-time correlation functions are quite
different.Comment: 10 pages, 4 figure
Nonequilibrium phase transition in a driven Potts model with friction
We consider magnetic friction between two systems of -state Potts spins
which are moving along their boundaries with a relative constant velocity .
Due to the interaction between the surface spins there is a permanent energy
flow and the system is in a steady state which is far from equilibrium. The
problem is treated analytically in the limit (in one dimension, as
well as in two dimensions for large- values) and for and finite by
Monte Carlo simulations in two dimensions. Exotic nonequilibrium phase
transitions take place, the properties of which depend on the type of phase
transition in equilibrium. When this latter transition is of first order, a
sequence of second- and first-order nonequilibrium transitions can be observed
when the interaction is varied.Comment: 13 pages, 9 figures, one journal reference adde
Disordered Potts model on the diamond hierarchical lattice: Numerically exact treatment in the large-q limit
We consider the critical behavior of the random q-state Potts model in the
large-q limit with different types of disorder leading to either the
nonfrustrated random ferromagnet regime or the frustrated spin glass regime.
The model is studied on the diamond hierarchical lattice for which the
Migdal-Kadanoff real-space renormalization is exact. It is shown to have a
ferromagnetic and a paramagnetic phase and the phase transition is controlled
by four different fixed points. The state of the system is characterized by the
distribution of the interface free energy P(I) which is shown to satisfy
different integral equations at the fixed points. By numerical integration we
have obtained the corresponding stable laws of nonlinear combination of random
numbers and obtained numerically exact values for the critical exponents.Comment: 8+ pages, 5 figure
Scaling properties at the interface between different critical subsystems: The Ashkin-Teller model
We consider two critical semi-infinite subsystems with different critical
exponents and couple them through their surfaces. The critical behavior at the
interface, influenced by the critical fluctuations of the two subsystems, can
be quite rich. In order to examine the various possibilities, we study a system
composed of two coupled Ashkin-Teller models with different four-spin couplings
epsilon, on the two sides of the junction. By varying epsilon, some bulk and
surface critical exponents of the two subsystems are continuously modified,
which in turn changes the interface critical behavior. In particular we study
the marginal situation, for which magnetic critical exponents at the interface
vary continuously with the strength of the interaction parameter. The behavior
expected from scaling arguments is checked by DMRG calculations.Comment: 10 pages, 9 figures. Minor correction
Critical behavior at the interface between two systems belonging to different universality classes
We consider the critical behavior at an interface which separates two
semi-infinite subsystems belonging to different universality classes, thus
having different set of critical exponents, but having a common transition
temperature. We solve this problem analytically in the frame of mean-field
theory, which is then generalized using phenomenological scaling
considerations. A large variety of interface critical behavior is obtained
which is checked numerically on the example of two-dimensional q-state Potts
models with q=2 to 4. Weak interface couplings are generally irrelevant,
resulting in the same critical behavior at the interface as for a free surface.
With strong interface couplings, the interface remains ordered at the bulk
transition temperature. More interesting is the intermediate situation, the
special interface transition, when the critical behavior at the interface
involves new critical exponents, which however can be expressed in terms of the
bulk and surface exponents of the two subsystems. We discuss also the smooth or
discontinuous nature of the order parameter profile.Comment: 16 pages, 9 figures, published version, minor changes, some
references adde
Anomalous Diffusion in Aperiodic Environments
We study the Brownian motion of a classical particle in one-dimensional
inhomogeneous environments where the transition probabilities follow
quasiperiodic or aperiodic distributions. Exploiting an exact correspondence
with the transverse-field Ising model with inhomogeneous couplings we obtain
many new analytical results for the random walk problem. In the absence of
global bias the qualitative behavior of the diffusive motion of the particle
and the corresponding persistence probability strongly depend on the
fluctuation properties of the environment. In environments with bounded
fluctuations the particle shows normal diffusive motion and the diffusion
constant is simply related to the persistence probability. On the other hand in
a medium with unbounded fluctuations the diffusion is ultra-slow, the
displacement of the particle grows on logarithmic time scales. For the
borderline situation with marginal fluctuations both the diffusion exponent and
the persistence exponent are continuously varying functions of the
aperiodicity. Extensions of the results to disordered media and to higher
dimensions are also discussed.Comment: 11 pages, RevTe