48 research outputs found
Self tolerance in a minimal model of the idiotypic network
We consider the problem of self tolerance in the frame of a minimalistic
model of the idiotypic network. A node of this network represents a population
of B lymphocytes of the same idiotype which is encoded by a bit string. The
links of the network connect nodes with (nearly) complementary strings. The
population of a node survives if the number of occupied neighbours is not too
small and not too large. There is an influx of lymphocytes with random idiotype
from the bone marrow. Previous investigations have shown that this system
evolves toward highly organized architectures, where the nodes can be
classified into groups according to their statistical properties. The building
principles of these architectures can be analytically described and the
statistical results of simulations agree very well with results of a modular
mean field theory. In this paper we present simulation results for the case
that one or several nodes, playing the role of self, are permanently occupied.
We observe that the group structure of the architecture is very similar to the
case without self antigen, but organized such that the neighbours of the self
are only weakly occupied, thus providing self tolerance. We also treat this
situation in mean field theory which give results in good agreement with data
from simulation.Comment: 7 pages, 6 figures, 1 tabl
Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation II: upper bound on the volume exponent
This paper continues a study on trajectories of Brownian Motion in a field of
soft trap whose radius distribution is unbounded. We show here for both
point-to-point and point-to-plane model the volume exponent (the exponent
associated to transversal fluctuation of the trajectories) is strictly less
than one and give an explicit upper bound that depends on the parameters of the
problem. In some specific cases, this upper bound matches the lower bound
proved in the first part of this work and we get the exact value of the volume
exponent.Comment: 28 page 4 figures, to appear in AIH
On-off Intermittency in Stochastically Driven Electrohydrodynamic Convection in Nematics
We report on-off intermittency in electroconvection of nematic liquid crystals driven by a dichotomous stochastic electric voltage. With increasing voltage amplitude we observe laminar phases of undistorted director state interrupted by shorter bursts of spatially regular stripes. Near a critical value of the amplitude the distribution of the duration of laminar phases is governed over several decades by a power law with exponent -3/2. The experimental findings agree with simulations of the linearized electrohydrodynamic equations near the sample stability threshold
Nonequilibrium phase transitions in finite arrays of globally coupled Stratonovich models: Strong coupling limit
A finite array of globally coupled Stratonovich models exhibits a
continuous nonequilibrium phase transition. In the limit of strong coupling
there is a clear separation of time scales of center of mass and relative
coordinates. The latter relax very fast to zero and the array behaves as a
single entity described by the center of mass coordinate. We compute
analytically the stationary probability and the moments of the center of mass
coordinate. The scaling behaviour of the moments near the critical value of the
control parameter is determined. We identify a crossover from linear
to square root scaling with increasing distance from . The crossover point
approaches in the limit which reproduces previous results
for infinite arrays. The results are obtained in both the Fokker-Planck and the
Langevin approach and are corroborated by numerical simulations. For a general
class of models we show that the transition manifold in the parameter space
depends on and is determined by the scaling behaviour near a fixed point of
the stochastic flow
Convolution of multifractals and the local magnetization in a random field Ising chain
The local magnetization in the one-dimensional random-field Ising model is
essentially the sum of two effective fields with multifractal probability
measure. The probability measure of the local magnetization is thus the
convolution of two multifractals. In this paper we prove relations between the
multifractal properties of two measures and the multifractal properties of
their convolution. The pointwise dimension at the boundary of the support of
the convolution is the sum of the pointwise dimensions at the boundary of the
support of the convoluted measures and the generalized box dimensions of the
convolution are bounded from above by the sum of the generalized box dimensions
of the convoluted measures. The generalized box dimensions of the convolution
of Cantor sets with weights can be calculated analytically for certain
parameter ranges and illustrate effects we also encounter in the case of the
measure of the local magnetization. Returning to the study of this measure we
apply the general inequalities and present numerical approximations of the
D_q-spectrum. For the first time we are able to obtain results on multifractal
properties of a physical quantity in the one-dimensional random-field Ising
model which in principle could be measured experimentally. The numerically
generated probability densities for the local magnetization show impressively
the gradual transition from a monomodal to a bimodal distribution for growing
random field strength h.Comment: An error in figure 1 was corrected, small additions were made to the
introduction and the conclusions, some typos were corrected, 24 pages,
LaTeX2e, 9 figure
Orbits and phase transitions in the multifractal spectrum
We consider the one dimensional classical Ising model in a symmetric
dichotomous random field. The problem is reduced to a random iterated function
system for an effective field. The D_q-spectrum of the invariant measure of
this effective field exhibits a sharp drop of all D_q with q < 0 at some
critical strength of the random field. We introduce the concept of orbits which
naturally group the points of the support of the invariant measure. We then
show that the pointwise dimension at all points of an orbit has the same value
and calculate it for a class of periodic orbits and their so-called offshoots
as well as for generic orbits in the non-overlapping case. The sharp drop in
the D_q-spectrum is analytically explained by a drastic change of the scaling
properties of the measure near the points of a certain periodic orbit at a
critical strength of the random field which is explicitly given. A similar
drastic change near the points of a special family of periodic orbits explains
a second, hitherto unnoticed transition in the D_q-spectrum. As it turns out, a
decisive role in this mechanism is played by a specific offshoot. We
furthermore give rigorous upper and/or lower bounds on all D_q in a wide
parameter range. In most cases the numerically obtained D_q coincide with
either the upper or the lower bound. The results in this paper are relevant for
the understanding of random iterated function systems in the case of moderate
overlap in which periodic orbits with weak singularity can play a decisive
role.Comment: The article has been completely rewritten; the title has changed; a
section about the typical pointwise dimension as well as several references
and remarks about more general systems have been added; to appear in J. Phys.
A; 25 pages, 11 figures, LaTeX2
Phase diagram of the random field Ising model on the Bethe lattice
The phase diagram of the random field Ising model on the Bethe lattice with a
symmetric dichotomous random field is closely investigated with respect to the
transition between the ferromagnetic and paramagnetic regime. Refining
arguments of Bleher, Ruiz and Zagrebnov [J. Stat. Phys. 93, 33 (1998)] an exact
upper bound for the existence of a unique paramagnetic phase is found which
considerably improves the earlier results. Several numerical estimates of
transition lines between a ferromagnetic and a paramagnetic regime are
presented. The obtained results do not coincide with a lower bound for the
onset of ferromagnetism proposed by Bruinsma [Phys. Rev. B 30, 289 (1984)]. If
the latter one proves correct this would hint to a region of coexistence of
stable ferromagnetic phases and a stable paramagnetic phase.Comment: Article has been condensed and reorganized; Figs 3,5,6 merged; Fig 4
omitted; Some discussion added at end of Sec. III; 9 pages, 5 figs, RevTeX4,
AMSTe