929 research outputs found
Delocalization transition of the selective interface model: distribution of pseudo-critical temperatures
According to recent progress in the finite size scaling theory of critical
disordered systems, the nature of the phase transition is reflected in the
distribution of pseudo-critical temperatures over the ensemble of
samples of size . In this paper, we apply this analysis to the
delocalization transition of an heteropolymeric chain at a selective
fluid-fluid interface. The width and the shift
are found to decay with the same exponent
, where . The distribution of
pseudo-critical temperatures is clearly asymmetric, and is well
fitted by a generalized Gumbel distribution of parameter . We also
consider the free energy distribution, which can also be fitted by a
generalized Gumbel distribution with a temperature dependent parameter, of
order in the critical region. Finally, the disorder averaged
number of contacts with the interface scales at like with
.Comment: 9 pages,6 figure
Smoothening of Depinning Transitions for Directed Polymers with Quenched Disorder
We consider disordered models of pinning of directed polymers on a defect
line, including (1+1)-dimensional interface wetting models, disordered
Poland--Scheraga models of DNA denaturation and other (1+d)-dimensional
polymers in interaction with columnar defects. We consider also random
copolymers at a selective interface. These models are known to have a
(de)pinning transition at some critical line in the phase diagram. In this work
we prove that, as soon as disorder is present, the transition is at least of
second order: the free energy is differentiable at the critical line, and the
order parameter (contact fraction) vanishes continuously at the transition. On
the other hand, it is known that the corresponding non-disordered models can
have a first order (de)pinning transition, with a jump in the order parameter.
Our results confirm predictions based on the Harris criterion.Comment: 4 pages, 1 figure. Version 2: references added, minor changes made.
To appear on Phys. Rev. Let
Numerical study of the disordered Poland-Scheraga model of DNA denaturation
We numerically study the binary disordered Poland-Scheraga model of DNA
denaturation, in the regime where the pure model displays a first order
transition (loop exponent ). We use a Fixman-Freire scheme for the
entropy of loops and consider chain length up to , with
averages over samples. We present in parallel the results of various
observables for two boundary conditions, namely bound-bound (bb) and
bound-unbound (bu), because they present very different finite-size behaviors,
both in the pure case and in the disordered case. Our main conclusion is that
the transition remains first order in the disordered case: in the (bu) case,
the disorder averaged energy and contact densities present crossings for
different values of without rescaling. In addition, we obtain that these
disorder averaged observables do not satisfy finite size scaling, as a
consequence of strong sample to sample fluctuations of the pseudo-critical
temperature. For a given sample, we propose a procedure to identify its
pseudo-critical temperature, and show that this sample then obeys first order
transition finite size scaling behavior. Finally, we obtain that the disorder
averaged critical loop distribution is still governed by in
the regime , as in the pure case.Comment: 12 pages, 13 figures. Revised versio
Directed polymer in a random medium of dimension 1+1 and 1+3: weights statistics in the low-temperature phase
We consider the low-temperature disorder-dominated phase of the
directed polymer in a random potentiel in dimension 1+1 (where )
and 1+3 (where ). To characterize the localization properties of
the polymer of length , we analyse the statistics of the weights of the last monomer as follows. We numerically compute the probability
distributions of the maximal weight , the probability distribution of the parameter as well as the average values of the higher order
moments . We find that there exists a
temperature such that (i) for , the distributions
and present the characteristic Derrida-Flyvbjerg
singularities at and for . In particular, there
exists a temperature-dependent exponent that governs the main
singularities and as well as the power-law decay of the moments . The exponent grows from the value
up to . (ii) for , the
distribution vanishes at some value , and accordingly the
moments decay exponentially as in . The
histograms of spatial correlations also display Derrida-Flyvbjerg singularities
for . Both below and above , the study of typical and
averaged correlations is in full agreement with the droplet scaling theory.Comment: 13 pages, 29 figure
Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules
We consider the non-equilibrium dynamics of disordered systems as defined by
a master equation involving transition rates between configurations (detailed
balance is not assumed). To compute the important dynamical time scales in
finite-size systems without simulating the actual time evolution which can be
extremely slow, we propose to focus on first-passage times that satisfy
'backward master equations'. Upon the iterative elimination of configurations,
we obtain the exact renormalization rules that can be followed numerically. To
test this approach, we study the statistics of some first-passage times for two
disordered models : (i) for the random walk in a two-dimensional self-affine
random potential of Hurst exponent , we focus on the first exit time from a
square of size if one starts at the square center. (ii) for the
dynamics of the ferromagnetic Sherrington-Kirkpatrick model of spins, we
consider the first passage time to zero-magnetization when starting from
a fully magnetized configuration. Besides the expected linear growth of the
averaged barrier , we find that the rescaled
distribution of the barrier decays as for large
with a tail exponent of order . This value can be simply
interpreted in terms of rare events if the sample-to-sample fluctuation
exponent for the barrier is .Comment: 8 pages, 4 figure
A simple model for DNA denaturation
Following Poland and Scheraga, we consider a simplified model for the
denaturation transition of DNA. The two strands are modeled as interacting
polymer chains. The attractive interactions, which mimic the pairing between
the four bases, are reduced to a single short range binding term. Furthermore,
base-pair misalignments are forbidden, implying that this binding term exists
only for corresponding (same curvilinear abscissae) monomers of the two chains.
We take into account the excluded volume repulsion between monomers of the two
chains, but neglect intra-chain repulsion. We find that the excluded volume
term generates an effective repulsive interaction between the chains, which
decays as . Due to this long-range repulsion between the chains, the
denaturation transition is first order in any dimension, in agreement with
previous studies.Comment: 10 page
Two-dimensional wetting with binary disorder: a numerical study of the loop statistics
We numerically study the wetting (adsorption) transition of a polymer chain
on a disordered substrate in 1+1 dimension.Following the Poland-Scheraga model
of DNA denaturation, we use a Fixman-Freire scheme for the entropy of loops.
This allows us to consider chain lengths of order to ,
with disorder realizations. Our study is based on the statistics of
loops between two contacts with the substrate, from which we define Binder-like
parameters: their crossings for various sizes allow a precise determination
of the critical temperature, and their finite size properties yields a
crossover exponent .We then analyse at
criticality the distribution of loop length in both regimes
and , as well as the finite-size properties of the contact
density and energy. Our conclusion is that the critical exponents for the
thermodynamics are the same as those of the pure case, except for strong
logarithmic corrections to scaling. The presence of these logarithmic
corrections in the thermodynamics is related to a disorder-dependent
logarithmic singularity that appears in the critical loop distribution in the
rescaled variable as .Comment: 12 pages, 13 figure
Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension via a Monte-Carlo procedure in the disorder
In order to probe with high precision the tails of the ground-state energy
distribution of disordered spin systems, K\"orner, Katzgraber and Hartmann
\cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-Carlo
Markov chain in the disorder. In this paper, we combine their Monte-Carlo
procedure in the disorder with exact transfer matrix calculations in each
sample to measure the negative tail of ground state energy distribution
for the directed polymer in a random medium of dimension .
In , we check the validity of the algorithm by a direct comparison with
the exact result, namely the Tracy-Widom distribution. In dimensions and
, we measure the negative tail up to ten standard deviations, which
correspond to probabilities of order . Our results are
in agreement with Zhang's argument, stating that the negative tail exponent
of the asymptotic behavior
as is directly related to the fluctuation exponent
(which governs the fluctuations
of the ground state energy for polymers of length ) via the simple
formula . Along the paper, we comment on the
similarities and differences with spin-glasses.Comment: 13 pages, 16 figure
A Solvable Model of Secondary Structure Formation in Random Hetero-Polymers
We propose and solve a simple model describing secondary structure formation
in random hetero-polymers. It describes monomers with a combination of
one-dimensional short-range interactions (representing steric forces and
hydrogen bonds) and infinite range interactions (representing polarity forces).
We solve our model using a combination of mean field and random field
techniques, leading to phase diagrams exhibiting second-order transitions
between folded, partially folded and unfolded states, including regions where
folding depends on initial conditions. Our theoretical results, which are in
excellent agreement with numerical simulations, lead to an appealing physical
picture of the folding process: the polarity forces drive the transition to a
collapsed state, the steric forces introduce monomer specificity, and the
hydrogen bonds stabilise the conformation by damping the frustration-induced
multiplicity of states.Comment: 24 pages, 14 figure
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