8 research outputs found
Height fluctuations of a contact line: a direct measurement of the renormalized disorder correlator
We have measured the center-of-mass fluctuations of the height of a contact
line at depinning for two different systems: liquid hydrogen on a rough cesium
substrate and isopropanol on a silicon wafer grafted with silanized patches.
The contact line is subject to a confining quadratic well, provided by gravity.
From the second cumulant of the height fluctuations, we measure the
renormalized disorder correlator Delta(u), predicted by the Functional RG
theory to attain a fixed point, as soon as the capillary length is large
compared to the Larkin length set by the microscopic disorder. The experiments
are consistent with the asymptotic form for Delta(u) predicted by Functional
RG, including a linear cusp at u=0. The observed small deviations could be used
as a probe of the underlying physical processes. The third moment, as well as
avalanche-size distributions are measured and compared to predictions from
Functional RG.Comment: 6 pages, 14 figure
Shock statistics in higher-dimensional Burgers turbulence
We conjecture the exact shock statistics in the inviscid decaying Burgers
equation in D>1 dimensions, with a special class of correlated initial
velocities, which reduce to Brownian for D=1. The prediction is based on a
field-theory argument, and receives support from our numerical calculations. We
find that, along any given direction, shocks sizes and locations are
uncorrelated.Comment: 4 pages, 8 figure
Free-energy distribution of the directed polymer at high temperature
We study the directed polymer of length in a random potential with fixed
endpoints in dimension 1+1 in the continuum and on the square lattice, by
analytical and numerical methods. The universal regime of high temperature
is described, upon scaling 'time' and space (with for the discrete model) by a continuum model with
-function disorder correlation. Using the Bethe Ansatz solution for the
attractive boson problem, we obtain all positive integer moments of the
partition function. The lowest cumulants of the free energy are predicted at
small time and found in agreement with numerics. We then obtain the exact
expression at any time for the generating function of the free energy
distribution, in terms of a Fredholm determinant. At large time we find that it
crosses over to the Tracy Widom distribution (TW) which describes the fixed
infinite limit. The exact free energy distribution is obtained for any time
and compared with very recent results on growth and exclusion models.Comment: 6 pages, 3 figures large time limit corrected and convergence to
Tracy Widom established, 1 figure changed
Freezing Transition in Decaying Burgers Turbulence and Random Matrix Dualities
We reveal a phase transition with decreasing viscosity at \nu=\nu_c>0
in one-dimensional decaying Burgers turbulence with a power-law correlated
random profile of Gaussian-distributed initial velocities
\sim|x-x'|^{-2}. The low-viscosity phase exhibits non-Gaussian
one-point probability density of velocities, continuously dependent on \nu,
reflecting a spontaneous one step replica symmetry breaking (RSB) in the
associated statistical mechanics problem. We obtain the low orders cumulants
analytically. Our results, which are checked numerically, are based on
combining insights in the mechanism of the freezing transition in random
logarithmic potentials with an extension of duality relations discovered
recently in Random Matrix Theory. They are essentially non mean-field in nature
as also demonstrated by the shock size distribution computed numerically and
different from the short range correlated Kida model, itself well described by
a mean field one step RSB ansatz. We also provide some insights for the finite
viscosity behaviour of velocities in the latter model.Comment: Published version, essentially restructured & misprints corrected. 6
pages, 5 figure
Functional Renormalization Group and the Field Theory of Disordered Elastic Systems
We study elastic systems such as interfaces or lattices, pinned by quenched
disorder. To escape triviality as a result of ``dimensional reduction'', we use
the functional renormalization group. Difficulties arise in the calculation of
the renormalization group functions beyond 1-loop order. Even worse,
observables such as the 2-point correlation function exhibit the same problem
already at 1-loop order. These difficulties are due to the non-analyticity of
the renormalized disorder correlator at zero temperature, which is inherent to
the physics beyond the Larkin length, characterized by many metastable states.
As a result, 2-loop diagrams, which involve derivatives of the disorder
correlator at the non-analytic point, are naively "ambiguous''. We examine
several routes out of this dilemma, which lead to a unique renormalizable
field-theory at 2-loop order. It is also the only theory consistent with the
potentiality of the problem. The beta-function differs from previous work and
the one at depinning by novel "anomalous terms''. For interfaces and random
bond disorder we find a roughness exponent zeta = 0.20829804 epsilon + 0.006858
epsilon^2, epsilon = 4-d. For random field disorder we find zeta = epsilon/3
and compute universal amplitudes to order epsilon^2. For periodic systems we
evaluate the universal amplitude of the 2-point function. We also clarify the
dependence of universal amplitudes on the boundary conditions at large scale.
All predictions are in good agreement with numerical and exact results, and an
improvement over one loop. Finally we calculate higher correlation functions,
which turn out to be equivalent to those at depinning to leading order in
epsilon.Comment: 42 pages, 41 figure
Functional renormalization group for anisotropic depinning and relation to branching processes
Using the functional renormalization group, we study the depinning of elastic
objects in presence of anisotropy. We explicitly demonstrate how the KPZ-term
is always generated, even in the limit of vanishing velocity, except where
excluded by symmetry. We compute the beta-function to one loop taking properly
into account the non-analyticity. This gives rise to additional terms, missed
in earlier studies. A crucial question is whether the non-renormalization of
the KPZ-coupling found at 1-loop order extends beyond the leading one. Using a
Cole-Hopf-transformed theory we argue that it is indeed uncorrected to all
orders. The resulting flow-equations describe a variety of physical situations.
A careful analysis of the flow yields several non-trivial fixed points. All
these fixed points are transient since they possess one unstable direction
towards a runaway flow, which leaves open the question of the upper critical
dimension. The runaway flow is dominated by a Landau-ghost-mode. For SR
elasticity, using the Cole-Hopf transformed theory we identify a non-trivial
3-dimensional subspace which is invariant to all orders and contains all above
fixed points as well as the Landau-mode. It belongs to a class of theories
which describe branching and reaction-diffusion processes, of which some have
been mapped onto directed percolation.Comment: 20 pages, 30 figures, revtex