10 research outputs found
Conformally Invariant Fractals and Potential Theory
The multifractal (MF) distribution of the electrostatic potential near any
conformally invariant fractal boundary, like a critical O(N) loop or a
-state Potts cluster, is solved in two dimensions. The dimension of the boundary set with local wedge angle is , with the central charge of the
model. As a corollary, the dimensions
of the external perimeter and of the hull of a Potts cluster obey
the duality equation . A related covariant
MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.Comment: 5 pages, 1 figur
"Screening" of universal van der Waals - Casimir terms by Coulomb gases in a fully-finite two-dimensional geometry
This paper is a continuation of a previous one [Jancovici and Samaj, 2004 J.
Stat. Mech. P08006] dealing with classical Casimir phenomena in semi-infinite
wall geometries. In that paper, using microscopic Coulomb systems, the
long-ranged Casimir force due to thermal fluctuations in conducting walls was
shown to be screened by the presence of an electrolyte between the walls into
some residual short-ranged force. Here, we aim to extend the study of the
screening (cancellation) phenomena to universal Casimir terms appearing in the
large-size expansions of the grand potentials for microscopic Coulomb systems
confined in fully-finite 2D geometries, in particular the disc geometry. Two
cases are solved exactly: the high-temperature (Debye-H\"uckel) limit and the
Thirring free-fermion point. Similarities and fundamental differences between
fully-finite and semi-infinite geometries are pointed out.Comment: 21 pages, 1 figur
Surface Polymer Network Model and Effective Membrane Curvature Elasticity
A microscopic model of a surface polymer network - membrane system is
introduced, with contact polymer surface interactions that can be either
repulsive or attractive and sliplinks of functionality four randomly
distributed over the supporting membrane surface anchoring the polymers to it.
For the supporting surface perturbed from a planar configuration and a small
relative number of surface sliplinks, we investigate an expansion of the free
energy in terms of the local curvatures of the surface and the surface density
of sliplinks, obtained through the application of the Balian - Bloch -
Duplantier multiple surface scattering method. As a result, the dependence of
the curvature elastic modulus, the Gaussian modulus as well as of the
spontaneous curvature of the "dressed" membrane, ~{\sl i.e.} polymer network
plus membrane matrix, is obtained on the mean polymer bulk end to end
separation and the surface density of sliplinks.Comment: 15 pages with one included compressed uuencoded figure
Screening of classical Casimir forces by electrolytes in semi-infinite geometries
We study the electrostatic Casimir effect and related phenomena in
equilibrium statistical mechanics of classical (non-quantum) charged fluids.
The prototype model consists of two identical dielectric slabs in empty space
(the pure Casimir effect) or in the presence of an electrolyte between the
slabs. In the latter case, it is generally believed that the long-ranged
Casimir force due to thermal fluctuations in the slabs is screened by the
electrolyte into some residual short-ranged force. The screening mechanism is
based on a "separation hypothesis": thermal fluctuations of the electrostatic
field in the slabs can be treated separately from the pure image effects of the
"inert" slabs on the electrolyte particles. In this paper, by using a
phenomenological approach under certain conditions, the separation hypothesis
is shown to be valid. The phenomenology is tested on a microscopic model in
which the conducting slabs and the electrolyte are modelled by the symmetric
Coulomb gases of point-like charges with different particle fugacities. The
model is solved in the high-temperature Debye-H\"uckel limit (in two and three
dimensions) and at the free fermion point of the Thirring representation of the
two-dimensional Coulomb gas. The Debye-H\"uckel theory of a Coulomb gas between
dielectric walls is also solved.Comment: 25 pages, 2 figure
Comment on the sign of the Casimir force
I show that reflection positivity implies that the force between any mirror
pair of charge-conjugate probes of the quantum vacuum is attractive. This
generalizes a recent theorem of Kenneth and Klich to interacting quantum
fields, to arbitrary semiclassical bodies, and to quantized probes with
non-overlapping wavefunctions. I also prove that the torques on
charge-conjugate probes tend always to rotate them into a mirror-symmetric
position.Comment: 13 pages, 1 figure, Latex file. Several points clarified and
expanded, two references added
Spanning Forests on Random Planar Lattices
The generating function for spanning forests on a lattice is related to the
q-state Potts model in a certain q -> 0 limit, and extends the analogous notion
for spanning trees, or dense self-avoiding branched polymers. Recent works have
found a combinatorial perturbative equivalence also with the (quadratic action)
O(n) model in the limit n -> -1, the expansion parameter t counting the number
of components in the forest. We give a random-matrix formulation of this model
on the ensemble of degree-k random planar lattices. For k = 3, a correspondence
is found with the Kostov solution of the loop-gas problem, which arise as a
reformulation of the (logarithmic action) O(n) model, at n = -2. Then, we show
how to perform an expansion around the t = 0 theory. In the thermodynamic
limit, at any order in t we have a finite sum of finite-dimensional Cauchy
integrals. The leading contribution comes from a peculiar class of terms, for
which a resummation can be performed exactly.Comment: 43 pages, Dedicated to Edouard Brezin and Giorgio Parisi, on the
occasion of their special birthda
Boundary Shape and Casimir Energy
Casimir energy changes are investigated for geometries obtained by small but
arbitrary deformations of a given geometry for which the vacuum energy is
already known for the massless scalar field. As a specific case, deformation of
a spherical shell is studied. From the deformation of the sphere we show that
the Casimir energy is a decreasing function of the surface to volume ratio. The
decreasing rate is higher for less smooth deformations.Comment: 12 page
Casimir energy and geometry : beyond the Proximity Force Approximation
We review the relation between Casimir effect and geometry, emphasizing
deviations from the commonly used Proximity Force Approximation (PFA). We use
to this aim the scattering formalism which is nowadays the best tool available
for accurate and reliable theory-experiment comparisons. We first recall the
main lines of this formalism when the mirrors can be considered to obey
specular reflection. We then discuss the more general case where non planar
mirrors give rise to non-specular reflection with wavevectors and field
polarisations mixed. The general formalism has already been fruitfully used for
evaluating the effect of roughness on the Casimir force as well as the lateral
Casimir force or Casimir torque appearing between corrugated surfaces. In this
short review, we focus our attention on the case of the lateral force which
should make possible in the future an experimental demonstration of the
nontrivial (i.e. beyond PFA) interplay of geometry and Casimir effect.Comment: corrected typos, added references, QFEXT'07 special issue in J. Phys.