2,106 research outputs found
Derivation of Hebb's rule
On the basis of the general form for the energy needed to adapt the
connection strengths of a network in which learning takes place, a local
learning rule is found for the changes of the weights. This biologically
realizable learning rule turns out to comply with Hebb's neuro-physiological
postulate, but is not of the form of any of the learning rules proposed in the
literature.
It is shown that, if a finite set of the same patterns is presented over and
over again to the network, the weights of the synapses converge to finite
values.
Furthermore, it is proved that the final values found in this biologically
realizable limit are the same as those found via a mathematical approach to the
problem of finding the weights of a partially connected neural network that can
store a collection of patterns. The mathematical solution is obtained via a
modified version of the so-called method of the pseudo-inverse, and has the
inverse of a reduced correlation matrix, rather than the usual correlation
matrix, as its basic ingredient. Thus, a biological network might realize the
final results of the mathematician by the energetically economic rule for the
adaption of the synapses found in this article.Comment: 29 pages, LaTeX, 3 figure
Fractionation effects in phase equilibria of polydisperse hard sphere colloids
The equilibrium phase behaviour of hard spheres with size polydispersity is
studied theoretically. We solve numerically the exact phase equilibrium
equations that result from accurate free energy expressions for the fluid and
solid phases, while accounting fully for size fractionation between coexisting
phases. Fluids up to the largest polydispersities that we can study (around
14%) can phase separate by splitting off a solid with a much narrower size
distribution. This shows that experimentally observed terminal polydispersities
above which phase separation no longer occurs must be due to non-equilibrium
effects. We find no evidence of re-entrant melting; instead, sufficiently
compressed solids phase separate into two or more solid phases. Under
appropriate conditions, coexistence of multiple solids with a fluid phase is
also predicted. The solids have smaller polydispersities than the parent phase
as expected, while the reverse is true for the fluid phase, which contains
predominantly smaller particles but also residual amounts of the larger ones.
The properties of the coexisting phases are studied in detail; mean diameter,
polydispersity and volume fraction of the phases all reveal marked
fractionation. We also propose a method for constructing quantities that
optimally distinguish between the coexisting phases, using Principal Component
Analysis in the space of density distributions. We conclude by comparing our
predictions to perturbative theories for near-monodisperse systems and to Monte
Carlo simulations at imposed chemical potential distribution, and find
excellent agreement.Comment: 21 pages, 23 figures, 2 table
Phase separation in mixtures of colloids and long ideal polymer coils
Colloidal suspensions with free polymer coils which are larger than the
colloidal particles are considered. The polymer-colloid interaction is modeled
by an extension of the Asakura-Oosawa model. Phase separation occurs into
dilute and dense fluid phases of colloidal particles when polymer is added. The
critical density of this transition tends to zero as the size of the polymer
coils diverges.Comment: 5 pages, 3 figure
Phase transitions in a ferrofluid at magnetic field induced microphase separation
In the presence of a magnetic field applied perpendicular to a thin sample
layer, a suspension of magnetic colloidal particles (ferrofluid) can form
spatially modulated phases with a characteristic length determined by the
competition between dipolar forces and short-range forces opposing density
variations. We introduce models for thin-film ferrofluids in which
magnetization and particle density are viewed as independent variables and in
which the non-magnetic properties of the colloidal particles are described
either by a lattice-gas entropy or by the Carnahan-Starling free energy. Our
description is particularly well suited to the low-particle density regions
studied in many experiments. Within mean-field theory, we find isotropic,
hexagonal and stripe phases, separated in general by first-order phase
boundaries.Comment: 12 pages, RevTex, to appear in PR
Density Functional for Anisotropic Fluids
We propose a density functional for anisotropic fluids of hard body
particles. It interpolates between the well-established geometrically based
Rosenfeld functional for hard spheres and the Onsager functional for elongated
rods. We test the new approach by calculating the location of the the
nematic-isotropic transition in systems of hard spherocylinders and hard
ellipsoids. The results are compared with existing simulation data. Our
functional predicts the location of the transition much more accurately than
the Onsager functional, and almost as good as the theory by Parsons and Lee. We
argue that it might be suited to study inhomogeneous systems.Comment: To appear in J. Physics: Condensed Matte
Mode-coupling theory for multiple-time correlation functions of tagged particle densities and dynamical filters designed for glassy systems
The theoretical framework for higher-order correlation functions involving
multiple times and multiple points in a classical, many-body system developed
by Van Zon and Schofield [Phys. Rev. E 65, 011106 (2002)] is extended here to
include tagged particle densities. Such densities have found an intriguing
application as proposed measures of dynamical heterogeneities in structural
glasses. The theoretical formalism is based upon projection operator techniques
which are used to isolate the slow time evolution of dynamical variables by
expanding the slowly-evolving component of arbitrary variables in an infinite
basis composed of the products of slow variables of the system. The resulting
formally exact mode-coupling expressions for multiple-point and multiple-time
correlation functions are made tractable by applying the so-called N-ordering
method. This theory is used to derive for moderate densities the leading mode
coupling expressions for indicators of relaxation type and domain relaxation,
which use dynamical filters that lead to multiple-time correlations of a tagged
particle density. The mode coupling expressions for higher order correlation
functions are also succesfully tested against simulations of a hard sphere
fluid at relatively low density.Comment: 15 pages, 2 figure
Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions
We evaluate the virial coefficients B_k for k<=10 for hard spheres in
dimensions D=2,...,8. Virial coefficients with k even are found to be negative
when D>=5. This provides strong evidence that the leading singularity for the
virial series lies away from the positive real axis when D>=5. Further analysis
provides evidence that negative virial coefficients will be seen for some k>10
for D=4, and there is a distinct possibility that negative virial coefficients
will also eventually occur for D=3.Comment: 33 pages, 12 figure
Phase behaviour of a model of colloidal particles with a fluctuating internal state
Colloidal particles are not simple rigid particles, in general an isolated
particle is a system with many degrees of freedom in its own right, e.g., the
counterions around a charged colloidal particle.The behaviour of model
colloidal particles, with a simple phenomenological model to account for these
degrees of freedom, is studied. It is found that the interaction between the
particles is not pairwise additive. It is even possible that the interaction
between a triplet of particles is attractive while the pair interaction is
repulsive. When this is so the liquid phase is either stable only in a small
region of the phase diagram or absent altogether.Comment: 12 pages including 4 figure
Parity forbidden excitations of Sr2CuO2Cl2 revealed by optical third-harmonic spectroscopy
We present the first study of nonlinear optical third harmonic generation in
the strongly correlated charge-transfer insulator Sr2CuO2Cl2. For fundamental
excitation in the near-infrared, the THG spectrum reveals a strongly resonant
response for photon energies near 0.7 eV. Polarization analysis reveals this
novel resonance to be only partially accounted for by three-photon excitation
to the optical charge-transfer exciton, and indicates that an even-parity
excitation at 2 eV, with a_1g symmetry, participates in the third harmonic
susceptibility.Comment: Requires RevTeX v4.0beta
Pressure Dependence of Born Effective Charges, Dielectric Constant and Lattice Dynamics in SiC
The pressure dependence of the Born effective charge, dielectric constant and
zone-center LO and TO phonons have been determined for -SiC by a linear
response method based on the linearized augmented plane wave calculations
within the local density approximation. The Born effective charges are found to
increase nearly linearly with decreasing volume down to the smallest volume
studied, , corresponding to a pressure of about 0.8 Mbar. This
seems to be in contradiction with the conclusion of the turnover behavior
recently reported by Liu and Vohra [Phys.\ Rev.\ Lett.\ {\bf 72}, 4105 (1994)]
for -SiC. Reanalyzing their procedure to extract the pressure dependence of
the Born effective charges, we suggest that the turnover behavior they obtained
is due to approximations in the assumed pressure dependence of the dielectric
constant , the use of a singular set of experimental data
for the equation of state, and the uncertainty in measured phonon frequencies,
especially at high pressure.Comment: 25 pages, revtex, 5 postscript figures appended, to be published in
Phys. Rev.
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