109 research outputs found
Generating functionals, consistency, and uniqueness in the integral equation theory of liquids
We discuss and illustrate through numerical examples the relations between
generating functionals, thermodynamic consistency (in particular the
virial-free energy one), and uniqueness of the solution, in the integral
equation theory of liquids. We propose a new approach for deriving closures
automatically satisfying such characteristics. Results from a first exploration
of this program are presented and discussed.Comment: 27 pages, 5 figure
Phase behavior of a fluid with competing attractive and repulsive interactions
Fluids in which the interparticle potential has a hard core, is attractive at
moderate separations, and repulsive at greater separations are known to exhibit
novel phase behavior, including stable inhomogeneous phases. Here we report a
joint simulation and theoretical study of such a fluid, focusing on the
relationship between the liquid-vapor transition line and any new phases. The
phase diagram is studied as a function of the amplitude of the attraction for a
certain fixed amplitude of the long ranged repulsion. We find that the effect
of the repulsion is to substitute the liquid-vapor critical point and a portion
of the associated liquid-vapor transition line, by two first order transitions.
One of these transitions separates the vapor from a fluid of spherical
liquidlike clusters; the other separates the liquid from a fluid of spherical
voids. At low temperature, the two transition lines intersect one another and a
vapor-liquid transition line at a triple point. While most integral equation
theories are unable to describe the new phase transitions, the Percus Yevick
approximation does succeed in capturing the vapor-cluster transition, as well
as aspects of the structure of the cluster fluid, in reasonable agreement with
the simulation results.Comment: 15 pages, 20 figure
A Homological Approach to Belief Propagation and Bethe Approximations
We introduce a differential complex of local observables given a
decomposition of a global set of random variables into subsets. Its boundary
operator allows us to define a transport equation equivalent to Belief
Propagation. This definition reveals a set of conserved quantities under Belief
Propagation and gives new insight on the relationship of its equilibria with
the critical points of Bethe free energy.Comment: 14 pages, submitted for the 2019 Geometric Science of Information
colloquiu
Structural Information in Two-Dimensional Patterns: Entropy Convergence and Excess Entropy
We develop information-theoretic measures of spatial structure and pattern in
more than one dimension. As is well known, the entropy density of a
two-dimensional configuration can be efficiently and accurately estimated via a
converging sequence of conditional entropies. We show that the manner in which
these conditional entropies converge to their asymptotic value serves as a
measure of global correlation and structure for spatial systems in any
dimension. We compare and contrast entropy-convergence with mutual-information
and structure-factor techniques for quantifying and detecting spatial
structure.Comment: 11 pages, 5 figures,
http://www.santafe.edu/projects/CompMech/papers/2dnnn.htm
Mean Field Renormalization Group for the Boundary Magnetization of Strip Clusters
We analyze in some detail a recently proposed transfer matrix mean field
approximation which yields the exact critical point for several two dimensional
nearest neighbor Ising models. For the square lattice model we show explicitly
that this approximation yields not only the exact critical point, but also the
exact boundary magnetization of a semi--infinite Ising model, independent of
the size of the strips used. Then we develop a new mean field renormalization
group strategy based on this approximation and make connections with finite
size scaling. Applying our strategy to the quadratic Ising and three--state
Potts models we obtain results for the critical exponents which are in
excellent agreement with the exact ones. In this way we also clarify some
advantages and limitations of the mean field renormalization group approach.Comment: 16 pages (plain TeX) + 8 figures (PostScript, appended),
POLFIS-TH.XX/9
- …