61 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
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
Direct entropy determination and application to artificial spin ice
From thermodynamic origins, the concept of entropy has expanded to a range of
statistical measures of uncertainty, which may still be thermodynamically
significant. However, laboratory measurements of entropy continue to rely on
direct measurements of heat. New technologies that can map out myriads of
microscopic degrees of freedom suggest direct determination of configurational
entropy by counting in systems where it is thermodynamically inaccessible, such
as granular and colloidal materials, proteins and lithographically fabricated
nanometre-scale arrays. Here, we demonstrate a conditional-probability
technique to calculate entropy densities of translation-invariant states on
lattices using limited configuration data on small clusters, and apply it to
arrays of interacting nanometre-scale magnetic islands (artificial spin ice).
Models for statistically disordered systems can be assessed by applying the
method to relative entropy densities. For artificial spin ice, this analysis
shows that nearest-neighbour correlations drive longer-range ones.Comment: 10 page
Cluster variation method and disorder varieties of two-dimensional Ising-like models
I show that the cluster variation method, long used as a powerful hierarchy
of approximations for discrete (Ising-like) two-dimensional lattice models,
yields exact results on the disorder varieties which appear when competitive
interactions are put into these models. I consider, as an example, the
plaquette approximation of the cluster variation method for the square lattice
Ising model with nearest-neighbor, next-nearest-neighbor and plaquette
interactions, and, after rederiving known results, report simple closed-form
expressions for the pair and plaquette correlation functions.Comment: 10 revtex pages, 1 postscript figur
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
Translation invariant extensions of finite volume measures
We investigate the following questions: Given a measure μΛ on configurations on a subset Λ of a lattice L, where a configuration is an element of ΩΛ for some fixed set Ω, does there exist a measure μ on configurations on all of L, invariant under some specified symme- try group of L, such that μΛ is its marginal on configurations on Λ? When the answer is yes, what are the properties, e.g., the entropies, of such measures? Our primary focus is the case in which L = Zd and the symmetries are the translations. For the case in which Λ is an interval in Z we give a simple necessary and sufficient condition, local translation invariance (LTI), for extendibility. For LTI measures we construct extensions having maximal entropy, which we show are Gibbs measures; this construction extends to the case in which L is the Bethe lattice. On Z we also consider extensions supported on periodic configurations, which are analyzed using de Bruijn graphs and which include the extensions with minimal entropy. When Λ ⊂ Z is not an interval, or when Λ ⊂ Zd with d > 1, the LTI condition is necessary but not sufficient for extendibility. For Zd with d > 1, extendibility is in some sense undecidable
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