15 research outputs found
Is the mean-field approximation so bad? A simple generalization yelding realistic critical indices for 3D Ising-class systems
Modification of the renormalization-group approach, invoking Stratonovich
transformation at each step, is proposed to describe phase transitions in 3D
Ising-class systems. The proposed method is closely related to the mean-field
approximation. The low-order scheme works well for a wide thermal range, is
consistent with a scaling hypothesis and predicts very reasonable values of
critical indices.Comment: 4 page
Melting as a String-Mediated Phase Transition
We present a theory of the melting of elemental solids as a
dislocation-mediated phase transition. We model dislocations near melt as
non-interacting closed strings on a lattice. In this framework we derive simple
expressions for the melting temperature and latent heat of fusion that depend
on the dislocation density at melt. We use experimental data for more than half
the elements in the Periodic Table to determine the dislocation density from
both relations. Melting temperatures yield a dislocation density of (0.61\pm
0.20) b^{-2}, in good agreement with the density obtained from latent heats,
(0.66\pm 0.11) b^{-2}, where b is the length of the smallest
perfect-dislocation Burgers vector. Melting corresponds to the situation where,
on average, half of the atoms are within a dislocation core.Comment: 18 pages, LaTeX, 3 eps figures, to appear in Phys. Rev.
More is the Same; Phase Transitions and Mean Field Theories
This paper looks at the early theory of phase transitions. It considers a
group of related concepts derived from condensed matter and statistical
physics. The key technical ideas here go under the names of "singularity",
"order parameter", "mean field theory", and "variational method".
In a less technical vein, the question here is how can matter, ordinary
matter, support a diversity of forms. We see this diversity each time we
observe ice in contact with liquid water or see water vapor, "steam", come up
from a pot of heated water. Different phases can be qualitatively different in
that walking on ice is well within human capacity, but walking on liquid water
is proverbially forbidden to ordinary humans. These differences have been
apparent to humankind for millennia, but only brought within the domain of
scientific understanding since the 1880s.
A phase transition is a change from one behavior to another. A first order
phase transition involves a discontinuous jump in a some statistical variable
of the system. The discontinuous property is called the order parameter. Each
phase transitions has its own order parameter that range over a tremendous
variety of physical properties. These properties include the density of a
liquid gas transition, the magnetization in a ferromagnet, the size of a
connected cluster in a percolation transition, and a condensate wave function
in a superfluid or superconductor. A continuous transition occurs when that
jump approaches zero. This note is about statistical mechanics and the
development of mean field theory as a basis for a partial understanding of this
phenomenon.Comment: 25 pages, 6 figure