📝 Original Info
- Title: More is the Same; Phase Transitions and Mean Field Theories
- ArXiv ID: 0906.0653
- Date: 2009-09-14
- Authors: ** Leo P. Kadano, The James Franck Institute, The University of Chicago **
📝 Abstract
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.
💡 Deep Analysis
Deep Dive into 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 stati
📄 Full Content
More is the Same;
Phase Transitions and Mean Field Theories
Leo P. Kadano
The James Franck Institute
The University of Chicago
email: leop@UChicago.edu
November 26, 2024
Abstract
This paper is the rst in a series that will look at the theory of phase
transitions from the perspectives of physics and the philosophy of science.
The series will consider a group of related concepts derived from con-
densed matter and statistical physics. The key technical ideas go under
the names of singularity, order parameter, mean eld theory, varia-
tional method, correlation length, universality class, scale changes,
and renormalization. The rst four of these will be considered here.
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. Dierent phases can be qualitatively dierent
in that walking on ice is well within human capacity, but walking on liquid
water is proverbially forbidden to ordinary humans. These dierences have
been apparent to humankind for millennia, but only brought within the
domain of scienti c understanding since the 1880s.
A phase transition is a change from one behavior to another. A rst
order phase transition involves a discontinuous jump in some statistical
variable. The discontinuous property is called the order parameter. Each
phase transition has its own order parameter. The possible order param-
eters range over a tremendous variety of physical properties. These prop-
erties 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 super uid or superconductor. A con-
tinuous transition occurs when the discontinuity in the jump approaches
zero. This article is about statistical mechanics and the development of
mean eld theory as a basis for a partial understanding of phase transition
phenomena.
Much of the material in this review was rst prepared for the Royal
Netherlands Academy of Arts and Sciences in 2006. It has appeared in
draft form on the authors' web site[1] since then.
1
arXiv:0906.0653v2 [physics.hist-ph] 14 Sep 2009
The title of this article is a hommage to Philip Anderson and his
essay More is Dierent,[2] [3] which describes how new concepts, not
applicable in ordinary classical or quantum mechanics, can arise from the
consideration of aggregates of large numbers of particles.
Since phase
transitions only occur in systems with an in nite number of degrees of
freedom, such transitions are a prime example of Anderson's thesis.
Contents
1
Introduction
3
1.1
Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
A phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
The beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4
The Ising model
. . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.5
More is the same; in nitely more is dierent . . . . . . . . . . . .
7
1.6
Ising-model results . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.7
Why study this model, or any model?
. . . . . . . . . . . . . . .
10
2
More is the same: Mean Field Theory
10
2.1
One spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Curie-Weiss many spins; mean elds . . . . . . . . . . . . . . . .
11
2.3
Meaning of the models . . . . . . . . . . . . . . . . . . . . . . . .
14
2.4
Johannes van der Waals and the theory of uids
. . . . . . . . .
16
2.5
Near-critical behavior of Curie-Weiss[23, 24] model . . . . . . . .
19
2.6
Critical indices . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3
Mean Field Theory Generalized
21
3.1
Many dierent mean eld theories
. . . . . . . . . . . . . . . . .
21
3.2
Landau's generalization
. . . . . . . . . . . . . . . . . . . . . . .
21
3.3
And onward... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
4
Summary
24
2
1
Introduction
The Universe was brought into being in a less than fully formed state, but was
gifted with the capacity to transform itself from unformed matter into a
truly marvelous array of physical structures ...
Saint Augustine of Hippo (354-430).
Translation by Howard J. van Till[4]
1.1
Phases
Matter exists in dierent thermodynamic phases, which are dierent states
of aggregation with qualitatively dierent properties. These phases provoked
studies that are instructive to the history of science. The phases themselves
are interesting to modern physics, and are provocative to modern philosophy.
For example, the philosopher might wish to note that, strictly speaking, no
phase transition can ever occur in a nite system. Thus, in some sense, phase
transitions are not exactly embedded in the nite world but, rather, are products
of the human imagination.
Condensed matter physics is a branch of physics deal
…(Full text truncated)…
Reference
This content is AI-processed based on ArXiv data.