© Creative Design and Text 1970–2004, K.
Kannemann
Klaus Kannemann
Studies
on the Linear Ising Model and Related Systems
M.
Sc. Thesis, University of Ottawa, 1970
Summary
The thesis is a study on the linear Ising model
and the related linear lattice fluid. After a brief discussion of the relevant formulae
from statistical mechanics, we focus our attention almost exclusively on these two types
of assembly. A methodology, using some aspects of set theory, is derived and then employed
throughout the thesis.
We obtain and compare results for models with
first- and finite-order interactions on the one hand, and models having modified
long-range interaction potentials on the other. The ferromagnetic behavior of the
straight-line version of the linear model with simplified long-range interaction potential
is demonstrated with mathematical rigor, whereby we use results from the theory of measure
and integration. By contrast, we show that the ring model with up to and inclusive
second-order interactions exhibits no spontaneous magnetization. Then, on extending the
matrix method employed for the second-order case, we motivate the conjecture that finite
order interaction potentials will not permit spontaneous magnetization at finite and
non-zero temperatures. This result agrees with a more general theorem on phase changes for
one-dimensional assemblies.
We derive the closed-form partition function
for the straight-line version of the linear lattice fluid having first-order interaction
potential only (square-well potential) and then we use this function to obtain the
partition function for an assembly having square well plus long-range tail interaction
potential. Again, in contrast to the former, the latter assembly can undergo a change of
phase. The mathematical mechanism of this change
of phase, if it occurs, is shown to be the modes of intersection of two floating curves. We consider this mechanism as rather interesting and, perhaps, novel — as compared to the
corresponding theorems of Yang and Lee. Using the same principles, we further propose a
method by which the closed-form partition function of an assembly having a finite-range step-potential is employed to derive the closed-form partition function for an
assembly having the sane finite-range step-potential plus long-range tail.
On using this method on the straight-line
version of the linear model with square-well plus long-range tail potentials, we again
show the existence of ferromagnetism — this would be expected anyway — but the more
interesting part of the result is that the critical temperature is now raised by a factor
greater than 2, using the same numerical value for the interaction potential. Our
conclusion is that the addition of a realistic first-order interaction potential to the
long-range-tail increases the thermal stability of the magnetic structure of the assembly.
Finally, we develop, from first principles, an
operator-formulation for the linear Ising model and the related lattice fluid. The
operator-formulation can be extended to higher dimensions and applies equally well to
periodic and non-periodic boundary conditions. We believe that the operator-formulation is
a novel approach to the Ising problem for arbitrary interactions. This principle is then
employed to derive in a brief and elegant manner a recurrence relation for the
straight-line model partition function, involving first-order interactions in the presence
of an external magnetic field. The recurrence-relation method was first published only in
1968.
We conclude the thesis by stating and proving a
curious little lemma for the partition function of the straight-line model having
interactions up to and inclusive second-order in the presence of an external magnetic
field.
Almost all of our work is built up right from
first principles and based on the methodology set forth at the beginning of the thesis.
Except for accepting the general formulation of the Ising problem, we have not attempted
to duplicate, or work in analogy to, results and methods already laid down in the current
intermediate literature on the subject. In view of this, we ask for the readers
understanding for our somewhat restricted references to published papers.
The thesis was written during the
summer of 1970, and publicly defended on September 9, 1970, before a large audience in the
old Physics and Mathematics Building at the University of Ottawa. The degree Master of
Science in Applied Mathematics was subsequently bestowed with magna cum laude.
Dr. R.G. Tross, then Department of Mathematics, now retired, was the supervisor.
The mathematical methods developed in
the thesis were considered novel at the time. Yet the venerable Ising Model
continues the be one of the leading paradigms of cooperative phenomena, and it is
precisely for its inherent similarity with neural nets that the model has been
adapted for the study and simulation of emergent phenomena and behavior in the
realm of Artificial Life -- those of the Connectionist School, of
which the author is an adherent. Further results can be expected.
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© Creative Design and Text 1970–2004,
K. Kannemann