Gibbs/equilibrium measures for functions of multidimensional shifts with countable alphabets

153 Pages·2011·0.59 MB·English

GIBBS/EQUILIBRIUM MEASURES FOR FUNCTIONS OF MULTIDIMENSIONAL
SHIFTS WITH COUNTABLE ALPHABETS.
Stephen R. Muir
Dissertation Prepared for the Degree of
DOCTOR OF PHILOSOPHY
UNIVERSITY OF NORTH TEXAS
May 2011
APPROVED:
Mariusz Urban´ski, Major Professor
Robert Kallman, Committee Member
John Quintanilla, Committee Member and
James Meernik, Acting Dean
of the Graduate School
CONTENTS
Acknowledgments v
CHAPTER 1. Introduction 1
1.1. Math or Science? 1
1.2. Roadmap to this paper. 2
1.3. Relation to the Extant Literature. 5
CHAPTER 2. Conceptual Introduction to Lattice Models. 9
2.1. Entropy of A Finite Dimensional Probability Vector. 9
2.2. Elementary Gibbs Ensembles and Variational Principle. 13
2.2.1. Review of Lagrange Multipliers. 13
2.2.2. Free Energy Characterization of the Gibbs State on an Abstract Finite
Configuration Space. 14
2.2.3. Maximum Entropy Question 16
2.3. Configuration Space and Interaction 17
2.4. Examples of Interactions. 19
2.5. Probability Measures i.e. States of the Configuration Space and their Entropy. 21
2.6. Hamiltonians and Gibbs States for an Interaction - The DLR Equations. 29
2.7. Relationship to Elementary Gibbs Ensembles. 30
2.8. Local Energy Formulation of Gibbs States 31
2.9. Physical Meaning of Equilibrium States, Characterization Theorems 33
2.10. Normalized versus Unnormalized Counting Measures 35
CHAPTER 3. The Space NZd 38
ii
3.1. The Topology of NZd; The Shift Action T 38
3.2. “Factoring” Finite Box Shaped Configurations 41
3.3. Relevant Spaces of Real Valued Functions on NZd 42
CHAPTER 4. The (Gibbs) Variational Principle 49
4.1. Defining the Shift Pressure. 49
4.2. Half the Variational Principle: Pressure is an Upper Bound on Negative Gibbs
Free Energy. 55
4.3. Completing the Variational Principle: The Pressure is the Least Upper Bound
on Negative Free Energy. 57
4.4. Variational Principle and Corollaries 60
CHAPTER 5. Gibbs States 63
5.1. Coordinate-wise Permutation within Λ and the associated Infinite Volume
n
Energy Loss. 63
5.2. Definition and DLR Characterization of Gibbs States. 67
5.3. The Precision with Which a Gibbs Measure of a Cylinder May be Approximated
by Partition Functions. Translation Invariant Gibbs States are Equilibrium
States. 75
5.4. Tight Sets of Measures, Existence of Gibbs States for exp-summable, regular
functions f. 79
5.5. Equilibrium States ⊇ Translation Invariant Gibbs States of Finite Entropy. 85
5.6. Equilibrium States ⊆ Translation Invariant Gibbs States. 88
CHAPTER 6. Constructive Equilibrium States 90
6.1. Definition and Tightness 91
6.2. Constructive Equilibrium States are Translation Invariant Gibbs States 97
6.3. Equilibrium States are Contained in the Closed Convex Hull of Constructive
Equilibrium States 107
iii
6.4. Extreme Equilibrium States. 113
CHAPTER 7. Conversion into Georgii’s setting for d-regular, exp-summable local
energy: normalizing and constructing a compatible interaction 117
APPENDIX A. 134
A.1. TopologiesforDualSpacesofNormedLinearSpaces, CovergenceofFunctionals,
Weak Convergence of Measures. 134
A.2. Alternate or More Elementary Proofs of Selected Facts. 135
A.3. Further Examples 144
BIBLIOGRAPHY 148
iv
ACKNOWLEDGMENTS
This paper is dedicated to all who take the time to read it.
With sincere thanks to my PhD advisor Mariusz Urban´ski, who introduced me to this
project and provided unwavering support during its course; to Robert Kallman and John
Quintanilla, who served on my committee; to Gerhard Keller and Hans-Otto Georgii, whose
books were a source of inspiration and whose responses to my early drafts led to significant
corrections and improvements; to all my teachers at UNT and Whitman, who set the foun-
dation on which this thesis was built; and to Mom, Dad, Eric, Mark, and Lauren, who need
no explanation.
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CHAPTER 1
INTRODUCTION
1.1. Math or Science?
The Ising model is the simplest and most famous example of a classical lattice model. It
makes sense in one, two, or three dimensions as a model of discrete pieces of matter arranged
on the integer lattice Zd, d = 1,2, or 3; the individual components of which may assume one
of two possible orientations. (E.g. tiny magnets, dipoles, which can only be oriented “up”
or ”down”; or two distinct metallic elements arranged in an alloy; or cells of empty space
which can be either “occupied” or “unoccupied” by a particle of gas.) In the Ising model
adjacent components of matter “interact” in the sense that there is a decrease in energy of
one unit (system becomes thermodynamically more stable) when two adjacent sites have the
same orientation, and an increase in energy of one unit (system becomes less stable) when
two adjacent sites have opposing orientations. It is intuitively clear that on any finite sub-
lattice the two energetically most favorable configurations are the unanimously down and the
unanimously up, respectively. With the probability measure that gives a weight of 1 to each
2
of these two unanimous configurations, the expected value of the orientation at each lattice
site is 0 and the correlation between the orientations at adjacent sites is 1. This is the ground
state. On the other hand, entropy attains a maximum on uniformly distributed measures.
Although the uniform distribution also gives expected orientation 0 to each individual lattice
site, it differs from the energetically most favorable state(ground state) in the correlation
between orientations at adjacent sites, which is 0 under the uniform distribution and 1 under
the ground state. The question arises: is there some intermediate probability measure which
minimizes the Gibbs free energy, i.e. the quantity (taking temperature to be 1) “expected
1
value of local energy- specific entropy”? This question points to the competition between
entropy and energy in determining the characteristics of thermodynamic equilibrium.
In this paper there are no in depth studies of specific models. Rather, we abstract the
model so much as to make it a purely mathematical project: the lattice has an unspecified
dimension, the individual sites may assume an infinite number of orientations (which we
make no attempt to describe), and the interaction/local energy is an arbitrary continuous
function (of infinite range) subject only to the weakest normalization (exp-summability)
and modulus of continuity (regularity) criteria we could get away with. Such abstraction
may seem excessive from a physical point of view, but tracking the technical mathematical
requirements in a physical model is one way a mathematician may attempt to collaborate
with the physics community. So, although interpretations in statistical mechanics will be
alluded to, this is truly a math paper - we aim to clarify the mathematical framework in
which the physical theory can take place.
1.2. Roadmap to this paper.
Section 2 begins by introducing the basic notion of entropy of a finite probability vector
and its connection with the classical “Gibbs ensemble”. The characterization of the Gibbs
ensemble as the measure of maximal negative free energy (i.e. the equilibrium measure),
proved by Lagrange multipliers in chapter 2.2.2, is the basic idea motivating the rest of
the paper. The remainder of Section 2 is devoted to a mostly non-rigorous exposition of
some mathematical ideas (and their broad physical interpretations) that can be used to
extend the idea of Gibbs and equilibrium measures to an “infinite volume” lattice model,
i.e. a product of an “individual state space” (or simply “alphabet”) over an infinite integer
lattice. Essentially this exposition summarizes the heuristics of three bibles ([12], [15], [4])
of thermodynamic formalism for lattice models.
The idea for our original theorems is to reproduce the same ideas in the configuration
space NZd. Section 1.3 below explains just how this extends the theory beyond what is
2
proved in the “bibles” just referenced. Section 3 introduces the basic machinery with which
our NZd theorems will be stated and proved, including the idea of cylinder sets and shift map
(Section 3.1), my original approach to adapt the “subadditivity” concept on the sequence
of origin centered cubes in Zd (Section 3.2), and some fundamental estimates for relevant
classes of functions (Section 3.3).
The first real goal (Section 4) is to describe a class of functions which can be thought of
as physically meaningful specific negative internal energies (we’ll call them local energies for
short, following [6]) in the sense that for such f the set of all expected specific negative free
energies (at temperature 1)
(cid:90)
fdµ+h(µ,T),
where µ is a translation invariant Borel probability measure on NZd having (cid:82) fdµ > −∞, is
bounded above. The sufficient conditions we present are exp-summability (Definition 4.3),
uniform contiunity, and finite oscillation (Definition 3.1). We’ll see that for such f the
well established “topological pressure” formula (Theorems 4.4, 4.13) for the supremum of
expected negative specific free energies carries over from the compact cases (where it is well
known to hold for all continuous functions) in essentially the same form.
However, as Example 2.11 shows, these conditions (exp-summability, uniform continuity,
and finite oscillation) do not suffice to guarantee the existence of an equilibrium state for
f, i.e. a translation invariant probability measure which gives expected negative specific
free energy equal to the supremum. The question of existence of equilibrium states can
be answered by considering Gibbs measures, i.e. measures which are roughly exponentially
distributed by local energy.
In an uncountably infinite configuration space the distribution of measure cannot be
expressed so simply as in the finite cardinality setting of Section 2.2.2. There are two
typical approaches to expressing a “Gibbsian” distribution of measure on countably infinite
products such as our NZd. One is via conditional measures - the DLR equations (named for
Dobrushin, Lanford and Ruelle). A slight complication arises because the DLR equations
3
are traditionally written in terms of interactions, whereas we want to present our therorems
strictly in terms of the local energy function. The classical “interaction” statement of the
DLR equations is Equation 1. Our local energy version of this (adapted from [6]) is Equation
4 in the statement of Theorem 5.9. The proof of the equivalence of the two formulations
under suitable conditions is addressed in Propositions 7.3 and 7.4.
The second common approach to describing an “infinite volume” Gibbs measure is to
use weak limits of countable Gibbs ensembles supported on finer and finer meshes of con-
figurations. We call these constructive equilibrium states (Definition 6.2); they are also
commonly called limiting Gibbs states. These will turn out to be significant in the sense
that all ergodic (w.r.t. the shift action) equilibrium measures must be constructive equilib-
rium measures (Theorem 6.14). By some general ergodic theory (see [16], e.g.) the ergodic
equilibrium states are exactly the extreme points of the convex set of all equilibrium states;
in physical models they correspond to to pure thermodynamic phases, i.e. states without a
phase transition.
OneofthecontributionsofthispaperistointroduceathirdapproachtodescribingGibbs
measures, and in fact we take this third one as our definition (Definition 5.7). It proscribes
how the measure must transfer under local permutations of the configuration space. This
idea is modeled after [6] and [3], but has never before been adapted to a noncompact space.
Section 5.1 contains basic estimates showing the definition is well posed for regular, exp-
summable functions. The definition has the advantage of admitting an easy proof of the
existence of Gibbs measures (Theorem 5.14) by invoking Prohorov’s theorem that tight
sets of probability measures are conditionally compact in the weak topology. Actually the
situation is slightly more complicated - first we must show via Theorem 5.9 that Estimates
11 hold, then we can demonstrate the tightness of the sets of measures relevant for existence
of Gibbs states, see Lemma 5.15 and Observations 5.16, 5.17.
The major remaing aim is to show that for regular, exp-summable local energy functions
thethreeapproachestoGibbsmeasuresareequivalentanddescribethefullsetofequilibrium
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measures. The first step is to show that Definition 5.7 of a Gibbs measure is equivalent to the
local energy version of the DLR equations (Theorem 5.9). Next the containment that shift
invariant Gibbs measures of finite entropy are necessarily equilibrium measures (Theorem
5.18) follows via what will be a staple estimate on the Gibbs measure of a cylinder set,
Equation 11. At this time we also describe another simple condition 12 on the local energy
that guarantees all its Gibbs measures to have finite entropy and thus prevents pathological
cases such as Example 2.11.
The opposite inclusion, that equilibrium states are necessarily Gibbs states, uses con-
structive equilibrium states as an intermediate step. Specifically, it is proved (Theorems
6.10, 6.11, via a long series of lemmas) that all constructive equilibria are necessarily Gibbs
measures in the sense of Definition 5.7, and shift invariant. Then it is proved (Theorem 6.12)
that all equilibrium states are in the weak closed convex hull of the constructive equilibrium
states. Because the set of Gibbs states is known to be weak closed and convex set (Observa-
tion 5.8), it follows that all equilibrium states are necessarily Gibbs states. For the sake of
completeness we include a short and direct proof (Theorem 5.21) adapted from Georgii [4],
attributed to C. Preston, that every equilibrium state is necessarily a Gibbs state.
This summarizes the original content of the paper, besides a few little facts relating to
the conversion between interaction and local energy formulations. Namely, in Section 7 I
believethecalculationsrelatingthelocalenergyformoftheDLRequationstothetraditional
interaction form and the partial results describing the set of all local energies arising from
absolutely summable interactions in terms of regularity have never been published. (Based
on some referee reports they may exist as “folklore” in very select circles.)
1.3. Relation to the Extant Literature.
The theory of Gibbs and Equilibrium states in the setting of the thermodynamic formalism
has been well developed since the 1960’s. Standard references for thermodynamic formal-
ism of lattice models include Ruelle’s [12] and it’s more physically oriented precursor [11];
5