.ig
:!cat %|tbl|eqn -Tps|sed -e 's/\\[(]co/\$/g'|troff -mm -i -U -Tps |grops -pletter >%.ps;gv %.ps&
:!cat %|eqn -Tps|tbl|sed -e 's/\\[(]co/\$/g'|troff -mm -i -U -Tps |grops -pletter >%.ps
:!cat %|tbl|eqn -Tps|sed -e 's/\\[(]co/\$/g'|troff -mm -i -U -Tps |grops -pletter >scr.ps;gv scr.ps&
:!cat %|eqn -Tps|tbl|sed -e 's/\\[(]co/\$/g'|troff -mm -i -U -Tps |grops -pletter >scr.ps
:!cat %|eqn -Tps|tbl|sed -e 's/\\[(]co/\$/g' >scr2
:!cat %|eqn -Tps|tbl >scr1
cat test2.mm | sed -e 1,5d -e "/----start reference/,/------end reference/d" -e '/^[.][\]["]/d' -e 's/[\][*][(]..//g' |blout >scr; cat scr|tbl|eqn -Tps|troff -mm -i -U -Tps |grops -pletter >scr.ps
..
.ig	\" porting comments
gnu groff + mm macros -> very close duplication of att dwbv3 formatting 
of following text == pseudo-scientific paper.
<291/3508 lines diff output -remarkably few changes (text like this 
and lines from both files are counted).
NOTE: 
problem with inline eqn expressions of type $"$"P$ == $P;
resolved by changing $-symbol bracketed by eqn $...$ delims to something
else,
e.g. \(co, then by use of filter reversing to $ just before pipe to troff;
alternative of changing all delimiters from $ to, e.g., | is more daunting.
NOTE:
each eqn must be within display block: .DS .EQ ..... .EN .DE
else eqn numbering (and more?) does not work
NOTE: 
references within references are handled differently than in dwbv3:
they are numbered starting after refs from main text, and presumably ad inf
for deeper nesting; this is counter-intuitive, imho - better, as done here,
.\" is to change ref within ref from \*(Rf to ident \*(xY, then up ref-cntr, 
.\" and give citation immediately following, in std format .RS xY.
setup
NOTE: 
figures are imported by grops under macro PSPIC - see: man grops;
.PSPIC [-L|-R|-I n] file [width [height]]
alternative: \X'ps: import file llx lly urx ury width [ height ]'
PIC and EPIC, found in m.tmac and described in man 7 groff_mm,
do not work (at least in my hands);
NOTE:
there is weakness in presentation of energy-$ equivalence in context of
development of Ising model: 
sign of $ values for parameters 
= sign of energy values for same parameters?
is system more stable if minimize total $ value?
sign of beta-sub-dollar and T-sub-dollar?
..
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.\" .fp 2 I HI
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.BS
'sp 0.425i	\" compensate for builtin vertical 0.425i offset (NEC feature?)
.BE		\" alternatively include sp -.425i in PH, but more complicated
.\"
.EQ
delim $$
.EN
.\"
.ds HF 3 3 3 3 3 3 3	\" 3 == bold font for heading levels ....
.ds HP 14 14 14 14 14 14 14
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.nr Hc 1	\" centered headings, through level nnn
.nr Hi 1	\" std paragraph indent, specified by Pt register
.nr Pt 1	\" indent paragraphs always
.\".nr Ej 1	\" page eject before top level headings
.\" .ds Rp "\fBFOOTNOTES AND REFERENCES\fP	\" fn ref page header
.ds Rp ".H 1 Footnotes\ \ and\ \ References \" fn ref page header
.\"
.ig	\" residue left in from thought to include cover letter to editor
DRAFT \h`1.8i`
DRAFT \h`1.8i`
DRAFT 
.SP2
.ad r
Tucson, Arizona
.br
May 21, 2002
.br
.ad l
Editor
.br
Zeitschrift f\o`u\(:a`r Ber\o`u\(:a`chtigtwissenschaft
.SP
Dear Sir:
.SP
.P
..
.\" end residue
.\".ps 16
.\".ps 18
.S 17
.nf
.ce
\fBIsing-Model Analysis of the Developer-Neighbor (D-N) Problem.\fP\*(Rf
.\"
.\"--------------start reference
.RS tI
[Editorial note: J. A. Rupley, 10/31/02.]
.br
I found this manuscript on my 
doorstep, wrapped in a blue blanket,
to which was pinned the following unsigned message, 
"Please take care of my child... I
have had to leave Tucson unexpectedly for a country with no Reciprocal 
Extradition Agreement."  
The author's name given on the manuscript is obviously a pseudonym.
However, one can deduce from the subject of the manuscript that the 
author is involved with planning and zoning, 
and from the apparent flight from our
City that the author is a lawyer or developer.
For several months I have attended,
as an observer,
meetings of a subcommittee of the
Planning and Zoning Commission
charged with review of the City's Big Box Ordinance, 
and it is possible, even likely, that the author, having met me there, 
saw fit to entrust me with his or her creation.
Wishing to avoid legal action,
I will not speculate more on the author's identity.
A draft cover letter indicated that the author wished the manuscript to be
submitted to a specific journal, 
the Zeitschrift f\[:u]r Ber\[:u]chtigtwissenschaft.
I have chosen not to honor this implied request.
Indeed, I have not been able to verify the existence of the 
so-apparently-well-named Zeitschrift.
.SP
The author may have been under considerable stress when creating this
manuscript: 
the footnotes referenced on the first three pages suggest
that the project was begun as
a vehicle for the author to vent discontent with an inimical environment.
However, the initial
cynical and acerbic tone vanishes entirely once the author has
described the model.
Was the author drawn into his own work at this point?
Did the author
decide to bury himself in this application of the theory of
phase transitions,
in order to escape briefly from a hostile world?
Did the author
come to believe that the Ising-model
picture of the D-N problem was not just an amusement, but was essentially
correct in isolating significant features of the problem?
This last possibility,
the author's belief in the correctness of the application,
is supported by the perhaps tiresome length of the manuscript.
The reader
who is about to judge the author's work,
is reminded of the deranged state of a person 
poised to flee our lovely City.
.RF
.\"--------------end reference
.\"
.fi
.S
.SP
.\".ps
.S 14
.SP
.ce
J. A. Velina
.sp .5v
.ce
Tucson, Arizona
.ce
October 31, 2002
.SP 2
.S
.\"
.H 1 "Introduction."
We report here a theoretical treatment of the well-known D-N, 
or Developer-Neighbor problem.\*(Rf
.\"
.\"--------------start reference within reference
.RS dN
The D-N problem is isomorphic to the CAVE problem.\*(cA
.RF
.\"--------------end reference
.\"
.\"
.\"--------------start reference within reference
.nr ref*nr +1
.RS cA
Citizens Against Virtually Everything.
.RF
.\"--------------end reference
.\"
\*(Rf
.\"
.\"--------------start reference
.RS uP
(a) Urban Political space is separable into the manifold of the 
Developer-Neighbor (D-N) problem 
and its orthogonal complement, the Developer-Politician (D-P) advantage.
.sp .5v
\fI(b)\fP The D-P advantage, a union of the Developer and Politician subspaces,
is not an advantage but in fact a problem,
in the view of neighbors and citizens at large.
In any case, the Developer-Politician advantage is isomorphic to
the Greed-and-Corruption problem.
.sp .5v
\fI(c)\fP The Neighbor-Politician (N-P) problem is equal to the D-N
problem.
Thus the CAVE problem is isomorphic to not only the D-N problem\*(dN 
but also the N-P problem, 
as is frequently remarked upon,
particularly during election years,
by elements of the D and P subspaces.\*(sP
.sp .5v
\fI(d)\fP N-P problems take a very long time to solve, which one speculates
has led to denoting a class of problems as NP-hard.
.sp .5v
\fI(e)\fP The Urban Political function space is complex and is rife with
imaginary time operators, which should surprise no one.
.RF
.\"--------------end reference
.\"
.\"
.\"--------------start reference within reference
.nr ref*nr +1
.RS sP
If you have a problem visualizing the Urban Political space
and its subspaces, 
be comforted,
I too have this problem.
.RF
.\"--------------end reference
.\"
\*(Rf
.\"
.\"--------------start reference
.RS wR
The D-N problem is widely recognized as central to 
the planning of cities, the management of 
population growth, and the loss of local elections.
.RF
.\"--------------end reference
.\"
Our analysis provides deep understanding of it 
and determines its optimal resolution.
.P
We assert that the principal features of the D-N
problem are displayed in the Residential-Commercial (R-C) 
system at equilibrium.
We analyze the Residential-Commercial equilibrium 
by use of the Ising model:\*(Rf
.\"
.\"--------------start reference
.RS iS
E. Ising,
\fIZ. Physik\fP
\fB31,\fP
253 (1925).
.RF
.\"--------------end reference
.\"
an area, such as
that of the City of Tucson, is divided into sites\*(Rf
.\"
.\"--------------start reference
.RS sI
In Planner-Developer language, a site is a land area consisting of a lot or
contiguous lots, designated for development as a single entity.  For the
present discussion, a site, i.e., a lattice site,
corresponds closely to a lot or fraction of a lot.
For clarity, "P-D site" denotes a development site;
"site" used without qualifier means a lattice site.
A P-D site is a cluster of lattice sites.\*(pD
.RF
.\"--------------end reference
.\"
.\"
.\"--------------start reference within reference
.nr ref*nr +1
.RS pD
We assume that P-D clusters are statistically independent.
.RF
.\"--------------end reference
.\"
on a square
grid; each site can be in one of two states, Commercial (C) or
Residential\ (R);\*(Rf
.\"
.\"--------------start reference
.RS pL
We assume that all land can be assigned objectively as being either 
Residential state or Commercial state,
e.g.: industrial use \(en Commercial state; ranch use \(en Residential state.
Unused land
is Residential state, 
without regard to its zoning or whether it is speculatively intended for
commercial use.
.RF
.\"--------------end reference
.\"
\*(Rf
.\"
.\"--------------start reference
.RS pR
In previous, unsuccessful treatments, 
these states have been denoted, respectively, paved (P) and saved (S), 
or profit ($ "\(co" $) and loss (()).
.RF
.\"--------------end reference
.\"
a site interacts with its nearest neighbors;
the strengths of the site-site interactions (RR, RC, CR, CC) and
the total number of sites (the size of the system)
are variable parameters; 
the values of these or equivalent parameters can be determined or set,
and when given can be used to calculate properties of the system.
.P
We can ask questions of this model:
(a) what fraction of the area (number of sites) are in the Commercial
or the Residential state for a particular set of 
parameter values (interaction strengths and total number of sites)?
(b) do sites of the same state cluster, and if so, what is the origin
of the clustering?
(c) are there ranges of parameter values for which the system is specially
sensitive (catastrophes)?
(d) how can we optimize the system?
.P
Our application of the Ising model to the D-N problem is 
analogous to application of the model
to the coil-helix transition of a protein.
The Ising model implies,
for a city as for a protein,
the possibility of a phase transition
and extreme sensitivity of the R-C equilibrium 
to changes in system conditions (the environment).
We note also that a city, like a protein, is a small system, in the sense it
comprises very many fewer than Avogadro's number of elements.
.P
.P
The discussion that follows is necessarily technical.
Partly in expiation,
the text is less succinct, somewhat pedagogical,
and more voluminously footnoted\*(Rf
.\"
.\"--------------start reference
.RS vF
The extensive footnotes are intended to comfort those who might need
solace when faced with distinguishing 
the trace from an eigenvalue,
or even
a logarithm from an exponential.
.RF
.\"--------------end reference
.\"
than the typical scientific communication. 
We suggest to the Reader that the footnotes be followed along with the text.
.SP 2
.\"
.H 1 "Theory" \*(Rf
.\"
.\"--------------start reference
.RS tH
If you disregard theory, please don't feel bad \(em nearly everyone
seems to disregard theory, which is fortunate for theorists, who
can be blunderers and yet be thought of 
(and think of themselves) as the top of the food chain.
In any event, if theory bores you or you have no use for it, 
please pass over this section
and go on to Results.
.RF
.\"--------------end reference
.\"
.P
In order to obtain a simple expression of the model, we treat
the one-dimensional case.
This can be solved by the 
matrix method.\*(Rf
.\"
.\"--------------start reference
.RS kR
H. A. Kramers and G. H. Wannier, \fI Phys. Rev.\fP \fB60,\fP 252 (1941).
.RF
.\"--------------end reference
.\"
\*(Rf
.\"
.\"--------------start reference
.RS kU
M. Toda, R. Kubo, and N. Saito,
\fIStatistical Physics: 1. Equilibrium Statistical Mechanics,\fP
2nd edition, Springer-Verlag, New York, 1992.
.RF
.\"--------------end reference
.\"
\*(Rf
.\"
.\"--------------start reference
.RS qU
Despite recent advances in nanotechnology, we ignore quantum effects.
.RF
.\"--------------end reference
.\"
Zimm and Bragg modeled in this way the phase transition between
coil and helix states of a polypeptide chain.\*(Rf
.\"
.\"--------------start reference
.RS zI
B. H. Zimm and J. K. Bragg, 
\fIJ. Chem. Phys.\fP
\fB31,\fP
526 (1959).
.RF
.\"--------------end reference
.\"
For convenience,\*(Rf
.\"
.\"--------------start reference
.RS cO
For \fIour\fP convenience, of course.  The sins of Pride, Greed, Envy,
Wrath, Lust, and Gluttony may not be among ours,\*(jA but Sloth is, and we
happen to have had already prepared in our lecture notes a tutorial 
on the Zimm-Bragg theory of the coil-helix transition.
[For a ridiculously small honorarium, we will deliver
a lecture on this subject that is deadly boring and suitable for terminating
your salon or soir\o`e\'`e by driving late-staying guests away 
at a moment of your choosing;
please contact the Lethe Agency.]
.RF
.\"--------------end reference
.\"--------------start reference within reference
.nr ref*nr +1
.RS jA
If you believe that, I have a London bridge on a beautiful site in the Arizona
desert, and it's for sale at a bargain price ("una ganga", 
as some have said).
.RF
.\"--------------end reference
.\"
we closely follow their treatment.\*(Rf
.\"
.\"--------------start reference
.RS yM
You well may wonder how I can call this section "Theory",
when it merely describes
the mindless application 
of someone else's theory (Zimm and Bragg's\*(zI) to a
congruent problem.
In defense, I rush to say that referees and editors 
virtually insist, to be in line with tradition, 
that there be a section titled "Theory"
in a communication of this kind.
Furthermore,
Zimm and Bragg did somewhat the same thing,
in their application of
Kramers and Wannier's\*(kR theory
to the coil\(emhelix transition.
So also did Kramers and Wannier, who used what is now obvious mathematics to
solve Ising's model,\*(iS
which ferromagnetism afficionados had been batting around already for
more than a decade.
But you get the idea,
and in any event,
the history of science
now appears to interest few
and to be ignored by nearly all.
.RF
.\"--------------end reference
.\"
.SP 2
.\"
.H 2 "Site-Site Interactions"
.P
Site-site interactions
are usually given as energies.
To attach physical meaning to this model of the
Residential-Commercial equilibrium,
we assert that a dollar value is equivalent to an
energy.\*(Rf
.\"
.\"--------------start reference
.RS eN
The electric bill demonstrates the proportionality of energy and dollars.
Furthermore,
as is well known, 
financial power is proportional to the rate of expenditure of
dollars, 
which parallels electric power being the rate of expenditure of energy and
gives independent support for the proportionality of energy and dollars.
.RF
.\"--------------end reference
.\"
.P
.\"(1)
.\"
.H 3 "Reference state."
We choose 
as reference state
a Residential site inside a region of contiguous
Residential sites.
The reference state is the zero of energy or of dollar value.
This selection of a reference state
eliminates one of the four interaction-energy parameters noted
above.
.P
.\"(2)
.\"
.H 3 "$bold {i sub CC}$
is the change in energy or dollar value\*(Rf
.\"
.\"--------------start reference
.RS mE
Measured in unitless scaled dollars.                               
.RF
.\"--------------end reference
.\"
.ig
of the
interaction of a Commercial site with its neighbors in a Commercial region,
for the same 
Residential region reference state.
Thus $i sub CC$
is the change in energy or dollars
..
associated with transformation of one Residential site 
inside a Residential region into a Commercial site
inside a Commercial region, i.e.,
expansion of a Commercial region by one site 
at the expense of a Residential region.
Using the monetary picture,
.DS I
.EQ (Eq.\ 1a)
i sub CC ~=~ "$"P ~+~ [ delta "$" sub CC ~-~  delta "$" sub RR ]
.EN
.DE
where
$ "\(co" P$ is the Profit/Commercial-site, 
$delta "\(co" sub CC$ is the contribution from interactions of a Commercial site with
its neighbors within a pure Commercial region,
and similarly,
$delta "\(co" sub RR$ is the contribution from interactions of a Residential site with
its neighbors within a pure Residential region.
Being the sum of a profit and a difference between two numbers of
comparable size, 
the value of $i sub CC$ typically is small.\*(Rf
.\"
.\"--------------start reference
.RS
If you have difficulty visualizing the relationships of this paragraph,
the following picture may help:
Consider a standard quantity\*(sQ of a gas of free Residential sites 
(free in the sense of independent \(em nothing is
free dollar-wise, of course); 
let this gas, R(g), 
condense to form
a Residential region, i.e., a Residential neighborhood, R(l),
with the establishment of Residential-Residential interactions and with a concomitant
change in energy,
.DS I \".SP
.EQ (Eq.\ A1a)
delta E sub RR ~=~ delta "$" sub RR 
~=~ "$" sub R(l) ~-~ "$" sub R(g)
.EN
.DE \".SP
We now transform an equal quantity of gaseous Residential sites into gaseous 
Commercial sites, noting that a profit may be derived from Commercial but not 
Residential sites,
.DS I \".SP
.EQ (Eq.\ A1b)
delta "$" sub {R(g) -> C(g)} 
~=~ "$" sub C(g) ~-~ "$" sub R(g) ~=~ "$"(Profit/C"-"site) ~=~ "$" P
.EN
.DE \".SP
and condense the Commercial-site gas to form a Commercial development, with
.DS I \".SP
.EQ (Eq.\ A1c)
delta E sub CC ~=~ delta "$" sub CC ~=~ "$" sub C(l) ~-~ "$" sub C(g)
.EN
.DE \".SP
By definition, 
for R(l) as the reference state,
the Commercial-site \(en Commercial-site energy
.DS I \".SP
.EQ (Eq.\ A1d)
i sub CC mark ~=~ "$" sub C(l) ~-~ "$" sub R(l) 
.EN
.DE
.DS I
.EQ (Eq.\ A1e)
lineup ~=~ "$"(Profit/C"-"site) ~+~ ( delta "$" sub CC ~-~ delta "$" sub RR )
.EN
.DE \".SP
that is, 
$i sub CC$ is the profit per Commercial site
plus the differential gain (loss) from establishing the interactions of
a Commercial site inside a Commercial region
versus those of a Residential site inside a Residential region.
.SP
If one still remains confused, 
pass on to the Results section, since the remainder of the Theory section
will confuse even more.
.RF
.\"--------------end reference
.\"--------------start reference within reference
.nr ref*nr +1
.RS sQ
The standard quantity of a substance typically is a mole, Avogadro's number,
$ N sub 0 ~=~ 6 x 10 sup 23$ particles.
This would be rather a lot of one-acre Residential sites, 
even hypothetical ones.
A more reasonable choice for a standard quantity would be a femtomole,
corresponding to a land area approximately 1000 miles square, about 5x the
territory of the four "Four Corners" states (AZ, CO, NM, UT).
Neo-Malthusian enviro-futurist Greens among us would view 
(with horror!) an area of this size as possibly being
"Mall-able" and certainly open to being zoned immediately
for Residential use.  
Others, mindful of world demographics and the 2050 population downturn,
might continue to consider a femtomole of Residential sites as somewhat excessive.
.RF
.\"--------------end reference
.\"
.P
To simplify notation and follow Zimm and Bragg, we define
the new parameter
.DS I
.EQ (Eq.\ 1b)
~~~~s mark ~=~ e sup {i sub CC}
.EN
.DE
.DS I \".SP .5v
.EQ 
i sub CC lineup ~=~ log s
.EN
.DE
.SP
.P
The parameter $~s~$ is the microscopic
equilibrium constant for interconversion of a Residential site
and a Commercial site, each being inside their respective regions.
Because $i sub CC$ is small,
$s$ will be close to unity.
.P
.\" (3) 
.\"
.H 3 "$bold {i sub RC}$
is the 
change in energy or dollar value
when 
initiating a region (or run) of Commercial sites 
adjacent to and following a region (or run) of Residential sites.
Equivalently, we can picture $i sub RC$ as the cost of
creating a \fInew\fP Residential-Commercial interface region; 
or as the cost of 
initiating
a \fInew\fP Commercial region by transforming a Residential site into a Commercial site 
inside a Residential region.
The value of $i sub RC$ is typically large and
negative, consisting of 
a small favorable contribution from $i sub CC$ 
plus a large cost of
initiating a new Commercial region within
a previously pristine Residential region.
The initiation cost is the sum of several contributions.\*(Rf
.\"
.\"--------------start reference
.RS sF
Mitigation of opposition of Residential 
neighbors to adjacent Commercial, 
fees (City, agents, lawyers, illegal-petition passers), 
interest on loans accruing as a result of bureaucratic delay
(whether by intent or by incompetence),
payments for politician and bureaucrat maintenance, etc. 
.RF
.\"--------------end reference
.\"
The cost need be paid only once per
Commercial region, but if it is large,
initiation of a new Commercial region is difficult.
.P
To isolate the initiation cost and so to simplify the model,
following Zimm and Bragg,
we define the parameter $sigma$ by
.DS I
.EQ (Eq.\ 2a)
sigma s mark ~=~ e sup {i sub RC}
.EN
.DE
Noting that $~s~=~e sup {i sub CC}$,
.DS I
.EQ (Eq.\ 2b)
log sigma mark ~=~ {i sub RC} ~-~ {i sub CC}
.EN
.DE
.DS I \".SP .5v
.EQ (Eq.\ 2c)
lineup ~=~ (cost~of~creation~of~R-C~interface)
.EN
.\"0 lineup ~<~ sigma ~<~ 1
.DE
.SP
.P
If the cost of creating the Residential-Commercial interface
is large and negative,
$sigma ~<<~ 1$.  
We use the value $sigma~=~ 10 sup -2$, unless stated otherwise.
.P
.\"(4) 
.\"
.H 3 "$bold {i sub CR}$.
If ${i sub RC}$ is the cost in
energy or dollar value associated with initiating a new Commercial region,
then we may picture 
${i sub CR}$ as a similar cost associated with terminating a Commercial
region.
By symmetry with ${i sub RC}$,
$i sub CR$ can be pictured as the cost of initiating a \fInew\fP
Residential region inside a Commercial region.
Following Zimm and Bragg, we set $i sub CR$ at zero.  
This is mathematically sound: 
one can choose to lump all of the interface
effect into $i sub RC$.\*(Rf
.\"
.\"--------------start reference
.RS iE
It also makes physical sense: mixed use
(Residential sites within a Commercial region) 
has been observed to be profitable.
.RF
.\"--------------end reference
.\"
.P
By setting ${i sub CR}$ at zero, we have eliminated a second of the four
interaction energy parameters noted above.
Thus only three variable parameters are sufficient to define the system:
$s$, the microscopic Residential-Commercial equilibrium constant,
$sigma$, the initiation cost for a new Commercial region,
and $n$, the total number of Residential and Commercial sites in the system.
.SP 2
.\"
.H 2 "Partition Function"
.P
Our aim is to calculate the partition function $Z$,
knowledge of which allows us to 
evaluate the equilibrium properties of a system.\*(Rf
.\"
.\"--------------start reference
.RS fE
A beautiful expression of the significance of the partition function,
or more precisely, of averaging in statistical mechanics,
has been given by Feynman.\*(sT
.RF
.\"--------------end reference
.\"
.\"
.\"--------------start reference within reference
.nr ref*nr +1
.RS sT
R. P. Feynman,
\fIStatistical Mechanics: A Set of Lectures,\fP
Notes taken by R. Kikuchi and H. A. Feiveson,
Edited by J. Shaham,
Addison-Wesley, New York, 1972.
.RF
.\"--------------end reference
.\"
The classical partition function is the sum of the statistical 
weights of all possible arrangements of the system, 
in this case, all possible ways of assigning Commercial and Residential states to each site.
.P
For the one-dimensional Ising model with the lattice (chain) 
wrapped and connected last
element to first, one finds easily by the matrix method that
.DS I
.EQ (Eq.\ 3)
Z ~=~ Tr "{"bold M sup up 20 n"}" ~=~ lambda sub down 20 0 sup  up 20 n
~+~ lambda sub down 20 1 sup up 20 n
.EN
.DE
where $n$ is the total number of sites, 
$bold M$ is the transfer matrix,\*(Rf
.\"
.\"--------------start reference
.RS tR
The transfer matrix is an operator that 
propagates the vector of the 
statistical weights of the joint
configurations of the system to the next site 
along the latticen chain.\*(kU\*(zI
.RF
.\"--------------end reference
.\"
.DS I
.EQ (Eq.\ 4)
bold M ~=~
left ( { ~~
pile {1 above sigma s} ~~~~
pile {1 above s} ~
} right )
.EN
.DE
and 
$lambda sub down 20 0 ~>~ lambda sub down 20 1$ are the two eigenvalues of 
$bold M$,
with 
.DS I
.EQ (Eq.\ 5)
lambda ~=~ 1 over 2 ( 1~+~s ~+-~ [ (1-s) sup 2 ~+~ 4 sigma s ] sup 1/2 )
.EN
.DE
.SP
.P
Note that $Z$ is a function only of the eigenvalues $lambda$ and of $n$.
The eigenvalues 
$lambda$ are functions only of $s$ and $sigma$.
Thus, $Z$ is a function only of the two interaction parameters,
$s$ and $sigma$, 
or equivalently $i sub CC$ and $i sub RC$,
and the system size, $n$:
.DS I
.EQ (Eq.\ 6)
Z ~=~ f(s, sigma , n)
.EN
.DE
.SP 2
.\"
.H 2 "Characterization of the System"
.P
Following Zimm and Bragg\*(zI\*(Rf
.\"
.\"--------------start reference
.RS z2
Zimm and Bragg obtain $Z$
for a linear one-dimensional lattice (not wrapped),
with the first three sites of the n-site chain fixed in the Residential 
state:
.DS I \".SP
.EQ (Eq.\ A2)
Z ~=~ {lambda sub down 20 0 sup up 20 {n-2} ( lambda sub down 20 0 - s) ~+~ 
lambda sub down 20 1 sup up 20 {n-2} (s - lambda sub down 20 1 ) }
over
{( lambda sub down 20 0 ~-~ lambda sub down 20 1 )}
.EN
.DE \".SP
The eigenvalues $lambda sub down 20 0 ~>~ lambda sub down 20 1$
are the same as 
given by Eq.\ 5.
This expression for $Z$ is closely similar to that of Eq.\ 3, 
and we use it for convenience, as noted.\*(cO
.RF
.\"--------------end reference
.\"
and armed with our knowledge of the partition function $Z$,
we first give, 
without proof and as
functions of $Z$ and the parameters $s, sigma$ and $n$,
expressions for variables that describe an urban 
Residential-Commercial equilibrium system;
then we estimate values for the variables and the parameters.
.SP
.\" (1) 
.\"
.H 3 "Fraction Commercial state, $theta$,  and Residential-Commercial equilibrium constant, $K sub C$.
The fraction of sites in the Commercial state,
.SP
.DS I
.EQ (Eq.\ 7a)
theta mark 
~=~ { n sub C} over {n sub C ~+~ n sub R}
.EN
.DE
.DS I \".sp .5v
.EQ (Eq.\ 7b)
lineup 
~=~ 1 over {n~-~3} {d~log~Z} over {d~log~s}
.EN
.DE
.P
The overall (macroscopic) equilibrium constant
describing the interconversion of Commercial and Residential
states,
.DS I
.EQ (Eq.\ 8)
K sub RC mark ~=~ { n sub C} over {n sub R}
~=~ { theta } over {1 ~-~ theta}
.EN
.DE
.P
Fig. 1 gives plots of the function $theta$ over the 
$log sub 10 n \(en log sub 10 s$ plane,
calculated for several values of $sigma$ (0.01, 0.1, and 1)
by use of Eq. 7b.
The low-$theta$ region at low $s$ corresponds to essentially pure
Residential, the high-$theta$ region at high $s$, 
to essentially pure Commercial.
The transition between regions becomes
increasingly sharp with smaller $sigma$.
For large $n$,
the Residential-Commercial 
transition is essentially independent of system size $n$
and the transition is centered at $s ~approx~ 1$.
.SP
.\"(2) 
.\"
.H 3 "Critical size, $ n sub crit$.
There is a size for the system, $n ~=~ n sub crit$, 
below which a Commercial region is unstable and 
not likely to exist.
This is a consequence of the cost associated with
initiation of a new Commercial region inside an existing Residential region.
For a Commercial region to form, this large cost,
given by $log sigma$ (Eq. 2c),
must be
balanced by the sum of many small gains in value, 
given by $n sub crit cdot log s$ (Eq. 1b),
for the transfer of $n sub crit$ sites from a
Residential region to a Commercial region, giving\*(Rf
.\"
.\"--------------start reference
.RS cR
This development of an expression for $n sub crit$ is heuristic.
It is intended to make plausible, through a simple argument, 
the clustering of sites of each type, Residential and Commercial.
$S sub C$, the average size of a cluster, defined in Sec. 2.3.4,
is a more theoretically-proper measure of clustering.
However, $S sub C$ depends on $nu$, defined by Eq. 10,
which is not as obvious in its physical meaning as Eq. 9.
Note that Eq. 9 fails for $s <= 1$.
.RF
.\"--------------end reference
.\"
.DS I
.EQ (Eq.\ 9)
n sub crit ~=~ - {log~sigma} over {log~s}
.EN
.DE
.SP
.ne 2
The value of $n sub crit$ is larger for smaller $sigma$ (more difficult initiation) 
and is smaller for larger $s$ (larger profit for each site transferred,
R \(-> C).
.P
There are two corollaries of Eq. 9:
(a) If there is a significant initiation cost
($sigma ~<~ 1$), 
Commercial sites are with high probability part of a
cluster of Commercial sites (a strip mall, etc.),
and if the initiation cost is large ($sigma~<<~1$), the clusters
will be large ($n sub crit$ large).
(b) If the system size $n$ is small, 
the Residential-Commercial transition occurs at a value of the
equilibrium constant $s$ significantly greater than $1$
(Fig. 1): 
$s$ must be sufficiently large
that agglomeration of a small number of sites 
overcomes the cost of cluster initiation.
.SP
.\"(3) 
.\"
.H 3 "Average number of Commercial regions in the system, $nu$.
The average number of Commercial regions
.DS I
.EQ (Eq.\ 10)
nu mark ~=~ {d~log~Z} over {d~log~sigma}
.EN
.DE
.SP
For $n$ sufficiently large and $s ~approx~ 1$,
.DS I
.EQ (Eq.\ 11a)
nu ~approx~ n over 2 { sqrt sigma }
.EN
.DE
and for $s ~>~ 1$,
.DS I
.EQ (Eq.\ 11b)
nu ~approx~ n over {s ~-~1} ~ { sigma }
.EN
.DE
.P
Fig. 2 is a contour plot of $log sub 10 nu$ over the
$log sub 10 n \(en log sub 10 s$ plane,
calculated for $sigma ~=~ 0.01$ 
by use of Eq. 10 (solid lines, contours for specified values of $log sub 10 nu$).
The dashed lines are the 0.1 and 0.9 contours for $theta$, limits which
define the transition region between pure Residential at small $s$ and pure
Commercial at high $s$.
.P
For fixed system size $n$ and $n$ large
(a horizontal cut across the graph at $log sub 10 n ~>~ 1.5$), 
the number of Commercial regions $nu$ is a maximum
in the central transition region (equilibrium constant $s ~approx~ 1$).
This is expected:
at high values of $s$, essentially all sites are Commercial
and joined into only one or a small number of separate large clusters;
at low values of $s$, there is essentially no Commercial, and thus only a
few separate small clusters.
.SP
.\"(4) 
.\"
.H 3 "Average size of a Commercial region, $S sub C$.
The average number of sites per Commercial region,
.DS I
.EQ (Eq.\ 12a)
S sub C mark ~=~ n ~cdot~ theta ~/~ nu
.EN
.DE
For $n$ sufficiently large and $s ~approx~ 1$,
.DS I
.EQ (Eq.\ 12b)
S sub C ~approx~ 1 over { sqrt sigma }
.EN
.DE
Eq. (12b) is a path to evaluation of $sigma$.
.P
.ne 2
Fig. 3 is a contour plot of $log sub 10 S sub C$ over the
$log sub 10 n \(en log sub 10 s$ plane, for $sigma ~=~ 0.01$.
For $n > 1/ sigma$, the approximation of Eq. 12b is seen to be accurate.
.P
A theoretically correct estimate of $n sub crit$ is the value of
$S sub C$ at the midpoint of the Residential-Commercial
transition (halfway between the dashed lines corresponding to contours for
$theta$ values of 0.1 and 0.9, Fig. 3).
.P
For $n$ small and $log sub 10 s ~>~ 0.5$, 
$n sub crit$ estimated by Eq. 9 is in agreement with the results of the
calculations shown in Fig. 3, based on Eq. 10 and 12a.
As noted, Eq. 9 fails for $s ~approx~ 1$, i.e., $log sub 10 s ~<~0.1$.
.P
For $n$ large, the cluster size at the midpoint of the transition,
$n sub crit$,
is independent of system size $n$.
For the parameter values used in the calculations of Fig. 3,
we find 
in the transition region
that the cluster size
$S sub C ~=~ sigma sup -2 ~=~ 10$ for $n ~>~ 20$.
As a consequence of constant cluster size,
the number of clusters in the system, $nu$, 
must for large $n$ increase
in direct proportion to $n$, as is found (Fig. 2).
.P
$S sub C$ is the correlation length.  
The correlation length is central to the
description of phase transitions
(scaling and power laws).
.SP 2
.\"
.H 2 "Estimation of Values 
In order to work with the above expressions based on the partition function,
we need 
a set of values for the parameters $sigma ,~ s$, and $n$
that are self-consistent 
and are plausibly grounded in observation.
For subsequent discussion of the Residential-Commercial equilibrium system, 
we need to evaluate also
several functions of the parameters:
$~theta ,~K sub RC ,~S sub C$, and $nu$.
.P
.\"
.H 3 "$bold theta ,~ bold K sub RC$."
The fraction of the land area of Tucson that is Commercial is ca. 20
percent,\*(Rf
.\"
.\"--------------start reference
.RS cF
The fraction of the full cash value that is Commercial was 17.88 percent for
Pima County in 2001 (data from Pima County Assessor's Office).
.RF
.\"--------------end reference
.\"
giving $theta~=~0.2$ and the equilibrium constant
$K sub RC ~=~0.25$.
.P
.\"
.H 3 "$bold s$."
For a Residential-Commercial system in the transition region\ \ ($0.1 ~<~ theta ~<~ 0.9$),\ 
and with $sigma ~<~ 0.1$ and $n ~>~ 1/ sqrt sigma$, 
we have
\ $s ~approx~ 1$\ \ ($log sub 10 s ~approx~ 0$)
\ (Fig. 1: $theta$ over $log sub 10 n \(en log sub 10 s$ plane; 
Fig. 2: dashed contours for $theta ~=~ 0.1, 0.9$).
This is a consequence 
of the transition region being sharp and centered at $s ~approx~ 1$
for small $sigma$.
Since for the Residential-Commercial equilibrium we observe\ $theta ~=~0.2$, 
which is within the transition region,
we may assert that $s ~approx~ 1$, 
provided $sigma$ is sufficiently small (see below).
.P
For small system size $n$,
$theta$ 
is dependent on $n$ as well as on $s$
(Fig. 1 and Fig. 2).
Thus the $theta$-dependent
Residential-Commercial equilibrium constant,
$K sub RC ~=~0.25$, 
is a lower limit for $s$, 
the microscopic Residential-Commercial equilibrium
constant.
$K sub RC$ and $s$ are equal at large $n$.
.P
.\"
.H 3 "$bold sigma ,~ bold S sub C$."
We observe that
Commercial land occurs as a cluster of
lattice sites, 
i.e., as a development site (a P-D site).\*(sI
The origin of clustering is the
cost for initiating a Commercial region (Sec. 2.3.2, Critical Size).
Consequently $sigma$ must be significantly less than unity.
.P
The dimensions of the lattice 
we use to model the system
must be physically plausible and sufficiently fine.
We set the 
distance between lattice points 
at 200 ft.,
corresponding to an
area per lattice site in two dimensions of one acre.\*(Rf
.\"
.\"--------------start reference
.RS tD
We move freely and without apology between 2-D and 1-D pictures.
Most readers should be familiar with 2-D representations of a city
(known to some as maps). 
We use this familiarity to explain aspects of the 1-D model.
If one requires rigor,
a 2-D lattice can be mapped onto a 1-D chain, e.g.,
by projection.
.RF
.\"--------------end reference
.\"
We estimate the average size of a P-D site as 10 acres,\*(Rf
.\"
.\"--------------start reference
.RS sC
Based only on personal observation and anecdotal information.
However, 
a crude estimate (order of magnitude) is sufficient for our intent,
to show the general features of the problem.
.RF
.\"--------------end reference
.\"
which
corresponds to a cluster of size $S sub C ~=~$ 10 lattice sites.
In one dimension, $sigma~approx~(S sub c ) sup -2$, and we obtain 
$sigma~approx~0.01$.
This estimate for $sigma$ validates the assertion made above
that $s ~approx~1$,
which was conditional upon a small value for $sigma$.
.P
.\"
.H 3 "$bold n ,~ bold nu$."
The system size $n$ is arbitrary as long as it is large,
i.e., larger than a P-D cluster,
$n ~>~ S sub C ~approx~ 1/ sqrt sigma $ (Fig. 1).
We set $n~=~$ 640, corresponding in two dimensions 
to one section of land.
The total number of Commercial sites, $n sub C ~=~ 0.2 cdot 640 ~=~ 128$, 
corresponds to a border ca. 260 ft. wide along the section boundaries.
The number of P-D-site clusters in a system of size
$n~=~640$ is $nu ~=~ n sub c / S sub C ~=~ 12.8$.
Metropolitan Tucson is about 1000 sections in area.
The City can be treated as an ensemble of 1000 systems of the type described
here.
.SP 2
.\"
.H 1 "Results\ \ and\ \ Discussion"
We have 
obtained in Sec. 2.4,
a self-consistent and physically plausible set of parameter estimates
that describe the
Residential-Commercial equilibrium of a typical urban system: 
$s ~approx~ 1;~ sigma ~approx~ 0.01;~ n ~approx~ 640$.
.P
This set of values corresponds to a single point 
(the state point or system point)
located at $log sub 10 n ~approx~ 2.8$ and $log sub 10 s ~approx~ 0$
on the top graph of Fig. 1 ($sigma ~=~ 0.01$) 
or on the plots of Fig. 2 and 3 (calculated for $sigma ~=~ 0.01$).
.P
We now analyze this system within the framework of the Ising model.
.SP 2
.\"
.H 2 "Equal cost of Residential and Commercial property; \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \  The Residential-Commercial Equilibrium.
For the above set of parameter values,
the microscopic equilibrium constant, $~s~$, for conversion of an interior
Residential-state site to an interior Commercial-state site, is close to unity.
Since the system size $n$ is large, 
a near-unit value for the equilibrium constant implies that
the system-wide average price for an acre of developed commercial property 
is closely equal to that for developed residential property.
.P
This result, equality of price for Residential and Commercial, is expected.
Market forces drive the system toward this equality:
if the equilibrium is displaced from unity and commercial land has
significantly greater value ($s~>~1$), 
then speculators will be rewarded 
for converting to commercial use parcels of residential land
adjacent to a commercial P-D site, 
until some factor dependent 
on the fraction of all sites that are Commercial sites,
such as an overabundance of commercial land driving down its value,
acts to restore the equilibrium value of $~s~approx~1$.
.P
From Eq. 1b, $~s~=~e sup {i sub CC}$.
The value of $s$ 
is determined by the values of
variables of the right-hand-side
of Eq.\ 1a for $~i sub CC$, which are:
the interaction of a Commercial site 
with its Commercial neighbors 
($delta "\(co" sub CC$)
compared with
interaction of a Residential site with its Residential neighbors
($delta "\(co" sub RR$), 
and the profit per Commercial site ($ "\(co" P$).
The profit per Commercial site
is likely to fluctuate and 
to respond quickly and strongly to
changes in conditions, 
accounting both for any transient displacement of the
system from equilibrium and 
also for subsequent restoration of the equilibrium.
.SP 2
.\"
.H 2 "Importance of the Initiation Parameter, $sigma$.
Eq. 2 states that the initiation parameter $sigma$ is determined by the 
cost of creating a Residential-Commercial interface.
Because this cost can be large, $sigma$ can be small,
as it is for the set of parameter values estimated above 
($sigma ~approx~ 0.01$).
.P
A value of $sigma < 1$ is at the heart of understanding the 
Residential-Commercial system.
As noted above,
the value of $s$ controls 
one important aspect of
the Residential-Commercial equilibrium,
the fraction of sites that are Residential or Commercial.
The value of $sigma$,
however,
controls perhaps more interesting 
and certainly more fundamental characteristics:
how the system is structured (clusters);
how it varies with change in $s$ (explosive growth);
the nature of catastrophes.
.P
Unlike $s$, $sigma$ should vary slowly, with a characteristic time
of several election cycles.
.\"
.H 3 "Clusters."
As noted in section 2.3.2, Critical Size, 
if there is a non-zero cost of initiation, 
associated with $sigma < 1$, 
then there must be contiguous Commercial sites sufficient in number 
$n ~>=~ n sub crit$
to collectively balance the initiation cost through the 
$n$ small favorable contributions
from transfer of sites from a Residential to a Commercial region.
Tucson like most cities in the West of this country,
shows local (on the section scale) segregation of 
Residential and Commercial sites, i.e.,
sites of each type are clustered,
consistent with the estimated value of $sigma ~approx~ 0.01$.
Not all areas of Metropolitan Tucson fit this picture
of relatively small clusters of Residential and Commercial.
Some areas are homogeneous
on the section scale 
(clusters of 640 sites or more): 
industrial areas; suburban residential areas, such as
the Foothills, without Commercial over several-mile distances;
undeveloped areas.
.P
Clustering is not a necessary characteristic of the structure of a city.
Mature, densely populated cities, many on the East Coast and some
on the West Coast, 
have large areas of contiguous sections with no clustering,
where Residential and Commercial interleave both horizontally and vertically.
Outside the city center, however, even in such cities,
clustering is found.
.P
.\"
.H 3 "Explosive growth."
Another consequence of their being an initiation cost for a Commercial region
is explosive growth of commercial regions in response to only a small
increase in the equilibrium constant
(the sharpness of the $theta$-$log s$ response).
Fig. 1 shows 
that a change in $s$ from 0.8 to 1.3
switches the character of
the system from 90-10 Residential-Commercial
to its opposite, 90-10 Commercial-Residential.
Only a small change in the value of Commercial relative to Residential
property has a disproportionate, nonlinear effect on the fraction of 
Commercial sites.
.P
The sharpness of the $theta$-$log s$ response increases
with smaller $sigma$ (greater cost of initiation),
the sharpness becoming infinite as $sigma$ approaches zero.
.P
Sharpness as a special characteristic vanishes
for $sigma ~=~ 1$, where sites do not cluster but interleave
freely.
For a $sigma ~=~ 1$ system, 
explosive growth, dependent upon a small value of $sigma$,
is not possible.
.P
.SP 2
.\"
.H 2 "Catastrophes."
We consider two possible catastrophes for a Residential-Commercial system:\*(Rf
.\"
.\"--------------start reference
.RS cT
There are other possible catastrophes.
Some would consider it a catastrophe were a Wal*Mart 
to be introduced into a Residential-Commercial system.
.RF
.\"--------------end reference
.\"
Greening of the City;
and Malling of it.
Both would be a consequence of the 
the sharpness of the transition, i.e.,
the sensitivity of the fraction Commercial, $theta$, 
to change in the relative value of Commercial property, $s$.
If the value of Commercial property were to be driven 
to 1/10 the value of Residential property
($log sub 10 s ~approx~ -1$), there would be essentially no Commercial remaining
in the system (Fig. 1), i.e., the City would have been Greened.
For a 10-fold greater value of Commercial compared to Residential, 
the result is the opposite,
and the City would have been Malled.\*(Rf
.\"
.\"--------------start reference
.RS nS
This is the night-sweat fear of the above-noted
neo-Malthusian enviro-futurist Green.
.RF
.\"--------------end reference
.\"
Once forced to an extreme, 
where there would be essentially either no Commercial or no Residential
(no jobs or no homes),
it is doubtful recovery could be fast.\*(Rf
.\"
.\"--------------start reference
.RS bB
Thus our citizenry would face 
either 
a bucolic life barren of civilizing contact
or 
the bitter life of the Commercial wasteland.
.RF
.\"--------------end reference
.\"
.P
Like explosive growth, 
catastrophes are a consequence of a 
large initiation cost 
leading to a small value of $sigma$ and a sharp transition.
Without a sharp transition,
an order-of-magnitude deviation of 
$s$ from its equilibrium value 
would not drive the system to an essentially pure Residential or pure Commercial state:
compare Fig.1 for $sigma ~=~ .01$ and $1$.
.P
What might be the origin of a large deviation in $s$?
Most likely are external events driving the
system out of equilibrium.\*(Rf
.\"
.\"--------------start reference
.RS wC
Consider the
effect of a massive inflow of wealthy Californians,
seeking refuge from the debacle impending for that state,
who could drive the price
of Residential property in Tucson higher than whence they came.
.RF
.\"--------------end reference
.\"
\*(Rf
.\"
.\"--------------start reference
.RS nC
Even without the Californians,
it is possible that Tucson politicians might run amok California-style and
impose surreal disadvantages upon Commercial land.
.RF
.\"--------------end reference
.\"
Alternatively,
random internal forces can drive the system.
Fluctuations in the relative price of Commercial land are to be
expected.  
Over a sufficiently long period of time,\*(Rf
.\"
.\"--------------start reference
.RS lT
Possibly a very, very, very long time.
.RF
.\"--------------end reference
.\"
the system will undergo
fluctuations much larger than the typical (RMS) deviation 
and so large as to be catastrophic.  
.SP 2
.\"
.H 2 "Optimization and the New Urbanism."
Can we optimize a Residential-Commercial equilibrium system?
It should not be difficult.
Three variable parameters determine the state of the system:
$s$, $sigma$, and $n$.
The equilibrium constant 
$s$ is restricted to be near unity by market forces (Sec. 3.1).
Since the system size $n$ is large relative to $n sub crit$ (Sec. 2.4.4),
the transition is insensitive to change in $n$ (Sec. 2.3.1).
Thus only $sigma$ can be adjusted.
.P
Since only $sigma$ can be adjusted, 
by optimization we mean 
minimization of the cost 
of creation of a Residential-Commercial interface, the initiation cost.
Since the cost owes to unfavorable interaction
between developer and neighborhoods adjacent to a Commercial site,
it should be subject to reduction through eliminating this conflict.
Reduction of the initiation cost to zero results in $sigma ~=~ 1$,
in which case 
there would be free interleaving of Residential and Commercial, 
without the clustering found for values of $sigma ~<~ 1$,
and with 
developer and neighbor as happy living together as separately.
.P
One can speculate that
as a city matures and develops large areas of high density Residential and
Commercial,
the natural evolution is toward the $sigma ~=~ 1$ state.
As one would expect for a mature city,
the $sigma ~=~ 1$ state does not show explosive growth (Sec. 3.2).
.P
Systems in this state ($sigma = 1$) with mixed Residential-Commercial
and no clustering
are found (Sec. 3.2.1), 
although not typically in the inland West of this country,
even where as in Tucson allowed by zoning.
.P
The paradigm of the New Urbanism corresponds to
the $sigma ~=~ 1$ state 
(mixed Residential-Commercial; no clustering).
We may conclude that this paradigm is inapplicable for Tucson,
at least for the foreseeable future 
and likely not until our City has evolved for several more generations
and the immense area of low-cost 
surrounding land is no longer available.\*(Rf
.\"
.\"--------------start reference
.RS nU
Designer fads, whether urban planning or clothes fashion,
typically are generated on one of our Coasts, or with even less 
appropriateness, in Europe.
The New Urbanism, like fashionable clothing, does not mix well with
one-ton pickups.
.RF
.\"--------------end reference
.\"
.SP 2
.\"
.H 2 "Phase Transitions."
Among the characteristics of phase transitions are:
(a) emergence of a new structure, 
with the measure of its extent being an order parameter,
here $theta$, the fraction of the system in the Commercial state;
(b) nonlinear response to change in a system parameter, 
reflected in explosive growth of the new structure 
within the transition region;
(c) cooperative interaction between elements, 
resulting in the above explosive growth, 
and also in clustering of elements of the same state
and segregation of the states into separate macroscopic phases;
(d) a correlation length, a characteristic size for clusters,
that varies strongly within the transition region;
(e) universality reflected in power laws,
for example, 
exponential dependence 
of the correlation length or of the order parameter on a system variable.\*(Rf
.\"
.\"--------------start reference
.RS oD
There are difficulties associated with a 1-D model: 
no true phase transition (segregation of phases) except at infinitely small
$sigma$;
no singularities in the power laws, e.g., 
no infinite correlation length at the critical point.
.RF
.\"--------------end reference
.\"
.P
The last of the above list, universality, explains why a phase-transition
model developed for inanimate ferromagnetic material can help us understand
the city, a product of human enterprise and 
of far greater complexity than a magnet.
The interacting elements
(humans, amino acids, molecules, atoms, spins)
may differ greatly in their nature,
but whatever their nature, 
interactions lead to clusters,
and associated with clusters,
there is a correlation length.
The basis of universality is the central importance of the 
correlation length, 
which for a system near a phase transition is the
only relevant measure of length.
Properties of a 
system undergoing a phase transition scale on the
correlation length.
As the view of the system is progressively coarse-grained,
more long-range,
the specific nature of the substances comprising the system 
no longer enters into the analysis.
Thus one expects the formalism of phase transitions to apply 
equally for cities and ferromagnets.\*(Rf
.\"
.\"--------------start reference
.RS uN
This paragraph is an attempt to discuss universality and scaling
with the limited intent of explaining
how phase transitions for different
substrates can be fundamentally similar,
and without any intent to consider
singularities at the critical point,
critical exponents, dimensionality, and other aspects of critical
phenomena and their analysis.
For a good treatment of phase transitions, see the text by Kubo,\*(kU
or the excellent treatment in the text by Huang.\*(hU
.RF
.\"--------------end reference
.\"
.\"
.\"--------------start reference within reference
.nr ref*nr +1
.RS hU
K, Huang, \fIStatistical Mechanics\fP,
2nd Edition, John Wiley, New York, 1987.
.RF
.\"--------------end reference
.\"
\*(Rf
.\"
.\"--------------start reference
.RS nD
Indeed, phase transitions can be thought of as endemic
in the natural and social sciences.
Wherever there is explosive growth and emergent structure, 
think, Phase Transition!
Such models have been applied widely:
ferromagnetism, 
protein folding, 
carriage of oxygen by hemoglobin,
membrane phenomena,
epidemics,
gelation,
liquid helium,
formation of galaxies,
etc.
.RF
.\"--------------end reference
.\"
\*(Rf
.\"
.\"--------------start reference
.RS bE
One might have difficulty persuading the officials of a large city to allow
a series of experiments in which system variables ($s$ and $sigma$)
are controlled over a sufficiently wide range of values 
to explore the transition region
and determine power-law behavior.
.RF
.\"--------------end reference
.\"
.P
The Residential-Commercial equilibrium displays self-organization:
(a) the state point characteristic of a city is firmly constrained
by market forces
to the center of the transition region ($s ~approx~ 1$);
(b) there is a
characteristic cluster size, determined by $sigma$, the initiation
cost.
.P
Complexity and complex systems
are faddishly discussed in science-related news.
Among the properties most closely associated with complexity are
explosive growth and nonlinearity, emergence, and self-organization.
It should be no surprise that 
the Residential-Commercial equilibrium system,
a product of human enterprise,
displays these diagnostic signs of complexity.
.SP 2
.\"
.EQ
define bkt '{size -2 beta}'
define bdol '{size -2 {beta sub "$"}}'
.EN
.H 2 "Temperature."
The two interaction parameters
of the Ising model as formulated by 
Zimm and Bragg\*(zI, 
$s$ and $sigma$, 
are defined by Eq. 1b and 2a,
which are exponentials in the energy (or for our purposes, the dollar value).
The exponents must be unitless.
.P
In the energy language,
the exponent is
of the form $i sub CC ~=~ - bkt E sub CC$,
where $E sub CC$ is the energy difference 
between the Commercial and Residential states and
the factor $bkt$ scales the energy according to its units.
$beta$ is a reciprocal temperature, proportional to $1/T$, where 
$T$ is the absolute temperature of the system.
.P
As implied in Sec. 2.1.2 (footnote 22),
there must be a similar scaling factor for the dollar value,
which we can call $bdol$.
.ig
The need to evaluate $bdol$
in describing the parameters 
was avoided by evaluating $s$ and $sigma$ directly
in Sec. 2.4, 
from a determination of the Residential-Commercial equilibrium constant and 
an estimate of the extent of clustering of Commercial sites.
..
We can evaluate $bdol$ by use of Eq. 2c.\*(Rf
.\"
.\"--------------start reference
.RS bD
From Eq. 2c,
for $| i sub RC |~>>~| i sub CC |$,
.in +.5i
$bdol ~=~ - log sigma / (cost~of~creation~of~R-C~interface)$,
.in
where the cost is in unscaled dollars.
We make the following estimates:
$log sigma ~approx~ -2$ (Sec. 2.4.3);
an interface cost equal to 10 percent of the value of
developed residential property in the interface;
an average over Metropolitan Tucson
of $ "\(co" $400,000 per developed residential acre 
(e.g., 4 residences/acre on 1/4-acre lots);
two Residential sites in the interface
of a Commercial site within a Residential region.
Thus
.in +.5i
$bdol ~approx~ - (-2) / ( 2 * 0.1 * "\(co"400,000) ~=~ 1/"\(co"40,000$.
.in
.RF
.\"--------------end reference
.\"
.P
By analogy with $bkt$,
it is plausible that 
$bdol$ is proportional to
$1/ T sub size -2 "\(co"$,
where $T sub size -2 "\(co"$ is 
a societal temperature
that has a role comparable
to the absolute temperature
in energy language.\*(Rf
.\"
.\"--------------start reference
.RS rT
$bdol$ determines what a dollar buys for you in the way of
developed Residential or Commercial property.
For example, 
because Tucson clearly has a higher 
societal temperature $T sub size -2 "\(co"$ than Benson,
one buys better, and so had better buy, in Benson. 
.RF
.\"--------------end reference
.\"
.SP 2
.\"
.H 2 "Concluding Summary."
We have shown that an analogy with phase transitions leads to 
understanding of the basic principles of the 
Residential-Commercial equilibrium,
and thus of its equivalent,
the D-N problem.
.P
We have answered
the four questions raised in Sec. 1, Introduction:
Fig. 1 shows 
the dependence of the fraction Commercial sites, $theta$, on system
parameters;
the clustering of sites has its origin in the initiation cost, $sigma$;
catastrophes would be associated with a large fluctuation in 
the 
microscopic
Residential-Commercial
equilibrium constant, $s$;
optimization of an immature Residential-Commercial system 
is equivalent to imposing the New Urbanism model.
.P
Can the D-N problem be solved?
Yes, but only for mature cities that have evolved to fit the
New Urbanism paradigm.
Those who live in the immature West are condemned to live also
with the D-N conflict.  But more cheaply.\*(Rf
.\"
.\"--------------start reference
.RS nP
In the New Urbanism model, $sigma ~=~ 1$ and there is no cost of
mitigating or otherwise overcoming developer-neighbor conflict.
One might believe that land, both Commercial and Residential, 
would be relatively less costly in a city that fit the New Urbanism model.
This naive expectation is of course not the case.
Because the New Urbanism model requires a mature city,
the multitude of
bureaucrats, lawyers, politicians, agents, and others of similar ilk
have available the refinements made by previous generations in the methods for
extracting ever greater blocks of money from both developer and residents.
No moral here, just depressing fact.
.RF
.\"--------------end reference
.\"
.P
We suggest several directions for future research. 
(a) \fIEvolution of a Residential-Commercial system with time\fP.
We have implied this evolution, in our discussions of fluctuations in $s$
and of the maturing of a city.
A rigorous approach would require construction of a time-dependent
probability density function.
(b)
\fIThe entropy of a Residential-Commercial system\fP.
Because simplification of a high-dimensional system by averaging, or projection,
introduces entropy,
there must be an entropic component for the energy (dollar value) terms of
Eq. 1 and 2.
(c)
\fINovel aproaches to
resolution of the D-N problem\fP,
e.g., through political and social realignment.\*(Rf
.\"
.\"--------------start reference
.RS nI
.\"
Both developer and neighbors suffer costs associated with participation of
the zoning bureaucracy and politicians in the D-N problem.
These costs are a transfer of funds from citizens (developers and neighbors)
to politicians, lawyers, bureaucrats, etc., over and above whatever costs are
required for actual mitigation at the development site.
We suggest seeking Foundation grants to explore the 
formation of a non-profit Institute,
created and administered jointly by developers and neighbors,
with the mission of compulsory mediation of D-N conflicts.
Participation of an entity of this kind could
reduce or eliminate the current substantial overhead costs.
Political action under the auspices of the Institute could 
eliminate planning departments (they do no planning anyway),
reduce graft, bribery and other aspects of politician maintenance,
shrink the size of local government,
reduce the number of lawyers,
foster a happy citizenry,
etc.
Developers would support such a plan 
\(em
they will vote for a yellow dog if it would improve the bottom line.
Neighbors will vote however they are told \(em 
but in such a way as never to agree.
.RF
.\"--------------end reference
.\"
.SK
.nr Pt 0	\" to get hanging indent on 1st reference
.		\" leave Pt=0 for legends also
.RP 1 3
.SK
.\"
.H 1 "Figures\ \ and\ \ Legends
.SK
.\".PI -C /usr/people/jar/words/demo_hc_tran/com2-01.ps "10.0i","","","",""
.\".EP
.pl 100i
\fIFigure 1\fP.
Fraction Commercial-sites, $theta$, over $log sub 10 n \(en log sub 10 s$ 
plane: 
top,
for 
$sigma ~=~ 0.01$;
middle,
for 
$sigma ~=~ 0.1$;
bottom,
for 
$sigma ~=~ 1$.
$n~$ is the system size (total number of sites, $~n~=~ n sub C ~+~ n sub R$) ;
$~s~$ is the microscopic Residential\(enCommercial equilibrium constant, with 
$log s ~=~ i sub CC$.
Values of $theta$ were calculated by use of Eq. 7b.
.PSPIC "/mntA/usr1/jar/words/demo_hc_tran/com2-01.ps" "5.3" "7.0"
.SK
.\".PI /usr/people/jar/words/demo_hc_tran/com1c-01.ps "10.0i","","","",""
.\".EP
\fIFigure 2\fP.
$log sub 10 nu$ over $log sub 10 n \(en log sub 10 s$ plane, for
$sigma ~=~ 0.01$.
$nu$ is the number of Commercial regions, i.e., clusters of Commercial sites,
in a system of size $n$.
Solid lines, contours for designated values of $log sub 10 nu$;
dashed lines, contours for $theta~=~0.1~and~0.9$, for lower and upper range,
respectively, of the Residential-Commercial transition region of Fig. 1 top graph (for $sigma ~=~
0.01$).
Values of $nu$ were calculated by use of Eq. 10.
.PSPIC /mntA/usr1/jar/words/demo_hc_tran/com1c-01.ps "5.3" "7.0" 
.SK
.\".PI /usr/people/jar/words/demo_hc_tran/com5-01.ps "10.0i","","","",""
.\".EP
\fIFigure 3\fP.
$log sub 10 S sub C$ over $log sub 10 n \(en log sub 10 s$ plane, for
$sigma ~=~ 0.01$.
$S sub C$ is the average size of a Commercial region.
Solid lines, contours for designated values of $log sub 10 S sub C$;
dashed lines, contours for $theta~=~0.1~and~0.9$ 
(bounds of the Residential-Commercial transition region for $sigma ~=~ 0.01$).
Values of $S sub C$ were calculated by use of Eq. 10 and 12a.
.PSPIC /mntA/usr1/jar/words/demo_hc_tran/com5-01.ps "5.6i" "7.4i"
