Chapter 14
## Time, Probability and ThermodynamicsA human being is part of a whole, called by us the Universe, a part limited in time and space. He experiences himself, his thoughts and feelings, as something separated from the rest--a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest us. Our task must be to free ourselves from this prison by widening our circles of compassion to embrace all living creatures and the whole of nature in its beauty. NOTE: The mathematics in this chapter may be difficult for some readers. The chapter is not essential to the general theme of the book. It is tangential. So feel free to skip it if it doesn't interest you.In 1977 I gave a talk about my research on size-density to an audience primarily composed of members of the science departments at Western Washington University. After the talk the Chairman of the Physics Department, Louis Barrett, walked with me in the direction of our offices.
My remarks about the forces shaping population distributions had
prompted him to think of analogies with the distributions of physical
particles. By the time we reached his office we were joking about
whether humans were "fermions" or "bosons" (explanation follows). Before
I left, he had me taking the joke seriously and, when I said I wanted to
pursue it further, he gave me a textbook
All physical particles are governed by one or the other of two
statistical distributions, the Fermi-Dirac distribution or the
Bose-Einstein.
where n _{i} is the number of particles
whose energy is ε_{i} and c_{i} is the
number of states that have the same energy ε_{i}. Under the
exclusion principle the "occupation index", n_{i}/c_{i}, cannot be
greater than 1.
At low temperatures virtually all the lower energy states are filled, and the two distributions differ from one another. But at higher temperatures the occupation index is sufficiently small at all energies for the effect of the exclusion principle to be unimportant. They each become similar to a distribution governing particles which, like gas molecules, are sufficiently widely separated to be distinguishable from one another:
I began to think about energy in physical systems as somehow
equivalent to time for living systems. Both can be used to do
work, both can be spent, wasted, manifested in a variety of ways. In the
textbook Lou gave me is the sentence: "A system of particles is stable
when its total energy is a minimum."^{[2]}
Perhaps that was the trigger; I don't remember.At any rate, I substituted time for energy in the Maxwell-Boltzmann distribution (humans can be distinguished from one another, by other humans at least), skipping any other direct physical analogies in the textbook derivation. The beauty of the derivation, to me, is the bare minimum of assumptions.
We begin by specifying a random variable We assume that the most probable distribution will be the one which can occur in the greatest number of ways. Pursuant to this we must determine the number of ways in which a given distribution can occur.
If there are c There are N! permutations among N individuals. But the n _{i}! permutations within the i-th level will not
affect the overall distribution. Since there are n_{1}!n_{2}! ...
n_{k}! of these irrelevant permutations,
the number of relevant ones is
The number of ways in which N individuals can be distributed among the possible levels of T is therefore The logarithmic transformation of which is By Stirling's formula, when n is very large so and, since Σ n _{i} = N,
We want to find the condition under which W is a maximum, but we have an expression for ln W rather than W; this is not a problem since (ln W) _{max} = ln W_{max}.
If W = W Since N ln N is a constant, it follows that Since so that and since, with N fixed, Σ δn _{i} = 0, it follows that
Eq. 4 thus becomes While Eq. 5 must be fulfilled by the most probable distribution, it does not fully specify that distribution. There are two constraints to take into account, namely, that the number of individuals and the total amount of time available to them are given: It follows that the variations δn _{i} in any of the n_{i}'s cannot be independent of one another but must
obey the relationships:
To incorporate these two conditions on the δn _{i}'s in Eq. 5 we
make use of Lagrange's method of undetermined multipliers. We multiply
Eq. 6 by -α and Eq. 7 by -β, where α and β are independent of the n_{i}'s, and add these expression to Eq. 5. We obtain
For each of the equations which enter into the sum in Eq. 8, the variation δn _{i} is effectively an independent variable. For Eq.
8 to hold, then the quantity in parentheses must be 0 for each value of
i. As a result
from which we obtain With time as a continuous variable and n(t) dt as the number of individuals between t and t+dt, Eq. 10 becomes I wanted to see if this distribution function could be applied to some sociological phenomena.
In Chapter 9 I suggested that time minimization might lead to the
traditional concentric zone model for cities. In the next section I
review some work in this area
Using small-area, census-tract type units for cities in Europe, North
America and Australia, Clark
D to permitting estimation of b from linear regression of ln D _{x} on x. This results in a direct
estimate of b (the slope) and an indirect estimate of D_{o} (the antilog of the intercept).
Clark computed values for b and D
Both tendencies were viewed as typical of urban expansion. With these
slight variations, he concluded that the negative exponential
distribution appeared to hold "for all times and places studied, from
1801 to the present day, and from Los Angeles to Budapest".
Later empirical work, summarized by Berry, et al. One weakness with Clark's curve, recognized from the very beginning, is that the central density, however necessary in order to fix the height of the curve, is fictional: It is found only by extrapolating the regression line inward from the outer residential areas. Theoretically, Clark's curve suggests that the density at dead center should be infinity; that is, the distance x can be made so small that the density created by a single person would approach infinity. If one person occupied one square foot at dead center, the density there would be the number of square feet in a square mile, 27,878,400. Actual residential densities, of course, are never this high. As Burgess' model (Fig. 9-2) suggests, actual central densities tend rather to be quite low, much lower than any extrapolated value. As in Fig. 14-2, the curve usually rises from a very low central value, peaks some distance from the center, and only then begins the exponential decline described by Clark's curve.
The problem with the concentric zone curve in Fig. 14-2 is that it is
only qualitative. It lacks the precision of an equation which could be
tested using real data. Newling
so that it could rise before beginning its decline, but this really is an exercise in curve fitting: if you keep adding enough parameters you can fit anything to any equation. What we want is a curve which can be fit to real-world data and which also has theoretical rationale behind it.
The general theoretical result, Eq. 11, can be applied to the problem of determining the distribution of population as a function of distance from the center of a city. The obvious link between distance x and time t is the velocity from which it follows that Making these substitutions in Eq. 11 we have The number of cells in the sample space, at the distance x, should be a function of the circumference at that distance With this substitution, Eq. 14 becomes We can evaluate the constant e ^{ -α}.
Integrating Eq. 16
Since the definite integral Eq. 17 becomes, with a = β/v and n = 1 so Substituting this in Eq. 16, To evaluate β, we compute the time T. From Eq. 13 we obtain so we can re-write Eq. 20 as Total time T is Substituting from Eq. 21, By the definite integral used to derive Eq. 18, this becomes
Since T/N is the mean time, and since x = v/t, it follows that the mean distance x _{μ} = vT/N, so
Substituting this in Eq. 20, Dividing Eq. 26 by N produces the probability distribution where λ = 2/x _{μ}.
This is the gamma distribution with c = 2. The mode of a gamma distribution is (c-1)/λ, so the mode of Eq. 27 is 1/λ and, therefore, λ = 1/mode. The gamma distribution parameters λ and c can be estimated from the mean and variance of x, by the method of matching moments, with
Random numbers fitting this distribution may be generated from where R _{1} and R_{
2} are random numbers between zero and one.
Fig. 14-3 shows the distribution of 1,000 random distances created
using Eq. 30 and the curve generated using Eqs. 28 and 29,
mode = 1.94, s
All that is needed to extend this technique to other distributions is to define the two time-components in Eq. 11 with reference to some specific time-expenditure. In the example here, time was related to distance, and the number of cells having the same time was given by the circumference. In a sense it looks as though Clark's distribution results from failing to think in two dimenisional (area) terms: you obtain his equation by moving linearly outward from the center, ignoring circumference, so that c(t) = 1. Each time-distance can be occupied by only one individual (the exclusion principle?) If time expenditure involved an initial startup cost and then decreased in importance over long distances, you might be able to set t = bx -cx ^{ 2}, i.e.,
Newling's distribution.
Many kinds of distributions
So far in this chapter I have resisted the temptation to draw direct analogies between Physics and Sociology. Analogy can be an intellectual trap. We call the interactance hypothesis the "gravity model" because it looks like the formula for gravity, but we know that there are differences between migration and gravitational attraction. People may or may not share a time level (be subject to the exclusion principle), but that doesn't literally make them bosons or fermions; we don't investigate their wave functions or spin properties. The only analogy employed in obtaining Eq. 11 was that between energy and time. Beyond that we only assumed a fixed number of individuals with the given amount of time, and that the distribution which could occur in the greatest number of ways was the most probable, assumptions which need not be limited to physical particles. At this point I am going to develop an analogy, or at least the structure for it, solely because I find it intriguing. It should have application in Sociology; at least I can't see any reason for restricting it to physical particles. But I don't know how to make the application.
For convenience, and because I can't suggest any immediate application of
the energy-time equivalence, I'm going to develop this derivation using
I rearrange the terms and multiply both sides by n _{i}
Then take the sum of both sides and add Σ n _{i} = N = 0
Combining terms and applying Stirling's formula Continuing to combine terms Since Σ ln x = ln Π x In Physics the constant β is defined where T is temperature and k is "Boltzmann's contant" (1.380 x 10 ^{-23} J/molecule-degree). Dividing Eq. 36 by β therefore yields
With the following identities, defined in Physics, we arrive at the "fundamental equation" ^{[12]} of
thermodynamics. It says that the total energy in a system is
entropy-times-temperature minus pressure-times-volume plus the chemical
potential of the N particles in the system.
temperature
is just velocity of movement. I sense an analogy between
temperature in physical systems and the technology of
transportation and communication in social systems. Modern
societies are "hotter". Big cities (high interactance centers)
are "hot": People, commodities and ideas move around faster.
"Hot" regions subdivide territory more thoroughly than "cool"
ones.
And what is NOTES:
[1] Arthur Beiser,
[2]
[3] begun with an undergraduate, Monte Jarvis, in 1988. He went to graduate school (University of Pennsylvania); we never completed it.
[4] Colin Clark, "Urban population densities",
[5]
[6] Brian J. L. Berry, James W. Simmons and Robert J.
Tennant, Urban Population Densities: Structure and Change",
[7]
[8] Bruce Newling, "The Spatial Variation of Urban
Population Densities",
[9] Samuel M. Selby, ed,
[10] for this and other properties of the gamma
distribution, see N. A. J. Hastings and J. B. Peacock,
[11] for these and many other distributions, see
Hastings and Peacock,
[12] Edward A. Desloge, |