Abstract
In elementary statistical mechanics, the entropy of a physical system of particles can be determined statistically by counting the number of possible microstates which correspond to a given macrostate. Although all of the early microstate descriptions were based upon Boltzmann statistics, the advance of quantum mechanics focused attention on three other statistical forms, namely Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions. Each of these forms corresponds to a particular assumption about the manner in which microstates are grouped into macrostates.
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Notes for Chapter 3
See, for example, Fisk and Brown (1975 a), Dacey and Norcliffe (1976), Lesse et al. (1978), or Roy and Lesse (1981).
In this instance, each macrostate corresponds to an interregional commodity distribution.
These particles are known as bosons or fermions, respectively.
In physical systems, these groups correspond to a set of energy states.
See Equation (2.1).
Achieved for example, by tossing a coin.
Box 1 is a fixed reference point.
See Fisk and Brown (1975 a).
In which there is no specific reference point.
See Jaynes (1957, p 620).
Shannon’s measure is given originally in Equation (2.15). For only two events (say E and not E), it can be expressed as which resembles the familiar Fermi-Dirac statistical form with i = 1. Differentiating U with respect to the probability p1, we get which is the inverse of the logrt function. Theil (1972) concludes that the log it model measures the sensitivity of uncertainty (or entropy) to variations in probabilities.
Spatial entropy is defined by Equation (2.22), which is the form proposed by Batty (1974 a, b; 1978 a). A similar derivation based on Equation (3.5) can be found in Snickars and Weibull (1977), but their interpretation of the ps distribution differs. They describe s as the a priori most probable distribution, which should be of the same order as the a posteriori prs distribution. Equation (3.13) is actually weighted according to the proportion of destinations in each region.
See, for example, Wilson (1973) or Fisk and Brown (1975 b).
See Fisk and Brown (1975 a).
See Equation (2.17).
See Jaynes (1968).
This procedure is currently under investigation at the Technical University of Delft.
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© 1983 Springer Science+Business Media New York
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Batten, D.F. (1983). Probability Distributions for Commodity Flows. In: Spatial Analysis of Interacting Economies. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3040-2_3
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DOI: https://doi.org/10.1007/978-94-017-3040-2_3
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