Abstract
The primary objectives of this chapter are twofold: first, to offer a review of progress in urban modelling using the methods of statistical mechanics; and second, to explore the possibility of using the thermodynamic analogy in addition to statistical mechanics. We can take stock of the “thermodynamics of the city” not in the sense of its physical states – interesting though that would be – but in terms of its daily functioning and its evolution over time. We will show that these methods of statistical mechanics and thermodynamics illustrate the contribution of urban modelling to complexity science and form the basis for understanding the evolution of urban structure.
It is becoming increasingly recognised that the mathematics underpinning thermodynamics and statistical mechanics have wide applicability. This is manifesting itself in two ways: broadening the range of systems for which these tools are relevant; and seeing that there are new mathematical insights that derive from this branch of Physics. Examples of these broader approaches are provided by Beck and Schlagel (1993) and Ruelle (1978, 2004). The recognition of the power of the method and its wider application goes back at least to the 1950s (Jaynes, 1957, for example) but understanding its role in complexity science is much more recent. However, these methods are now being seen as offering a major contribution. In general, the applications have mainly been in fields closely related to the physical sciences. The purpose of this chapter is to demonstrate the relevance of the methods in a field that has had less publicity but which is obviously important: the development of mathematical models of cities. The urban modelling field can be seen, in its early manifestation, as a precursor of complexity science; and, increasingly, as an important application within it (Wilson 2000).
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Notes
- 1.
The detailed justification for this is well known and not presented here.
- 2.
There are many possible definitions of entropy that can be used here, but for present purposes, they can all be considered to be equivalent.
- 3.
For simplicity, we will henceforth drop the quotation marks and let them be understood when concepts are being used through analogies.
- 4.
This model, in more detailed form, has been widely and successfully applied.
- 5.
We should explore whether we can determine a measure of A from the topology of the {c ij }.
- 6.
Note that P appears to have the dimensions of “density” x’money’.
- 7.
Can we take A i B j as an i–j partition function? Can we work backwards and ask what we would like the free energy be for this system? If (2.11) specifies the energy and β (=1/kT) the temperature, then F = U – TS becomes F = C – S/kβ? Then if F = NkT log Z, what is Z?
- 8.
ter Haar (1995, p. 202) does show that each subsystem within an ensemble can itself be treated as an ensemble provided there is a common β value.
- 9.
The following equations can be derived from (2.31) with A substituted for V and T = 1/kβ.
- 10.
- 11.
It is possible to introduce a β i rather than a β which reinforces this idea.
- 12.
We elaborate the notion of phase changes in the next section. Essentially, in this case, they would be discrete “jumps” in the {T ij } or {S ij } arrays at critical values of parameters such as β.
- 13.
K could be j-dependent as K j (and indeed, usually would be) but we retain K for simplicity of illustration.
- 14.
Clarke and Wilson (1985).
- 15.
It would be interesting to calculate the derivatives of the free energy – the F-derivatives – to see whether there is a way of constructing N[W j > x] out of F. Are we looking at first or second order phase transitions?
- 16.
I am grateful to Aura Reggiani for this suggestion.
- 17.
It can be shown that we can carry out an entropy maximizing calculation on {S ij } simultaneously and that leads to a conventional a spatial interaction model and the same model for {W j }. The implication of this argument is that if we obtain a {W j } model with the method given here, we should then recalculate {S ij } from an spatial interaction model and then iterate with {W j }.
References
Ashby WR (1956) Design for a brain. Chapman and Hall, London
Beck C, Schlagal F (1993) Thermodynamics of chaotic systems. Cambridge University Press, Cambridge
Clarke M, Wilson AG (1985) The dynamics of urban spatial structure: the progress of a research programme. Trans Inst Br Geog 10: 427–451
Dearden J, Wilson AG (2008) An analysis system for exploring urban retail system transitions, Working Paper, 141 Centre for Advanced Spatial Analysis, University College London
Evans SP (1973) A relationship between the gravity model for trip distribution and the transportation model of linear programming, Transp Res 7:39–61
Finn CBP (1993) Thermal Physics. Nelson Thornes, Cheltenham
Friston K, Mattout J, Trujillo-Barreto N et al. (2006) Variational free energy and the Laplace approximation. Neuroimage 34:220–234
Friston K, Stephan KE (2007) Free energy and the brain. Synthèse 159:417–458
ter Haar D (1995) Elements of statistical mechanics (3rd ed.). Butterworth-Heinemann, London
Harris B, Wilson AG (1978) Equilibrium values and dynamics of attractiveness terms in production-constrained spatial-interaction models. Environ Plan A 10:371–88
Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620–630
Martin P (1991) Potts models and related problems in statistical mechanics. World Scientific, Singapore
Pippard AB (1957) The elements of classical thermodynamics. Cambridge University press, Cambridge
Ruelle D (1978/2004) Thermodynamic formalism, 2nd edn. Cambridge University Press, Cambridge
Wilson AG (1967) A statistical theory of spatial distribution models. Transp Res 1:253–69
Wilson AG (1970) Entropy in urban and regional modelling. Pion, London
Wilson AG (1976) Catastrophe theory and urban modelling: an application to modal choice, Environ Plan A 8:351–356
Wilson AG (2000) Complex spatial systems: the modelling foundations of urban and regional analysis. Prentice Hall, Harlow
Wilson AG (2008) Boltzmann, lotka and volterra and spatial structural evolution: an integrated methodology for some dynamical systems. J R Soc Interface 5:865–871
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Wilson, A. (2009). The “Thermodynamics” of the City. In: Reggiani, A., Nijkamp, P. (eds) Complexity and Spatial Networks. Advances in Spatial Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01554-0_2
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