A Spatial Interaction Model Based on Statistical Mechanics

Abstract

In this chapter, a new model of spatial interaction is proposed with a view of statistical mechanics. It argues that spatial interaction is microscopically independent of macroscopic phenomenon and that patterns of interaction based upon individual agents emerge in space through Brownian motion. Using principles of statistical mechanics, we solve this complex process problem. This contrasts with Wilson’s model in which spatial interaction exponentially decays in geographical space and the maximum entropy method is used to obtain the form of distance decay. In this paper, the empirical method is used to test the model. The model shows that the coefficient of the spatial interaction cannot be simply treated as a Lagrange parameter, which is related to both the average spatial scale of an agent particle migration and the likelihood of agent impulses. Finally, this model suggests that the free flow of population (or capital) enhances spatial interactions.

Appendix

For integration

$$ G(r)=\underset{0}{\overset{\infty}{\int}}\frac{k}{2\pi D\sqrt{t}}\exp \left[-\left(at+\frac{r^2}{2 Dt}\right)\right] dt $$

(8.17)

$$ x=\sqrt{at} $$

(8.18)

Let

$$ dx=\frac{\sqrt{a}}{2\sqrt{t}} dt $$

(8.19)

$$ G(r)=\underset{0}{\overset{\infty}{\int}}\frac{k}{\pi D\sqrt{a}}\exp \left[-\left({x}^2+\frac{r^2a}{2{Dx}^2}\right)\right] dx $$

(8.20)

$$ \underset{0}{\overset{\infty}{\int}}\exp \left[-\left({x}^2+\frac{b}{x^2}\right) dx\right]=\frac{\exp \left(-2b\right)}{2}\sqrt{\pi} $$

(8.21)

We have

The second integration is a general integration. Comparing (8.20) with (8.21), we could obtain

$$ b=r\sqrt{\frac{a}{2D}}\equiv \beta r $$

(8.22)

$$ G(r)=\frac{k}{2\sqrt{\pi {D}^2}}\exp \left(-\sqrt{\frac{2a}{D}}r\right) $$

(8.23)

$$ or\kern2.00em K=\frac{k}{\delta \sqrt{\pi {D}^2}} $$

(8.24)

So, we obtain Formulas (8.11) and (8.12).