The diffusion equation is one of the three great partial differential equations of classical physics. It describes the flow or diffusion of heat in the presence of temperature gradients, fluid flow in porous media in the presence of pressure gradients, and the diffusion of molecules in the presence of chemical gradients. [The other two equations are the wave equation, which describes the propagation of electromagnetic waves (including light), acoustic (sound) waves, and elastic (seismic) waves radiated from earthquakes; and LaPlace’s equation, which describes the behavior of electric, gravitational, and fluid potentials, all part of potential field theory. The diffusion equation reduces to LaPlace’s equation at steady state, when the field of interest does not depend on t. Poisson’s equation is LaPlace’s equation with a source term.]
Joseph Fourier developed the diffusion equation for heat conduction in 1807, and it has significant associations with probability theory (Narasimhan, 2009), as we will see shortly. In a novel and fascinating application, Gene Humphreys has employed solutions of the diffusion equation to describe the density of desert tortoises in the presence of population gradients caused by new dirt roads cut in the Mojave Desert. These new dirt roads induce an immediate line sink for unsuspecting tortoises. As of this writing in early September, I am not sure whether Gene has published this work.
Most of us here know that the diffusion equation has also been used to describe the evolution through time of scarp-like landforms, including fault scarps, shoreline scarps, or a set of marine terraces. The methods, models, and data employed in such studies have been described in the literature many times over the past 25 years. For most situations, everything you will ever need (or want) to know can be found in Hanks et al. (1984) and Hanks (2000), the latter being a review of numerous studies of the 1980s and 1990s and a summary of available estimates of the mass diffusivity κ. The geometric parameterization of scarp-like landforms is shown in Figure 1.
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