Relation of Ficks Second Law to the Heat Equation

For evolution of a temperature field during heat flow, an equation with the same form as Eq. 4.2 arises  

Therefore, any result that follows from considerations of the form of Fick s second law applies to evolution of heat as well as concentration. However, the thermal and mass diffusion equations differ physically. The mass diffusion equation, dc/dt = V DVc, is a partial-differential equation for the density of an extensive quantity, and in the thermal case, dT/dt = V kVT is a partial-differential equation for an intensive quantity. The difference arises because for mass diffusion, the driving force is converted from a gradient in a potential V/u to a gradient in concentration Vc, which is easier to measure. For thermal diffusion, the time-dependent temperature arises because the enthalpy density is inferred from a temperature measurement. 

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