Can we make the heat equation covariant ?

Question

Good morning! 

I was wondering if one can extend the heat equation in order to make it covariant. 

In other words, in there some kind of operator \(D_\mu\)and a vector field (or perhaps another representation of the Lorentz group) \(A^\mu\)such that one can derive the heat equation \((\partial_0 - \partial^i \partial_i)\phi = 0\) as a projection or a special case of \(D_\mu A^\mu = 0\)? 

All I can find has a special treatment for time that I want to avoid. 

Here are some random thoughts that did not actually help me : 

• Perhaps I should add a length-dimensional factor \(\alpha\) to \(\partial^i \partial_i\) which has a certain transformation behaviour ; 
• There is a problem with the number of Lorentz indices in the expression of the heat equation ; 
• I somehow expect the heat/energy to diffuse with a wave-like front replacing the instant transmission of heat of classical solutions ; 
• Heat is energy which is part of the 4-momentum : we should be able to find another 4-vector with heat as its timelike component ; 
• Using the substitution \(t\mapsto i\tau\), one can transform the heat equation into a Schrödinger equation ; the covariant heat equation thus might be found using a covariant extension of the Schrödinger equation, such as the Dirac equation (but I do not expect the Dirac equation to be the right one : what would spinors do here ?!) 

Comments

Using the substitution t↦iτ, one can transform the heat equation into a Schrödinger equation ; the covariant heat equation thus might be found using a covariant extension of the Schrödinger equation, such as the Dirac equation (but I do not expect the Dirac equation to be the right one : what would spinors do here ?!) 

Such a way of "obtaining" ("deriving") a Schrödinger equation is wrong. You should not rely on appearance because time stays time - it does not become "imaginary", but the equation variable $T(x,\tau)$ changes its meaning completely. And vice versa: such variable changes are not a right way of obtaining correct "heat equations".

Answer 1

There is a lot of literature on covariant versions of the heat equation, and more generally the Navier-Stokes equation. A convenient entry point for a search is the book  Rational extended thermodynamics by 
Müller and Ruggeri.