On The Acausality of Heat Propagation

In many physics and chemistry courses, one is typically taught that heat propagates according to the heat equation, which is a parabolic partial differential equation:

$\boxed{u_t = \alpha u_{xx}}$,

where $\alpha$ is the thermal diffusivity and is material dependent. Note also, we are considering the one-dimensional case for simplicity.

Now, let $f(x-at)$ be a solution to this problem, which represents a wave travelling at speed $a$. We get that

$\boxed{-a f' = \alpha f''}$.

This implies that

$\boxed{u(x,t) = -\frac{\alpha c_1}{a} \exp\left[-\frac{a (x-at)}{\alpha}\right] + c_{2}}$,

where $c_{1}, c_{2}$ are constants determined by appropriate boundary conditions. We can see that as $a \to \infty$, $u(x,t) < \infty$! That is, that even under an infinite propagation speed (greater than the speed of light), the solution to the heat equation remains bounded. PDE folks will also say that solutions to the heat equation have characteristics that propagate at an infinite speed. Thus, the heat equation is fundamentally acausal, indeed, all such distribution propagations from Brownian motions to simple diffusions are fundamentally acausal, and violate relativity theory.

Some efforts have been made, and it is still an active area of mathematical physics research to form a relativistic heat conduction theory, see here, for more information.

What we really need are hyperbolic partial differential equations to maintain causality. That is why, Einstein’s field equations, Maxwell’s equations, and the Schrodinger equation are hyperbolic partial differential equations, to maintain causality. This can be seen by considering an analogous methodology to the wave equation in 1-D:

$\boxed{u_{tt} = c^2 u_{xx}}$.

Now, consider a travelling wave solution as before $f(x-at)$. Substituting this into this wave equation, we obtain that

$\boxed{a^2 f'' = c^2 f'' \Rightarrow a^2 = c^2 \Rightarrow a = \pm c}$.

That is, all solutions to the wave equation travel at the speed of light, i.e., $a = c$! Therefore, wave equations are fundamentally causal, and all dynamical laws of nature, must be given in terms of hyperbolic partial differential equations, as to be consistent with Relativity theory.