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Citing from Wikipedia's article on relativistic heat conduction:

For most of the last century, it was recognized that Fourier equation (and its more general Fick's law of diffusion) is in contradiction with the theory of relativity, for at least one reason: it admits infinite speed of propagation of heat signals within the continuum field. [...] To overcome this contradiction, workers such as Cattaneo, Vernotte, Chester, and others proposed that Fourier equation should be upgraded from the parabolic to a hyperbolic form,

$$\frac{1}{C^2}\frac{\partial^2 \theta}{\partial t^2} +\frac{1}{\alpha}\frac{\partial \theta}{\partial t}=\nabla^2\theta$$ also known as the Telegrapher's equation. Interestingly, the form of this equation traces its origins to Maxwell’s equations of electrodynamics; hence, the wave nature of heat is implied.

It appears to me that the PDEs describing any other diffusion process –for instance, the Fokker–Planck equation for Brownian motion– will also assume an infinite speed of propagation. Then, if my intuition is correct, they'll be incompatible with SR, and will have to be "upgraded" to hyperbolic, wave-like equations.

If this were a general rule, would we have, for instance, a relativistic wave equation for Brownian motion? It appears unlikely... Is there, then, any example of diffusion-like/dispersive equation whose form "survives" into a relativity-compatible description?

Edit:

I'll add a broader reformulation of the question, as suggested by a @CuriousOne comment:

Can we find a first order equation that models the finite velocity limits or are we automatically being thrown back to second order equations? Is there a general mathematical theorem at play here about the solutions of first vs. second order equations?

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    $\begingroup$ Brownian motion typically arises from the scaling limit of a random walk, which involves 'kicks' from some ambient fluid. A relativistic version would involve transforming to the (local) rest frame of the fluid before integrating the trajectory further. As long as the velocity at each step of the original random walk is less than $c$, it will still behave like ordinary Brownian motion at large scales, with probability satisfying a local diffusion equation. The approximate diffusion process would predict some amount of superluminal transport, but the amount would be negligible. $\endgroup$ – TotallyRhombus May 15 '16 at 15:00
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This is a subtle and somewhat complicated question, but I think the basic answer is ``no''.

1) The relativistic Boltzmann equation is $$ p^\mu\partial_\mu f = C[f] $$ which has the same structure as the non-relativistic Boltzmann equation. This equation can be used to derive relativistic Fokker-Planck equations. One example is the Landau collision term, which describes the scattering of charged particles in a relativistic plasma. The resulting FP equation has the same structure as the non-relativistic FP equation, see, for example http://www.sciencedirect.com/science/article/pii/0378437180901570 .

2) Also note that the Cattaneo equation (and similar equations for other diffusive problems) are not ``fundamental'' equations. Take the equation of current conservation $$ \partial_0 n +\vec\nabla\cdot\vec\jmath = 0 . $$ Fick's law is that $\vec\jmath$ is instantaneously equal to the diffusive flux $-D\vec\nabla n$. This is incompatible with relativity. We can try to fix things by writing down a relaxation time model for the current, $$ \tau\partial_0 \vec\jmath = -(\vec\jmath+D\vec\nabla n) , $$ which gives the Cattaneo equation $$ \tau\partial_0^2 n + \partial_0 n - D\nabla^2 n = 0 \, . $$ But, in general there could be a much more complicated memory kernel $$ \vec\jmath (r,t) =\int dr' dt' \, G(r,t;r' ,t' )\nabla n(r' ,t' ) $$ and the relaxation time model is an approximation that follows from simple kinetic models in the limit $\partial_0n \ll n/\tau$.

3) Also note that the issue is not just related to relativistic invariance and causality. In a non-relativistic gas it is also impossible for the current to be instantaneously equal to the diffusive flux. Take an ultracold gas in which the atoms move at speeds $\sim cm/s$. Then any diffusive front that moves at $m/s$ (nowhere near the speed of light) is clearly unphysical, and the Cattaneo equation is more appropriate than Fick's law. What is happening here is that we took Fick's law, which is a long-wavelength (coarse grained) approximation, and pushed it to distances that are too short.

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    $\begingroup$ While I like your answer (and I agree that it's not a matter of relativity, to begin with), the Cattaneo equation is second order (am I wrong about that?), so in a broad sense the OP's question remains: can we find a first order equation that models the finite velocity limits or are we automatically being thrown back to second order equations? Is there a general mathematical theorem at play here about the solutions of first vs. second order equations? $\endgroup$ – CuriousOne May 15 '16 at 21:28
  • $\begingroup$ Thank you! That was an informative answer. @CuriousOne: I didn't remark there were issues even with non-relativistic gases, but I'd say there is still an incompatibility between relativity and diffusion-like equations (take Schrödinger equation, for instance). I suscribe the rest of your comment, though. $\endgroup$ – David Herrero Martí May 16 '16 at 13:21
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    $\begingroup$ @DavidHerreroMartí: Absolutely. I was merely trying to say that relativity is not the first thing that makes diffusion equations "unphysical", but I can't see any compatibility between first order equations and relativity, either. I do believe that there are some strong mathematical statements about this, but I don't know, anymore, where I saw them. $\endgroup$ – CuriousOne May 16 '16 at 16:52
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The root cause is ultimately because of the assumption or use of Fick's law, or Fourier's law which is not appropriate at relativistic scales. The hyperbolic heat equation is simply a neat fix to diffusion problems that satisfies relativity. By appropriately going into a relativistic Fourier's law of Fick's law, you can always derive the new "correct" PDE. Note that this doesn't necessarily yield the hyperbolic heat equation since it just simply accounts for diffusion and wave-propagation.

Depending on the physics of the particular property, we know that advection is another form of transport. Radiation (spontaneous medium-less propagation) is also possible. So in general the actual physically correct PDE could be just about anything.

Regarding a general theorem, I believe the classification of problems as parabolic/hyperbolic/elliptic/other already captures these effects.

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  • $\begingroup$ Why is Fick's law not valid in the relativistic domain? Standard relativistic hydro (as used in Astrophysics, Heavy Ion Physics, AdS/CFT) is based on Fick's and Fourier's law. $\endgroup$ – Thomas Oct 29 '16 at 1:21

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