Often, the analogy between the Schrödinger equation and a diffusion equation is pointed out, because of the first order in the time derivative. A diffusion equation describes heat propagation as well. But the imaginary unit makes the Schrödinger equation of a different nature, more like the propagation of a wave. Now in a superfluid, under some circonstances, heat propagates very quickly, in a way that has the full features of a wave. That is called the second sound. By analogy, is it possible to model the motion of a quantum particle as a second sound in some superfluid medium?
The idea of the Schrödinger equation as a diffusion equation has been investigated by many authors: Nelson, Böhm etc. The connection with the Brownian motion and other Markov processes has been done. The Wick rotation is well known. There have also been many mechanical models of the Maxwell equations, including in a perfect fluid. But the Schrödinger equation remains a wave equation because the wave function is complex. There are destructive interferences which are not possible with a diffusion equation where the function is real. That's why a non-local potential must be introduced. Actually, the Schrödinger equation is the non relativistic approximation of the Klein-Gordon equation, which is second order in space and time dérivatives. So, all that is but mathematical curiosity, there is no physical content.
In contrast, the second sound in a superfluid is really physical, and it is the propagation mode of heat that is normally described by a diffusion equation. If I understand correctly, there is a superfluid and a normal phase, and there is collective (circular or alternating) motion.
Of course the many exotic properties of a superfluid such as ⁴He below 2°K are quantum phenomena. The superfluid is a Bose condensate that acts as if all of the atoms are in their quantum ground state with zero energy and zero entropy. One might think of a container of such superfluid as one giant molecule containing all of the atoms. Heat is transported through the superfluid by propagation of quantized excitations in the superfluid called phonons (also rotons above 0.5°K), just as in the case of such heat propagation through the lattice of an ideal crystal. The duality between an intuitive picture of the phonon "particle" as a "wave packet" of vibrations is analogous to that between the quantum particle we call a photon and a propagating electromagnetic wave packet, but neither are described by the Schrödinger equation, but rather by "second quantization". One big difference is that the phonons have a large probability to scatter from one another in the superfluid, with a mean free path of less than 0.1mm at temperatures above 1°K. The speed of"first sound" is just the ballistic velocity of a phonon in the superfluid. Perhaps ironically, the population of phonons and rotons behaves like an ideal gas at such temperatures, and can be treated classically, with "second sound" being familiar pressure sound waves in this gas, transporting heat. Only at temperatures below 0.4°K does the phonon mean free path become of order 10 cm, so that their flow becomes ballistic and thermal transport can be considered as diffusion. So I am unaware of the benefit of considering the Schrödinger equation in this context.
May I offer two points in this regard:
1) Apart from first sound, which is just ordinary sound, all other sound modes in quantum fluids -- zeroth sound (in Fermi fluids such as liquid 3He) as well as second, third, fourth and fifth sounds (are there any more?!) -- are deeply rooted in QM.
2) One line of thought worth exploring is the quantum-hydrodynamic formulation of the Schrodinger equation. According to this approach, (psi) is written as (amplitude) x , (exp[iphase]), which is often referred to as the Madelung transformation. Equating then real and imaginary parts on both sides of the equation yields a continuity-like equation as well as a Bernoulli-like equation. These are rich in content and may shed some light on the issue at hand.
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