Emergence and Collapse of Quantum Mechanical Superposition: Orthogonality of Reversible Dynamics and Irreversible Diffusion

Based on the modelling of quantum systems with the aid of (classical) non-equilibrium thermodynamics, both the emergence and the collapse of the superposition principle are understood within one and the same framework. Both are shown to depend in crucial ways on whether or not an average orthogonality is maintained between reversible Schrödinger dynamics and irreversible processes of diffusion. Moreover, said orthogonality is already in full operation when dealing with a single free Gaussian wave packet.
In an application, the quantum mechanical “decay of the wave packet” is shown to simply result from sub-quantum diffusion with a specific diffusivity varying in time due to a particle’s changing thermal environment. The exact quantum mechanical trajectory distributions and the velocity field of the Gaussian wave packet, as well as Born’s rule, are thus all derived solely from classical physics.

Dispersion of a free Gaussian wave packet: Considering the particles of a source as oscillating “bouncers”, they can be shown to “heat up” their (generally non-local) environment in such a way that the particles leaving the source (and thus becoming “walkers”) are guided through the thus created thermal “landscape”. In the Figures, the classically simulated evolution of exemplary averaged trajectories is shown (i.e., averaged over many single trajectories of Brownian-type motions). These trajectories are thus no “real” trajectories, but they only represent the averaged behaviour of a statistical ensemble. The results are in full agreement with quantum theory, and in particular with Bohmian trajectories.
This is so despite the fact that no quantum mechanics is used in the calculations (i.e., neither a quantum mechanical wave function, nor a guiding wave equation, nor a quantum potential), but purely classical physics.

Simulation with coupled map lattices of classical diffusion and a time-dependent diffusivity. Two examples are shown, with different halfwidths of the initial Gaussian distribution, respectively: (1+1)-dimensional space-time diagrams (time axis from bottom to top), with the intensity field and nine exemplary averaged trajectories. In a, the initial half width is twice as large as in b. Note that the narrower the Gaussian distribution is concentrated initially around the central position, the more the thus “stored” heat energy tends to push trajectories apart.

Gerhard Groessing, Siegfried Fussy, Johannes Mesa Pascasio, Herbert Schwabl
Emergence and Collapse of Quantum Mechanical Superposition: Orthogonality of Reversible Dynamics and Irreversible Diffusion
Physica A 389, 21 (2010) 4473-4484