Two weeks ago I asked to Prof. Friedwardt Winterberg about his 1995 paper, where he derived quantum mechanics from Boltzmann equation (Z. Naturforsch. 50a). This paper is interesting because it seems to be more consistent compared to Nelson's derivation of QM. Furthermore Winterberg's paper starts with Planck aether assumption.
In physics, specifically non-equilibrium statistical mechanics, the Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behavior of a thermodynamic system not in thermodynamic equilibrium. It was devised by Ludwig Boltzmann in 1872. See http://en.wikipedia.org/wiki/Boltzmann_equation.
Therefore it appears very interesting to find theoretical connection between Boltzmann equation and quantum mechanics. Other paper which discusses link between Boltzmann equation and quantum relativistic kinetic theory, is Escobedo et al., EJDE, 2003. See: https://www.ceremade.dauphine.fr/~mischler/articles/18BQRelectronic.pdf
So do you agree that quantum mechanics can be derived from Boltzmann equation? Your comments are welcome.
I think there are two different issues addressed in your question, if I understand it correctly. The last part is if you can write a Boltzmann equation for quantum mechanical systems, about which there is an extensive literature. There is a paper by Morita with a derivation, Am. J. Phys., 7, (1966), 1039, who starts from the quantum Liouville equation.
The other issue is if you can derive quantum mechanics from classical statistical mechanics, I am not sure if this would not be considered by Planck himself as "mathematical jugglery" (he also viewed his own derivation of the energy levels of the quantum mechanical oscillator in that way). Without having studied this paper in depth, I can raise the objection Winterberg himself addresses about other derivations. The Boltzmann equation he uses (eq. 10) is not the full Boltzmann equation, but an approximation in which only two-particle collisions are taken into account, (dilute gas approximation), and more importantly, the two-particle distribution function in the original integral is approximated by a product of one-particle distribution functions, the so-called Stosszahlansatz. In classical systems this can be used to derive an H theorem, which identifies a function that always decreases in time, and consequently dissipative dynamics. So, if this derivation is correct, it should derive some form of a dissipative Schrodinger equation, which it apparently does not. As Winterberg himself points out, the Schrodinger equation is time reversal invariant. The Boltzmann equation he uses is not. Just as it is impossible to derive dissipative equations from time reversible ones (you need the Stosszahlansatz), it is also impossible to derive reversible equations from irreversible ones (apart from just leaving out the dissipative terms of course, but then there is nothing left). Besides, where is the evidence for the existence of negative masses essential for this to work? In addition quantum statistics is different in essential ways from classical statistics, and I do not see how things like entanglement could possible result from such a derivation.
Dear Dr. Gert Van der Zwan, thank you for your answer. Yes perhaps Winterberg's derivation of quantum mechanics needs improvement, but alas i am no expert in boltzmann equation. Best wishes
Even if everything is perfectly rigorous, he's got one approximate equation (Schrodinger) through an approximation, with no hope to recover more accurate forms such as Dirac's equation or QED. I think some of Gert's point can be summarized as such: he derived something more accurate starting from a less accurate equation through a not well-understood approximation, but a logical person should normally try to do it the other way around.
Something much better is Valentini's "subquantum H-theorem". http://arxiv.org/pdf/quant-ph/0403034.pdf
Here we have a different, subquantum theory (which is de Broglie-Bohm theory, but without the quantum equilibrium assumption). Then we have the subquantum H-theorem, which proves that, similar to the original H-theorem, the entropy is increasing.
From a time-symmetric fundamental theory, but using a time-asymmetric coarse-graining. It is this coarse-graining which leads to an increasing entropy. And, in the limit, it gives quantum equilibrium - as the extremal limiting state, which is also time-symmetric.
The answer is no. Have a look at review articles on Bell's Theorem elaborating on this question and why there cannot be a local classical statistical explanation of quantum mechanics. See, e.g., http://arxiv.org/abs/quant-ph/0107093.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Quantum Physics
- Quantum Mechanics