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Computing heat capacities in Bohmian Mechanics
@user7348, I'm not clear how such a calculation would be done. The standard way in dBB is to "piggyback" on normal QM. The way it goes is as follows. Suppose you want to know position of the particle, when it hits a detector, say. It's guided there by the pilot wave with its classical portion and the quantum potential. First you assume the particle starts in a certain probabilistic distribution called "equilibrium" which has the property of satisfying Born probability. Then you look how it will behave under the influence of the pilot wave. You notice that its position, as it evolves, will still be given by Born probability applied to the pilot wave! As shown by the equivalence theorem. You can consider detailed trajectories as much as you want - you'll still come up with this fact. So, naturally, you just do the standard QM calculation (psi norm gives probability) and you're done. Even if standard QM never existed, this would still be the natural and easiest approach in dBB.

That's for position. Similar considerations apply for other observables, equivalence has been shown for all of them (AFAIK).

So it's not clear to me what you're asking for. One possibility is, when the calculating scientists gets to the place where he'd like to invoke the equivalence theorem, don't. Instead insert its proof; then go ahead and use it. That would, perhaps, make it more of a calculation "from the framework of dBB"?

Please outline, just roughly, how you imagine the dBB calculation should go. Then we can see how it differs from the normal, easy way via equivalence theorem.
For an update on the Hardy debate with Motl, see our comments here: http://physics.stackexchange.com/questio...ys-paradox
Hardy's "paradox" is no big deal, LM's right - at least, righter than Hardy - on this one.

You didn't address my request.
I was thinking some more about computing heat capacities in dBB. As Schmelzer has pointed out, Bohm (1952) showed that the usual Born rule is applied to the pilot wave so that one obtains the usual predictions as in quantum theory. That is, the statistical predictions based on the Born rule are completely reproduced. The problem is that the heat capacities are computed by taking the log(#possible distinct states). This has nothing to do with the Born rule, so Bohm (1952) isn't relevant here. In fact, the phase spaces of the two theories are indeed quite distinct, and only QM allows one to compute the right heat capacities as Motl clearly demonstrated.
I disagree. In quantum equilibrium, the state in dBB is completely defined the the quantum wave function. So the "number of possible distinct states" is also the same.

Of course, to clarify this, one also has to take a look at the foundations of thermodynamics. Because this is nothing which starts with some nicely invented axioms or postulates that one, for undefined reasons, has to count some number of possible distinct states. All this has a base in the microscopic physics, is derived from it. What is, in particular, essential, is that there is some microscopic definition of entropy, which is conserved. As for classical thermodynamics, where this is \( S = - \int \rho \ln \rho dp \land dq\), as in quantum theory, where it is the von Neumann entropy.

BTW, the word "phase space" has nothing to do nor with the space of states of quantum theory, nor of dBB theory, it is about a particular (Hamiltonian) formalism of classical theory.
I'm not a physicist, but I used "phase space" because Lubos used it in the same context:

the entropy of an atom (the logarithm of the volume of the phase space of states that are accessible at the thermal equilibrium) -- LM

Unfortunately, I'm just beginning to learn some basic physics. It will be a long time before I'm really able to follow this conversation, if ever.
Thanks for the discussion.
Nothing wrong with not being a physicist! I'm not either, although as a mathematician with a lot of physics experience I'm in the ballpark, especially with QM, since my degree is in functional analysis (Hilbert Spaces etc). LM and Schmelzer are at much higher level but I can follow the conversation pretty well.

But you can still contribute and, as a layman, judge, to some extent.

Imagine you were wondering which of two boxers was the best. Imagine one of them ("A") was ready to fight, anywhere, anytime. And, that he never said a word against the other ("B"), just: "bring it on". Imagine that B adamantly refused to get in the ring with A. He'd gone a round or two a while ago, no clear decision, just enough to get a taste. Then he suddenly stopped the match and banned "A", forever, from his Boxing Club - for a reason we can only guess. Imagine B talked big - as long as A wasn't around - bad-mouthing him behind his back. But absolutely refused to get in the ring with him ever again.

Ok, which would you guess is probably the better, more confident, fighter?

You see, you can judge physics issues, tentatively at least, without knowing any physics!
(06-01-2016, 02:15 AM)user7348 Wrote: I'm not a physicist, but I used "phase space" because Lubos used it in the same context:

the entropy of an atom (the logarithm of the volume of the phase space of states that are accessible at the thermal equilibrium) -- LM
This is, indeed, the definition of entropy in classical Hamiltonian physics, where we have a phase space.

And this seems indeed to be the point where Lubos makes the error. He presents dBB theory as a sort of classical theory, thus, one would have to apply formulas known from classical thermodynamics. He seems to ignore that these formulas are not a priori postulates but follow from Hamiltonian physics.

Just to add a warning: The foundations of thermodynamics are a domain were misunderstandings and confusion are quite common even among professionals, so that it really difficult for laymen to understand it.
(06-01-2016, 01:55 PM)schmelzer Wrote: This is, indeed, the definition of entropy in classical Hamiltonian physics, where we have a phase space.  And this seems indeed to be the point where Lubos makes the error.  He presents dBB theory as a sort of classical theory, thus, one would have to apply formulas known from classical thermodynamics.  He seems to ignore that these formulas are not a priori postulates but follow from Hamiltonian physics.'
Aren't we really talking about momentum space wrt dBB and phase space wrt to thermodynamics? So the pilot wave of dBB remains constant in momentum space while thermodynamics infers a change ?
Sorry, I was unable to interpret this in a meaningful way. DBB theory does not work in the phase space but in configuration space.

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