Haggard Rovelli thermodynamics paper says what time is

Prathyush

Equation 7 essentially follows from Schrodinger's equation(please correct me on this if i am wrong), though i haven't verified the algebra it, seems to be correct.

Equation 11 he introduces temperature and uses ΔE=kT. This is what I find to be strange.

What is the logic behind this? The original ΔE is to do with energy eigenstates of the Hamiltonian under consideration.

He asserts that variance is kT, firstly this need not be true for all systems. the general expression will depend on the specific Hamiltonian under consideration.
Secondly assuming that the variance is kT, how can you associate this variance with ΔE in equation 7 that deals with energy eigenstates of the Hamiltonian.

It is possible that I did not understand the authors intentions, in the derivation of equation 11. But I think it needs further explanation.

marcus

Thanks! This gives me something to focus on and think about. Maybe the identification of ΔE with kT is shaky. It seemed solid to me, but I will have another look. (Tomorrow when I wake up, it's bedtime here )

For readers new to the thread, here's the paper being discussed:

http://arxiv.org/abs/1302.0724
Death and resurrection of the zeroth principle of thermodynamics
Hal M. Haggard, Carlo Rovelli
(Submitted on 4 Feb 2013)
The zeroth principle of thermodynamics in the form "temperature is uniform at equilibrium" is notoriously violated in relativistic gravity. Temperature uniformity is often derived from the maximization of the total number of microstates of two interacting systems under energy exchanges. Here we discuss a generalized version of this derivation, based on informational notions, which remains valid in the general context. The result is based on the observation that the time taken by any system to move to a distinguishable (nearly orthogonal) quantum state is a universal quantity that depends solely on the temperature. At equilibrium the net information flow between two systems must vanish, and this happens when two systems transit the same number of distinguishable states in the course of their interaction.
5 pages, 2 figures

DimReg

Take ΔEΔt ≈ hbar. It's just the heisenberg uncertainty principle for energy time. Compare it to equation 7, and note that in the paragraph above equation 11 ΔE is derived to be ≈ kT.

Edit: it looks like Marcus already responded before I got here. oops

dx

Interesting. It seems likely that the universal time scale h/kT associated with a temperature T has some significance.

Paulibus

DX: Yes, for me it is interesting. See my post #9, where I wrote Just substitute for kT, the change in energy for the step or quantum jump, and the relation step energy-change = h times the frequency of the emitted photon gives the result I've emphasized in the above quote. I thought this was interesting.

The 'universal time scale' you mention is a scale where time is counted in steps of (photon frequency)^-1, at least for single atoms.

fzero

It might be interesting to look at a degenerate Fermi gas (i.e., when [itex]kT \ll E_F[/itex], the Fermi energy). Then the average internal energy is

[tex]E \equiv \frac{U}{N} \sim \frac{3 E_F}{5} \left[ 1 + \frac{5}{12} \left( \frac{\pi k T}{E_F}\right)^2 \right].[/tex]

This contains the leading order correction in an expansion in [itex]kT/E_F[/itex] (see, for example, eq (8.30) of http://www.physics.udel.edu/~glyde/P.../chapter_8.pdf). We can compute the variance in the energy using ([itex]\beta = 1/(kT)[/itex])

[tex](\Delta U)^2 = - \frac{\partial U}{\partial \beta},[/tex]

so that

[tex]\Delta E \sim \pi \sqrt{ \frac{(kT)^3}{2E_F}}.[/tex]

This is very different from [itex]\sim kT[/itex], because the leading term in the energy was independent of the temperature. There is obviously some issue with the proposed "universal time step" when you apply it to the simplest fermionic system.

marcus

Interesting, a system where ΔE ~ T^1.5 instead of the more typical ΔE ~ T¹ As a reminder for anyone reading the thread, here's the paper being discussed:

http://arxiv.org/abs/1302.0724
Death and resurrection of the zeroth principle of thermodynamics
Hal M. Haggard, Carlo Rovelli
(Submitted on 4 Feb 2013)
The zeroth principle of thermodynamics in the form "temperature is uniform at equilibrium" is notoriously violated in relativistic gravity. Temperature uniformity is often derived from the maximization of the total number of microstates of two interacting systems under energy exchanges. Here we discuss a generalized version of this derivation, based on informational notions, which remains valid in the general context. The result is based on the observation that the time taken by any system to move to a distinguishable (nearly orthogonal) quantum state is a universal quantity that depends solely on the temperature. At equilibrium the net information flow between two systems must vanish, and this happens when two systems transit the same number of distinguishable states in the course of their interaction.
5 pages, 2 figures
=======================

One thing to note about this topic is that the overall aim is to develop general covariant thermodynamics (among other things, invariant under change of coordinates) so that "state" at a particular time may be the wrong approach to defining equilibrium. One may need to define equilibrium between processes or histories rather than between states.

Defining a state at a particular time appears to break general covariance, at least at first sight. There may be some way to get around this. But in any case one of the first things one needs to do is generalize the idea of equilibrium to a general covariant setup, where you put two processes in contact. Equilibrium corresponds to no net flow (of something: heat, information...) between the two.

I've been absorbed with other matters for the past few days, but this paper is intriguing and I want to get back to it. So maybe we can gradually get refocused on it.

marcus

All the paper does, in a certain sense, is motivate and propose a general covariant idea of equilibrium. The non-relativistic examples and discussion leading up to section IV are heuristic.
==quote==
IV. EQUILIBRIUM BETWEEN HISTORIES
Let us come to the notion of equilibrium. Consider two systems, System 1 and System 2, that are in interaction during a certain interval. This interaction can be quite general but should allow the exchange of energy between the two systems. During the interaction interval the first system transits N₁ states, and the second N₂, in the sense illustrated above. Since an interaction channel is open, each system has access to the information about the states the other has transited through the physical exchanges of the interaction.
The notion of information used here is purely physical, with no relation to semantics, meaning, significance, consciousness, records, storage, or mental, cognitive, idealistic or subjectivistic ideas. Information is simply a measure of a number of states, as is defined in the classic text by Shannon [17].
System 2 has access to an amount of information I₁ = logN₁ about System 1, and System 1 has access to an amount of information I₂ = log N₂ about System 2. Let us define the net flow of information in the course of the interaction as δI = I₂ − I₁. Equilibrium is by definition invariant under time reversal, and therefore any flow must vanish. It is therefore interesting to postulate that also the information flow δI vanishes at equilibrium. Let us do so, and study the consequences of this assumption. That is, we consider the possibility of taking the vanishing of the information flow
δI = 0 (15)
as a general condition for equilibrium, generalizing the maximization of the number of microstates of the non-relativistic formalism.³
==endquote==

You can see that the paper is still in a heuristic mode because in thinking about information we fall back on the idea of state. I expect that a mathematically rigorous treatment of the same subject might employ Tomita time. What is being set out here is an intuitive basis---how to think about equilibrium in general covariant context. But I could be wrong and the idea of state could be rigorously defined at this point.

==quote from Conclusions==
VI. CONCLUSIONS
We have suggested a generalized statistical principle for equilibrium in statistical mechanics. We expect that this will be of use going towards a genuine foundation for general covariant statistical mechanics.
The principle is formulated in terms of histories rather than states and expressed in terms of information. It reads: Two histories are in equilibrium if the net information flow between them vanishes, namely if they transit the same number of states during the interaction period.
This is equivalent to saying that the thermal time τ elapsed for the two systems is the same,..
==endquote==

That, I think, is the key statement of the paper. However you think about it, whatever your intuitive grasp, a DEFINITION of gen. cov. equilibrium is being proposed.
Two processes or histories are in equilibrium if during an interval of contact the thermal time elapsed in each is the same.

Drakkith

Marcus, any chance you could explain this to me?

marcus

Well there was a guy at Caltech, named Richard Tolman, who wrote a book (published 1934) about General Relativity.
http://en.wikipedia.org/wiki/Richard_C._Tolman
He found that in curved spacetime a column of material at equilibrium would be at different temperature. It was a very slight effect. Temperature was naturally higher when you were lower down in a gravitational field.

If you ignore GR, and the Tolman Effect, then temperature is a good indicator of equilibrium. Two systems are in equilibrium if they are the same temperature. ("Zeroth Law") Put them in contact and there is no net flow of heat between.

But if you take account of GR, and the Tolman Effect, then that is not true. Upstairs and downstairs can be in contact and have come into equilibrium, but downstairs is a tiny bit higher temperature. So ever since 1930s it has been known that the Zeroth Law notoriously fails if you allow for GR.

EDIT: I didn't know the name of the book, so looked it up:
Relativity, Thermodynamics, and Cosmology. Oxford: Clarendon Press. 1934.

Drakkith

Ah, that does seem like a tiny problem. Thanks!

Paulibus

I'm looking for a really simple way to consider covariant thermal equilibrium, and have got to wondering whether the information exchange by two observers in black-body cavities, differently situated in a spacetime pervaded by gravity, couldn't be quantified by simply counting the black-body photons each observer receives from the other, through small windows.

Perhaps equilibrium could be judged to prevail when each observer finds the locally measured flux of black-body photons coming from the other to be the same? Such measured flux depends on measured space dimensions and on measured time intervals which, for Tomita or thermal time, seem to me to be a count of time-steps of size (reciprocal of measured photon frequency).

Since both perceived space dimensions and perceived time step-lengths vary over gravity-pervaded spacetime, could this provide a covariant procedure?

marcus

That sounds like a way to prove the Tolman effect! Have an upstairs and a downstairs cavity. And a small hole connecting the two. Thermal radiation from upstairs would gain energy (be blueshifted ) by falling into the downstairs cavity. The observer downstairs would think that he was getting the same inflow as he was losing as an outflow.

the two observers would think they were in equilibrium, although they would actually be in slightly different temperatures.

I've never bothered to look up how Richard Tolman proved that effect. I'm lazy I guess and tend to just wait for the next paper rather than looking ahead--I expect other people to do the work

but actually what you are talking about does sound like ingredients for a math proof of the Tolman effect.

BTW one way people have of talking about the Tolman effect is to say "Energy weighs." I'm not sure if that is a good way to think about it, or if it helps much, but I've seen the phrase used. Maybe there's some intuition in it.
Getting late here, so I'd better be off to bed.