# Haggard Rovelli thermodynamics paper says what time is

by marcus
Tags: haggard, paper, rovelli, thermodynamics, time
 PF Patron Sci Advisor P: 22,332 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
P: 172
 Quote by Prathyush I read parts of the paper, and I find equation 11 weakly motivated. without adequate motivation the whole theory quickly falls apart. I wonder if anyone has something to say about it.
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
PF Patron
HW Helper
P: 1,955
 Quote by marcus 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)
Interesting. It seems likely that the universal time scale h/kT associated with a temperature T has some significance.
P: 151
DX: Yes, for me it is interesting. See my post #9, where I wrote
 Quote by Paulibus ....Consider also the case of a non-thermal system, say a single atom. Here transitions involve the emission/absorption of a photon, and a 'step' or 'quantum jump' is, for any observer of the process, just her/his proper time for a single photon oscillation.
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.
PF Patron
HW Helper
P: 2,595
 Quote by 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.
It might be interesting to look at a degenerate Fermi gas (i.e., when $kT \ll E_F$, the Fermi energy). Then the average internal energy is

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

This contains the leading order correction in an expansion in $kT/E_F$ (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 ($\beta = 1/(kT)$)

$$(\Delta U)^2 = - \frac{\partial U}{\partial \beta},$$

so that

$$\Delta E \sim \pi \sqrt{ \frac{(kT)^3}{2E_F}}.$$

This is very different from $\sim kT$, 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.
 PF Patron Sci Advisor P: 22,332 Interesting, a system where ΔE ~ T1.5 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.
PF Patron
P: 10,391
Marcus, any chance you could explain this to me?
 The zeroth principle of thermodynamics in the form "temperature is uniform at equilibrium" is notoriously violated in relativistic gravity.
PF Patron
P: 22,332
 Quote by Drakkith Marcus, any chance you could explain this to me?
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.
 PF Patron P: 10,391 Ah, that does seem like a tiny problem. Thanks!
 P: 151 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?
 PF Patron Sci Advisor P: 22,332 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.
 PF Patron Sci Advisor P: 22,332 I've been reviewing the Haggard Rovelli "Zeroth Law" paper, and now see it as a truly basic one. I think it provides the conceptual framework for how general covariant statistical mechanics will be done. Notice that because the idea of the "state of a system a given time" is not a covariant notion, we shift our focus from instantaneous state to protracted process. "The core idea is to focus on histories rather than states. Two systems placed in contact are described as two histories joined for a given interaction period. In this conceptual framework, the paper shows how natural ideas of time, temperature, and equilibrium arise in a generally covariant way. As an example, the authors give an elementary derivation of Wien's displacement law. (Section 5, page 4). Thermal time turns out to be connected to the Heisenberg uncertainty principle, which thereby acquires new concrete meaning. See page 3, right before equation (14): "In a sense it is 'time counted in natural elementary steps', which exist because the Heisenberg principle implies an effective granularity of the phase space." 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 As it happened this paper did quite well on our first quarter MIP poll (over a quarter of us voted for it).
 PF Patron Sci Advisor P: 22,332 There has been some followup to this paper, and some related work has appeared. I'll try to bring the references up to date. http://arxiv.org/abs/1306.5206 The boundary is mixed Eugenio Bianchi, Hal M. Haggard, Carlo Rovelli (Submitted on 21 Jun 2013) We show that Oeckl's boundary formalism incorporates quantum statistical mechanics naturally, and we formulate general-covariant quantum statistical mechanics in this language. We illustrate the formalism by showing how it accounts for the Unruh effect. We observe that the distinction between pure and mixed states weakens in the general covariant context, and surmise that local gravitational processes are indivisibly statistical with no possible quantal versus probabilistic distinction. 8 pages, 2 figures As far as I can see this clinches the choice of formalism. The problem of quantum gravity is that of finding a general covariant QFT describing the behavior of geometry. And we know that GR has a deep relation to statistical mechanics. Ultimately that means quantum statistical mechanics or QSM So the goal involves finding a single general covariant formalism for both covariant QFT and QSM. In a general covariant setting there is no preferred time, so time-flow will probably need to arise Tomita-style from the quantum descriptor of the process enclosed in the boundary---that is by an element (or mix of elements) of the boundary Hilbertspace.
 PF Patron Sci Advisor P: 22,332 There is (what I believe is) a very interesting related development by Laurent Freidel. Together with students/collaborators such as Ziprick and Yokokura. Freidel uses the term "screen" for the boundary of a spacetime region containing a process. He also calls it a "time-like world-tube". Freidel makes the telling distinction between a truncation (e.g. a finite dimensional Fock space) and an approximation (the sort of thing one might expect to have a continuum limit.) At the same time he is proposing a new kind of truncation for geometry: a continuous cell-decompostion into spiral-edge cells with flat interior. See the first talk of http://pirsa.org/13070057 , by Ziprick. This seems a substantial improvement over previous cellular decompositions used in QG, and generalizes Regge action. The edges of the spatial cell do not HAVE to be helical, they can be straight, but they are allowed to corkscrew or roll a little if they need to. Freidel's talk is the first one of http://pirsa.org/13070042. You might, as I did, find some of the concepts unfamiliar and difficult to grasp, but nevertheless could find it worth watching (perhaps more than once.) He insists on concentrating the physics in the boundary as much as possible (surface tension, entropy production, internal energy, relaxation to equilibrium...everything is happening in boundary, or as he says "screen"). BTW the boundary can have several topological components and usual ideas of inside/outside can be reversed. The observer can be surrounded by process, looking out from his own world-tube. One reason the video talks, and the slides PDF, are valuable is because in many cases more pictorial. E.g. Ziprick shows a sample picture of a spiral-edge cell.