# Haggard Rovelli thermodynamics paper says what time is

by marcus
Tags: haggard, paper, rovelli, thermodynamics, time
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 Astronomy Sci Advisor PF Gold P: 23,087 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: 184
 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
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PF Gold
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 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.
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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.
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PF Gold
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 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.
 Astronomy Sci Advisor PF Gold P: 23,087 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.
 Astronomy Sci Advisor PF Gold P: 23,087 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.
Mentor
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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.
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PF Gold
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 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.
 Mentor P: 11,503 Ah, that does seem like a tiny problem. Thanks!
 P: 175 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?
 Astronomy Sci Advisor PF Gold P: 23,087 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.
 Astronomy Sci Advisor PF Gold P: 23,087 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).
 Astronomy Sci Advisor PF Gold P: 23,087 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.
 Astronomy Sci Advisor PF Gold P: 23,087 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.
 Astronomy Sci Advisor PF Gold P: 23,087 We are seeing a paradigm take shape, I think. Made of separately familiar ideas in a possibly new configuration. A process has a boundary (Oeckl gives the axiomatics). A boundary is an interface for information flow---one could say a "channel". Freidel says "screen". Two adjoining processes are in equilibrium if the net information flow is zero during their interface contact. This is kind of interesting. During their contact the two processes could be experiencing different rate of TIME and different subjective TEMPERATURE but if they are in equilibrium the effects somehow balance out. They each see the other going through the same number of changes, the same number of phasespace cells. The quantum descriptor of a process lives in a Hilbertspace defined on the BOUNDARY of the process. I will refrain from calling the descriptor a "state" because that has the usual connotation of a "state at a given instant of time". There is no time: no objective time external to the process which can be referenced independently of the process descriptor. The boundary Hilbertspace vectors describe accessible initial-during-final information about the process. If it is a deep-rooted unalterable habit to call certain elements of a Hilbertspace by the name of "states" then you should, but I am calling them "descriptors" of the process interfaced by the boundary mainly just to teach myself to think differently, namely in process or history terms. One can ask the amplitude of a given description on the process boundary. It is a general covariant version of "transition amplitude", and the theory should give this. One can ask about the time-flow subjective to the process, as described by a given element or mix of elements in the boundary Hilbertspace. Tomita told us how to get an idea of "time" from such a descriptor, that is a flow on the observable algebra, or a one-parameter group of automorphisms. That's kind of interesting. Still lots of gaps and questions in the paradigm. I understand only a tiny percentage of it. In Oeckl's talk he said that if you want to include FERMIONIC information in the boundary Hilbertspace the you have to generalize the Hilbertspace to have a negative definite as well as a positive definite piece. A "Krein" space is the direct sum of an ordinary (pos) Hilbert and a kind of inverted (neg) Hilbert. Strange, if true. If it is true, then can one carry through with the Tomita construction? I'm totally in the dark about this. Which is why it's interesting. Apparently there was a Mr. Krein who lived in the Ukraine, someone who will be famous if Oeckl has his way. Google it if you like. So there is a kind of reading list (or "watching list") to lay out http://arxiv.org/abs/1302.0724 Death and resurrection of the zeroth principle of thermodynamics http://arxiv.org/abs/1306.5206 The boundary is mixed Chirco's lead talk on http://pirsa.org/13070085 Freidel's lead talk on http://pirsa.org/13070042 Oeckl's lead talk on http://pirsa.org/13070084 Ziprick's lead talk on http://pirsa.org/13070057. It surprised me to find that all four talks I wanted to cite were the first on their respective session recordings. That's lucky because it makes it easier to start watching. You don't have to wait for buffering before you drag to the desired segment. You just click on the link, select flash, and it starts. Goffredo Chirco is a postdoc at Marseille who is interested in general covariant QSM (quantum stat. mech.). In a way I am repeating, in this post, the viewpoint presented in his talk. The adoption of Oeckl boundary formalism aims at getting both QFT and QSM in the same general covariant setup. Freidel's current work on "screens" seems to me like parallel evolution (which has turned up some very interesting new things). Repeating some comment: It may be shallow of me but I like Freidel's 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.) One can ask about the conditional amplitude of something on the condition that there are N particles. One does not have to take the direct sum of all the Fock spaces for every N. Also even though it's risky to adopt something this novel, Freidel's radically new truncation for geometry appeals to me. It is a continuous cell-decompostion into spiral-edge cells, each with flat interior. 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. Loops 13 talks are an important resource to keep handy.http://pirsa.org/C13029 Here are abstracts of parallel session talks: http://www.perimeterinstitute.ca/sit...bstracts_7.pdf Here are links to the parallel session talks: http://www.physicsforums.com/showthr...21#post4461021
 Astronomy Sci Advisor PF Gold P: 23,087 Since we've turned a page I'll bring forward the papers we are discussing in this thread. Please keep in mind that the topic is the idea of time that arises in these particular papers. If someone has a different idea perhaps connected with some other research, they are welcome to start their own thread about it in the appropriate forum. But in this thread let's please stay focused on what is presented here: 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 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/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 To recall the essentials of what I said earlier, this seems to clinch 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.
 Astronomy Sci Advisor PF Gold P: 23,087 The statement in the Bianchi Haggard Rovelli abstract highlighted above "...Oeckl's boundary formalism incorporates QSM naturally, and we formulate general-covariant QSM in this language." makes it urgent to ask questions about Oeckl's formulation of quantum theory. He has recently come out with a radically different alternative version which requires fewer axioms. I gather it really is proposed as an optional alternative, not as a replacement. The earlier axioms are included in an appendix. The new alternative version is apt to strike people as conceptually unfamiliar---it uses positive real numbers (a generalized notion of probability) in place of complex amplitudes (!) but promises to be able to recover conventional quantum mechanical results. Lucian Hardy is credited with having inspired this seemingly risky gambit. On the other hand this alternative Oeckl formulation is IMHO aesthetically appealing. It certainly is not the version being used by the Loop gravity authors but I don't want to ignore it. http://arxiv.org/abs/1212.5571 A positive formalism for quantum theory in the general boundary formulation Robert Oeckl (CCM-UNAM) (Submitted on 21 Dec 2012) We introduce a new "positive formalism" for encoding quantum theories in the general boundary formulation, somewhat analogous to the mixed state formalism of the standard formulation. This makes the probability interpretation more natural and elegant, eliminates operationally irrelevant structure and opens the general boundary formulation to quantum information theory. 28 pages A recent exposition of the more familiar older form of Oeckl's formulation of quantum theory is here: http://arxiv.org/abs/1201.1877 Schrödinger-Feynman quantization and composition of observables in general boundary quantum field theory Robert Oeckl (UNAM) (Submitted on 9 Jan 2012) We show that the Feynman path integral together with the Schrödinger representation gives rise to a rigorous and functorial quantization scheme for linear and affine field theories. Since our target framework is the general boundary formulation, the class of field theories that can be quantized in this way includes theories without a metric spacetime background. We also show that this quantization scheme is equivalent to a holomorphic quantization scheme proposed earlier and based on geometric quantization. We proceed to include observables into the scheme, quantized also through the path integral. We show that the quantized observables satisfy the canonical commutation relations, a feature shared with other quantization schemes also discussed. However, in contrast to other schemes the presented quantization also satisfies a correspondence between the composition of classical observables through their product and the composition of their quantized counterparts through spacetime gluing. In the special case of quantum field theory in Minkowski space this reproduces the operationally correct composition of observables encoded in the time-ordered product. We show that the quantization scheme also generalizes other features of quantum field theory such as the generating function of the S-matrix. 47 pages One slight inconsistency of terminology: in the more recent paper an infinitesimally thin region is called a "slice". What is now called a slice region was called an "empty region" in the earlier paper. This change is pointed out by the author. In any case confusion is unlikely to result. Overall the style is conveniently thorough and clear.

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