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On the Origin of Species Thermodynamics and the Black Hole - Tower Correspondence
by Alvaro Herráez, Dieter Lüst, Joaquin Masias, Marco Scalisi
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Submission summary
| Authors (as registered SciPost users): | Alvaro Herraez · Joaquin Masias |
| Submission information | |
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| Preprint Link: | scipost_202411_00052v1 (pdf) |
| Date submitted: | 2024-11-25 11:19 |
| Submitted by: | Herraez, Alvaro |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Species thermodynamics has been proposed in analogy to black hole thermodynamics. The entropy scales like an area and is given by the mere counting of the number of the species. In this work, we {\it derive} the constitutive relations of species thermodynamics and explain how those {\it originate} from standard thermodynamics. We consider configurations of species in thermal equilibrium inside a box of size L, and show that the temperature T of the system, which plays a crucial role, is always upper bounded above by the species scale Λsp. We highlight three relevant regimes: (i) when L−1<T<Λsp, and gravitational collapse is avoided, the system exhibits standard thermodynamics features, for example, with the entropy scaling like the volume of the box; (ii) in the limit L−1≃T→Λsp we recover the rules of species thermodynamics with the entropy scaling like the area of the box; (iii) an intermediate regime with L−1≃T<Λsp that avoids gravitational collapse and fulfills the Covariant Entropy Bound; this interpolates between the previous two regimes and its entropy is given simply in terms of the counting of the species contributing to the thermodynamic ensemble. This study also allows us to find a novel and independent bottom-up rationale for the Emergent String Conjecture. Finally, we present the {\it Black Hole - Tower Correspondence} as a generalization of the celebrated Black Hole - String Correspondence. This provides us with a robust framework to interpret the results of our thermodynamic investigation. Moreover, it allows us to qualitatively account for the entropy of black holes in terms of the degrees of freedom of the weakly coupled species in the tower.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-1-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202411_00052v1, delivered 2025-01-17, doi: 10.21468/SciPost.Report.10504
Report
The paper discusses several aspects of the role that many species play in the definition of EFTs and within the context of the Swampland program. The authors discuss "species thermodynamics", derive the constitutive relations from standard thermodynamics, and see where the species cutoff comes in the context of the covariant entropy bound. They argue that there are only two possible cutoffs that are consistent with their analysis, namely the KK decompactification scale and the string scale, and as evidence, they present an analysis that gets rid of all other possibilities. Finally, they generalize the black hole-string correspondence principle to include the KK scale as well, transforming the principle into that of black hole-tower correspondence.
The paper is interesting, and it has a lot of consistency checks. The paper certainly meets this journal's criteria, and therefore I recommend it be published.
However, it would be beneficial for the readers if the authors would address the following questions and remarks:
1) The last sentence on page 1 mentions several arguments for why the species scale should be the final EFT cutoff, but no elaboration is provided nor a reference. Even though these might be familiar arguments, at least a reference would be in place.
2) The concept of "species thermodynamics" is never defined in the paper. At least the introduction should have the definition of it so that the reader is clear on what is being discussed. In particular, the difference between standard thermodynamics and species thermodynamics is not explained well: if we are simply introducing a new scale, why is there a need for a different type of thermodynamics etc. Please elaborate on this.
3) I am a little bit confused about where the black hole-tower correspondence point lies. Usually for the bh-string correspondence, the string coupling is inversely proportional to the entropy; is the same being done here? Please elaborate.
4) There is a typo in equation 2.10: it should be 2-d, not d-2.
5) In equation 3.6, it is not defined what is meant by P_q (perhaps it is in the appendix, but it should be readable from the main text).
6) On page 22, when the authors discuss the conditions of an appropriate tower, first, it is not clear why these two conditions are put and not some others (seems arbitrary), and since it is the crux on which the constitutive relations are derived from, the authors should better and in a clearer way motivate this notion of appropriate towers; second, the second condition is almost unreadable-please rewrite it less confusingly.
7) Above equation 3.87, \Delta is not defined (I assume it is the distance, but please be consistent). Likewise, on the same page 32, SDC is not defined (again, I can assume what it is, but still).
8) Typo in the second sentence in the Conclusions section (twice derived deriving).
9) In the second paragraph on page 48, the authors discuss some subtleties but they do not explicitly say what are they about; please provide an explanation and/or a reference (why is there a mismatch, how does this new paper resolve it explicitly, etc.). I understand that an explanation was intended for that paragraph, but it was, unfortunately, written in an unclear way.
10) Did I understand correctly from your footnote 20 that the black holes of size m^{-1}_{KK} are never dominant thermodynamically (either in the microcanonical or canonical)? This seems to be different compared to the standard bh-string correspondence picture in which a bh transitions into a stringy solution. Could you comment more on this, if possible?
Questions:
1) In the string thermodynamics framework, one deals with the thermal scalar formalism, as defined by Atick and Witten in the 80s. Can you comment on a possible "species thermal scalar" or if such a thing even makes sense? If not, please explain why not.
2) A crucial part of the black hole-string correspondence is the need for adiabaticity as one goes from one description to another. Do you need to impose a similar notion here? Please comment.
3) Does the new correspondence play a role in the Gregory-Laflamme instability? If so, in what way? There were several recent papers on this topic within the bh-string correspondence (https://inspirehep.net/literature/2844283, https://inspirehep.net/literature/2865831, https://inspirehep.net/literature/2851290). Do you see a connection with their approach?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Anonymous (Referee 1) on 2025-1-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202411_00052v1, delivered 2025-01-15, doi: 10.21468/SciPost.Report.10501
Report
This paper derives the constitutive relations of species thermodynamics — a notion of thermodynamics proposed for systems of large numbers of species at thermal equilibrium. So far, species thermodynamics had been formulated in analogy to black hole thermodynamics in the literature. This work takes an important step by deriving these laws of species thermodynamics from first principles.
The authors carefully analyze configurations of species in a box of size L by separating into several regimes for the temperature T. The rules of species thermodynamics are recovered in the limit where L−1=T approaches the species scale Λsp, with the entropy scaling as the area of the box. Away from this limit the entropy scales in the usual way as the volume of the box, or a mixture between these two cases.
In turn, these rules of species thermodynamics are subsequently applied to configurations of species given by towers of particles. It is found that the only consistent towers must have a degeneracy that grows at least polynomially and at most exponentially. This provides important evidence for the emergent string conjecture, as these degeneracy rates correspond to Kaluza-Klein (KK) towers and string excitations respectively.
Finally, the authors propose a black hole-tower correspondence, which is a generalization of the well-established black hole-string correspondence. This extends the black hole-string correspondence, where usually the string coupling is varied, to KK-like limits with varying radii of the extra dimensions. They do not investigate the details of this transition, but rather leave it as a compelling direction for future research.
The paper is well-written and I recommend it for publication in SciPost. It easily fulfills the journal expectations listed by the authors, as it connects several research areas and concepts (swampland, covariant entropy bound, black hole-string correspondence), and opens up interesting follow-up opportunities for research (species thermodynamics, black hole-tower correspondence).
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Alvaro Herraez on 2025-02-04 [id 5184]
(in reply to Report 2 on 2025-01-17)First of all we would like to thank the referee for their careful report, positive feedback and for highlighting several improvements to be made. Let us now address the different points one by one:
1) This sentence simply serves as an introduction for the rest of the paragraph, where all the references (from [12] to [24]) are cited in a more detailed way, explaining the context in which they are relevant. We plan to rephrase this in a future version to make it clearer.
2) The laws of species thermodynamics are explicitly presented in section 3.4, as well as our derivation of them. We agree with the referee, though, that a brief explanation in the introduction would be convenient for the reader, and plan to add it below eq. (1.3), where species thermodynamics is mentioned for the first time, and the original references are provided.
3) The key observation for the formulation of the Black Hole - Tower correspondence, in analogy to the Black Hole String correspondence, is the fact that the latter can be formulated in the language of species. Upon identifying the species length with the string length, ℓsp∼ℓstr, the relation between the Planck scale and the species scale is controlled by ℓd−2Pl,d∼g2s,dℓd−2sp (see eq. (5.1) in the manuscript), and the correspondence takes place at very weak gravitational coupling, namely when gs,d≪1 (since then ℓPl,d/ℓsp≪1), at the mass M∗ℓstr∼1/g2s,d∼S (see eq. (5.6) ). Precisely at that point, the entropies and masses coincide. Similarly, this can be performed for a KK tower associated to p-dimensions, upon the right identification of the species scale, ℓsp∼ℓPl,d+p, and the modulus controling the ratio between the latter and the d-dimensional Planck mass, which is nothing but the volume of the internal dimensions, i.e., ℓd−2Pl,d∼ℓd−2Pl,d+pVp(see eq. (5.10)). Thus, the very weak (d-dimensional) gravitational coupling happens for Vp≫1, and the transition takes place at M∗ℓsp∼Vp∼S, where the entropies and masses also coincide. The general idea is explained in section 5.3, where identifying ϕ in eq. (5.21) with either g−2s,d or Vp one recovers the the correspondence points of both cases. A detailed explanation of all this is the content of section 5.
4)We thank the referee for spotting this typo, we will correct it in a future version.
5) The definition of Pq is indeed presented in the appendix, but we completely agree with the referee that it would be clearer if also stated below (3.6). We plan to add it there in a future version.
6) As explained in the text, the motivation for these two conditions is twofold. First, they are fulfillied in all examples we know in top-down constructions, namely for KK towers and for towers of weakly coupled string oscilators. Second, these conditions are the minimal set of conditions that recover the thermodynamic properties of schwarzschild BHs for temperatures in the limit T→Λsp, and towers that do not fulfill them would never be able to match the energy and entropy of black holes in such limit, which is the key property of “appropriate towers”, as explained in section 3. Additionally, we will restate the second condition in a clearer way in a future version, as suggested by the referee.
7) We will clarify that Δ refers to the distance above (3.87) in a future version. We will also replace "SDC"->"Distance Conjecture" in page 32 for clarity.
8) We thank the referee for spotting the typo, we will correct it.
9) The subtlety lies in the fact that the identification Λsp∼mstr naively implies Nsp∼1, and this is inconsistent with Λsp≪MPl according to the definition of species scale (c.f. eq. (1.1) in the manuscript). As explained in the text, a self-consistent counting can then be performed but yields extra logarithmig factors that do not match the species scale identified from higher-curvature corrections, raising a puzzle. The canonical ensemble interpretation allows for a consistent resolution of such puzzle, since it allows for an arbitrarily high-number of species to contribute, even if their masses are above Λsp. This is the case only if their degeneracy can compesate the Boltzman suppression factor, which is precisely the case for a string tower, since it possesses an exponential degeneracy. We plan to state this in a clearer way in the corresponding paragraph in the conclusions.
10) The object that we study for the black hole-tower transition in decompactification limits is the one that wraps the extra dimensions. Once its horizon in the d non-compact dimensions becomes RBH≲m−1KK, for a given mass there might be a more entropic object —even though this is not fully clear according to recent discussions on the Gregory-Laflamme instability in string theory, as also pointed by the referee in his Question 3, namely the fully localized spherical black hole in D-dimensions. However, as dicussed in section 5.2.—the pargraph below eq. (5.12)—and footnote 20 (see also the reply to question 3 below), this does not play a crucial role for our argument, since for us it is enought o have meta-stable black brane solution. Furthremore, these are the only ones that allow us to probe the species scale in the non-compact dimensions while still following constant entropy lines as the extra dimensions decompactify.
Questions: 1) This is indeed the next natural step in establishing the black hole-tower correspondence, as mentioned in the manuscript, but it is beyond the scope of the present work— we are currently working on it as a separate project.
2) Adiabaticity is indeed a crucial part for the general argument that motivates the black hole-tower correspondence. In section 5 we refer to “constant entropy lines” and “adiabatic trajectories” indistinguishably, and all the diagrams depicted there follow such adiabatic trajectories.
3) In the context of decompactification limits, the Gregory-Laflamme instability can definitely play an important role when studying dynamical transitions of black hole solutions. However, for the arguments in this paper (and as explained in section 5.2.—the pargraph below eq. (5.12)—and footnote 20) it does not play a crucial role, given that we focus on the space of (meta-)stable black hole solutions and follow constant entropy lines, as opossed to studying the dynamical decays of such objects. Additionally, the references pointed out by the referee were not included simply because they appeared after the last version of our manuscript was uploaded to arXiv, but we agree they could be relevant for future extensions of our work.