On the origin of species thermodynamics and the black hole - tower correspondence

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 derive the constitutive relations of species thermodynamics and explain how those originate from standard thermodynamics. We consider configurations of species in thermal equilibrium inside a box of size L <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>L</mml:mi> </mml:math> , and show that the temperature T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>T</mml:mi> </mml:math> of the system, which plays a crucial role, is always upper bounded above by the species scale \Lambda_\mathrm{sp} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>s</mml:mi> <mml:mi>p</mml:mi> </mml:mstyle> </mml:msub> </mml:math> . We highlight three relevant regimes: (i) when L^{-1}&lt; T&lt;\Lambda_\mathrm{sp} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>&lt;</mml:mo> <mml:mi>T</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>s</mml:mi> <mml:mi>p</mml:mi> </mml:mstyle> </mml:msub> </mml:mrow> </mml:math> , 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}\simeq T→ \Lambda_\mathrm{sp} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>≃</mml:mo> <mml:mi>T</mml:mi> <mml:mo>→</mml:mo> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>s</mml:mi> <mml:mi>p</mml:mi> </mml:mstyle> </mml:msub> </mml:mrow> </mml:math> we recover the rules of species thermodynamics with the entropy scaling like the area of the box; (iii) an intermediate regime with L^{-1}\simeq T→ \Lambda_\mathrm{sp} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>≃</mml:mo> <mml:mi>T</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>s</mml:mi> <mml:mi>p</mml:mi> </mml:mstyle> </mml:msub> </mml:mrow> </mml:math> 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 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.