The Hagedorn Temperature as a Nonequilibrium Dynamical Bottleneck in String Thermodynamics
We revisit the Hagedorn regime of string theory from a nonequilibrium perspective using Steepest-Entropy-Ascent Quantum Thermodynamics (SEAQT), which formulates thermodynamic evolution directly on the state manifold without requiring a globally well-defined canonical ensemble. Treating the inverse temperature as an instantaneous, state-dependent quantity, we derive its exact scalar evolution equation, governed in the commuting limit by higher-order fluctuation moments. In an open-system extension we prove that the inverse-temperature mobility equals unity identically on the quasi-canonical family: a divergent Hagedorn variance is necessary but not sufficient for slowing-down, and a reservoir coupled democratically to the whole spectrum drives the system across the Hagedorn scale unimpeded. Slowing-down requires instead a hierarchy of injection rates, in which direct production of highly excited strings by the reservoir is slow compared with a cascade through light quasi-particle modes. The mobility then collapses to a floor set by the ratio of the two rates, the intensive variable is pinned near the Hagedorn scale, and energy condenses into the long-string sector at a finite rate; the pinning is a long-lived transient of computable duration. Promoting the Hagedorn scale to a moduli-dependent quantity along emergent-string limits yields a finite critical distance $Δ_*$ at which the thermodynamic inertia diverges as $(Δ_*-Δ)^{a-3}$ for $1<a<3$, and beyond which pinning can activate. Since Kaluza--Klein towers admit no such divergence, this provides a thermodynamic discriminant between emergent-string and decompactification limits.