[Paper Review] Can a Hagedorn system have a temperature other than $T_C$ or can a thermostat have a temperature other than its own?

L. G. Moretto, K. A. Bugaev+2 more|ArXiv.org|Jan 4, 2006

Complex Systems and Time Series Analysis3 citations

TL;DR

This paper argues that Hagedorn systems—characterized by an exponential level density ρ∝exp(aE)—can only exist at a single, intrinsic temperature TC=1/a, contrary to the conventional use of a partition function Z(T)=TC T/(TC−T) that implies arbitrary temperatures below TC. The authors demonstrate that such a partition function is thermodynamically inconsistent because it assumes a system can be coupled to a thermostat at any T<TC, which is impossible for a Hagedorn system whose entropy is linear in energy and thus has a fixed, unique temperature T_C.

ABSTRACT

This is a note intended to complement our paper (nucl-th/0504010) and addressed to the attention of QGP workers interested in bag models, Hagedorn spectra, and the like. It tries to show that with a Hagedorn-like experimental spectrum the partition function can not be calculated and that a canonical description derived for the microcanonical ensemble exists only for a single, fixed temperature.

Motivation & Objective

• To challenge the conventional use of a temperature-dependent partition function Z(T) = TC T/(TC−T) for Hagedorn systems.
• To clarify that systems with exponential level densities ρ∝exp(aE) are intrinsically thermostats at a single fixed temperature TC=1/a.
• To show that the standard partition function approach fails because it assumes arbitrary coupling to a thermostat, which is not physically valid for such systems.
• To establish that thermal equilibrium is only possible when the logarithmic derivative of the entropy (i.e., inverse temperature) of both systems match, which is only possible at T=TC for Hagedorn systems.

Proposed method

• Analyzes the level density ρ(E) = exp(E/TC) for a Hagedorn system, showing it leads to a linear entropy S(E) = E/TC.
• Derives the partition function Z(T) via Laplace transform: Z(T) = ∫ exp(E/TC) exp(−E/T) dE = TC T/(TC−T), but shows this is physically invalid.
• Uses the microcanonical-to-canonical transition to show that the partition function only makes sense if the system's entropy S(E) allows matching of inverse temperatures with a thermostat.
• Applies the condition for thermal equilibrium: ∂S_A/∂E = ∂S_B/∂E, which requires matching of 1/T_A and 1/T_B, only possible if both systems have compatible temperature-dependent entropy derivatives.
• Demonstrates that for a Hagedorn system, S(E) is linear, so T_A is constant and independent of energy, making it a true thermostat at T_C.
• Concludes that a system with linear S(E) cannot be assigned a temperature T≠T_C, and thus Z(T) for T≠T_C is not thermodynamically meaningful.

Experimental results

Research questions

• RQ1 Can a Hagedorn system be assigned a temperature other than its intrinsic critical temperature TC=1/a?
• RQ2 Is the standard partition function Z(T)=TC T/(TC−T) valid for Hagedorn systems, or does it violate thermodynamic consistency?
• RQ3 Can a system with an exponential level density ρ∝exp(aE) be coupled to a thermostat at any temperature T<TC?
• RQ4 What is the physical meaning of the partition function Z(T) when the entropy S(E) is linear in energy?
• RQ5 Under what conditions is thermal equilibrium possible between two systems, especially when one is a Hagedorn system?

Key findings

• A Hagedorn system with level density ρ∝exp(aE) has a fixed, intrinsic temperature TC=1/a, and cannot exist at any other temperature.
• The standard partition function Z(T)=TC T/(TC−T) is thermodynamically inconsistent because it implies the system can be in thermal equilibrium at any T<TC, which contradicts the system's fixed temperature.
• The partition function of a system is only physically meaningful if its entropy S(E)=lnρ(E) allows matching of the inverse temperature (dS/dE) with a thermostat; for Hagedorn systems, this is only possible at T=TC.
• Thermal equilibrium between two systems requires matching of their inverse temperatures; for a Hagedorn system, this is only possible if the second system also has T=TC.
• A system with linear S(E) cannot be forced into a temperature T≠T_C, as its temperature is an intrinsic property determined by the slope of S(E).
• The use of Z(T) for Hagedorn systems leads to erroneous results in resonance gas models and bag model calculations, as it violates the principle that a thermostat can only have its own intrinsic temperature.