[Paper Review] Thermodynamics of exotic matter with constant w=P/E

Ernst Trojan, George V. Vlasov|arXiv (Cornell University)|Aug 3, 2011

Advanced Thermodynamics and Statistical Mechanics7 references3 citations

TL;DR

This paper proposes a thermodynamic model for exotic matter with a constant equation of state $ P = wE $, showing that such matter can be described as an ideal gas of quasi-particles with energy spectrum $ \varepsilon_p \sim p^{wq} $ in $ q $-dimensional space. The key result is that fermionic exotic matter with $ w < 0 $ exhibits negative entropy and negative heat capacity, a signature of non-standard thermodynamic behavior.

ABSTRACT

We consider a substance with equation of state $P=wE$ at constant $w$ and find that it is an ideal gas of quasi-particles with the energy spectrum $ε_p\sim p^{wq}$ that can constitute either regular matter (when $w&gt;0$) or exotic matter (when $w&lt;0$) in a $q$-dimensional space. Particularly, an ideal gas of fermions or bosons with the energy spectrum $ε_p=m^4/p^3$ in 3-dimensional space will have the pressure $P=-E$. Exotic material, associated with the dark energy at $E+P&lt;0$, is also included in analysis. We determine the properties of regular and exotic ideal Fermi gas at zero temperature and derive a low-temperature expansion of its thermodynamical functions at finite temperature. The Fermi level of exotic matter is shifted below the Fermi energy at zero temperature, while the Fermi level of regular matter is always above it. The heat capacity of any fermionic substance is always linear dependent on temperature, but exotic matter has negative entropy and negative heat capacity.

Motivation & Objective

• To derive a thermodynamic description of matter with a constant equation of state $ P = wE $, including both regular and exotic forms.
• To identify the energy spectrum of quasi-particles that reproduce the given equation of state in $ q $-dimensional space.
• To analyze the zero-temperature and finite-temperature thermodynamics of ideal Fermi gases composed of such quasi-particles.
• To determine the behavior of the Fermi level, entropy, and heat capacity for both regular ($ w > 0 $) and exotic ($ w < 0 $) matter.
• To explore the implications for astrophysical systems, particularly compact stars and dark energy models.

Proposed method

• Derives the energy spectrum $ \varepsilon_p = a p^{wq} $ that reproduces the equation of state $ P = wE $ for an ideal gas of quasi-particles in $ q $-dimensional space.
• Applies standard statistical mechanics formalism using Fermi-Dirac and Bose-Einstein distribution functions in momentum space.
• Uses dimensional integration over $ q $-dimensional momentum space to compute particle number density, energy density, pressure, and free energy.
• Performs low-temperature expansions of thermodynamic functions, including the Fermi level and heat capacity, using asymptotic analysis.
• Introduces a momentum cutoff $ p_0 $ to regularize divergent energy densities when $ w \leq -1 $.
• Analyzes the Hagedorn-like behavior at $ w = -1 $, where energy and particle density exhibit a logarithmic relation.

Experimental results

Research questions

• RQ1 What energy spectrum of quasi-particles reproduces a constant equation of state $ P = wE $ in $ q $-dimensional space?
• RQ2 How do the thermodynamic functions—especially the Fermi level, entropy, and heat capacity—behave for fermionic systems with $ w < 0 $?
• RQ3 What is the low-temperature behavior of the Fermi gas, and how does the Fermi level shift relative to the zero-temperature Fermi energy?
• RQ4 How does the model handle divergences in energy density when $ w \leq -1 $, and what role does a momentum cutoff play?
• RQ5 Can the model describe both exotic matter with $ P < 0 $, $ E > 0 $ and matter with $ P > 0 $, $ E < 0 $, and what are the thermodynamic implications?

Key findings

• The equation of state $ P = wE $ is realized by an ideal gas of quasi-particles with energy spectrum $ \varepsilon_p = a p^{wq} $, where $ a $ is a constant that can be positive or negative.
• For $ w < 0 $, the Fermi level at finite temperature lies below the zero-temperature Fermi energy, in contrast to regular matter where it lies above.
• The heat capacity of fermionic exotic matter with $ w < 0 $ is negative, as shown explicitly for $ w = -1 $, where $ C_V = -\frac{\pi^2}{3} \Sigma_q \frac{|a|T}{|\varepsilon_F|^2} $.
• Exotic matter with $ w < 0 $ has negative entropy, a non-standard thermodynamic feature confirmed by the low-temperature expansion $ S = \mathrm{sign}(w) \frac{\pi^2}{3} \Sigma_q \frac{T}{|\varepsilon_F|} n_F $.
• At $ w = -1 $, the energy density and particle number density are logarithmically related, resembling the Hagedorn equation of state.
• When $ w \leq -1 $, the energy density diverges without a momentum cutoff $ p_0 $, necessitating regularization for physical consistency.