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Statistical Mechanics of Quarkyonic Matter

Marcus Bluhm, Yuki Fujimoto, Marlene Nahrgang

Abstract

We extend the theoretical formulation of Quarkyonic Matter within the IdylliQ model framework proposed in [Y. Fujimoto et al., Phys. Rev. Lett. 132, 112701 (2024) [1]] for zero temperature to non-zero temperatures. To this end, we develop a consistent statistical mechanics and grand canonical ensemble description of Quarkyonic Matter as a quantum system subject to additional inequality constraints due to the Pauli exclusion principle acting simultaneously on baryons and their constituent quarks. These constraints result in a significant reduction in the number of physically available baryon states compared to an ideal Fermi gas. As a consequence, the one-particle baryon distribution function factorizes into a thermal Fermi-Dirac distribution and a momentum-dependent density of states. This separation allows us to derive a proper definition of the entropy density that satisfies the third law of thermodynamics in the zero-temperature limit. Moreover, we find that inside Quarkyonic Matter the physical temperature and the physical baryon chemical potential differ from the Lagrange multipliers appearing in the Fermi-Dirac distribution which may have important consequences for the thermodynamics of Quarkyonic Matter.

Statistical Mechanics of Quarkyonic Matter

Abstract

We extend the theoretical formulation of Quarkyonic Matter within the IdylliQ model framework proposed in [Y. Fujimoto et al., Phys. Rev. Lett. 132, 112701 (2024) [1]] for zero temperature to non-zero temperatures. To this end, we develop a consistent statistical mechanics and grand canonical ensemble description of Quarkyonic Matter as a quantum system subject to additional inequality constraints due to the Pauli exclusion principle acting simultaneously on baryons and their constituent quarks. These constraints result in a significant reduction in the number of physically available baryon states compared to an ideal Fermi gas. As a consequence, the one-particle baryon distribution function factorizes into a thermal Fermi-Dirac distribution and a momentum-dependent density of states. This separation allows us to derive a proper definition of the entropy density that satisfies the third law of thermodynamics in the zero-temperature limit. Moreover, we find that inside Quarkyonic Matter the physical temperature and the physical baryon chemical potential differ from the Lagrange multipliers appearing in the Fermi-Dirac distribution which may have important consequences for the thermodynamics of Quarkyonic Matter.

Paper Structure

This paper contains 17 sections, 96 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The problem of non-vanishing entropy at zero temperature and its solution
  3. Grand canonical ensemble in the IdylliQ picture
  4. Modification by the Pauli exclusion principle for quarks
  5. Thermodynamic limit
  6. The zero temperature limit
  7. Physical temperature
  8. Physical pressure, the grand potential and the partition function
  9. Derivation of the grand canonical ensemble for a restricted Fock space
  10. Determination of the density of states
  11. Method of Lagrange multipliers with inequality constraints
  12. Determination of $g(k)$ at zero temperature
  13. Determination of $g(k)$ at non-zero temperature
  14. Physical chemical potential and temperature
  15. Summary and conclusions
  16. ...and 2 more sections

Figures (3)

  • Figure 1: The baryon distribution function $f_{\rm B}(k)$ (left) as a function of baryon momentum $k$ and the corresponding quark distribution function per color degree of freedom $f_{\rm Q}(q)$ (right) as a function of quark momentum $q$ at a fixed $\hat{T}=0.01$ GeV (upper panels) and at a fixed $\hat{\mu}=M_N$ (lower panels). The dashed vertical lines indicate the position of the bulk momentum ${k_{\rm bu}}$ and ${q_{\rm bu}}$ in the Quarkyonic Matter regime. With $\Lambda = 0.3\,\text{GeV}$ and $M_N = 0.94\,\text{GeV}$, we find $\mu_{\rm sat} = 0.9784$ GeV and $T_{\rm sat} = 0.0621$ GeV.
  • Figure 2: Scaled bulk momentum ${k_{\rm bu}}/\Lambda$ as a function of $\hat{\mu}/M_N$ for different values of $\hat{T}/\Lambda$.
  • Figure 3: The scaled physical temperature $T/\Lambda$ (left panel) and physical baryon chemical potential $\mu_{B}/\Lambda$ (right panel) as functions of the scaled Lagrange multiplier $\hat{\mu}/M_N$ for different values of $\hat{T}/\Lambda$.