# Phase Transitions in the Interacting Relativistic Boson Systems

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## Abstract

**:**

## 1. Introduction

## 2. Canonical Ensemble: Condensation in Ideal Boson Gas

## 3. Self-Interacting Scalar Field

#### 3.1. The Effective Lagrangian in the Mean-Field Approximation

#### 3.2. Hamiltonian Density in the Mean-Field Approximation

#### 3.3. Bosonic System with ${\phi}^{4}+{\phi}^{6}$ Self-Interaction

## 4. An Interacting Boson System within the Thermodynamic Mean-Field Model

#### 4.1. Parametrization of the Interaction

## 5. Condensation of Interacting Bosons at Finite Temperatures

## 6. Particle–Antiparticle System with Conserved Isospin (Charge) Density

#### 6.1. Derivation of Basic Equations

#### 6.2. Numerical Results: Second-Order Phase Transitions Generated by the Particles That Carry Dominant Charge

## 7. Canonical Ensemble vs. Grand Canonical Ensemble: Description of the Boson Systems in the Presence of a Condensate

#### 7.1. Particle-Number Conservation in an Ideal Single-Component Bosonic System

#### 7.2. Charge Conservation in an Interacting Particle–Antiparticle Boson System

- (a)
- (b)
- when both components, i.e., mesons ${\pi}^{-}$ and ${\pi}^{+}$, are in the condensate phase, it is necessary to modify set (a), see hereinafter;
- (c)

#### 7.3. Other Examples

## 8. Conclusions

- The intersections of the particle density curves with the critical curve indicate second-order phase transitions in the system.
- At the point where the particle density of ${\pi}^{-}$ mesons touches the critical curve, the virtual phase transition of second order, i.e., a phase transition without setting the order parameter, appears.
- The meson system develops a first-order phase transition for sufficiently strong attractive interactions via forming a Bose condensate, thus releasing the latent heat. The model predicts that the condensed phase is characterized by a constant total density of particles.
- The grand canonical ensemble cannot describe the state of the condensate since the chemical potential ${\mu}_{I}$ is significantly affected by the conditions of condensate formation, so it cannot be used as a free variable if the system is in the condensed phase. That is why the grand canonical ensemble is not suitable for describing a multi-component system in the condensate phase, even if only one of the components is in the condensate.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SMF | Scalar mean-field model |

TMF | Thermodynamic mean-field model |

FED | Free energy density |

## Appendix A. Thermodynamically Consistent Mean-Field Model for the Interacting Particle–Antiparticle System

#### The System of Particles and Antiparticles

## Notes

1 | In the nonrelativistic case, where ${\mu}_{\mathrm{nonrel}}=\mu -m$, the maximum value of the thermal-particle density is achieved at zero chemical potential. |

2 | It should be noted that we just conventionally say “condensate phase”. In fact, it is the thermodynamic state of a system that contains thermal particles and condensed particles at the same time. |

3 | In fact, the name “condensate phase” is just a conventional one because this phase is a mixture of the thermal (kinetic) particles and the condensed particles. |

4 | For our choice of the total charge of the system, it is the ${\pi}^{-}$ mesons. |

5 | Because we named it as the virtual phase transition of the second order. |

6 | For our choice of the total charge of the system, these are ${\pi}^{-}$ mesons. |

7 |

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**Figure 1.**

**Left panel:**particle-number density versus temperature in ideal single-component gas. The horizontal lines represent two constant particle density samples, $n=0.1,\phantom{\rule{0.166667em}{0ex}}0.2$ fm${}^{-3}$, which correspond to critical temperatures ${T}_{\mathrm{c}}^{\left(0.1\right)}$ and ${T}_{\mathrm{c}}^{\left(0.2\right)}$, respectively. Here, the critical curve ${n}_{\mathrm{lim}}^{\left(\mathrm{id}\right)}\left(T\right)$ is defined in (1).

**Right panel:**normalized critical temperature $\tilde{T}=T/m$ vs. normalized particle density $\tilde{n}=n/{m}^{3}$ in ideal single-component gas.

**Figure 2.**

**Left panel:**energy density versus temperature for the same system and conditions as in the left panel. The red dashed line marked as ${\epsilon}_{\mathrm{lim}}^{\left(\mathrm{id}\right)}$ represents the energy density of the states that belong to the critical curve ${n}_{\mathrm{lim}}^{\left(\mathrm{id}\right)}$ depicted in the upper panel.

**Right panel:**heat capacity normalized to ${T}^{3}$ as a function of temperature in the ideal single-component gas where the particle-number density is kept constant.

**Figure 3.**

**Left panel:**scalar density vs. temperature, $b=25\phantom{\rule{0.166667em}{0ex}}{m}_{\pi}^{-2}$, $a\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\kappa \phantom{\rule{0.166667em}{0ex}}{a}_{\mathrm{c}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{a}_{\mathrm{c}}=2m\sqrt{b}$.

**Right panel:**particle-number density vs. temperature, $B=10{m}_{\pi}{v}_{0}^{2}$, $A=\kappa \phantom{\rule{0.166667em}{0ex}}{A}_{\mathrm{c}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{A}_{\mathrm{c}}=2\sqrt{mB}$. In both panels, the shaded area indicates the states of the Bose–Einstein condensate. Crosses on both panels separate metastable and non-physical states.

**Figure 4.**

**In both panels:**Pressure vs. temperature for the supercritical attraction, $\kappa =1.1$. The solid blue line that consists of two segments, ${p}_{\mathrm{lg}}$ and ${p}_{\mathrm{mix}}^{\left(2\right)}$, is the final equation of state, ${T}_{\mathrm{c}}$ is the critical temperature that indicates the phase transition of the first order. Crosses on both panels separate metastable and non-physical states.

**Left panel:**The scalar mean-field model. The pressure ${p}_{\mathrm{mix}}^{\left(1\right)}$ corresponds to the scalar density ${\sigma}_{1}$.

**Right panel:**The thermodynamic mean-field model. The pressure ${p}_{\mathrm{mix}}^{\left(1\right)}$ corresponds to the particle-number density ${n}_{1}$.

**Figure 5.**

**Left panel:**The particle-number densities ${n}^{(-)}$ of ${\pi}^{-}$ mesons versus temperature for the system of interacting ${\pi}^{+}$ - ${\pi}^{-}$ mesons at fixed isospin density ${n}_{\mathrm{I}}=0.1$ fm${}^{-3}$ and the set of “weak” attraction parameters $\kappa =0,\phantom{\rule{0.166667em}{0ex}}0.6,\phantom{\rule{0.166667em}{0ex}}0.85,\phantom{\rule{0.166667em}{0ex}}0.96,\phantom{\rule{0.166667em}{0ex}}1$. The red dashed curve ${n}_{\mathrm{lim}}$ reflects the maximal density of thermal ${\pi}^{-}$ mesons (or ${\pi}^{+}$ mesons) in the ideal ${\pi}^{+}$ - ${\pi}^{-}$ gas. The dashed area indicates the phase with the condensed particles. The open stars show the onset of phase transition of the second order of the ${\pi}^{-}$ mesons.

**Right panel:**The particle-number densities ${n}^{(+)}$ of ${\pi}^{+}$ mesons versus temperature at the same set of parameters as in the left panel. The “dark” star corresponding to the ${T}_{\ast}$ temperature indicates a virtual second-order phase transition of the ${\pi}^{+}$ component without condensate formation.

**Figure 6.**

**Left panel:**Energy density versus temperature in the interacting particle–antiparticle system of pions at $\kappa =0,\phantom{\rule{0.166667em}{0ex}}0.6,\phantom{\rule{0.166667em}{0ex}}0.85,\phantom{\rule{0.166667em}{0ex}}0.96,\phantom{\rule{0.166667em}{0ex}}1$. The isospin (charge) density is kept constant, ${n}_{\mathrm{I}}=0.1$ fm${}^{-3}$. The points of the phase transition of the second order are indicated by the corresponding temperatures ${T}_{\mathrm{c}1}^{(-)},\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{c}2}^{(-)},\phantom{\rule{0.166667em}{0ex}}{T}_{\ast},\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{c}}^{(-)}$.

**Right panel:**heat capacity as a function of temperature for the same system and conditions as in the left panel.

**Figure 7.**

**Left panel:**Heat capacity normalized by ${T}^{3}$ as a function of temperature in the interacting particle–antiparticle system at $\kappa =1$ (black solid curve). The isospin (charge) density is kept constant, ${n}_{\mathrm{I}}=0.1$ fm${}^{-3}$. The derivative of heat capacity is shown in a small window.

**Right panel:**Energy density normalized by ${T}^{4}$ versus temperature for the same system and conditions as in the left panel (black solid curve). The enlarged central area of the graphic is shown in a small window.

**Figure 8.**

**Left panel:**Density of the condensate of ${\pi}^{-}$ mesons as a function of temperature in the interacting particle–antiparticle gas at $\kappa =1$. The isospin (charge) density is kept constant, ${n}_{\mathrm{I}}=0.1$ fm${}^{-3}$. Shaded blue areas show the condensate states of ${\pi}^{-}$ mesons. The blue solid line shows the behavior of the chemical potential.

**Right panel:**The same as in the left panel but for $\kappa =1.1$. The sail-like shaded area indicates the condensate states created by ${\pi}^{-}$ mesons and by ${\pi}^{+}$ mesons at the same time. The gap of the chemical potential at $T={T}_{\mathrm{cd}}$ reflects phase transition of the first order, which creates the condensate of ${\pi}^{-}$ and ${\pi}^{+}$ mesons.

**Figure 9.**

**Left panel:**Chemical potential vs. temperature in an ideal single-component boson gas at conserved mean value n of the particle-number density for two samples: $n=0.1$ fm${}^{-3}$ with ${T}_{\mathrm{c}}^{\left(0.1\right)}$ (the black solid line) and $n=0.2$ fm${}^{-3}$ with ${T}_{\mathrm{c}}^{\left(0.2\right)}$ (the black dashed line). The segment $\mu =m$ belongs to the condensate phase.

**Right panel:**Density of thermal particles vs. temperature in an ideal single-component boson gas. The critical curve ${n}_{\mathrm{lim}}^{\left(\mathrm{id}\right)}$ is defined in (3). (The same notations as in the left panel).

**Figure 10.**Interacting particle-antiparticle boson system in the thermodynamic mean-field model.

**Left panel:**Particle densities vs. temperature at conserved isospin (charge) density ${n}_{\mathrm{I}}=0.1$ fm${}^{-3}$ as the solid blue line consisting of several segments (${\pi}^{-}$ mesons) and the dashed blue line consisting of several segments (${\pi}^{+}$ mesons). The vertical segment for both dependencies indicates a phase transition of the first order with the creation of the condensate. In the condensate phase, ${\mu}_{I}=0$. A dashed red line is the critical curve ${n}_{\mathrm{lim}}\left(T\right)$, see Equation (63).

**Right panel:**Particle-number densities vs. temperature at ${n}_{\mathrm{I}}=0$ (or at ${\mu}_{I}=0$): $\left(1\right)$ the supercritical attraction $\kappa =1.1$ is shown as a solid blue line consisting of several segments; the vertical segment (solid blue line) indicates a phase transition of the first order with the creation of the condensate; $\left(2\right)$ particle densities at “weak” attraction $\kappa \le 1$ are shown as solid black lines in the thermal phase. A dashed red line is the critical curve. Crosses on both panels separate metastable and non-physical states.

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**MDPI and ACS Style**

Anchishkin, D.; Gnatovskyy, V.; Zhuravel, D.; Karpenko, V.; Mishustin, I.; Stoecker, H.
Phase Transitions in the Interacting Relativistic Boson Systems. *Universe* **2023**, *9*, 411.
https://doi.org/10.3390/universe9090411

**AMA Style**

Anchishkin D, Gnatovskyy V, Zhuravel D, Karpenko V, Mishustin I, Stoecker H.
Phase Transitions in the Interacting Relativistic Boson Systems. *Universe*. 2023; 9(9):411.
https://doi.org/10.3390/universe9090411

**Chicago/Turabian Style**

Anchishkin, Dmitry, Volodymyr Gnatovskyy, Denys Zhuravel, Vladyslav Karpenko, Igor Mishustin, and Horst Stoecker.
2023. "Phase Transitions in the Interacting Relativistic Boson Systems" *Universe* 9, no. 9: 411.
https://doi.org/10.3390/universe9090411