The Principle of Maximum Entropy and the Distribution of Mass in Galaxies
Abstract
:1. Purpose and Organization of This Review
2. Self-Gravitating Systems of Maximum Entropy
2.1. Solution Using the Classical Boltzmann–Gibbs Entropy: Isothermal Sphere
2.2. Solution Using the Tsallis Entropy: Polytropes
2.2.1. Properties of the Polytropes
- Analytic solutions. The case is known as the Schuster [59] sphere or Plummer [60] model and has an analytic solution,
- Connection with the isothermal sphere. Since the Boltzmann–Gibbs entropy approaches the Tsallis entropy in the limit , and this limit corresponds to (see Equation (19)), one can recover the isothermal sphere (Equations (10) and (12)) taking the limit of the Lane–Emden equation (Equation (21)) when [61].
- Analytic approximations. Most polytropic indexes do not admit an analytic solution and either Equation (21) has to be integrated numerically or one has to resort to the analytic approximations existing in the literature. The original approximations date back to pre-computer days when polytropes were fundamental to model stellar structures [50]. However, these approximations may still be useful nowadays to fit large datasets of galaxies. Fitting requires evaluating polytropes many times, and this computation usually determines the total speed of the algorithm. Chandrasekhar [50] expands as a polynomial around . The first terms are
- Central cores. All politropes have a core (a central plateau in the density distribution) with the same shape independently of the index m [18,48]. Equations (23) and (22), and the first two terms in the approximation (30) lead to,The second approximate identity in the right-hand side of Equation (33), which holds for , indicates that after normalization by and , all polytropes collapse to a single shape independently of m. In other words, except for a trivial normalization, all the polytropes look the same in their cores. This property is followed by halos from numerical simulations of self-gravitating systems (Section 4) as well as by real galaxies (Section 5).
- Relationship with polytropic gases. As we explain above, the properties of polytropes are well known because of the formal equivalence with polytropic gases, traditionally used to model stars analytically [50]. They are formally identical provided that the exponent that defines the relation between pressure (P) and ,In this equivalence, plays the role of gas temperature (T) so that the perfect gas law, , turns into,
2.2.2. Plane-of-the-Sky Projected Polytropes
3. Relationship between Polytropes and Sérsic Profiles
4. Relationship between Polytropes and CDM Mass Density Profiles
5. Observational Support for Galaxies Reaching Tsallis Maximum Entropy
6. Pathways to Thermalization
7. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
BH | Black Hole |
CDM | Cold Dark Matter |
DF | Distribution Function |
DM | Dark Matter |
MW | Milky Way |
NFW | Navarro, Frenk, and White [22] |
SMBH | Supermassive Black Hole |
TE | Thermodynamic Equilibrium |
1 | This is precisely the meaning given in the paper to the term thermodynamic equilibrium. It is used for distributions that maximize the function describing the entropy of the system. |
2 | Consider the identity , expand exp in its Taylor series, and take the limit . |
3 | Plugging into Equation (21) the expansion of around , , one ends up with the identity , which implies that either or when . |
4 | |
5 | For an MW-like halo of mass and radius and 100 kpc, respectively, the free-fall time is only 0.2 Gyr. |
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Sánchez Almeida, J. The Principle of Maximum Entropy and the Distribution of Mass in Galaxies. Universe 2022, 8, 214. https://doi.org/10.3390/universe8040214
Sánchez Almeida J. The Principle of Maximum Entropy and the Distribution of Mass in Galaxies. Universe. 2022; 8(4):214. https://doi.org/10.3390/universe8040214
Chicago/Turabian StyleSánchez Almeida, Jorge. 2022. "The Principle of Maximum Entropy and the Distribution of Mass in Galaxies" Universe 8, no. 4: 214. https://doi.org/10.3390/universe8040214