A Model of Financial Pyramid with Quasi-Rational Participants

A model of a financial pyramid is proposed in which each participant makes a decision about entering and exiting the pyramid based on the maximin principle using his (or her) ideas about the characteristics of other participants. If the pyramid organizers are able to carry out the whole process quickly enough (so that the payoffs to agents who participated in the pyramid and left it in time do not matter too much), then exactly those agents who overestimated the share of losers in the total mass of agents will lose.

Context. Traditional studies of stellar streams typically involve phenomenological ΛCDM halos or ad hoc dark matter (DM) profiles with different degrees of triaxiality, which preclude us from gaining insights into the nature and mass of the DM particles. Recently, the maximum entropy principle of halo formation has been applied to provide a DM halo model that incorporates the fermionic (quantum) nature of the particles while leading to DM profiles that depend on the fermion mass. These profiles develop a more general “dense core – diluted halo” morphology that can explain the Galactic rotation curve, while the degenerate fermion core can mimic the central massive black hole (BH). Aims. We model the GD-1 stellar stream using a spherical core-halo DM distribution for the host that simultaneously explains the dynamics of the S-cluster stars through its degenerate fermion core without a central BH. Methods. We used two optimization algorithms in order to fit both the initial conditions of the stream orbit and the fermionic model. We modeled the baryonic potential with a bulge and two disks (thin and thick) with fixed parameters according to the recent literature. The stream observables were 5D phase-space data from the Gaia DR2 survey. Results. We were able to find good fits for both the GD-1 stream and the S-stars for a family of fermionic core-halo profiles parameterized by the fermion mass. The particle masses are constrained in the range 56 keV c−2, with a corresponding DM core of ∼103 Schwarzschild radii, to 360 keV c−2, which corresponds to the most compact core of 5 Schwarzschild radii prior to the gravitational collapse into a BH of about 4 × 106 M⊙. Conclusions. This work provides evidence that the fermionic profile is a reliable model for the massive central object and for the DM of the Galaxy. Remarkably, this model predicts a total Milky Way mass of 2.3 × 1011 M⊙, which agrees with recent mass estimates obtained from Gaia DR3 rotation curves (Gaia RC). In summary, with one single fermionic model for the DM distribution of the Milky Way, we obtain a good fit on three totally different distance scales of the Galaxy: ∼10−6 kpc (central, S-stars), ∼14 kpc (middle, GD-1), and ∼30 kpc (boundary, Gaia RC mass estimate).

This paper introduces a class of galactic models which extend Jaffe's spherical models to axisymmetric systems, and then studies the properties of their densities and two-integral even distribution functions. The models have finite total mass and finite densities which, at large distances, decay radially like r−4 except on the major axis, and like r−3 on the major axis. The more flattened the galaxy, the stronger is the dependence of the even distribution functions on the angular momenta of its stars. Their distribution functions can be obtained by using the maximum entropy principle or assuming the anisotropy of the models. In particular, some formulae analogous to those of Hunter & Qian are obtained to calculate two-integral odd distribution functions, and they can be applied to obtain the distribution functions under the assumption of anisotropy for the oblate models.

The Gross–Pitaevskii–Poisson equations that govern the evolution of self-gravitating Bose–Einstein condensates, possibly representing dark matter halos, experience a process of gravitational cooling and violent relaxation. We propose a heuristic parametrization of this complicated process in the spirit of Lynden-Bell’s theory of violent relaxation for collisionless stellar systems. We derive a generalized wave equation that was introduced phenomenologically in Chavanis (Eur Phys J Plus 132:248, 2017) involving a logarithmic nonlinearity associated with an effective temperature $$T_\mathrm{eff}$$ and a damping term associated with a friction $$\xi $$ . These terms can be obtained from a maximum entropy production principle and are linked by a form of Einstein relation expressing the fluctuation-dissipation theorem. The wave equation satisfies an H-theorem for the Lynden-Bell entropy and relaxes towards a stable equilibrium state which is a maximum of entropy at fixed mass and energy. This equilibrium state represents the most probable state of a Bose–Einstein condensate dark matter halo. It generically has a core-halo structure. The quantum core prevents gravitational collapse and may solve the core-cusp problem. The isothermal halo leads to flat rotation curves in agreement with the observations. These results are consistent with the phenomenology of dark matter halos. Furthermore, as shown in a previous paper (Chavanis in Phys Rev D 100:123506, 2019), the maximization of entropy with respect to the core mass at fixed total mass and total energy determines a core mass–halo mass relation which agrees with the relation obtained in direct numerical simulations. We stress the importance of using a microcanonical description instead of a canonical one. We also explain how our formalism can be applied to the case of fermionic dark matter halos.