**8. Discussion and Conclusions**

In this note, we included the force that an inhomogeneous entropy might exert on the dynamics of particles in phase space. This is not an obvious step. It is a purely collective effect. So far, any such force has not yet been included in the dynamics of large numbers of particles and kinetic theory.

Collective effects of this kind are known from ponderomotive forces and pressure forces. However, the inclusion of a separate ponderomotive force or a pressure force on the microscopic level is not necessary. Ponderomotive forces are taken care of by the interaction of the electromagnetic field with the particles. They arise from correlations. Similarly, the pressure force, which is directly proportional to the gradient of the particle density on any level, is already included in the evolution of the phase space density.

The entropy force is different in the sense that it is not obvious that it is given by the gradient of density. The entropy force takes care of the chaotic disorder that is produced by the dynamics itself. Neither the pressure, nor the temperature account for it. Entropy is a function of the phase space density, not its moment like pressure and temperature. It is the cause of chaotic expansion of the phase space in the course of the dynamics. It therefore gives rise to a collective effect felt on the level of the microscopic phase space density. In many processes, this effect may be very small and negligible. This will depend on scales. Elucidating those effects requires investigation of particular cases. The present work presents the theory on which such attempts must be based.

We derived the basic kinetic equation for the one particle phase space density including the global entropy force. Since this force is a global integral one, it has no microscopic source, while it affects the dynamics of the one-particle phase space density through the interaction Hamiltonian. This gives rise to the construction of a kinetic equation for the entropic phase space density, which turns out to be of similar structure, like the Liouville (Klimontovich) equation for the phase space density. There is, however, a fundamental difference in that the entropic kinetic equation is self-referential. It determines the dynamics of the entropy density in phase space by reference to the integrated entropy of the entire system itself. This is a very important finding because it shows that the entropy density in phase space evolves holographically. It is determined by itself. It takes care of its own evolution, which in addition depends on the evolution of the matter phase space density. This can be interpreted as a self-regulation of the evolution of entropy on the microscopic level, which takes care of the total produced entropy. According to this finding, any many-particle system that in the course of its dynamics generates entropy in an inhomogeneous and time-dependent way is subject to the entropy that controls its own evolution. This we feel is the most important insight we arrived at in our analysis. Entropy generates and controls itself in this highly nonlinear way already on the microscopic level of phase space density. We conjecture that these properties remain when including quantum effects in kinetic theory on the microscopic quantum level without the need to introduce a seed source of entropy.

In order to demonstrate an effect, we in the introductory part of this note treated the astrophysical example of a Schwarzschild black hole. This led us to the definition of the Schwarzschild constant, a universal constant, which is essentially the Planck force, which so far had been formally postulated without finding a physical interpretation. Its meaning lies in the Schwarzschild constant as the entropy force at the black hole horizon. Some of its implications we have discussed briefly. The entropy force at the horizon, being independent of charge, is of only one sign. It causes a repulsive force. This is similar to the gravitational force, though counteracting it. It thus in large space may compete with the gravitational attraction, where it may become kind of an anti-gravity. Investigation of its effect on the universal expansion might also be of interest, as also the role it may play in primordial black holes and planckions, which may have been generated in the early universe, the Big Bang, and if surviving, possibly due to the anti-gravity action of the entropy force, could provide all or part of the mysterious dark matter on the scales of clusters of galaxies.

**Author Contributions:** Conceptualization, W.B. and R.T., formal analysis, R.T., writing—review and editing, W.B. and R.T.

**Funding:** This research received no external funding.

**Acknowledgments:** This work was part of a Visiting Scientist Programme at the International Space Science Institute (ISSI) Bern. The interest of the ISSI Directorate is acknowledged, as is in particular the friendly hospitality of the ISSI staff. Thanks are directed to the ISSI system administrator Saliba F. Saliba for technical support and to the librarians Andrea Fischer and Irmela Schweizer for access to the library and literature.

**Conflicts of Interest:** The authors declare no conflict of interest.
