**7. Evolution of Entropic Phase Space Density**

Having defined the entropic phase space density in Equation (17,) the question arises how it possibly evolves in phase space. Since the entropy is given as the phase space integral with respect to the entropic phase space density, an always positive quantity, this question is not senseless. Once knowing its evolution, the entropy can be calculated by integration. Moreover, if an equation for the phase space density can be obtained, its entropic momentum should yield an evolution equation for the entropy, which, essentially, in the long-term limit should be the fundamental thermodynamic laws, while in

the short term, it should give the evolution equation of entropy with time. In order to construct the entropic phase space equation, we multiply Equation (12) by <sup>−</sup> log <sup>N</sup> *<sup>m</sup> <sup>a</sup>* to obtain:

$$\left\{ \partial\_t \mathcal{S}\_a^{\rm m} + \left[ \mathcal{H}\_{N\_{a'}} \mathcal{S}\_a^{\rm m} \right] - \left\{ \partial\_t \mathcal{N}\_a^{\rm m} + \left[ \mathcal{H}\_{N\_{a'}} \mathcal{N}\_a^{\rm m} \right] \right\} \right\} = 0 \tag{48}$$

The term in the braces vanishes identically, yielding ultimately:

$$\left[\partial\_t \mathcal{S}\_a^m + \left[\mathcal{H}\_{N\_{a'}} \mathcal{S}\_a^m\right]\right] = 0\tag{49}$$

We thus find the almost trivial result that the microscopic entropic phase-space density <sup>S</sup>*<sup>m</sup> <sup>a</sup>* itself satisfies the Liouville equation, i.e., the continuity equation for the entropic phase space density in the phase space. It thus evolves in the microscopic particle phase space like a dissipationless fluid. This is just another expression for the fact that (classically), there are no microscopic sources of entropy, nor is there any entropic field. Entropy is just disorder in the particles.

However, the microscopic entropic density in phase space is not entirely independent of any disorder. It acts back on itself via the integral entropy force term contained in the above Hamiltonian H*Na* . It provides an entropy potential contribution *US* = *TS* to the Hamiltonian with *S*, the momentum space integral of the entropy phase space density <sup>S</sup>*<sup>m</sup> <sup>a</sup>* {N *<sup>m</sup> <sup>a</sup>* }, which itself is a function of the phase space density <sup>N</sup> *<sup>m</sup> <sup>a</sup>* . Though the structure of the kinetic equation for the entropy density <sup>S</sup>*<sup>m</sup> <sup>a</sup>* remains the same as that of the phase space density <sup>N</sup> *<sup>m</sup> <sup>a</sup>* , both containing the entropy force term and becoming integro-differential equations, the phase space density is determined by the integral entropy density through the entropy force. This force is obtained by adding up all contributions over all phase space. This shows that both the kinetic equation for the particle density and the kinetic equation for the entropy density in phase space must be solved together as both are intimately related.

One can interpret this result in the way that the holographic reaction of the integral entropy on the evolution of the phase space density of the entropy appears like an elementary entropy source. This is not unsatisfactory, because it can hardly be expected that a microscopic source of entropy would exist as there are no entropy charges and no entropy fields in the world, at least not classically. Instead, the elementary source arises from the non-linear self-interaction of the entropy. This is a rather important conclusion in that, presumably, little will be changed when including quantum effects in a quantum mechanical treatment, making the transition to quantum statistical mechanics.
