**3. Microscopic Phase-Space Density and the Entropy Force**

So far, we just discussed the effect of the entropy force on a particular object: Schwarzschild black holes in astrophysics. We now turn to the general kinetic problem of the microscopic evolution of the particle distribution in many-particle physics. We restrict solely to classical systems, i.e., to systems that are described on the microscopic level by the classical Liouville equation, respectively its Klimontovich [21] equivalent in Equation (12) given below, and its hydrodynamic generalization [22].

Liouville's equation describes the evolution of the microscopic phase space density in *N*-dimensional phase space. On the classical elementary level of indistinguishable point charges, which have some properties like mass *ma*, possibly some charge *e* of different sign, and are distributed over a spatial volume *V* with volume element *d*3*q* and the momentum volume of element *d*<sup>3</sup> *p* can be described alternatively [21,22] by an exact known phase space density:

$$\begin{aligned} \mathcal{N}\_{a}^{m}(\mathbf{p}, \mathbf{q}, t) &= \sum\_{i=1}^{N\_{a}} \delta \left( \mathbf{p} - \mathbf{p}\_{ai}(t) \right) \delta \left( \mathbf{q} - \mathbf{q}\_{ai}(t) \right) \\ &\equiv \sum\_{i=1}^{N\_{a}} \delta \left( \mathbf{x} - \mathbf{x}\_{ai}(t) \right) \end{aligned} \tag{10}$$

which simply counts the number of particles of sort *a* in the entire 6*D*-phase space volume, such that it is normalized as: 

$$N\_a = \int d^3p \, d^3q \, \mathcal{N}\_a^m(\mathbf{p}, \mathbf{q}, t) \tag{11}$$

For a constant particle number, the time dependence is implicit in the particle trajectories **p***ai*(*t*), **q***ai*(*t*) such that integration has to be performed along all of them. One may note that the microscopic phase space density <sup>N</sup> *<sup>m</sup> <sup>a</sup>* is otherwise dimensionless. This is seen from the definition of the delta-functions, which in the integration over phase space simply count numbers, which of course means that the normalization to space and momentum is implicit to them. Later, we will make the normalization more explicit, as this will be required by reference to the entropy.

Since the assumption is that the particles are classical, then in the absence of any particle sources or losses, the particle number in phase space is conserved along all the dynamical trajectories of the particles under their mutual, as well as external forces. In this case, the continuity equation of the particles, i.e., the microscopic Liouville equation in the *Na*-particle 6*D*-phase space [21], reads simply:

$$\dot{\mathcal{N}}\_{a}^{m} \equiv \frac{\partial \mathcal{N}\_{a}^{\prime m}}{\partial t} + \frac{\mathbf{p}}{m\_{a}} \cdot \nabla\_{\mathbf{q}} \mathcal{N}\_{a}^{m} + \frac{d\mathbf{p}}{dt} \cdot \frac{\partial \mathcal{N}\_{a}^{\prime m}}{\partial \mathbf{p}} = \boldsymbol{0} \tag{12}$$

Of course, here, **p˙** = **F** is the total force that acts on the particles at their location **q** = **q***ai*(*t*) and thus on the phase space density, and the two last terms together constitute the Poisson bracket [... ] in the Liouville equation, which in *Na*-phase space, the 6*D*-phase space that within, the *Na* particles perform their trajectories, is a tautology.

The entropy force can be compared with other more conventional forces. Let, for simplicity, the total force **F** = **F***<sup>Q</sup>* + **F***<sup>S</sup>* be the sum of the entropy force and of another potential force **F***<sup>Q</sup>* = −∇*qU*, where *U*(**q**, *t*) is the force potential. The entropy force just adds the potential *TS*(**q**, *t*) to the force potential *U*. Note that the entropy-force potential is always positive, as already made use of above, because there are neither negative temperatures [23] (The absence of negative temperatures is immediately clear from the definition of the temperature *T* as proportional to the mean ensemble averaged square of the momentum fluctuations *<sup>T</sup>* ∼ (*δ***p**)2 of all particles in the volume, which clearly, is a positive definite quantity. Negative temperatures would require imaginary momenta or particle mass. There are no candidates for such particles, though experiments of neutrino oscillations provide negative mean square masses, which, however, are interpreted differently.), nor are there negative entropies. Clearly, for strong forces acting on the particles and weak entropy gradients, the entropy force is negligible. This might be the usual case. On the other hand, if on the large scale the

inter-particle forces compensate, the entropy force will remain because there is no obvious counterpart that could compensate it. For instance, when dealing with electrostatic interactions only in the absence of any external fields and forces, the microscopic force **F***<sup>m</sup> <sup>Q</sup>*(**q**, *<sup>t</sup>*) = <sup>∑</sup>*<sup>a</sup> ea*E*m*(**q**, *<sup>t</sup>*) is the Coulomb force acting on the charges *ea* <sup>=</sup> *ae* with *<sup>a</sup>* = +, <sup>−</sup> in the microscopic electrostatic field <sup>E</sup>*m*(**q**, *<sup>t</sup>*) obeying Maxwell–Poisson's equations:

$$\nabla\_{\mathbf{q}} \cdot \mathcal{E}^{\mathbf{m}} = \frac{1}{\epsilon\_0} \sum\_{\mathbf{q}} \rho\_{\mathbf{ac}}^{\mathbf{m}}(\mathbf{q}, t), \quad \mathcal{E}^{\mathbf{m}} = -\nabla\_{\mathbf{q}} \Phi\_{\mathbf{c}}^{\mathbf{m}}(\mathbf{q}, t) \tag{13}$$

with electrostatic potential Φ*<sup>m</sup> <sup>e</sup>* , and thus, *U* = ∑*<sup>a</sup> ea*Φ*<sup>m</sup> <sup>e</sup>* . On the microscopic level in phase space, the microscopic electric space charge (not the charge density) of species *a* is:

$$
\rho\_{\rm ac}^{m}(\mathbf{q},t) = \varepsilon\_{\rm a} \int d^3 p \, \mathcal{N}\_a^m(\mathbf{p}, \mathbf{q}, t), \quad \int d^3 q \rho\_{\rm ac}^m = \varepsilon\_a \mathcal{N}\_a \tag{14}
$$

It simply counts all charges in the total volume not relating them to the spatial volume *Va* yet. Summing over all species *a*, the total space charge is obtained. On average, it will be zero. The charges are moving, and there is a microscopic current:

$$\mathbf{j}\_{ae}^{m}(\mathbf{q}\_{\prime},t) = \frac{e\_{a}}{m\_{a}} \int d^{3}p \,\mathbf{p} \,\mathcal{N}\_{a}^{m}(\mathbf{p},\mathbf{q},t) \,\, = \,\ e\_{a} \mathbf{v}\_{a}(\mathbf{q},t)\mathcal{N}\_{a} \tag{15}$$

with **v***a*(**q**, *t*) the average velocity of particles in group *a*. It gives rise to an internal magnetic field, which, in the electrostatic approximation, is relativistically small and is thus neglected (e.g., [21]), though this is not completely correct, because in a linear theory of fluctuations, it should be taken into account.

Electrostatic interactions have been the subject of exhaustive investigations in the literature. Here, they serve only as another force field against which the entropy force can be compared. The striking difference is that for the entropy force, no field is generated because there is no entropy charge comparable to *ea* and, hence, no singularity that would act as the source of the entropy field. In other words, the entropy field is, in contrast to the electric field, not related to field equations and thus lacks a field theory. Disorder lacks any elementary source not being a field, at least in classical physics.

The entropy of a system entering the first law of total energy conservation is an integral quantity. In order to refer to it on the elementary level of the microscopic kinetic equation in 6*D*-phase space, one has to return to its microscopic definition as the phase space average of the probability distribution.

It is convenient to define an entropy phase space density by referring to Gibbs–Boltzmann's definition of entropy through the probability density. Entropy density will then be obtained by integrating out the momentum space coordinates in the usual way. In the definition of the phase space density of entropy, we will at this point not yet make the assumption that the phase space volume is constant, but include the spatial dependence as well. This is advantageous because it allows making use of phase space densities. Integrating out the volume can be done at a later stage. With this in mind, the "microscopic Boltzmann entropy phase-space density" of species *a* becomes (A number of other definitions or generalizations of entropy different from Boltzmann–Gibbs have been put forward in the near past [24–30], the physical, not the statistical meaning of which is not entirely clear. Though the theory could be extended to include those, we will neither refer to, nor use them in this note.):

$$\mathcal{S}\_{a\mathcal{B}}^{m}(\mathbf{p}, \mathbf{q}, t) = -\log \mathcal{N}\_{a}^{\prime m}(\mathbf{p}, \mathbf{q}, t), \quad \mathcal{N}\_{a}^{\prime m} = \mathcal{N}\_{a}^{m} / \mathcal{N}\_{a} \tag{16}$$

where the phase space density has been normalized to the total number *Na* of particles of species *a*. This makes the argument of the logarithm smaller than one, of which the negative sign takes care. A definition like this leans on Boltzmann's proposal. It is incomplete on the microscopic level because the entropy is a collective quantity, which is obtained by integrating over momentum space with the

microscopic phase space density N*<sup>a</sup>* as the weight. One thus has as microscopic *N*-particle entropy phase-space density in (**p**, **q**)-space:

$$\mathcal{S}\_a^m(\mathbf{p}, \mathbf{q}, t) = -\mathcal{N}\_a^m(\mathbf{p}, \mathbf{q}, t) \text{ log } \mathcal{N}\_a^{\prime m}(\mathbf{p}, \mathbf{q}, t) \tag{17}$$

This microscopic phase-space density of the entropy is not the entropy itself, which is a function solely of the space coordinates. The *Na*-particle entropy of sort *a* is explicitly obtained by the integration of (17) over the entire momentum phase space:

$$\begin{aligned} S\_a^m(\mathbf{q}, t) &= \int d^3 p \, S\_a^m(\mathbf{q}, \mathbf{p}, t) \\ &\equiv \int d^3 p' d^3 q' \delta(\mathbf{q} - \mathbf{q}') \, S\_a^m(\mathbf{q}', \mathbf{p}', t) \end{aligned} \tag{18}$$

and is always positive, as is easily seen from the definition of the microscopic phase space density, a positive quantity, and the above choice of entropy. Summing over all particle species *a* then gives the total entropy. Moreover, because information is transported via some field, for instance the electromagnetic field, the time under the integral in Equation (17) is the retarded time *t<sup>R</sup>* = *t* − |**q** − **q** |/*c* where *c* is the velocity of signal/information transport between the particles at locations **q** and **q** in the real-space subspace of the 6*D*-phase space [31,32]. In conventional kinetic theory, retardation is neglected because *c* is the velocity of light, and the distances between particles are usually less than *ct*. This is also assumed in the following.

There is a direct correspondence between this real space microscopic entropy density and the real space charge density *ρm*(**q**, *t*). Both enter the force term via taking the spatial gradient. The difference is that for the electric charge density, this step passes through the electric field <sup>E</sup>*m*, which is generated by the space charges. Repeated again, for the entropy, there is no such field, nor field equation in classical physics. Entropy is not a charge of some entity and thus does not generate a field. Taking its spatial gradient directly provides the force that acts on the particle at location **q**.
