**4. Kinetic Equation with Entropy Force**

The entropy force acting on species *a* is the negative gradient of Equation (18). This leads to a repulsive force, independent of any charge. It adds to the potential *U* in Klimontovich's equation. Thus, taking it into account in Equation (12), it becomes clear that it does not affect the particle number and thus does not imply any important change in the microscopic *Na*-particle phase space density <sup>N</sup> *<sup>m</sup> a* . The main interest is in its effect on the one-particle kinetic phase space distribution function *fa*(**x**, *t*). This is defined through the ensemble-averaged *Na*-particle phase space density:

$$\frac{N\_a}{V\_d} f\_a(\mathbf{x}, t) = \left< \mathcal{N}\_d^m(\mathbf{x}, t) \right>\tag{19}$$

where ... indicates the ensemble average, and explicitly for the one-particle distribution:

$$f\_a(\mathbf{x}\_{a1}, t) \quad = \quad V\_a \int f\_N d^6 \mathbf{x}\_{a2} \dots d^6 \mathbf{x}\_{aN\_a} \prod\_{b \ne a} d^6 \mathbf{x}\_{b1} \dots d^6 \mathbf{x}\_{bN\_b} \tag{20}$$

$$\left< N\_a^m(\mathbf{x}, t) \right> \quad = \quad N\_a \int \delta(\mathbf{x} - \mathbf{x}\_{a1}) f\_N \prod\_a d^6 \mathbf{x}\_{a1} \dots d^6 \mathbf{x}\_{aN\_d} \tag{21}$$

with *Va* ≡ *V* the spatial volume occupied by the indistinguishable particle sort *a*. *fN* is the *N*-particle distribution function, and the integration is with respect to all indistinguishable particles *N* − 1, but one, the particle with coordinates **x***a*1, as has been defined by Klimontovich [21]. In fact, the distribution function *fN* is not explicitly given. It can be resolved on the way of sequentially stepping up the ladder from the one-particle distribution function to higher order distribution functions, which depend on one, two, three, or more indistinguishable particles.

Before proceeding to rewriting the Klimontovich Equation (12), it is necessary to investigate what happens to the entropy when performing the ensemble average implied in the former equations. We do actually not need the entropy itself, rather its spatial gradient, i.e., we need the spatial gradient of the entropy-phase space density (17). For this, we have:

$$-\nabla\_{\mathbf{q}} \mathcal{S}\_{a}^{m} = \left(1 + \log \mathcal{N}\_{a}^{\prime m}\right) \nabla\_{\mathbf{q}} \mathcal{N}\_{a}^{m} \tag{22}$$

The entropy force is the integral of the gradient of Equation (18) over the primed phase space:

$$\begin{split} \mathbf{F}\_{S}^{a}(\mathbf{q},t) &= \ & T\nabla\_{\mathbf{q}} \int d^{3}p'd^{3}q'\delta(\mathbf{q}-\mathbf{q}') \times \\ & \times \ & \mathcal{N}\_{a}^{\prime m}(\mathbf{p}',\mathbf{q}',t) \text{ log } \mathcal{N}\_{a}^{\prime m}(\mathbf{p}',\mathbf{q}',t) \end{split} \tag{23}$$

Reduction of the entropy-force term in Klimontovich's equation requires performing the ensemble average of the term:

$$\mathbf{F}\_S^d(\mathbf{q}, t) \cdot \frac{\partial}{\partial \mathbf{p}} \mathcal{N}\_d^m(\mathbf{q}, \mathbf{p}, t) \tag{24}$$

In order to do so, we need to consider different groups of particles such that:

$$\begin{split} \mathbf{F}\_{S}(\mathbf{q},t) &= \ & T \sum\_{b} \nabla\_{\mathbf{q}} \int d^{3}p' d^{3}q' \delta(\mathbf{q}-\mathbf{q}') \times \\ & \times \ & \mathcal{N}\_{b}^{m}(\mathbf{p}',\mathbf{q}',t) \text{ log } \mathcal{N}\_{b}^{\prime m}(\mathbf{p}',\mathbf{q}',t) \end{split} \tag{25}$$

This produces formally the entropy force contribution to the Klimontovich equation:

$$\begin{aligned} \mathbf{F}\_{\mathbf{S}} & \quad \frac{\partial}{\partial \mathbf{p}} \mathcal{N}\_{a}^{m} = T \nabla\_{\mathbf{q}} \sum\_{b} \int d^{3}p' d^{3}q' \delta(\mathbf{q} - \mathbf{q}') \cdot \\ & \quad \cdot \quad \frac{\partial}{\partial \mathbf{p}} \left\langle \mathcal{N}\_{a}^{m}(\mathbf{p}, \mathbf{q}, t) \mathcal{N}\_{b}^{m}(\mathbf{p}', \mathbf{q}', t) \log \mathcal{N}\_{b}^{\prime m}(\mathbf{p}', \mathbf{q}', t) \right\rangle \end{aligned} \tag{26}$$

where ... indicates the ensemble average, and we have used Equation (22). The momentum differentiation affects only terms containing the phase space density. This leads to the appearance of the logarithmic term on the right and introduces a third-order correlation term. The (*N* − 1)-particle ensemble-averaged term provides problems because it contains the logarithm of the phase space density. In a somewhat severe approximation, we may assume that the logarithm is a slowly-varying function. Its argument is smaller than one such that it can be expanded, which yields:

$$\begin{aligned} \left\langle \mathcal{N}\_a^m(\mathbf{p}, \mathbf{q}, t) \mathcal{N}\_b^m(\mathbf{p'}, \mathbf{q'}, t) \log \mathcal{N}\_b^{\prime m}(\mathbf{p'}, \mathbf{q'}, t) \right\rangle \\ \approx \left\langle \mathcal{N}\_a^m(\mathbf{p}, \mathbf{q}, t) \mathcal{N}\_b^m(\mathbf{p'}, \mathbf{q'}, t) \left( \mathcal{N}\_b^{\prime m}(\mathbf{p'}, \mathbf{q'}, t) - 1 \right) \right\rangle \end{aligned} \tag{27}$$

This generates the ensemble-averaged Klimontovich equation:

$$\begin{aligned} \frac{\partial \langle \mathcal{N}\_a^m \rangle}{\partial t} &+ \quad \frac{\mathbf{p}}{m\_a} \cdot \nabla\_q \langle \mathcal{N}\_a^m \rangle - T \nabla\_q \sum\_{b \neq a} \int d^b \mathbf{x}' \,\delta(\mathbf{q} - \mathbf{q}') \\ &\cdot \quad \frac{\partial}{\partial \mathbf{p}} \Big< \mathcal{N}\_a^m(\mathbf{x}, t) \,\mathcal{N}\_b^m(\mathbf{x}', t) \Big> = - \Big< \mathcal{C}\_a^S(\mathbf{x}, t) \Big> \end{aligned} \tag{28}$$

The average purely entropic collision term on the right collects the third-order correlations:

$$\begin{aligned} \left< \mathcal{C}\_{a}^{S}(\mathbf{x},t) \right> &= \left. T \nabla\_{q} \sum\_{b \neq a} \int d^{6} \mathbf{x}' \,\delta(\mathbf{q} - \mathbf{q}') \\ & \cdot \quad \frac{\partial}{\partial \mathbf{p}} \left< \mathcal{N}\_{a}^{m}(\mathbf{x},t) \,\mathcal{N}\_{b}^{m}(\mathbf{x}',t) \,\mathcal{N}\_{b}^{\prime m}(\mathbf{x}',t) \right> \end{aligned}$$

In these expressions, we have, for simplicity of writing, only included the entropy force term. One trivially adds the microscopic Coulomb or any other force term to this if required (Note, however, that adding the microscopic gravitational force causes problems because it remains uncompensated (As in any kinetic theory, this is an important difference between gravitation and any other force. It implies that in kinetic theory gravitation can only be included consistently in a general relativistic quantum gravitation where gravitation is balanced by quantum fluctuations.).). The entropy force term resembles the latter, but lacks a charge singularity. This is replaced by the spatial derivative of the delta-function, which appears under the integral.

The main difference is that already on this very basic level, the presence of the entropy force contributes a purely entropic dissipative term C*<sup>S</sup> <sup>a</sup>* (**x**, *t*), which has been transferred to the right in the above expression. This term arises due to the generation of entropy in the system. It is a three-particle correlation term, as will become clear below. It is caused by the logarithm in the entropy, the continuous growth of entropy in a many-particle system. Whether it can be neglected as being of higher order is a subtle question. It causes collisionless dissipation in the presence of entropy. Since this effect is non-collisional, when neglecting particle collisions, one must take care whether its neglect is allowed. Below, we show that, however, dissipation is proportional to the inverse particle number *N*−<sup>1</sup> *<sup>a</sup>* and can in most cases for very large numbers of particles be neglected.

The next step in this theory is to relate the last equation to the one-particle distribution function defined in Equation (19). Following Klimontovich [21], this is achieved via considering the fluctuations:

$$
\delta \mathcal{N}\_a^m(\mathbf{x}, t) = \mathcal{N}\_a^m(\mathbf{x}, t) - \left\langle \mathcal{N}\_a^m(\mathbf{x}, t) \right\rangle \tag{29}
$$

When ensemble-averaged, these deviations from the mean phase-space density vanish, and we have:

$$
\begin{array}{rcl}
\left< \mathcal{N}\_{a}^{m}(\mathbf{x},t) \mathcal{N}\_{b}^{m}(\mathbf{x}',t) \right> &=& \left< \mathcal{N}\_{a}^{m}(\mathbf{x},t) \right> \Big< \mathcal{N}\_{b}^{m}(\mathbf{x}',t) \Big> \\
&+& \left< \delta \mathcal{N}\_{a}^{m}(\mathbf{x},t) \, \delta \mathcal{N}\_{b}^{m}(\mathbf{x}',t) \right>
\end{array} \tag{30}
$$

We can now make use of the definition of the one-particle distribution function *fa*(**x**, *t*) by Klimontovich [21]. Define the particle density *na* = *Na*/*Va* to obtain:

$$\begin{aligned} \left< \mathcal{N}\_a^m(\mathbf{x}, t) \mathcal{N}\_b^m(\mathbf{x'}, t) \right> &= \begin{aligned} \left. n\_a n\_b \right[ f\_a(\mathbf{x}, t) f\_b(\mathbf{x'}, t) \\ + \left. \delta\_{ab}(\mathbf{x}, \mathbf{x'}, t) \right] &+ \left. \delta\_{ab} \delta(\mathbf{x} - \mathbf{x'}) n\_a f\_a(\mathbf{x}, t) \end{aligned} \tag{31} $$

Here, *gab*(**x**, **x** , *t*) is the two-particle correlation function, which results from the ensemble-averaged product of the fluctuations *<sup>δ</sup>*<sup>N</sup> *<sup>m</sup>* of the phase space density in the last term on the right in Equation (30). The three terms in the expression (31) are of the same structure as in the ordinary one-particle kinetic theory [21]. One may note that the last term, which is linear in the distribution function, simply becomes absorbed in the convective term in the kinetic equation. In non-relativistic, theory it just causes a translation. We can immediately write down the one-particle kinetic equation including the entropy force. One must, however, take care of to which terms the gradient and momentum operations apply. This yields the result:

$$\begin{split} & \frac{\partial f\_{a}}{\partial t} + \frac{\mathbf{p}}{m\_{a}} \cdot \nabla\_{\mathbf{q}} f\_{a} - T \, \nabla\_{\mathbf{q}}^{b} \sum\_{b} n\_{b} \int d^{6} \mathbf{x'} \\ & \times \delta(\mathbf{q} - \mathbf{q'}) \cdot \frac{\partial}{\partial \mathbf{p}} \left[ f\_{a}(\mathbf{x}, t) f\_{b}(\mathbf{x'}, t) \right] = \left( \mathcal{G}\_{ab}^{\mathrm{S}(\mathbf{x}, t)} - \left< \mathcal{C}\_{a}^{\mathrm{S}(\mathbf{x}, t)} \right> \right)(\mathbf{x}, t) \end{split} \tag{32}$$

which, when integrating over the primed spatial coordinate, simplifies to:

$$\begin{split} \frac{\partial f\_{a}}{\partial t} &+ \frac{\mathbf{p}}{m\_{a}} \cdot \nabla\_{\mathbf{q}} f\_{a} - T \, \frac{\partial}{\partial \mathbf{p}} f\_{a}(\mathbf{q}, \mathbf{p}, t) \\ &+ \nabla\_{\mathbf{q}} \sum\_{b} n\_{b} \int d^{3}p' \, f\_{b}(\mathbf{q}, \mathbf{p}', t) = \left( \mathcal{G}\_{ab}^{S(\mathbf{x}, t)} - \left< \mathcal{C}\_{a}^{S(\mathbf{x}, t)} \right> \right)(\mathbf{x}, t) \end{split} \tag{33}$$

In this expression on the right, the term G*ab* results from the two-particle correlation term *gab*(**x**, **x** , *t*). It corresponds to what in kinetic theory is understood as direct particle collisions. The last expression contains the integral over the primed momentum space. Only the distribution *fb* depends on this integration. It is therefore convenient to define the number density *ρ<sup>a</sup>* of species *a* as:

$$\rho\_a(\mathbf{q}, t) = n\_a \int d^3 p \, f\_a(\mathbf{q}, \mathbf{p}\_\prime t) \tag{34}$$

and the last expression just includes the global entropy force term:

$$\begin{aligned} \frac{\partial f\_a}{\partial t} &\quad + \quad \frac{\mathbf{p}}{m\_a} \cdot \nabla\_\mathbf{q} f\_a - T \left(\nabla\_\mathbf{q} \sum\_b \rho\_b(\mathbf{q}, t)\right) \cdot \frac{\partial f\_a}{\partial \mathbf{p}} \\ &= \quad \left(\mathcal{G}\_{ab}^{S(\mathbf{x}, t)} - \left<\mathcal{C}\_a^{S(\mathbf{x}, t)}\right>\right)(\mathbf{x}, t) \end{aligned} \tag{35}$$

The sum is over all particle components, implying the total number density. Thus, the entropy force term simply adds to any other potential force term in the kinetic equation. This is true already to first order in the expansion of the logarithmic term in the definition of the entropy. In the case of charged particles, such a force is the Coulomb force or the Lorentz force, when including magnetic fields. The difference is, however, that this force term does not depend on charge while acting on the microscopic particle phase space distribution. It resembles the gravitational force, but does not contain its inverse square dependence on the inter-particle distance. This is advantageous as it releases from the necessity of compensation. On the other hand, the new force term introduces another non-linearity contained in the density, which itself is the integral of the distribution function.

The entropy force resembles a pressure force on the kinetic level. With zero right-hand side in the kinetic equation, it conserves particle number. This is a rather simple result, which, of course, could have been anticipated, without reference to any complicated derivation from first principles as done here, by adding a macroscopic entropy force to the force terms.

The ensemble-averaged term C*S a* is a purely entropic lowest order in the smallness of <sup>N</sup> *<sup>m</sup> <sup>a</sup>* ) dissipation term for which, in conventional kinetic theory, no equivalence arises. This term is, however, small and thus negligible, as will be shown in the next section.

In the collisionless kinetic theory of forces between particles, any non-collisional dissipation term caused by particle interactions via their fields yields correlations, which can be neglected, respectively discussed away by comparing dissipation and collisionless scales. The entropic dissipation term instead remains because it is not caused by particle collisions, nor wave–particle interactions. There is no entropy source field that leads to the correlations between particles. Rather, it is the inhomogeneity in the macroscopic disorder that is responsible for the fluctuations and the appearance of the dissipative entropic correlations between the fluctuations leading to the dissipation term. Hence, this term remains

even under completely collisionless conditions. Once disorder exhibits spatial structure, it will always be present. In the next section, we provide the explicit versions of these two terms.
