**6. Kinetic Equation for Fluctuations**

The remaining problem is the behavior of fluctuations. These are defined as deviations in the one-particle phase space distribution *fa* from its mean "equilibrium" value ¯ *fa* as:

$$
\delta f\_a = f\_a - \bar{f}\_{a\prime} \qquad \overline{\delta f}\_a = 0 \tag{44}
$$

The evolution equation of the fluctuations is obtained from Equation (33) via subtracting the averaged kinetic equation:

$$\begin{aligned} \frac{\partial \bar{f}\_a}{\partial t} &+ \frac{\mathbf{p}}{m\_a} \cdot \nabla\_\mathbf{q} \bar{f}\_a - T \sum\_b n\_b \int d^3 p' \times \\ \times \nabla\_\mathbf{q}^b &\cdot \frac{\partial}{\partial \mathbf{p}} \left[ \bar{f}\_a(\mathbf{q}\_\prime \mathbf{p}\_\prime t) \bar{f}\_b(\mathbf{q}\_\prime \mathbf{p}', t) + \overline{\delta f\_a \delta f\_b} \right] = \overline{\mathcal{G}\_{ab}^S} \end{aligned} \tag{45}$$

The mean collision term on the right contains all the contributions of the correlations of the mean and fluctuating quantities. Being interested only in linear fluctuations and assuming that the collisions are weak enough to not contribute to the evolution of fluctuations, we drop this term in the following. Subtracting from the complete kinetic equation, the fluctuations obey the non-collisional equation:

$$\begin{split} &\frac{\partial \delta f\_{a}}{\partial t} + \frac{\mathbf{p}}{m\_{a}} \cdot \nabla\_{\mathbf{q}} \, \delta f\_{a} - T \sum\_{b} n\_{b} \int d^{3}p' \\ &\times \nabla\_{\mathbf{q}}^{b} \cdot \frac{\partial}{\partial \mathbf{p}} \Big[ \delta f\_{a}(\mathbf{q}\_{\nu} \mathbf{p}, t) \bar{f}\_{b}(\mathbf{q}\_{\nu} \mathbf{p}', t) \Big] = \\ &T \sum\_{b} n\_{b} \int d^{3}p' \nabla\_{\mathbf{q}}^{b} \cdot \frac{\partial}{\partial \mathbf{p}} \Big[ \bar{f}\_{a}(\mathbf{q}, \mathbf{p}, t) \delta f\_{b}(\mathbf{q}, \mathbf{p}', t) - \overline{\delta f\_{a} \delta f\_{b}} \Big] \end{split} \tag{46}$$

This expression still contains the average *δ faδ fb* of the squared fluctuations. If this is a constant on the fluctuation time scale, then the equation can be rescaled. In linear theory, it would be neglected to first order and taken into account to second order in a quasi-linear approach.

Again, carrying out the integration with respect to **p** , the last expression simplifies to:

$$\begin{aligned} \frac{\partial \delta f\_a}{\partial t} &+ \frac{\mathbf{p}}{m\_a} \cdot \nabla\_\mathbf{q} \,\delta f\_a - T \, \frac{\partial \delta f\_a}{\partial \mathbf{p}} \cdot \nabla\_\mathbf{q} \left(\sum\_b \bar{\rho}\_b(\mathbf{q}, t)\right) \\\\ \mathbf{q} &= T \sum\_{b \neq a} n\_b \int d^3 p' \nabla\_\mathbf{q}^b \cdot \frac{\partial}{\partial \mathbf{p}} \left[f\_a(\mathbf{x}, t) \delta f\_b(\mathbf{q}, \mathbf{p}', t) - \overline{\delta f\_a \delta f\_b}\right] \end{aligned} \tag{47}$$

where in the term on the right-hand side, we retained the fluctuation in the distribution function, not replacing it with the density fluctuation *δρb*(**q**, *t*) for the obvious reason that this is an equation for the fluctuations in the distribution function itself. The term on the right couples all fluctuations in the different particle components *<sup>b</sup>* <sup>=</sup> *<sup>a</sup>* to ¯ *fa*.

The linear theory is still complicated by the fact that it contains the sum over the particle correlations. Here, one must include all particles contained in the medium. Moreover, we have written here only those terms that result from the inclusion of the entropy force. To these terms, one must add the electromagnetic force terms. Since the electromagnetic and entropy forces superimpose, this does not produce any additional mixing, but simply adds those common and well-known terms that take care of the electromagnetic interactions. In this sense, the formal theory is complete. The ranges of the two different forces acting on the particle populations are vastly different because the entropy force is a collective force, which does not originate from any elementary charge. There is no singularity of the entropy that could give rise to an entropy field. The search for such singularities is outside classical physics.
