**5. Dissipative Terms**

In order to complete the theory, one needs to express the two dissipative terms in the final kinetic Equation (32). The collision term G*ab* is of the same structure as the Coulomb collision term [21,33]. It adds to the latter:

$$\mathcal{G}\_{ab}^{\mathbb{S}} = T \sum\_{b} n\_{b} \nabla\_{\mathbf{q}} \cdot \int d^{6} \mathbf{x}' \delta(\mathbf{q} - \mathbf{q}') \frac{\partial}{\partial \mathbf{p}} \mathbf{g}\_{ab}(\mathbf{x}, \mathbf{x}', t) \tag{36}$$

In the particular case that the correlation *gab* does not contain any singularity, this expression has no singularity at **q** = **q** other than that in the derivative of the delta-function, which replaces **q** → **q** in the correlation function *gab* when the integration is carried out. The remaining expression becomes:

$$\mathcal{G}\_{ab}^{S}(\mathbf{q}, \mathbf{p}, t) = T \frac{\partial}{\partial \mathbf{p}} \cdot \sum\_{b} n\_{b} \int d^{3}p' \,\nabla\_{\mathbf{q}} \mathcal{g}\_{ab}(\mathbf{q}, \mathbf{p}, \mathbf{p}', t) \tag{37}$$

The presence of the entropy force thus contributes to the collision term via the particle correlation function. Clearly, due to the strong Coulomb force at short distances, the particle interaction on the short scales is dominated by the Coulomb force. However, at distances larger than the Coulomb collision length, the entropic collisional interaction remains. In a charge collisionless plasma, for instance, the Coulomb term causes charge screening felt inside the Debye sphere and eliminates the microscopic electric field between the charges on scales larger than the Debye length *λD*. On such scales, entropic dissipation might enter the scene. Since it is proportional to temperature *T* and also number density *ρ*, it has the character of a collisional contribution of the pressure in the inhomogeneities of the entropy.

One may take notice that the spatial gradient operator can be taken out of the integral in the last expression. This allows writing:

$$\mathcal{G}\_{ab}^{S}(\mathbf{q}\_{\prime}\mathbf{p}\_{\prime}t) = T\,\nabla\_{\mathbf{q}} \cdot \frac{\partial}{\partial\mathbf{p}} G\_{a}^{S}(\mathbf{q}\_{\prime}\mathbf{p}\_{\prime}t) \tag{38}$$

where we introduced the entropic correlation integral:

$$G\_a^S(\mathbf{q}, \mathbf{p}, t) = \sum\_b n\_b \int d^3 p' \, g\_{ab}(\mathbf{q}, \mathbf{p}, \mathbf{p}', t) \tag{39}$$

We will return to the discussion of this correlation integral below, because it contains the most interesting effect introduced by the entropy force.

In general, the correlation will contain contributions from the forces that depend on local charges. This leads from the field equations to singularities in the correlation function, as the example of the Coulomb force suggests. The simplified form of the correlation term in the dissipation function retains the form in Equation (36) and must be explicitly spelled out. Exchanging the derivatives, this can be written as:

$$\mathcal{G}\_{ab}^{S} = T \frac{\partial}{\partial \mathbf{p}} \cdot \sum\_{b} n\_{b} \nabla\_{\mathbf{q}} \int d^{6} \mathbf{x}' \delta(\mathbf{q} - \mathbf{q}') \mathcal{g}\_{ab}(\mathbf{x}, \mathbf{x}', t) \tag{40}$$

Let us now turn to the remaining term C*S a* . From Equations (27) and (28), one realizes that it contains the product of three microscopic phase space densities before taking the ensemble average. This complicates its calculation. In analogy to Equations (30) and (31), it requires the introduction of higher order correlations. Formally, this is quite simple as it has been pioneered by Klimontovich [21]

how one would have to deal with it in this case. We need the third-order ensemble average, which becomes in the same way as (31):

$$\begin{aligned} \left< \mathcal{N}\_a^m \mathcal{N}\_b^m \mathcal{N}\_c^m \right> &= n\_a n\_b n\_c f\_{abc}(\mathbf{x}, \mathbf{x}', \mathbf{x}') \\ &+ \delta\_{ab} n\_a n\_c \delta(\mathbf{x} - \mathbf{x}') f\_{ac}(\mathbf{x}, \mathbf{x}'') \\ &+ \delta\_{ac} n\_a n\_b \delta(\mathbf{x} - \mathbf{x}'') f\_{ab}(\mathbf{x}, \mathbf{x}') \\ &+ \delta\_{bc} n\_a n\_c \delta(\mathbf{x}' - \mathbf{x}'') f\_{ac}(\mathbf{x}, \mathbf{x}'') \\ &+ \delta\_{ab} \delta\_{bc} \delta(\mathbf{x} - \mathbf{x}') \delta(\mathbf{x} - \mathbf{x}'') f\_a(\mathbf{x}) \end{aligned} \tag{41}$$

For convenience, we dropped the common variable *t*. The two- and three-particle distribution functions *fab*, *fabc* read:

$$\begin{aligned} f\_{ab}(\mathbf{x}, \mathbf{x}') &= & f\_{a}(\mathbf{x}) f\_{b}(\mathbf{x}') + g\_{ab}(\mathbf{x}, \mathbf{x}') \\ f\_{abc}(\mathbf{x}, \mathbf{x}', \mathbf{x}'') &= & f\_{a}(\mathbf{x}) f\_{b}(\mathbf{x}') f\_{c}(\mathbf{x}'') \\ &+ & f\_{a}(\mathbf{x}) g\_{bc}(\mathbf{x}', \mathbf{x}'') + f\_{b}(\mathbf{x}') g\_{ac}(\mathbf{x}, \mathbf{x}'') \\ &+ & f\_{c}(\mathbf{x}'') g\_{ab}(\mathbf{x}, \mathbf{x}') + g\_{abc}(\mathbf{x}, \mathbf{x}', \mathbf{x}'') \end{aligned} \tag{42}$$

With their help, the dissipation function can be constructed. However, its structure simplifies substantially because in our case, on the left in the first line in Equation (41), the microscopic phase space densities of index *b* and *c* are identical, and only the *fab* contributes. Hence, the ensemble average becomes:

$$
\left< \mathcal{N}\_a^m \mathcal{N}\_b^m \mathcal{N}\_c^{\prime m} \delta\_{bc} \delta(\mathbf{x}^{\prime} - \mathbf{x}^{\prime\prime}) \right> = n\_a n\_b f\_{ab}(\mathbf{x}, \mathbf{x}^{\prime}) \frac{1}{N\_b} \tag{43}
$$

Therefore, only the two-particle distribution function *fab* would be relevant in the determination of the dissipative term, i.e., in the first line in Equation (42). This is, however, the same as what we already used in Equation (31). To this result, one has to apply the operation of space and momentum differentiation. Hence, the collisionless dissipative term contributed by the entropy force is of the same kind as the collision term <sup>G</sup>*<sup>S</sup> ab* it contributes, though being of a different sign. In other words, the two terms would cancel to first order if there were not the normalization to particle number *Nb*. Since *Nb* <sup>≈</sup> *Na* 1, we find that the collisionless dissipation due to the entropy force C*<sup>S</sup> <sup>a</sup>* G*<sup>S</sup> ab* is small and can be neglected in comparison with the entropic collision term.

Thus, it is the collisional correlation term <sup>G</sup>*<sup>S</sup> ab*(**x**, *t*) (37) that is retained as a long-range collisional dissipation introduced by the presence of the entropy force. It adds to the Coulomb collisions and might become important on the large scales much larger than the Debye scale or any other inter-particle interaction scale. this important result suggests that the entropic dissipation is a mesoscale, respectively macro effect. We should, however, point out here that we are still dealing with a non-relativistic theory. At large scales, transport and propagation of information, respectively entropy, cannot be neglected anymore, and the theory has to given a covariant relativistic formulation.
