*5.2. Fermions-Like Quons*

The kinetic of fermion-like quons is depicted by the nonlinear Fokker-Planck equation

$$\frac{\partial f}{\partial t} = \nabla \left( D \, m \, \beta \, \gamma\_{\text{Fermi}}(f) \, \overline{\nu} + D \, \Omega\_{\text{Fermi}}(f) \, \nabla f \right), \tag{61}$$

with a nonlinear drift term

$$
\gamma\_{\text{Fermi}}(f) = \gamma\_+ \, q^f + \gamma\_- \, q^{-f} + \gamma\_0 \, \, \, \, \, \tag{62}
$$

where

$$\gamma\_{+} = -\frac{1}{(q-1)^{2}} \, , \qquad \gamma\_{-} = -\frac{q}{(q-1)^{2}} \, , \qquad \gamma\_{0} = -\gamma\_{+} - \gamma\_{-} \, , \tag{63}$$

and a nonlinear diffusive term

$$
\Omega\_{\text{Fourier}}(f) = \omega\_0 + \omega\_+ \neq^f + \omega\_- \neq^{-f},\tag{64}
$$

*Entropy* **2019**, *21*, 841

where

$$
\omega\_0 = \frac{2\,q \,\ln q}{(q-1)^2}, \qquad \omega\_+ = -\frac{\ln q}{(q-1)^2}, \qquad \omega\_- = -\frac{q \,\ln q}{(q-1)^2}.\tag{65}
$$

and reduces to a constant ΩFermi(*f*) → 1 in the *q* → 1 limit. Again, type I quons undergo classical diffusive process governed by a nonlinear diffusion current.

The steady state now follows from relation

$$\frac{q^f - 1}{q^{1-f} - 1} = e^{-c},\tag{66}$$

that solved for *f*() gives

$$f\_{\text{Fourier}}(\varepsilon) = \frac{1}{\ln q} \ln \left( \frac{1 - e^{-\varepsilon}}{2} + \sqrt{\left(\frac{1 - e^{-\varepsilon}}{2}\right)^2 + q \, e^{-\varepsilon}} \right) \,. \tag{67}$$

and reduces to the stationary distribution of fermion particles in the *q* → 1 limit.
