*5.1. Bosons-Like Quons*

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

$$\frac{\partial f}{\partial t} = \nabla \left( D \, m \, \beta \, \gamma\_{\text{Box}}(f) \, \text{v} + D \, \Omega\_{\text{Box}}(f) \, \nabla f \right) \, , \tag{55}$$

with a nonlinear drift term

$$
\gamma\_{\text{Ros}}(f) = \gamma\_2 \, q^{2f} + \gamma\_1 \, q^f + \gamma\_0 \, \, \, \, \, \tag{56}
$$

where

$$\gamma\_2 = \frac{q}{(q-1)^2}, \qquad \gamma\_1 = -\frac{1+q}{(q-1)^2}, \qquad \gamma\_0 = -\gamma\_2 - \gamma\_1 \tag{57}$$

and a nonlinear diffusive term

$$
\Omega\_{\text{Row}}(f) = \frac{\ln q}{q - 1} q^f \, \Big|\,\,\,\,\,\tag{58}
$$

that reduces to a constant ΩBose (*f*) → 1 in the *q* → 1 limit. Therefore, differently from the symmetric case, Boson-like type I quons undergo classical diffusive process governed by a nonlinear diffusion current.

The steady state classical Boson-like quons follows from Equation (11) that in this case reads

$$\frac{q^f - 1}{q^{1+f} - 1} = e^{-x} \, , \tag{59}$$

that solved for *f*() gives

$$f\_{\text{Rows}}(\epsilon) = \frac{1}{\ln q} \ln \frac{\epsilon^c - 1}{\epsilon^c - q} \,. \tag{60}$$

As expected, in the *q* → 1 limit, the steady state of the Boson-like quons (60) reduces to the stationary distribution of bosons.
