**4. Type II Quons**

Within the formalism introduced above, let us now derive some statistics starting from deformed algebras already proposed in literature.

To start with, let us consider the *κ*-sum [43] defined in

$$\mathbf{x} \oplus \mathbf{y} = \mathbf{x}\sqrt{1 + \mathbf{x}^2 \mathbf{y}^2} + \mathbf{y}\sqrt{1 + \mathbf{x}^2 \mathbf{x}^2} \,\mathrm{,}\tag{26}$$

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

whose deformation parameter *ξ* ≡ *κ* is limited to |*κ*| ≤ 1 and the *κ*-sum recovers the standard sum in the *κ* → 0 limit.

The above *κ*-sum is the momenta relativistic additivity law of special relativity and plays a central role in the construction of *κ*-statistical mechanics [45].

According to Equation (21), the function *a*(*x*) should be determined from the relation

$$a^{-1}(\mathfrak{x}\oplus\mathfrak{y}) = a^{-1}(\mathfrak{x}) + a^{-1}(\mathfrak{y})\,. \tag{27}$$

In order to solve this functional equation we use the following identity

$$\text{If } \mathbf{x} = \frac{1}{\kappa} \sinh\left(\operatorname{arcsinh}\left(\kappa \mathbf{x}\right)\right) \text{ ,}\tag{28}$$

in the r.h.s. of Equation (26) that becomes

$$\begin{aligned} \left(\mathbf{x} \oplus \mathbf{y}\right) &= \frac{1}{\kappa} \sinh\left(\operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{x}\right)\right) \sqrt{1 + \left(\sinh\left(\operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{y}\right)\right)\right)^2} \\ &+ \frac{1}{\kappa} \sinh\left(\operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{y}\right)\right) \sqrt{1 + \left(\sinh\left(\operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{x}\right)\right)\right)^2} \\ &= \frac{1}{\kappa} \sinh\left(\operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{x}\right)\right) \cosh\left(\operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{y}\right)\right) + \frac{1}{\kappa} \sinh\left(\operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{y}\right)\right) \cosh\left(\operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{y}\right)\right) \\ &= \frac{1}{\kappa} \sinh\left(\operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{x}\right) + \operatorname{arcsinh}\left(\mathbf{x} \cdot \mathbf{y}\right)\right). \end{aligned} \tag{29}$$

This means that

$$\operatorname{arcsinh}\left(\kappa(\mathbf{x}\oplus y)\right) = \operatorname{arcsinh}\left(\kappa\mathbf{x}\right) + \operatorname{arcsinh}\left(\kappa y\right). \tag{30}$$

which forces us to define

$$a^{-1}(\mathbf{x}) = \frac{1}{\kappa} \arcsin \mathbf{h}(\kappa \mathbf{x}) \quad \Rightarrow \quad a(\mathbf{x}) = \frac{\sinh(\kappa \mathbf{x})}{\kappa} \,. \tag{31}$$

It is worth observing that function *a*(*x*), derived in our approach within the *κ*-algebra, has already been studied in literature starting from [26,27] where quon statistics of type II has been introduced from a Hermitian version of the *q*-oscillator algebra of creation and annihilation operators.

In fact, recalling that algebra of type II quons is based on the symmetric *q*-numbers

$$f[\mathbf{x}] = \frac{q^{\mathbf{x}} - q^{-\mathbf{x}}}{q - q^{-1}} \; . \tag{32}$$

which are invariant under the exchange *q* → 1/*q*; it is easy to verify as definition (32) is related to function (31) according to

$$a[\mathbf{x}] = \frac{a(\mathbf{x})}{a(\mathbf{1})}.\tag{33}$$

with *κ* = ln *q*.

Within the *κ*-algebra the generalized exponential reads *E*(*x*) ≡ exp*κ*(*x*) and analogously the generalized logarithm reads *L*(*x*) ≡ ln*κ*(*x*), where

$$\exp\_{\mathbf{x}}(\mathbf{x}) = \left(\sqrt{1 + \mathbf{x}^2 \mathbf{x}^2} + \mathbf{x} \,\mathbf{x}\right)^{1/\kappa},\tag{34}$$

$$\ln\_{\mathbb{K}}(\mathbf{x}) = \frac{\mathbf{x}^{\kappa} - \mathbf{x}^{-\kappa}}{2\kappa},\tag{35}$$

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

and fulfill relations (23) and (25), respectively, with the *κ*-sum given in (26).

The deformed-subtraction is given in

$$\mathbf{x} \ominus y \equiv \mathbf{x} \sqrt{1 + \mathbf{x}^2 y^2} - y \sqrt{1 + \mathbf{x}^2 \mathbf{x}^2} \, , \tag{36}$$

being, in this case, 0<sup>∗</sup> ≡ 0 and *x* ≡ −*x*.

The function *a*(*f*) given in Equation (31) defines univocally the function *b*(*f*) throughout Equations (18) or (19) with *c* = arcsinh(*κ*)/*κ*.

Therefore, the nonlinear kinetic underling type II quons statistics is depicted by a linear Fick diffusive current

$$j\_{\rm diff} = D \,\nabla f \,,\tag{37}$$

with a constant diffusive coefficient. In fact, it is straightforward to verify from Equation (8) that in this case Ω = 1 [46]. Thus, like standard bosons and fermions, type II quons also undergo classical diffusive process governed by a linear diffusion current.

The corresponding nonlinear Fokker-Planck equation becomes

$$\frac{\partial f}{\partial t} = \nabla \left( D \, m \, \beta \, \gamma(f) \, \overline{\nu} + D \, \nabla f \right),\tag{38}$$

where

$$
\gamma(f) = \gamma\_+ \, e^{2\kappa f} + \gamma\_- \, e^{-2\kappa f} + \gamma\_0 \, \, \, \, \tag{39}
$$

with

$$\gamma\_{+} = \frac{\pm\sqrt{1+\kappa^{2}} + \kappa}{4\kappa^{2}} \; , \qquad \gamma\_{-} = \frac{\pm\sqrt{1+\kappa^{2}} - \kappa}{4\kappa^{2}} \; , \qquad \gamma\_{0} = -\gamma\_{+} - \gamma\_{-} \; . \tag{40}$$

The steady state follows from Equation (11), that in this case reads

$$\frac{\sinh(\kappa f)}{\sinh(\operatorname{arcsinh}(\kappa) \pm \kappa f)} = \epsilon^{-\varepsilon} \, , \tag{41}$$

that solved for *f*() gives

$$f(\epsilon) = \frac{1}{\kappa} \operatorname{arctanh} \left( \frac{\kappa}{\epsilon^c \mp \sqrt{1 + \kappa^2}} \right) \,. \tag{42}$$

As easy check, in the *κ* → 0 limit, functions *a*(*f*) and *b*(*f*) reduce to the one of standard bosons and fermions (12) and (13), as well as the nonlinear Fokker-Planck equation (38) reduces to Equation (14) and, in the same limit, the steady state (42) reproduces distribution (15).
