Kaniadakis Deformed Distribution and Statistics of Non-Linear Kinematics and Function - Inverse Function Pairs
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint Journal |
| Terbitan: |
, 2020
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/3631385 |
Daftar Isi:
- Addendum Jan. 30, 2020 On p. 13 of (1), an equation equivalent to ln(k(f1)) + ln(k(f2)) = ln(k(f1')) + ln(k(f2')) is listed and kinetic energy conservation and particle number conservation assumed. Thus, if ln(k) is the inverse of f1 and f1 is a function of e-u, one immediately obtains the energy conservation equation. This is the same process used for the Maxwell-Boltzmann case where k is the identity function. Kaniadakis (1) has introduced a statistics of non-linear kinematics based on a generalized Fokker-Planck type equation. He introduces a function: K= Sk - 1/T (E-uN) ((1)) where:Sk =- Integral phase space Integral df ln(k(f(x)) ((2)) Here f(x) is the distribution with x=1/T(e-u), T being the temperature and u the chemical potential. “k” is a function related another function G of f1 and f2, where G(f,f’) is related to the transition rate. For Maxwell-Boltzmann two body scattering, k is the identity function. Kaniadakis condition for “equilibrium” is: d/df K = 0. He further postulates that: Sk= Integral phase Integral df [lnk] (af). Here lnk(af)= ln(k(f(x), with lnk being a new function of f, in particular a generalized log function 1/2k (fk-f-k). The equilibrium solution satisfying dK/df=0 is expk(x) where expk is the generalized log function. In performing dK/df, -1/T (E-uN) is written as: -1/T Integral phase f f(e-u) (e-u) so it appears as a “constraint”. In an earlier note, we argued that in Maxwell-Boltzmann statistical mechanics, one often sees the maximization of entropy subject to the energy constraint i.e. d/df [ - f ln(f) - b1 (e-u)f ] = 0 ((3)) This is solved to obtain: f= exp(-1/T (e-u)) We argue, however, that an f which is the inverse of ln is being found with the argument of f being 1/T(e-u). Then, one immediately has a solution. We argue in this note, that the same thought process is occurring in the Kaniadakis approach. In other words: dK/df = 0 = d/df [ df ln(k(f)) - 1/T ( (e-u) f ] ((4)) Thus, one finds a function ln(k) which is the inverse of f so ln(k(f))= (e-u). The generalized log i.e. ln-k = 1/2k (fk-f-k) is already known from mathematics. For k->0, it becomes the regular log. Thus, one need only find its inverse function to obtain the equilibrium distribution. In Kaniadakis theory, it is postulated that ln(k(f)) is the same as ln-k(af). Here we consider a=1.