Consider first the parameters \(\alpha\) and \(\beta\) for two colliding particles. Assume that the colliding particles have initial charges \(q_{j0}\), and \(q_{i0}\), and work function difference \(\Delta \phi\). Denote the transfered charge by \(q_i(t)\) and \(q_j(t)\), and let \(t=0\) stand for the beginning of the collision. The charge transfer rate is then given by
\[\frac{dq_i(t)}{dt} \propto \frac{\varepsilon}{\delta_c e} \left( \Delta \phi - \frac{\delta_c e}{4 \pi \varepsilon} \left( \frac{ q_{i0}+q_i(t) }{r_i^2} - \frac{ q_{j0}+q_j(t) }{r_j^2} \right) \right). \label{eq:ap1}\]
For the charge rates we have
\[\begin{aligned} \frac{d}{dt}\left( q_i + q_j \right) &=& \frac{dq_i}{dt} + \frac{dq_j}{dt} \\ &=& \frac{dq_i}{dt} - \frac{dq_i}{dt} = 0.\end{aligned}\]
Hence, the charge is conserved and we may write
\[q_j(t) = q_i(0) + q_j(0) - q_i(t) = -q_i(t). \label{eq:ap2}\]
Inserting Eq. \eqref{eq:ap2} to Eq. \eqref{eq:ap1} we obtain
\[\frac{dq_i}{dt} \propto \frac{\varepsilon}{\delta_c e} \left( \Delta \phi - \frac{\delta_c e}{4 \pi \varepsilon} \left( \frac{ q_{i0} }{r_i^2} - \frac{ q_{j0} }{r_j^2} \right) - \frac{\delta_c e}{4 \pi \varepsilon} \left( \frac{1}{r_i^2} + \frac{1}{r_j^2} \right) q_i(t) \right).\]
We recognize that the \(\alpha\) and \(\beta\) are now given by
\[\beta = \frac{\varepsilon}{\delta_c e} \left( \Delta \phi - \frac{\delta_c e}{4 \pi \varepsilon} \left( \frac{ q_{i0} }{r_i^2} - \frac{ q_{j0} }{r_j^2} \right) \right)\]
and
\[\alpha = \frac{1}{4 \pi } \left( \frac{1}{r_i^2} + \frac{1}{r_j^2} \right).\]
For the walls the charge transfer rate is obtained by mirroring the particle respect to wall, and assuming an imaginary charge that has equal charge with opposite sign in the other side of the wall. Hence we substitute for \(q_{j0}\)
\[q_{j0} = -q_{i0}.\]
Therefore, the \(\alpha\) and \(\beta\) for the wall-particle collisions become
\[\beta = \frac{\varepsilon}{\delta_c e} \left( \Delta \phi - \frac{\delta_c e q_{i0}}{2 \pi \varepsilon r_i^2} \right)\]
and
\[\alpha = \frac{1}{2 \pi r_i^2 }.\]
The contact area in the softsphere model is given by
\[A_{max} = \pi \delta_{max} \frac{r_i r_j}{ r_i+r_j },\]
where \(\delta_{max}\) is the maximum overlap distance, \(r_i\) and \(r_j\) are the particle radiuses. The overlap \(\delta_{max}\) for similarly sized particles satisfies
\[\delta_{max} \ll (r_i+r_j) \frac{r_i}{r_j}.\]
The coefficient \(\alpha\) can be writen as
\[\alpha = \frac{1}{4\pi} \frac{r_i^2 + r_j^2}{r_i^2 r_j^2}.\]
Consider first a collision between two particles
\[\begin{aligned} \alpha A_{max} &=& \frac{1}{4\pi} \frac{r_i^2 + r_j^2}{r_i^2 r_j^2} \pi \delta_{max} \frac{r_ir_j}{r_i+r_j} \\ &=& \frac{1}{4} \left ( \frac{r_i \delta_{max} }{r_j (r_i+r_j)} + \frac{r_j \delta_{max} }{r_i (r_i+r_j)} \right) \\ &\ll& \frac{1}{4}\left( 1 + 1 \right) < 1.\end{aligned}\]
For wall collisions the the overlap distance satisfies \(\delta_{max} \ll r_i\), hence
\[\begin{aligned} \alpha A_{max} &=& \frac{1}{2 \pi r_i^2 } \pi \delta_{max} r_i \\ &=& \frac{\delta_{max}}{2r_i} \\ &\ll& \frac{1}{2} < 1.\end{aligned}\]