= Solution
Here $\mathbf j$ is electric <current density>, $n$ the <number density> of independently mobile <ions>, $Q$ their charge, $T$ absolute temperature and $k_B$ the <Boltzmann constant>. The sum runs over all nearest-neighbour destinations $j$ of site $i$, and $\mathbf r_{ij}$ is the displacement from $i$ to $j$. The applied field is $\boldsymbol{\mathcal E}$, $\omega_0$ is an attempt <frequency>, and $\Delta E_{ij}$ is the zero-field <reaction energy barrier> for the indicated jump. With the normalization of the supplied equation,
$$
\boxed{f(\omega_0,\Delta E_{ij},T)=\omega_0\exp[-\Delta E_{ij}/(k_BT)]}.
$$
It has units s$^{-1}$; it is not an equilibrium occupation <probability>. The current formula is the linear-response result of a small field-induced asymmetry in forward and reverse rates, requiring $|Q\mathbf r_{ij}\cdot\boldsymbol{\mathcal E}|\ll k_BT$. Use its explicit factor $1/2$ consistently rather than absorbing it into a redefined jump rate.
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