and $\delta \mu_i \in \mathbb{R}$ are some “small” perturbations. For definiteness, we assume that all subsequent expressions are first order expansions in $\delta T_i$ and $\delta \mu_i$.\[Assumption Linear transport\]

To study the transport, we consider the first order expansion of any distribution function: $$g(E; T_j, \mu_j) = g(E; T, \mu) + \frac{\partial g}{\partial T_j}(E;
T, \mu) \ \delta T_j + \frac{\partial g}{\partial \mu_j}(E;
T, \mu)  \ \delta \mu_j~.$$ Note that the three considered distribution functions – are of the form: $$\label{form g}
g(E; T_j, \mu_j) = g\left(\frac{E-\mu_j}{k_\B
  T_j}\right)~.$$ Hence, one can rewrite the derivatives of $g$, with respect to $T_j$ and $\mu_j$, in terms of the derivative of $g$ with respect to $E$, and obtain the following result: $$\label{First Order Exp}
f_j(E) \equiv f(E; T_j, \mu_j) = f(E; T, \mu) - \frac{\partial f}{\partial E}(E;
T, \mu) \ \left[\left(\frac{E-\mu}{T}\right) \delta T_j + \delta \mu_j
  \right]~.$$ Substituting this first order expansion into the expressions – for the currents, and using the relations , one sees that all terms containing $f(E; T, \mu)$ vanish and one is left with $$\begin{aligned}
I_i &=& \sum_j L_{ij}^{(0)} \, \frac{\delta \mu_j}{e} + L_{ij}^{(1)}
\, \frac{\delta T_j}{T}~,\label{The electric current}\\
J_i &=& \sum_j L_{ij}^{(1)} \, \frac{\delta \mu_j}{e} + L_{ij}^{(2)}
