Physical meaning of phenomenological equations for transport coefficients Luttinger in his publication about the theory of thermal transport coefficients write phenomenological equations in the form
$$
j_\alpha(r) = L_{\alpha\gamma}^{(1)}\bigg[E_\gamma-\frac{1}{e}T\nabla_\gamma(\frac{\mu}{T})\bigg]+L_{\alpha\gamma}^{(2)}T\nabla_\gamma(\frac{1}{T}) + \tilde{L}_{\alpha\gamma}^{(2)}(-\nabla_\gamma\psi) \tag{2.13a}
$$
$$
j_\alpha^E(r) = L_{\alpha\gamma}^{(3)}\bigg[E_\gamma-\frac{1}{e}T\nabla_\gamma(\frac{\mu}{T})\bigg]+L_{\alpha\gamma}^{(4)}T\nabla_\gamma(\frac{1}{T}) + \tilde{L}_{\alpha\gamma}^{(4)}(-
\nabla_\gamma\psi) \tag{2.13b}
$$
where $j_\alpha$ is an electrical current density and $j_\alpha^E$ is a thermal current density in $\alpha$ direction. $E_\gamma$ is externally applied electrical field in $\gamma$ direction. $\mu$ is chemical potential. $\psi$ is the gravitational potential field (defined by Luttinger to take into account the externally applied thermal gradient). And $L_{\alpha\gamma}^{(i)}$ are the transportation coefficients.
I want to understand the meaning of these two equations. I know that the first term in square brackets of $(2.13a)$ equation means the electrical current in $\alpha$ direction is proportional to the electric field in $\gamma$ direction. A similar argument is true for thermal current in the $(2.13b)$ equation.
What is the physical meaning of the rest of the terms in both equations?
 A: You should have a look at the Onsager Reciprocal Relations, see also this question.
They are basically the generalization of the Fick's law of diffusion, or the Fourier law of thermal conduction, or Ohm's law...
It is not possible to "derive" any non-equilibrium rate law (Fick's, Fourier's, etc...) from equilibrium thermodynamics, simply because they are beyond the scope of the theory: this is why your relations are defined "phenomenological".
It is postulated that there is a linear relation between the "fluxes" $J^i_a$ and the "forces" $\partial_i X_a$, which are gradients of some generalized potentials $X_a$:
$$
J^i_a = L^{ij}_{ab} \partial_j X_b \, ,
$$
where $i$ is a spatial index and $a$ is the index counting the number of different fluxes.
In your case you have the spatial indexes $\alpha, \gamma$ but your matrix is not a square one in $a,b$: you have 3 potentials + 1 electric field (so, 4 potentials, as the electric field should arise from a potential as well), but only 2 fluxes. To really understand why it has been constructed in that way one should go trough all the details of the paper.
The original article of Onsager (1931) is quite clear and has some concrete examples, but I personally suggest also the one of Casimir (1945).
