2.4.2 The Onsager Relations

The thermodynamics of irreversible processes postulates that, when local entropy production is represented as a sum of products of fluxes and forces, both must be linearly related to each other [22]. This can be expressed in the form

$$\frac{\delta\alpha_i}{\delta t} = \sum_j L_{i,j}\gamma_j$$ (2.1)

where the $\alpha_i$ are a set of measurable parameters of the system, the $L_{i,j}$, or "kinetic coefficients", are a function of the state of the system, depending from the external fields, and

$$\gamma_j\equiv\frac{\delta S}{\delta\alpha_j}$$ (2.2)

with $S$ being the entropy of the system.

In equilibrium the entropy S is a maximum and the $\gamma_j$ are therefore zero. Thus the $\gamma_j$ are a measure of the deviation from the equilibrium state. If the $L_{i,j}$ were to be zero for all $i\neq j$, each flux $\delta\alpha_j/\delta t$ would depend only on its own driving force $\gamma_j$, and the various processes could be considered as independent. Thus the quantities $L_{i,j}$ are a measure of the interference of the $j$th process with the $i$th process [7].

Onsager's theorem states that

$$L_{i,j}(\mathbf{B}) = L^T_{j,i}(-\mathbf{B})$$ (2.3)

where $\mathbf{B}$ is the applied magnetic field. Even if the magnetic field is reversed, the symmetry in the mutual interference of two or more irreversible processes will prevail [7,28,29].

The structure of the $L_{i,j}$ highly depends on the system. If the medium is anisotropic, the $L_{i,j}$ are tensors, and the $L_{i,j}$ become scalars if the medium is isotropic.