Onsager Symmetry Postulate Explained

The Onsager symmetry postulate, also known as Onsager's reciprocal relations, is a key principle in irreversible thermodynamics. It deals with systems that are slightly away from thermodynamic equilibrium, where there are simultaneous flows (fluxes) of different quantities, such as heat, matter, or electric charge, driven by generalized forces, such as temperature gradients, chemical potential gradients, or electric potential gradients.

According to linear irreversible thermodynamics, the fluxes ($J_i$) are linearly related to the forces ($X_k$) by phenomenological coefficients ($L_{ik}$):

\( J_i = \sum_k L_{ik} X_k \)

Here:

• \( J_i \) represents the flux of quantity \( i \).
• \( X_k \) represents the generalized force driving the flow of quantity \( k \).
• \( L_{ik} \) are the phenomenological coefficients which describe how much the force \( X_k \) contributes to the flux \( J_i \).

The coefficients \( L_{ii} \) are the direct coefficients, relating a flux to its conjugate force (e.g., heat flux to temperature gradient). The coefficients \( L_{ik} \) where \( i \neq k \) are the cross-coupling coefficients, describing the coupling between different flows and forces (e.g., heat flow driven by a concentration gradient, which is the Soret effect, or diffusion driven by a temperature gradient, which is the Dufour effect).

Onsager's symmetry postulate states that, in the absence of magnetic fields and Coriolis forces, the matrix of phenomenological coefficients is symmetric:

\( L_{ik} = L_{ki} \)

This means the coefficient relating the force \( X_k \) to the flux \( J_i \) is equal to the coefficient relating the force \( X_i \) to the flux \( J_k \). This has profound implications for coupled transport phenomena.

We can also express the phenomenological coefficient \( L_{ik} \) as the partial derivative of the flux \( J_i \) with respect to the force \( X_k \), assuming all other forces are held constant:

\( L_{ik} = \left( \frac{\partial J_i}{\partial X_k} \right)_{X_j, j \neq k} \)

Similarly,

\( L_{ki} = \left( \frac{\partial J_k}{\partial X_i} \right)_{X_j, j \neq i} \)

Therefore, Onsager's symmetry postulate \( L_{ik} = L_{ki} \) can be written in terms of partial derivatives:

\( \left( \frac{\partial J_i}{\partial X_k} \right) = \left( \frac{\partial J_k}{\partial X_i} \right) \)

Matching this general form to the notation used in the options (using 'q' and 'Q' as indices for different types of fluxes/forces), the correct expression for Onsager symmetry postulate is:

\( \frac{\partial J_q}{\partial X_Q} = \frac{\partial J_Q}{\partial X_q} \)

This corresponds directly to the first option provided.