Derivation of the Onsager symmetry

[Efil_Kei]

Derivation of the Onsager symmetry in many textbooks and papers is as follows: First, assume that the correlation function of two state variables,$a_i$ and $a_j$ satifsies for sufficiently small time interval $t$ that
$$\langle a_i(t) a_j(0) \rangle = \langle a_i(-t) a_j(0) \rangle = \langle a_i(0) a_j(t) \rangle. $$
Then, transforming leftmost and rightmost expressions yields
$$\langle a_i(t) a_j(0) \rangle = \langle a_i(0) a_j(0) \rangle -k_B L_{ij},$$
and
$$\langle a_i(0) a_j(t) \rangle = \langle a_i(0) a_j(0) \rangle -k_B L_{ji},$$
respectively, since $\langle \dot{a_i}a_j \rangle= -k_B L_{ij}$, where $k_B$ is the Boltzmann constant. It follows from the two expressions that
$$L_{ij}=L_{ji},$$
I have a question here. If we equate the leftmost one with the one in the center, not with the rightmost one, in the first equation, it can be obtained that
$$\langle a_i(0) a_j(0) \rangle -k_B L_{ij}=\langle a_i(0) a_j(0) \rangle +k_B L_{ij},$$
then this leads to
$$L_{ij}=-L_{ij}.$$
This means $L_{ij}$ must be 0 and contradicts the Onsager's general results.

Am I mathematically wrong somewhere above or am I missing some physical logic?

 

[TeethWhitener]

This is very odd. In Onsager's original paper:
https://journals.aps.org/pr/abstract/10.1103/PhysRev.38.2265
he simply uses the relation:
$$\langle a_i(t) a_j(0) \rangle = \langle a_i(0) a_j(t) \rangle$$
to denote the condition of microscopic reversibility (equation 1.2, ibid). The relation:
$$\langle a_i(t) a_j(0) \rangle = \langle a_i(-t) a_j(0) \rangle $$
for small $t$ is only true at a local extremum; that is, where the derivative vanishes. In fact, looking at your derivation of Onsager's relations in post 1, this second relation is not used anywhere. I'm not sure why it would have been incorporated into the papers you're reading.