- #1
Efil_Kei
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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?
$$\langle a_i(t) a_j(0) \rangle = \langle a_i(-t) a_j(0) \rangle = \langle a_i(0) a_j(t) \rangle. $$
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?