Derivation of the not-so-common fluctuation theorems

The detailed fluctuation theorems of the exact form $P(A)/P(-A)=e^A$ exist only for a handful of variables $A$, namely for work (Crooks theorem), for total entropy change (Seifert's theorem), etc. However, the so-called modified detailed fluctuation theorems can be formulated for several other thermodynamic variables as well. The difference is that the modified relations contain an extra factor, which is dependent on $A$. This factor is usually an average of a quantity $e^{-B}$, where $B\neq A$, with repect to the conditional probability distribution $P(B|A)$. The corresponding modified integral fluctuation theorems also differ from their original counterparts, by not having the usual form $\left<e^{-A}\right>=1$. The generalization of these relations in presence of feedback has been discussed briefly. The results derived here serve to complement the already existing results in fluctuation theorems. The steps leading to the quantum version of these derivations have been outlined in the appendix.