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Active microscopic objects, e.g., an enzyme molecule, are modeled by the oddLangevin system, in which energy injection from the substrate to the enzyme isdescribed by the antisymmetric part of the elastic matrix. By applying theOnsager-Machlup integral and large deviation theory to the odd Langevin system,we can calculate the cumulant generating function of the entropy change of athermal bath due to the state transition. For an $N$-component system, weobtain a formal expression of the cumulant generating function and demonstratethat the oddness $\lambda$, which quantifies the antisymmetric part of theelastic matrix, leads to higher-order cumulants that do not appear in a passiveelastic system. To demonstrate the effect of the oddness under the concreteparameter, we analyze the simplest two-component system and obtain the optimaltransition path and cumulant generating function. The cumulants obtained fromexpansion of the cumulant generating function increase monotonically with theoddness. This implies that the oddness causes the uncertainty of stochasticstate transitions.

Entropy change due to stochastic state transitions of odd Langevin system