Fluctuation theorem for hidden entropy production

Elimination of seemingly unnecessary variables in Markovian models may cause a difference in the value of irreversible entropy production between the original and reduced dynamics. We show that such difference, which we call the hidden entropy production, obeys an integral fluctuation theorem if all variables are time-reversal invariant, or if the density function is symmetric with respect to the change of sign of the time-reversal antisymmetric variables. The theorem has wide applicability, since the proposed condition is mostly satisfied in the case where the hidden fast variables are equilibrated. The main consequence of this theorem is that the entropy production decreases by the coarse-graining procedure. By contrast, in the case where a stochastic process is obtained by coarse-graining a deterministic and reversible dynamics, the entropy production may increase, implying that the integral fluctuation theorem should not hold for such reductions. We reveal, with an explicit example, that the nonequilibrated time-reversal antisymmetric variables play a crucial role in distinguishing these two cases, thus guaranteeing the consistency of the presented theorem.

Figure 1

Scheme of the multibaker map. The map considers the label of squares ($r$), two-dimensional coordinates inside the square ($\xi ,\eta$), and the discretized velocity ($v$). The map is area-preserving and time-reversal invariant since the transition rules in the $v=+$ and $v=-$ systems are exactly opposite to each other [Eqs. (17), (19)].