Thermodynamic uncertainty relation for systems with unidirectional transitions

We derive a thermodynamic uncertainty relation (TUR) for systems with unidirectional transitions. The uncertainty relation involves a mixture of thermodynamic and dynamic terms, namely the entropy production from bidirectional transitions, and the net flux of unidirectional transitions. The derivation does not assume a steady state, and the results apply equally well to transient processes with arbitrary initial conditions. As every bidirectional transition can also be seen as a pair of separate unidirectional ones, our approach is equipped with an inherent degree of freedom. Thus, for any given system, an ensemble of valid TURs can be derived. However, we find that choosing a representation that best matches the system's dynamics over the observation time will yield a TUR with a tighter bound on fluctuations. More precisely, we show that a bidirectional representation should be replaced by a unidirectional one when the entropy production associated with the transitions between two states is larger than the sum of the net fluxes between them. Thus, in addition to offering TURs for systems in which such relations were previously unavailable, the results presented herein also provide a systematic method to improve TUR bounds via physically motivated replacement of bidirectional transitions with pairs of unidirectional transitions. The power of our approach and its implementation are demonstrated on a model for a random walk with stochastic resetting and on the Michaelis-Menten model of enzymatic catalysis.