We compare dynamical fluctuations in stochastic processes with and without detailed balance. For general Markov chains, there are explicit rate functions for the joint large deviations of the empirical current and density, which can be derived either in a long-time limit (so-called level-2.5) or by considering many copies of the Markov chain (an ensemble) . For some Markov processes, such as exclusion processes, the behaviour on large length and time scales can be described by macroscopic fluctuation theory (MFT) . This theory also provides formulae for rate functions of the current and density, as well as a decomposition of the current into two orthogonal contributions that are symmetric and anti-symmetric under time reversal. In the context of sampling by Markov chain Monte Carlo (MCMC), a number of recent results indicate that systems where detailed balance is broken generically converge more rapidly to their steady states, leading to improved sampling efficiency . We show how this result is related to the existence of two orthogonal currents within MFT . Then, we show how a similar pair of currents can be defined in general Markov chains, together with their conjugate forces . This last result shows how some aspects of MFT are already present within generic Markov chains, as well as providing new insight into the improved sampling efficiency that is available on breaking detailed balance.
 Maes, Netocny, Wynants, Markov Proc. Rel. Fields 14, 445 (2008)
 Bertini, De Sole, Gabrielli, Jona-Lasinio, Landim, Rev Mod Phys 87, 593 (2016)
 Rey-Bellet, Spiliopoulos J Stat Phys 164, 472 (2016)
 Kaiser, Jack, Zimmer, J Stat Phys 168, 259 (2017)
 Kaiser, Jack, ZImmer, in preparation.