Coarse-Graining via Lumping: Exact Calculations and Fundamental Limitations

Fluctuations are central to stochastic thermodynamics and form the foundation of fluctuation theorems as well as thermodynamic and kinetic uncertainty relations. These results rely on accurately characterizing the full probability distributions of quantities such as entropy production and currents. In realistic systems, particularly biological ones, limited experimental resolution restricts access to microscopic dynamics, motivating the use of coarse-grained descriptions. A key question is how much information about dynamical fluctuations and time-reversal symmetry breaking survives such coarse-graining. In the case of decimation, where selected microscopic states are removed while the connectivity among the remaining ones is preserved, the full entropy production distribution can remain unaltered. Such cases are often regarded as exact coarse grainings because the reduced dynamics faithfully reproduce the statistics of the underlying process. Whether this property extends to more general forms of coarse graining remains unclear. Here we study lumping, a coarse-graining procedure that merges distinct microstates into effective states while preserving transition probabilities among the retained ones. This procedure can be carried out without approximation in the calculation of coarse-grained observables within a semi-markovian framework, and in some cases forms an exact representation of the underlying system. In other cases, the resulting dynamics are no longer exact representations of the original process. We find that while some signatures of irreversibility persist, the full entropy production distribution is generally not preserved. When lumping cuts through cyclic pathways, such as collapsing a three-state loop into two states, the mean entropy production remains but the fluctuations increase, leading to violations of the Gallavotti–Cohen fluctuation theorem and thermodynamic uncertainty relations even though the underlying Markov process obeys them. These results identify general conditions under which exact coarse-grained representations can be achieved and clarify the intrinsic limits of coarse graining, showing that formally exact procedures may still produce inexact reduced models.