Coarse-Graining via Lumping: Exact Calculations and Fundamental Limitations

Abstract

Detecting broken time-reversibility at micro- and nanoscale is often difficult when experiments offer limited state resolution. We introduce a lumping method that builds an effective semi-Markov model able to reproduce exactly the full entropy-production statistics of the microscopic dynamics. The mean entropy production stays accurate even when hidden current-carrying cycles are merged, though higher-order information can be unavoidably lost. In these cases, we capture violations of fluctuation theorems consistent with experiments, opening a path to novel inference strategies out of equilibrium.

Figures (3)

• Figure 1: Lumping pairs of states on a one-dimensional ring introduces memory. (a) A one-dimensional Markov ring with right and left hopping rates $r$ and $l$, respectively. Adjacent sites are lumped in pairs, producing a non-Markovian coarse-grained system where memory arises from unresolved microscopic transitions. (b) Example of a coarse-grained trajectory (green) obtained by lumping a microscopic trajectory (blue) on a six-state ring with $r=4$ and $l=1$. (c) Waiting-time distributions of the microscopic (blue) and lumped (green) systems. The lumped distribution $\Psi(t)$ (Eq. \ref{['eq:Psi_uniform_ring']}) deviates from the exponential form of the Markov process, reflecting memory effects; the inset shows the same curves on a log-linear scale.
• Figure 2: Lumping through loops and the breakdown of Gallavotti-Cohen Symmetry. (a) Three-state system coarse-grained into an effective two-state model. (b) Scaled cumulant generating functions (SCGFs) $\varepsilon(\lambda)$ of the microscopic (blue) and lumped (green) systems. The parameters are $W_{12}=3$, $W_{23}=2$, $W_{31}=1$ (counterclockwise) and $W_{13}=2$, $W_{32}=1$, $W_{21}=2$ (clockwise), placing the system out of equilibrium. Both SCGFs share the same slope at $\lambda=0$, $\partial_\lambda \varepsilon(\lambda)|_{\lambda=0} \equiv \langle\sigma\rangle$, showing that the coarse-grained model reproduces the exact mean EPR. (c) Broken Gallavotti--Cohen symmetry in the lumped system: the probability of observing negative entropy fluctuations deviates from the linear relation obeyed by the microscopic dynamics (blue). (d) Entropy-production fluctuations in the lumped system as a function of the parameter $\alpha$ in $W_{21}=\alpha\, W_{31}W_{12}/W_{13}$. (e) Difference in variances, $\mathrm{Var}[\sigma]-\mathrm{Var}'[\sigma]$, between the microscopic and lumped descriptions. At $\alpha=1$, the lumping becomes exact ($\varepsilon(\lambda)\equiv\varepsilon'(\lambda)$), and all higher-order statistics are recovered exactly.
• Figure 3: Sketch illustrating the lumping procedure of a secondary loops in a linear network. Every secondary loop is composed by a triplet of $(X,Y,Z)$ states, which are coarse-grained together in a single lumped state $\Xi$. The lumping results in a non-Markovian dynamics in which we are able to identify "self-loop" transitions of the lumped state to itself. A proper accounting of these transitions is what enables us to retain a completely exact description of the EPR in the lumped dynamics.