Effective Thermodynamics for a Marginal Observer

Thermodynamics is usually formulated on the presumption that the observer has complete information about the system he or she deals with: no parasitic current, exact evaluation of the forces that drive the system. For example, the acclaimed fluctuation relation (FR), relating the probability of time-forward and time-reversed trajectories, assumes that the measurable transitions suffice to characterize the process as Markovian (in our case, a continuous-time jump process). However, most often the observer only measures a marginal current. We show that he or she will nonetheless produce an effective description that does not dispense with the fundamentals of thermodynamics, including the FR and the 2nd law. Our results stand on the mathematical construction of a hidden time reversal of the dynamics, and on the physical requirement that the observed current only accounts for a single transition in the configuration space of the system. We employ a simple abstract example to illustrate our results and to discuss the feasibility of generalizations.

Figure 1

We consider the model in Eq. (6) with rates $1=w_{31}=w_{14}=w_{32}=w_{43}=w_{21}/(1+x)$ and $10=w_{13}=w_{41}=w_{23}=w_{34}{e}^{-y}=w_{12}{e}^{-x}/(1+x)$. Main frame: For $x=y=0$, we plot the SCGF $\lambda (q)$ and its symmetric $\lambda (\tilde{F}-q)$; the two do not coincide and only meet at 0, $\tilde{F}/2$, and $\tilde{F}$, showing that the FR does not hold. We also plot (dashed) $\tilde{\lambda }(\tilde{F}-q)$, which perfectly coincides with $\lambda (q)$, showing that the FR is restored by the hidden-TR dynamics. Inset: When only transition 1–2 is counted, for $y=0$, straight lines are the loci where $\lambda (q,x)\equiv 0$; apart from the trivial $q=0$ and $x=-1$ lines, the diagonal line corresponds to the effective affinity $\tilde{F}(x)$, showing that it is linear in $x$. The curvy lines are obtained when we also count 34 [by setting ${\phi }_{ij}={\delta }_{i,1}{\delta }_{j,2}+{\delta }_{i,3}{\delta }_{j,4}-(i\leftrightarrow j)$ and $y=x$]. The effective affinity is no longer linear in $x$.