Stochastic thermodynamics of interacting degrees of freedom: Fluctuation theorems for detached path probabilities

Systems with interacting degrees of freedom play a prominent role in stochastic thermodynamics. Our aim is to use the concept of detached path probabilities and detached entropy production for bipartite Markov processes and elaborate on a series of special cases including measurement-feedback systems, sensors, and hidden Markov models. For these special cases we show that fluctuation theorems involving the detached entropy production recover known results which have been obtained separately before. Additionally, we show that the fluctuation relation for the detached entropy production can be used in model selection for data stemming from a hidden Markov model. We discuss the relation to previous approaches including those which use information flow or learning rate to quantify the influence of one subsystem on the other. In conclusion, we present a complete framework with which to find fluctuation relations for coupled systems.

Figure 1

Setup of a joint trajectory. Subsystems $x$ and $y$ update one after the other.

Figure 2

Decomposition of the joint trajectory into two subtrajectories with the other process acting as a time-dependent protocol.

Figure 3

Reverse joint trajectory.

Figure 4

Setup of a measurement-feedback system. Top: the system $x$ is under feedback control from the controller $y$. In every time step, $y$ represents a measurement of the current state of $x$ which is then used to influence its dynamics. Bottom: time-reversed process.

Figure 5

Setup of a sensor. Top: sensor $x$ is influenced by the environment $y$ which is modeled as a stochastic process. Bottom: time-reversed process.

Figure 6

Setup of a hidden Markov model. Top: $y$ is a stochastic process that can only be observed through noisy measurements $x$. Bottom: time-reversed process.

Figure 7

Convergence of the IFT for the auxiliary entropy production for our specific hidden Markov model. The trajectories were generated using the following parameters: $T=5$, $a=0.3$, $b=0.2$, $p_0=0.1$, and $\epsilon =0.2$. We analyzed the IFT using the generating parameters (black line/ third from the top) and three sets of parameters in which one model parameter differs from the generating parameter (other lines). In the interest of clarity, error bars are only shown for larger sample sizes.