### Applied Semi-Markov Processes

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Add to Clipboard. Add to Collections. Order articles. Fetching bibliography My Bibliography Add to Bibliography. Generate a file for use with external citation management software. Create File. Published by Oxford University Press. Images from this publication. See all images 5 Free text. Illustration of pooled semi-Markov process. A and E are the states and have different length distributions in the two semi-Markov processes. Inferring parental genomic ancestries using pooled semi-Markov processes.

Comparisons of the estimated genomic ancestry of each parent with the ground truth.

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The top row is for Mexican samples, each dot corresponding to one parent: European proportions left , Native American proportions middle and African proportions right. The bottom row is for Puerto Rican samples: European proportions left , Native American proportions middle and African proportions right. Comparisons of the estimated and true average block length of each ancestry type.

The top row is for Mexican samples, each dot is one parent: average European block length left , average Native American block length middle and average African block length right. The bottom row is the average block length in Puerto Ricans for European left , Native American middle and African right ancestries.

On simulated Mexican trios, comparisons of the estimated genomic ancestry of each parent with the ground truth. The total entropy production rate at steady-state is We would like to assign a partial entropy production rate when one only has access to a limited set of states and transitions. To be concrete, we focus on the scenario depicted in Fig. Previously, two approaches for assigning partial entropy production rate in such a case have been defined in the literature, both of which provide a lower bound on the total entropy production rate 56 : the passive partial entropy production rate due to Shiraishi and Sagawa 52 , and the informed partial entropy production rate due to Polettini and Esposito 53 , Although the informed partial entropy production was proven to produce a better estimation of the total dissipation compared with the passive partial entropy production, i.

Hence, even if the system is in a nonequilibrium steady-state, when the current over the observed link is zero, these estimators cannot give a nontrivial lower bound on the total entropy production.

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To be fair, we point out that each estimator uses different information. For the KLD estimator, we assume that the observer can record whether the system is in states 1 or 2, or in the hidden part of the network, H , which is a coarse-grained state representing the unobserved subsystem.

Jumps between states 1 and 2 follow Poissonian statistics, as in a general continuous-time Markov process, with the same rates as in the original network. On the other hand, jumps from H to 1 or 2 are not Poissonian and depend on the state just prior to entering the hidden part.

Similar to Eq. In this case, the affinity estimator, Eq. As expected, this term vanishes when detailed balance holds and the current is zero see Methods. Applying Eq. Let us apply the hidden semi-Markov entropy production framework to a specific example of a network with four states, two of which are hidden Fig. For each value of F , we contracted the dynamics to the three states, 1, 2, and H Fig. We further calculated both the passive- and informed-partial entropy production rates to compare all the estimators to the total entropy production rate Fig.

Our results clearly demonstrate the advantage of using the waiting time distributions for bounding the total entropy production rate compared with the two other previous approaches. Our framework can reveal the irreversibility and the underlying dissipation, even when the observed current vanishes, without the need of manipulating the system.

Hidden network. The resulting observed semi-Markov dynamics c. The KLD entropy production rate was also estimated from simulated experimental data, obtained by sampling random trajectories of 10 7 jumps using the Gillespie algorithm The simulated trajectories Fig.

## Applied Semi-Markov Processes | SpringerLink

A slight modification of the case analyzed in the previous section allows us to study molecular motors with hidden internal states. We are interested in the schemes previously sketched in Fig. Analogous to the previous example, the observed dynamics is a second-order semi-Markov process. Notice that the probabilities are normalized as. The contribution due to the relative entropy between waiting time distributions is.

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Let us apply the molecular motor semi-Markov entropy production framework to a specific example. We consider the following two-state molecular motor model of a power stroke engine that works by hydrolizing ATP against an external force F , see Fig. Molecular motor. The chemical energy is used to power the motor against and external force F. Notice that the distributions are only different in the presence of a chemical drive. The state of the motor is described by its physical position and its internal state, which can be either active, that is, capable of hydrolyzing ATP, or passive.

Owing to the translational symmetry in the system, all the spatial positions are essentially equivalent. An example of a trajectory is illustrated in Fig.

## Inferring broken detailed balance in the absence of observable currents

The resulting waiting time distributions are shown in Fig. We have analytically derived an estimator of the total entropy production rate using the framework of semi-Markov processes. The novelty of our approach is the utilization of the waiting time distributions, which can be non-Poissonian, allowing us to unravel irreversibility in hidden degrees of freedom arising in any time-series measurement of an arbitrary experimental setup.

Our estimator can thus provide a lower bound on the total entropy production rate even in the absence of observable currents. Hence, it can be applied to reveal an underlying nonequilibrium process, even if no net current, flow, or drift, are present. We stress that our method fully quantifies irreversibility. Owing to the direct link between the entropy production rate and the relative entropy between a trajectory and its time reversal, as manifested in Eq. One can consider utilizing other properties of the waiting time distribution to bound the entropy production, through the thermodynamics uncertainty relations 4 , 61 , 62 , for example.

We have illustrated our method with two possible applications: a situation where only a subsystem is accessible to an external observer and a molecular motor whose internal degrees of freedom cannot be resolved. Using these examples, we have demonstrated the advantage of our semi-Markov estimator compared with other entropy production bounds, namely, the passive- and informed-partial entropy production rates, both of which vanish at stalling conditions. In summary, we have developed an analytic tool that can expose irreversibility otherwise undetectable, and distinguish between equilibrium and nonequilibrium processes.

This framework is completely generic and thus opens opportunities in numerous experimental scenarios by providing a new perspective for data analysis. When the system is a particle jumping between the sites of a lattice, the semi-Markov process is also called a CTRW. It is also called the waiting time distribution.

Its average. Semi-Markov processes arise in a natural way when states are removed or decimated from Markov processes with certain topologies.

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## Inferring parental genomic ancestries using pooled semi-Markov processes.

For this purpose, we have to sum over all possible paths from 1 to 2 through the decimated network. The probability that such a path occurs with an exact duration t is. This is a convolution.