This module implements a cloning algorithm similar to those described in Refs. [1], [2].

The cloning algorithm allows to measure probabilities of a class of rare events of Markov processes. More explicitly, the cloning algorithm can be used to measure probabilities of rare events that can be formulated as survival probabilities.

For example, consider the following question: For a freely diffusion particle (Wiener process) starting at $X_{t=0} = 0$, what is the survival probability $P(T)$ to never leave the interval $[-1,1]$ until some large time $t = T$?

One would naively measure this survival probability by generating a large number of sample time series (realizations of the stochastic process), and evaluating the fraction of time series that fulfill the survival condition. In our example this means first generating $N$ realizations of the Wiener process (each of which starts at the origin), and then counting how many sample time series of duration $T$ have never left the interval $[-1,1]$. If of the $N$ generated realizations, $M$ have never the interval, the ratio $M/N$ is a reasonable estimate for the survival probability, $P(T) \approx M/N$.

The problem with this approach is that the survival probability often decays exponentially. In our example it decays as $\log(P) \sim -T$, so that one needs an exponentially large number of sample time series to get an acceptable estimate of the survival probability. It is often infeasible to generate and analyze such a large number of time series.

The cloning algorithm is a means for overcoming the exponential decay of the number of surviving samples.

To do this, the algorithm divides the time interval $I = [0,T]$ into $n_{\mathrm{iterations}}$ shorter time intervals of duration $\Delta T := T/n_{\mathrm{iterations}}$, and treats the survival probability for each interval separately:

1. For the first interval $I_0 = [0, \Delta T]$, $N^{(0)}_{\mathrm{clones}}$ time series are generated using the initial conditions of the problem. Those time series are then used to estimate the survival probability naively (as discussed in the previous section) for the duration $I_0$.
2. For the second time interval $I_1 = [\Delta T, 2 \Delta T]$, $N^{(1)}_{\mathrm{clones}}$ time series are then generated, with initial conditions drawn from the final positions of the survivors from the previous iteration. The resulting time series are then used to estimate the survival probability in the time interval $I_1$.
3. Step 2 is repeated until the survival probability has been obtained up to the final time $T$.

The "cloning" in the name of the algorithm comes from the fact that at the end of each iteration, we take the surviving final states and "clone" them by drawing initial conditions for the next iteration. Note that to be able to use the final positions of each batch of trajectories as initial conditions for the next batch, we assume that the time series is Markovian.

By considering the survival problem on each interval $I_i$ separately, and periodically increasing the number of sample trajectories, we overcome the exponential temporal decay of sample time series that fulfill the survival condition.

For chosen duration $\Delta T$, the implemented cloning algorithm uses the recent decay of the survival probabiliy to estimates a reasonable value for the number of samples $N^{(i)}_{\mathrm{clones}}$ at each iteration.

For a more detailed explanation for the cloning algorithm can be found in Fig. 2 and Appendix B of Ref. [1], as well as Fig. 1 and Appendix B of Ref. Ref. [2].

Example notebooks are provided in the folder examples/:

• sojourn probability.ipynb: In this notebook, we measure the survivial probability for freely diffusive stochastic dynamics to remain within a tube of radius $R$ around a given reference path $\varphi(t)$ (in this context the surviving probability is also called sojourn probability). We evaluate the decay rate corresponding to the survival probability, and compare the measured decay rate to theoretical predictions based on the Onsager-Machlup stochastic action.
• tubular exit rate from recorded time series.ipynb: In this notebook we, from recorded time series, infer the tubular exit rate for a finite-radius tube around a reference path. The time series samples that we use are generated using the file generate samples.py, which is also located in the examples folder.

More notebooks to be added.

To install the module cloning_algorithm, as well as its requirements (NumPy, multiprocess, and h5py, clone this repository and run the installation script:

>> git clone https://github.com/juliankappler/cloning_algorithm.git
>> cd cloning_algorithm
>> pip install -r requirements.txt
>> pip install .

[1] Experimental Measurement of Relative Path Probabilities and Stochastic Actions. J. Gladrow, U. F. Keyser, R. Adhikari, and J. Kappler. Physical Review X, vol. 11, p. 031022, 2021. DOI: 10.1103/PhysRevX.11.031022.

[2] Measurement of irreversibility and entropy production via the tubular ensemble. J. Kappler and R. Adhikari. Physical Review E, vol. 105 (4), p. 044107, 2022. DOI: 10.1103/PhysRevE.105.044107.