This article provides a hands-on introduction to the main concepts of our PRL paper Consistent Time Reversal and Reliable and Accurate Inference in the Presence of Memory.
In statistical physics, we often care about how much energy a microscopic system dissipates into its environment. This can be a measure of efficiency in microscopic biological systems, or a way to quantify how far a system is from equilibrium. The dissipation is directly related to the entropy-production rate, which can be estimated from a time series of the system’s states.
We describe a path of microscopic states by \(\Gamma_t=(x_\tau)_{0\leq\tau\leq t}\), and its time-reversed path by \(\theta\Gamma_t\). Reversing the path means reading \(x_\tau\) backwards as \(x_{t-\tau}\), including the durations spent in each state, as illustrated in Figure 1.
We are interested in the asymmetry between the likelihoods of observing the forward path \(\Gamma_t\) and the backward path \(\theta\Gamma_t\). More formally, this asymmetry is captured by the steady-state entropy-production rate:
\[ \dot S = k_B \lim_{t\rightarrow\infty}\frac{1}{t} \Bigg\langle\ln\frac{d\mathbb{P}[\Gamma_t]}{d\mathbb{P}[\theta\Gamma_t]}\Bigg\rangle. \]
Here the expectation is over forward paths. Thus, the more time-asymmetric the path statistics are per unit time, the larger the entropy-production rate. There is a rich literature applying \(\dot S\) to biophysical systems, such as neural activity or molecular motors; see our paper for more references.
Simplifications in the Microscopic Setting
If we can observe a stationary Markovian system’s microscopic dynamics, estimating \(\dot S\) is straightforward. The transition rates and stationary probabilities determine the entropy-production rate:
\[ \dot S_{\mathrm{microscopic}} = k_B\sum_{i,j : i\ne j}p_i w_{ij}\ln\frac{w_{ij}}{w_{ji}}, \]
where \(p_i\) is the stationary probability of state \(i\), and \(w_{ij}\) is the transition rate from \(i\) to \(j\). (For this to be well-defined, we assume thermodynamic consistency: i.e. whenever \(w_{ij}>0\), we also have \(w_{ji}>0\).) We can therefore compute \(\dot S\) just by inferring transition rates and the steady state from microscopic paths.
The Reality: Mesoscopic Observations
In practice, however, we often cannot observe the microscopic path \(\Gamma_t\) directly. Instead, we can only observe a coarse-grained version of the system: rather than resolving each microscopic state, we observe which group of microstates the system is in. We refer to those grouped, mesoscopic states as lumps, and write the observed path as \(\hat\Gamma_t\).
Strikingly, even if the microscopic process is Markovian, the coarse-grained process is generally not. The probability of leaving a lump in a given direction can depend on how the system entered it. This means that memory effects must be accounted for when estimating \(\dot S\) from mesoscopic observations.
Accounting for Memory
To account for memory, we use the \(k\)-th order estimator (Eq. (3) in our paper [1]). Setting the constant \(k_B=1\) for simplicity, the estimator is
\[ \begin{aligned} \dot S_k^{\mathrm{est}} &=\frac{1}{T}\sum_{\hat\gamma_k} \mathbb P\!\left[\hat\gamma_k^{(1:k+1)}\right]\,\ln\frac{ \mathbb P\!\left[\hat\gamma_k^{(f)}\mid\hat\gamma_k^{(1:k)}\right] }{ \mathbb P\!\left[(\theta\hat\gamma_k)^{(f)}\mid(\theta\hat\gamma_k)^{(1:k)}\right] }, \end{aligned} \tag{1}\]
where
- \(k\geq1\) is the estimator order. Each term uses \(k+1\) consecutive observed lump visits
- \(\hat\gamma_k=(\ell_1,\ldots,\ell_{k+1})\) is one such observed tuple of lump visits
- \(\mathbb P\!\left[\hat\gamma_k^{(1:k+1)}\right]\) is the stationary probability of observing that tuple
- \(\mathbb P\!\left[\hat\gamma_k^{(f)}\mid\hat\gamma_k^{(1:k)}\right]\) is the conditional probability of transitioning to the final lump given the preceding \(k\) lumps
- \(\mathbb P\!\left[(\theta\hat\gamma_k)^{(f)}\mid(\theta\hat\gamma_k)^{(1:k)}\right]\) is the corresponding conditional probability in the reversed sequence
- \(T\) is the mean residence time per observed lump visit.
We next demonstrate the estimator on an instructive example. You can do the same with our provided Python code, implemented efficiently in JAX.
Example: A driven ring
Consider the cycle graph depicted in Figure 2. Here we take six microscopic states, labelled \(1,\ldots,6\), with clockwise rate \(\omega\) and anticlockwise rate \(\kappa\), and lump neighbouring states into pairs.
By symmetry, the stationary distribution is uniform. There is a single dissipative cycle, and its exact entropy-production rate is
\[ \dot S=(\omega-\kappa)\ln\frac{\omega}{\kappa}. \]
The observed process retains memory: entering a lump from its clockwise neighbour puts the system into a different microscopic state than entering from its anticlockwise neighbour. Importantly, this system has 2nd order memory: The preceding lump together with the next lump determines the microscopic state uniquely. Thus, this example is second-order semi-Markov.
Numerical simulation of the ring
For a given \(\omega\), we simulate 512 paths with 8,192 recorded steps per path, after 256 warmup steps. We use our JAX implementation of the \(k\)-th order estimator to compute estimates for \(k=1,2,3,4\). See our code at GitHub.
Move the slider to change the driving \(\omega\). The plot shows the finite-sample estimates for orders \(k=\) 1, 2, 3, and 4. The dashed line is the exact microscopic rate, and the table compares the estimates with their analytic infinite-data limits.
As you can see, even this small simulation gives very accurate estimates. You further see that order-2 suffices, while the classical order-1 estimator loses half of the dissipation. At \(\omega=\kappa=1\) the system is at equilibrium and no entropy is being produced.
More complicated Networks
This estimator can be applied to more complicated networks, including those with hidden dissipative cycles, such as the Sierpinski-type network and the Brusselator in Figure 3.
Properties of the estimator
What do we know about the estimator’s properties? If we have an observed semi-Markov process of order \(n\), and use a \(k\)-th order estimator, how much can we “trust” the estimate? Can we at least get bounds on the true dissipation?
As we show in the paper, the answer depends on the relationship between \(k\) and \(n\). We first make the following general observation: If the lumps hide some dissipative cycles, it is not possible to directly observe “how often” the system traverses such a cycle; the observer just sees the system remaining in the same lump. Even after memory is accounted for, dissipative cycles hidden by the coarse graining generally prevent full recovery of the microscopic dissipation. However, we may hope that the estimator can provide a bound on the dissipation. Depending on \(k\) and \(n\), we may indeed have this desirable property:
- If \(k=1\), the estimator equals the memoryless estimate constructed from stationary coarse-grained transition fluxes and gives a lower bound on the microscopic dissipation, as shown by Esposito [2].
- If \(k = n\), the estimator has the same order as the process, and it gives a lower bound on the dissipation (we prove this in Theorem 1 of our Supplemental Material [3]).
- If \(k > n\), the estimator still yields the same lower bound, i.e. the estimator plateaus at \(\dot S_n^{\mathrm{est}}\) for \(k \ge n\) (we prove this in Theorem 2 of our Supplemental Material [3]).
- If \(1<k < n\), we cannot make any general statement. Interestingly, the estimator may overestimate the dissipation.
To illustrate those properties, we show in Figure 4 the estimates in dependence of the estimator’s order \(k\). Indeed, too little order \(e.g. k=2\) may overestimate the dissipation, whereas for sufficient order \(k\ge n\) the estimator plateaus and provides a lower bound (in this case equality) on the dissipation.
(No) Implications between power laws and self-similarity
Beyond the analysis of the estimators, our paper examines apparent power-law scaling of the memoryless estimate \(\dot S_M=\dot S_1^{\mathrm{est}}\) with the scale of coarse graining. Such a power law, or its exponent alone, does not identify the scales on which energy is dissipated, as we show in the paper. Insteady, accounting for memory is essential as the examples in this post and more complex examples in our paper show.
For more results and further details, read our open-access paper here.
I want to thank Tolya Kolomeisky and my supervisor Aljaž Godec for their collaboration and support. If you use our work and insights, please cite:
@article{schwarzConsistentTimeReversal,
title = {Consistent Time Reversal and Reliable and Accurate Inference in the Presence of Memory},
author = {Schwarz, Tassilo and Kolomeisky, Anatoly B. and Godec, Alja\v{z}},
journal = {Phys. Rev. Lett.},
volume = {137},
issue = {18},
pages = {187401},
numpages = {10},
year = {2026},
month = {Oct},
publisher = {American Physical Society},
doi = {10.1103/gwf7-6kvr},
url = {https://link.aps.org/doi/10.1103/gwf7-6kvr}
}This blogpost is written entirely by humans. The animation and some figures in this blog post were created with the support of LLMs.
References
[1]
T. Schwarz, A. B. Kolomeisky, and A. Godec, Consistent time reversal and reliable and accurate inference in the presence of memory, Phys. Rev. Lett. (2026).
[2]
M. Esposito, Stochastic thermodynamics under coarse graining, Phys. Rev. E 85, 041125 (2012).
[3]
T. Schwarz, A. B. Kolomeisky, and A. Godec, Supplemental material for ‘consistent time reversal and reliable and accurate inference in the presence of memory’, Phys. Rev. Lett. (2026).