On the emergence of memory in equilibrium versus non-equilibrium systems

Experiments often probe observables that correspond to low-dimensional projections of highdimensional dynamics. In such situations distinct microscopic configurations become lumped into the same observable state. It is well known that correlations between the observable and the hidden degrees of freedom give rise to memory effects. However, how and under which conditions these correlations emerge remains poorly understood. Here we shed light on two fundamentally different scenarios of the emergence of memory in minimal stationary systems, where observed and hidden degrees of freedom evolve either cooperatively or are coupled by a hidden non-equilibrium current. In the reversible setting strongest memory manifests when the time-scales of hidden and observed dynamics overlap, whereas, strikingly, in the driven setting maximal memory emerges under a clear time-scale separation. Our results hint at the possibility of fundamental differences in the way memory emerges in equilibrium versus driven systems that may be utilized as a"diagnostic"of the underlying hidden transport mechanism.

Considering in particular ergodic dynamics in the sense that the probability distribution to be found in a given microscopic state at long times relaxes to a unique stationary, equilibrium or non-equilibrium, steady state from any initial condition, the extent of memory is necessarily finite and is more prominent if the hidden degrees of freedom are slow [1,2,58].Yet, even in this "well behaved", thermodynamically consistent [59] setting quite little is known about the possible ways in which the dynamics of observables can become correlated with that of hidden degrees of freedom on different time scales.A particularly intriguing question is whether there are any characteristic differences between how memory emerges in reversible versus irreversible, driven systems when observed dynamics is much faster than the hidden one.
Addressing this problem in full generality is a daunting task.Here we focus on the minimal "cooperative" setting The upper green states are lumped to the observed "ON" state and the lower orange ones to the observed "OFF" state.In order to separate the time scale of horizontal and vertical dynamics, we choose the rates (in arbitrary units) a = 0.5 and b = 25 to make vertical process always much faster than horizontal one.In the driven model (see Fig. 2) we set γ1 > 1 and γ2 = 1, whereas in the reversible "allosteric" model (Fig. 3) we chose γ1 = γ2 ≡ γ ≥ 1.A trajectory of the full dynamics (blue) and its corresponding projection (gray) are shown in panel (b).[60][61][62][63], where the microscopic dynamics is a Markov process on a planar network and we observe only the vertical coordinate whereas the horizontal transitions are hidden (see Fig. 1).This particular setting is important for understanding "active secondary transport"-transporter proteins exploiting the energy stored in transmembrane gradient of one type of molecules to transport another type against their gradients [64][65][66].Moreover, it is also relevant as the physical basis of the sensitivity of the flagellar motor in E. coli in sensing concentrations of its regulator [60][61][62]67], where two fundamentally different explanations were proposed to explain the "ultrasensitivity" of the motor's response, an equilibrium allosteric and a dissipative non-equilibrium model [61,67,68].
We are not interested in the biophysical implications of the model.Instead, we use the above two distinct settings merely as a minimal model of non-Markovian 2-state dynamics, where we can "turn on" memory in a controlled manner via an equilibrium versus non-equilibrium mechanism.We consider the scenario, where the observed dynamics is faster than the hidden (see Fig. 1a) but there is not necessarily a large time-scale separation present.Interestingly, in the driven setting we only change the fastest time scales, whereas in the reversible cooperative model we alter all time scales.In both cases we find a maximal capacity for memory, i.e. the maximal magnitude of memory saturates at a finite coupling and nonequilibrium driving, respectively.Interestingly, in the reversible setting the memory manifests strongest when the time-scale separation between hidden and observed transition becomes partially lifted and the hidden and observable time-scales overlap, whereas in the driven setting maximal memory occurs in the presence of a time-scale separation.Our results provide deeper insight into the emergence of memory in the distinct situations when observed and hidden dynamics either evolve cooperatively or become coupled by a hidden non-equilibrium current.
Setup.-We consider a 6-state continuous-time Markov process (see Fig. 1a) with generator L, whose elements L n,m are transition rates between states m → n given by where we choose a = 1/2 and b = 25 such that in the "baseline" model horizontal transitions are slower than vertical ones, i.e. the hidden dynamics relax slower than the observable ON⇄OFF transitions.We adopt the Dirac bra-ket notation and denote the transition probability from microscopic state i to microscopic state j in time t as G(j, t|i) ≡ ⟨j|e Lt |i⟩ and the stationary probability of state j as P s (j) = lim t→∞ G(j, t|i).In the baseline model with γ 1 = γ 2 = 1 the dynamics is reversible and the ON/OFF states are equi-probable in the steady state.The parameters γ 1 and γ 2 are acceleration factors when they are larger than 1.In the cooperative allosteric regime with γ 1 = γ 2 ≡ γ > 1 the dynamics obeys detailed balance, i.e.P s (i)L j,i = P s (j)L j,i , ∀i, j.Conversely, in the driven model we set γ 1 > 1, γ 2 = 1; here detailed balance is violated, i.e. ∃i, j for which P s (i)L j,i ̸ = P s (j)L j,i .
(that is, the model is irreversible yet thermodynamically consistent [59]).It turns out that L defined this way is diagonalizable, i.e. we can find a bi-orthonormal basis In this notation the steady-state probability of state i is given by We can also determine the eigenspectrum analytically for the driven model, which has eigenvalues . The reversible allosteric model (γ > 1) cannot be diagonalized analytically and we therefore provide numerical results instead.
We assume that the full system is prepared in a steady state P s (j) and only vertical ON⇄OFF transitions are observed with observable sets ON = {2, 4, 6} and OFF = {1, 3, 5}.We determine the non-Markovian transition probability of the observed process kt , Q Ps (n, t| m) with m, n ∈ {ON, OFF} as [1] where 1 Ω is the indicator function of the set Ω.The non-Markovian transition probability between two fixed observed states m → n as well as the observable return probability m → m depend on the preparation of the full system [69].Moreover, in spite of the full system being prepared in the stationary state P s , by specifying the initial observed state (here either "ON" or "OFF") we "quench" the full system out of the steady state by conditioning on the state of the observable [1,69].Without loss of generality we will focus on the scenario where the observable is initially in the ON state, i.e. k0 = ON.
To quantify the magnitude of memory in the projected dynamics, we follow [55] and construct the auxiliary Chapman-Kolmogorov (CK) transition density Note that for a non-Markovian process Q CK depends on both t 1 and t 2 .The Chapman-Kolmogorov construction Q CK (n, t 1 + t 2 | m) corresponds to a fictitious dynamics where we force at time t 1 all hidden degrees of freedom to their stationary distribution and thereby erase all memory of their initial condition.When the observed ON⇄OFF dynamics is Markovian we have the converse is not true in general [1,55].As soon as for some n, the observable at time t 1 + t 2 "remembers" the state of hidden degrees of freedom at time t 1 .
We use the Kullback-Leibler divergence D k [p||q] ≡ k p(k) ln[p(k)/q(k)] ≥ 0 to quantify the difference between Q Ps and Q CK [55] ) where the superscript k in D k [p||q] denotes the independent dummy variable of the measures p and q.In the absence of memory D CK m (t 1 , t 2 ) = 0, ∀t 1 , t 2 .Conversely, as we are interested in ergodic dynamics prepared in a steady state, we have that D CK m → 0 whenever t 1 +t 2 → 0 or t 1 + t 2 → ∞ [55].Therefore, by the positivity of D CK m we will have at least one maximum in the half-space to t 1 , t 2 > 0. We quantify the magnitude of memory in terms of the global maximum on t 1 , t 2 > 0 In the baseline setting (γ 1 = γ 2 = 1) the observed and hidden dynamics are decoupled (i.e.all microscopic pathways are equivalent).As a result, the observed dynamics is Markovian and D CK max ( m) = 0. We are interested in the dependence of D CK max ( m) as we couple the vertical and horizontal dynamics cooperatively or by a dissipative current, that is, on γ 1 and γ in the driven and cooperative model, respectively.
Driven setting.-Wefirst consider the driven scenario with γ 1 > 1 and γ 2 = 1.Instead of γ 1 we use the steadystate entropy production rate of the microscopic dynamics Ṡ(γ 1 ) = i,j P s (j)L i,j ln[P s (j)L i,j /(P s (i)L j,i )] to indicate how far the system is driven out of equilibrium and we change γ in equidistant units of the chemical potential ln(γ 1 /γ 2 ) = ln γ 1 that drives the system out of equilibrium, i.e. γ increases exponentially.We first provide some intuition about the microscopic dynamics.
Since λ 1,2 are independent of γ 1 and because λ 3 ≃ 2(a + b) for b ≫ a and γ 1 ≥ 2, by increasing γ 1 we only alter λ 4,5 (see Fig. 2a and Appendix B).That is, we are essentially only tuning the fastest time scales, whereas the slow time scales remain unaffected by the driving.We also alter the stationary distribution P s ( m).Because b ≫ a the system (even without driving) tends to first explore vertical paths to reach a "quasi-steady state" between observed ON-OFF states within a time scale of approximately ∼ 1/b = 0.04 (see Fig. 2b).Afterwards, the probability redistributes horizontally within observable states.However, because the transition rates from state 1 to 2 and from 6 to 5 are accelerated by a factor of γ 1 , transitions 2 → 1 and 6 → 5 will be instantly followed by the reverse transitions 1 → 2 and 5 → 6.Thus, the probability distribution dominantly redistributes along the microscopic path 2 ↔ 4 ↔ 3 ↔ 5, and finally reaches a steady state "skewed" in the hidden direction (see Fig. 2b).We now address the magnitude of the emerging memory via D CK max in Eq. ( 5).We find that D CK max monotonically increases with γ 1 (note that Ṡ is a monotonically increasing function of γ 1 ), and eventually at ≈ 0.02324, where the location of the supremum approaches t 1 = t 2 ≈ 0.08 as γ 1 → ∞.Note that the maximal memory is attained on a time scale that is longer than the local vertical equilibration time ≈ 1/b = 0.04.The saturation may be explained by noticing that as γ 1 → ∞, only states 2, 3, 4, 5 have a non-zero probability and accelerated paths are almost never traversed.As a result, D CK max no longer changes with γ 1 .
Note that in this driven setting vertical transitions are always much faster than horizontal ones, which maintains a separation of time scales between observed and hidden dynamics.The memory we observe is thus "only" a manifestation of the relaxation of hidden degrees of freedom.
Reversible cooperative setting.-Wenow inspect the reversible scenario where γ 1 = γ 2 ≡ γ ≥ 1.As before, we first give some insight into the microscopic dynamics.As time evolves, in the first stage the system initially populates microscopic states 2 and 4, when γ is large especially state 2. The accelerated transition paths do not instantly play a role.Thus, similar to the driven model, the system tends to first explore vertical paths (paths 2 → 1 and 4 → 3) on a time scale of 1/b ∼ 0.04.In the second stage, also similar to the driven model, transitions 2 → 1 will instantly go back to state 2.During this stage, the probability redistributes in the horizontal direction.However, since the transition rates inside the ON state (L 2,4 , L 4,6 ) are also accelerated by a factor of γ, the time scale of the horizontal redistribution of probability is not necessarily larger than that of vertical dynamics.Here, the two main frequently visited vertical paths are 2 → 1 and 4 → 3.
Note that when we increase γ in this reversible setting, such that aγ > b, part of the horizontal transition rates exceeds vertical ones, i.e. the time scales of observable dynamics and hidden dynamics overlap and there is no time-scale separation.This overlapping (and "mixing") of hidden and observable time scales may contribute to the appearance of two shoulders in Fig. 3d  Conclusion.-In this Letter we addressed the emergence of memory in a minimal setting, where the microscopic dynamics corresponds to a Markov process on a planar network and we observe only the vertical ON⇄OFF dynamics, whereby the horizontal dynamics are hidden.Our aim was to gain insight into how correlations between hidden and observed dynamics emerge, in particular if and how the nature of these correlations de-pends on whether the microscopic dynamics is reversible (i.e.obeys detailed balance) or instead is driven.In the former scenario, the observed and hidden degrees of freedom are coupled cooperatively, whereas in the latter scenario the coupling emerges due to a non-equilibrium current.We focused on quantifying the magnitude of memory while tuning cooperativity or irreversible driving.Many features were found to be similar in both setting.However, in the reversible setting the strongest memory was found in the expected situation, when the time-scales of hidden and observed dynamics overlap.Conversely, in the driven setting maximal memory is reached under a clear time-scale separation.Our work therefore unravels qualitative differences in the way memory can emerge in equilibrium versus driven systems.While we focused on a simple model, our findings pave the way for more systematic studies.From a practical, "diagnostic" perspective, our results imply the possibility to gain insight about the dynamic coupling underlying active secondary transport [64][65][66]  ON reaches its maximum at a given γ in the reversible cooperative scenario (see Fig. 3c).As stated in the main text, the two shoulders in Fig. 3d are a result of discontinuities in the shift of peak position.To visualize this, we show in Fig. A1 the dependence of the peak position on γ.Note that G CK in Eq. ( 3) is a symmetric function of t 1 and t 2 , so the peak of D CK ON always occurs at t * 1 = t * 2 .

FIG. 1 .
FIG. 1. (a)Schematics of the full Markov network for the respective models.The upper green states are lumped to the observed "ON" state and the lower orange ones to the observed "OFF" state.In order to separate the time scale of horizontal and vertical dynamics, we choose the rates (in arbitrary units) a = 0.5 and b = 25 to make vertical process always much faster than horizontal one.In the driven model (see Fig.2) we set γ1 > 1 and γ2 = 1, whereas in the reversible "allosteric" model (Fig.3) we chose γ1 = γ2 ≡ γ ≥ 1.A trajectory of the full dynamics (blue) and its corresponding projection (gray) are shown in panel (b).
at γ ≈ 38 and γ ≈ 443.The shoulders are a result of the shift in position of the peak of D CK m (t 1 , t 2 ) (for details see Fig. A1 in the Appendix A).As γ tends to become very large, D CK max saturates to ≈ 0.08471, and the location of peak approaches t 1 = t 2 ≈ 0.13.

2 FIG
FIG. A1.Dependence of the peak position of D CKON , (t * 1 , t * 2 ), on the cooperativity parameter γ in the reversible model.Note the two discontinuities.
from observations of memory in the transmembrane transport of either species.Acknowledgments.-Financialsupport from the Max Planck School Matter to Life supported by the German Federal Ministry of Education and Research (BMBF) in collaboration with the Max Planck Society (fellowship to XZ), and from the European Research Council (ERC) under the European Union's Horizon Europe research and innovation programme (grant agreement No 101086182 to AG) is gratefully acknowledged.Appendix A: Peak position of D CK in the reversible setting.-Let(t * 1 , t * 2 ) ≡ arg max t1,t2 D CK m (t 1 , t 2 ) be the values of t 1 and t 2 when D CK