Non-Arrhenius barrier crossing dynamics of non-equilibrium non-Markovian systems

The non-equilibrium non-Markovian barrier crossing dynamics of a one-dimensional massive coordinate, described by the non-equilibrium version of the generalized Langevin equation with unequal random and friction relaxation times, is studied by simulations and analytical methods. Within a harmonic approximation, a general formula for the barrier crossing time is derived which agrees favorably with simulations. Non-equilibrium random forces with a relaxation time longer than the friction relaxation time induce non-Arrhenius behavior and dramatically increase the barrier crossing time; within the harmonic theory this corresponds to a reduced effective temperature which also modifies the spatial and velocity distributions.

Of particular interest are non-equilibrium reaction kinetics, i.e., the influence of non-thermal forces on barrier crossing dynamics in some physical or abstract reaction coordinate space, because here the fields of non-equilibrium statistical mechanics, non-Markovian dynamics and rare phenomena meet [20,21]. As examples we mention metastable evolutionary dynamics [22], barrier effects on motion patterns of mammals [23] and cancer cells [24], the effects of active fluctuations on the zipping/unzipping dynamics of RNA hairpins [25], protein folding under nonequilibrium conditions [26] and chemical reactions in the presence of strong time-dependent electric fields [27].
We study the barrier crossing dynamics of a one dimensional non-equilibrium non-Markovian model, where the single-exponential relaxation times of the friction memory kernel and the random force are different [18,20,21] and thus the fluctuation-dissipation theorem is broken. This is a rather general model that by projection and dimensional reduction can be derived from a wide class of different microscopic non-equilibrium systems [15,18]. Our analytical calculations show that this particular form of non-equilibrium not only changes the pre-exponential factor but drastically modifies the exponential Arrhenius 51001-p1 factor itself. Our derived closed-form expression for the MFPT demonstrates that the modification of the Arrhenius factor can be described by an effective temperature, that is proportional to the square of the friction and random relaxation-time ratio, in good agreement with simulations. The same effective temperature also describes the non-equilibrium position and velocity distributions. In a previous study on non-equilibrium barrier crossing, a similarly defined effective temperature was shown to capture the non-equilibrium modification of the transitionpath time [53]. The effect of the random-force correlation time on the barrier crossing dynamics is illustrated by two non-equilibrium simulation trajectories in fig. 1, where the random-force correlation time is shorter (red trajectory) and longer (blue trajectory) than the friction correlation time, with all other parameters being the same: the red trajectory shows many barrier crossing events, while the blue trajectory does not show a single barrier crossing over the same simulation time. From these sample trajectories, we conclude that an increase of the random-force correlation time increases the mean barrier crossing time.
Our model is based on the approximate generalized Langevin equation (GLE) with linear friction (1) where m is the particle mass, Γ V (t) the friction memory kernel, t 0 some initial time and F R (t) denotes the random force characterized by the general autocorrelation where β is a numerical constant. This version of the GLE has been recently shown to follow from an exact form of the GLE in the limit when the correlations between particle velocity and random force are homogeneous in reaction coordinate space and thus corresponds to a physically sound and realizable asymptotic form [54], the main restriction is that non-linear friction effects are neglected. The potential U (x) is time-independent and thus corresponds to the potential of mean force that describes the stationary equilibrium distribution of the reaction coordinate. As a potential that leads to barrier crossing events, we choose a double well where 2L is the separation between the minima and U 0 is the barrier height.
In equilibrium, the fluctuation-dissipation theorem predicts Γ R (t) = Γ V (t) and β −1 = k B T equals the heat bath thermal energy. In the general non-equilibrium scenario, Γ R (t) and Γ V (t) are independent functions, which for simplicity we choose as single exponentials, and β does not correspond to the heat-bath temperature. We normalize Γ R (t) and Γ V (t) such that their integral is independent of the memory times, τ R and τ V , which allows us to decouple memory time effects from friction effects (note that in a previous treatment of a similar model a different normalization of these memory functions was chosen [53]). With no loss of generality, we consider equal prefactors γ R = γ V = γ, since different prefactors can be absorbed into the definition of β in eq. (2). For different random and friction memory times, τ R = τ V , the fluctuation-dissipation theorem is irrevocably violated and thus the system is out of equilibrium [55][56][57]. The non-equilibrium version of the GLE in eq. (1) describes non-equilibrium systems in terms of a one-dimensional reaction coordinate that is coupled to a non-thermal random force and subject to a time-independent potential, it can be derived by dimensional reduction of microscopic non-equilibrium models that are coupled to multiple heat baths at different temperatures [15]. Two other time scales characterize the model, namely the inertial time τ m = m/γ and the diffusion time τ D = L 2 γβ. We first present analytical results for the MFPT based on the positional autocorrelation function C(t) = x(t)x(0) . For this we employ a harmonic approximation of eq. (1) and use U har (x) = Kx 2 /2, where K is the second derivative of the double-well potential at the minima, K = U (L) = 8U 0 /L 2 . The calculation we present here significantly generalizes our previous derivation [42], as will be pointed out below. Fourier transforming eq. (1) for t 0 → −∞ and solving forx(ω), we obtaiñ which defines the response functionχ(ω). The half-sided 51001-p2 Non-Arrhenius barrier crossing dynamics of non-equilibrium non-Markovian systems while the Fourier transform of the symmetric random force correlation Γ R (t) is This can be rewritten in a form that corresponds to the standard result for the memory-less harmonic oscillator in equilibrium (i.e., τ V = τ R = 0), where we have introduced effective frequency-dependent friction, mass and temperature The mapping introduced here is valid for arbitrary frequencies and for non-equilibrium situations, whereas in our previous work we considered the equilibrium case and the asymptotic limits of high and low frequencies separately [42]. We have introduced a numerical constant that in the harmonic case takes the exact value c 1 = 1 and will be used as a fit parameter to account for non-harmonic corrections. For τ R = τ V equilibrium is recovered and β eff = β, for τ R = τ V = 0 the equilibrium Markovian limit is recovered and additionally γ eff = γ and m eff = m. Note that the potential curvature K is not renormalized. In the overdamped case, characterized by a vanishing bare mass m = 0, the effective mass m eff would become negative. Thus, a finite bare mass m is needed in order to regularize the dynamics of the system in our analytical approach. In the low and high friction limits, eq. (10) is dominated by the poles ω 2 L = K/m eff for Km eff > γ 2 eff and ω 2 H = −K 2 /γ 2 eff for Km eff < γ 2 eff , respectively (see the SM). From these characteristic frequencies we obtain from eq. (11a) the effective friction in the low and high friction limits (see the SM), from which the effective mass m eff and the effective temperature β eff in the low and high friction limits follow from eqs. (11b) and (11c) by insertion. We introduced two more numerical constants c 2 and c 3 that will be used as fit parameters for describing our numerical data.
Using the Markovian mapping in eq. (10), we now construct an expression for the mean-first passage time τ MFP between the two potential wells at ±L for the non-Markovian non-equilibrium case. First we consider the equilibrium scenario, defined by τ R = τ V , in which case β eff = β, as follows from eq. (11c), and start with the Markovian limit, τ R = τ V = 0. From the classical Kramers results in the high and low-friction limits, τ MFP = e βU0 2 √ 2πγ/K and τ MFP = e βU0 3πm/(8 √ 2βU 0 γ), respectively [29,58], we construct the heuristic interpolating expression Note that we have simply summed the two asymptotic limits and added a crossover term. In fig. 2(a) we compare simulation results for τ MFP (data points) with the heuristic formula eq. (13) (dashed line) and with Melnikov-Meshkov theory [30] (orange line), the latter becoming exact in the high-barrier limit (see the SM), for three different barrier heights. Details on the simulation methods are given in the SM. The agreement is very good, which means that the numerical simulations are converged and that the heuristic expression eq. (13) works. The Kramers limits are denoted by red lines in fig. 2(a).
We for now stay in equilibrium but consider the more general non-Markovian case and replace the friction coefficient and mass in eq. (13) by their effective expressions in eqs. (11). This only has to be done for the low-friction term proportional to m/γ in eq. (13), because here the resulting effects are dominant compared to the high-friction term proportional to γ/K (or the crossover term) in eq. (13). This is also reflected by the fact that fitting the simulation data for the MFPT with the heuristic formula, we obtain a very small value c 3 = 0.01 so that γ H eff ≈ γ 51001-p3 (see the SM). From eqs. (11) and (12) we find in the lowfriction limit m L eff /γ L eff = m/γ − c 1 τ V + c 2 τ 2 V K/γ, which shows that memory has two opposing effects: For small τ V the MFPT eq. (13) goes down due to mass renormalization, thus short memory accelerates barrier crossing, while for long memory the MFPT goes up due to friction renormalization, i.e., long memory slows down barrier crossing [38,40,42]. We also see that the effect memory has on the MFPT survives in the overdamped scenario, m = 0, since the memory correction terms are independent of the bare mass. We combine the memory acceleration term with the overdamped contribution in eq. (13) according to γ/K − c 1 τ V /(βU 0 ) ≈ (γ/K)/(1 + c 1 Kτ V /(γβU 0 )), which ensures that τ MFP stays positive and improves the description of the numerical data. We thereby derive the final expression for the MFPT in the equilibrium non-Markovian case, which is equivalent to a previously suggested heuristic formula [42], The harmonic result is recovered for c 1 = 1 = c 2 = 1. By fitting to simulation data in fig. 2(b) for fixed barrier height βU 0 = 3 and a few different rescaled masses τ m /τ D (symbols), we obtain c 1 = 64/3 and c 2 = 2/3, which accounts for the fact that the simulated potential in eq. (3) is not harmonic. We see that the formula eq. (14) is very accurate for general mass/friction ratios and rescaled memory times τ V /τ D . We now turn to the non-equilibrium case τ V = τ R . From eq. (11c) and the poles ω L and ω H , we obtain In the limits τ m /τ V < 1, τ m /τ R < 1, γ/(Kτ V ) < 1, γ/(Kτ R ) < 1, which hold for the simulated systems, as we demonstrate below, we can write We see that the effective temperature β −1 NEQ depends quadratically on the ratio of the memory friction time τ V and the random relaxation time τ R . Replacing β by β NEQ , we obtain from eq. (14) our final expression for τ MFP in the non-Markovian non-equilibrium case, Before comparing eq. (18) with simulation results for τ MFP , we consider the joint position-velocity distribution, which, according to our mapping, is predicted as Note that we assume the mass prefactor of the kinetic energy not to be renormalized. From this point on, we focus on the overdamped case and choose τ m /τ D = 0.1, because this is the most relevant regime for biophysical applications. In fig. 3(a) we show logarithmic positional distributions − ln P (x) for various values of τ V /τ D and τ R /τ D , shifted vertically so that they agree for x = ±L. The simulation data for the equilibrium case τ R = τ V is denoted by a black line and agrees perfectly with the expected Boltzmann distribution − ln P (x) = βU (x), denoted by black spheres; the non-equilibrium simulation results are shown as colored lines. In fig. 3(b) we show −(β/β eff ) ln P (x) with the prefactor β/β eff fitted such that the data superimpose. We see that all distributions agree perfectly, meaning that an effective temperature describes the nonequilibrium position distributions well. In figs. 3(c) and (d) we plot the fit results for β eff /β (red stars) as a function of τ R /τ D and τ V /τ D , respectively, and obtain perfect agreement with the predictions in eqs. (15), (16), (17). This in particular means that the asymptotic expression for β NEQ in eq. (17) is accurate for the employed parameters.
In fig. 4(a) we show the logarithmic velocity distributions − ln P (v) for various values of τ V /τ D and τ R /τ D , shifted vertically so that they agree for v = 0. Again, the simulation data for the equilibrium case τ R = τ V is denoted by a black line and agrees perfectly with the expected Maxwell-Boltzmann distribution − ln P (v) = βmv 2 /2 (black spheres), the non-equilibrium simulation results are shown as colored lines. In fig. 4(b) we show −(β/β eff ) ln P (v) with fitted prefactors β/β eff and again find the data to perfectly superimpose. In figs. 4(c) and (d) we favorably compare the fit results for β eff /β (red stars) with the predictions in eqs. (15), (16), (17). In essence, the position and velocity distributions from our non-equilibrium simulations are perfectly described by an effective temperature that depends on the ratio of memory and random relaxation times according to eq. (17).
Finally, in fig. 5 we compare τ MFP from simulations (symbols) with eq. (18) as a function of τ R /τ V for three different values of τ V /τ D , note that equilibrium is recovered for τ R /τ V = 1. We see that as the ratio τ R /τ V increases, τ MFP grows dramatically (note the logarithmic vertical scale), which reflects the decrease of the effective temperature in eq. (17) and the divergence of the exponential Arrhenius factor in eq. (18). In contrast, for decreasing ratio τ R /τ V the MFPT levels off, which reflects a competition of the pre-exponential memory slow-down factor proportional to τ 2 V with the exponential factor in eq. (18).
In conclusion, by solving the non-equilibrium non-Markovian harmonic oscillator in terms of its effective friction, mass and temperature, and using the Kramers results for the Markovian barrier crossing dynamics, we analytically derive a closed-form expression for the nonequilibrium non-Markovian barrier crossing dynamics in a double well that holds in the low and high friction regimes. Our derived expression for the MFPT agrees with a previously suggested heuristic formula for the MFPT in the non-Markovian equilibrium limit [42,43] and describes our non-equilibrium simulation results rather well without additional fitting parameters. Whereas memory and inertial effects only influence the exponential prefactor of the MFPT, non-equilibrium effects modify the exponential Arrhenius factor itself and thus drastically modify barrier crossing times. The random correlation time τ R and the memory friction time τ V influence the non-equilibrium MFPT in a markedly asymmetric fashion, the MFPT increases steeply for τ R /τ V > 1 but rather levels off for τ R /τ V < 1. These findings are possibly relevant for in vivo non-equilibrium protein folding, since in recent experiments on the folding speed of a nascent polypeptide during translation a delay of the folding has been measured [26]. Whether co-translational folding is characterized by increased random-force relaxation times seems plausible but of course has to be substantiated by microscopic modeling. We note that our derived result for the MFPT depends crucially on the form of our friction and random-force memory functions, who are taken as singleexponential functions and whose integrals are defined to be independent of the memory times. In passing we note that our analytically derived effective temperature also describes position and velocity distribution of our non-linear barrier crossing system very accurately even far away from equilibrium. * * * We acknowledge support by Deutsche Forschungsgemeinschaft, Grant CRC 1114, code 235221301, Project B03 and by the European Research Council Advanced Grant NoMaMemo Grant 835117.
Data availability statement: All data that support the findings of this study are included within the article (and any supplementary files).