Short-time scales in the Kramers problem: past, present, future (review and roadmap dedicated to the 95th birthday of Emmanuel Rashba)

Let us first consider the case of the bottom initial state:

First, consider the case when the absorbing wall is close to the bottom: $U(q)$ may then be approximated by a parabola (Fig.2(a))

The instants $t_{n}$ mark intervals corresponding to different topologies of the MPEP: for $t\leq t_{1}$, $[q(\tau)]$ is monotonic while, for $t_{n}<t\leq t_{n+1}$ ($n=1,2,3,...$), $[q(\tau)]$ possesses $n$ turning points. As $t$ changes, the MPEP varies continuously for any $t$, including $t=t_{n}$. The exit velocity is non-zero unless $t=t_{n}$ (Fig.2(a)).

Apart from a quantitative description of the case when the wall is close to the bottom of the well, the parabolic approximation provides qualitative estimates of the time and energy scales of the steps in the general case. However, some features of the steps $S_{\rm min}(t)$ and of the associated evolution of the MPEP change qualitatively as the absorbing wall moves towards the saddle.

Let us move the absorbing wall $q_{aw}$ to a distinctly non-parabolic region of $U(q)$, but still not too close to the saddle ($<q_{aw}^{(c)}$). One can reduce the 4th-order differential equation (15) to a 2nd-order equation for $q$ plus a 1st-order one for the auxiliary variable $\Gamma^{\prime}$ soskin :

The system (20), in addition to providing an algorithm footf that is faster in some ranges of parameters than solving Eq. (15), has a remarkable feature: if $\tilde{E}=0$, the equation for $\Gamma^{\prime}$ can be integrated explicitly soskin . So, the 4th-order equation (15) reduces to a closed 2nd-order equation footg . Allowing for $\dot{q}_{i}=0$, the equation for the time-reversed trajectory $[\tilde{q}(\tau)]\equiv[q(t-\tau)]$ becomes

One can show (cf. soskin ) that the number of possible finite values of $t$ in eq.(23), such that $\tilde{q}(t)=q_{b}$, equals the number $N$ of turning points in the noise-free ($t=\infty$) trajectory. Labelling such times $t$ as $t_{n}\equiv t_{n}(q_{aw})$ ($n=1,2,...,N$), one may relate $n$ to the number $n_{tp}$ of turning points in the trajectory (23)-(24): $n=n_{tp}+1$. $t_{n}$ increases with $n$ and, if $N=\infty$, the trajectory (23)-(24) for $t=t_{n}$ with $n\rightarrow\infty$ coincides with the noise-free trajectory. If

Thus, if $\Gamma<2\omega_{0}$, then $S_{\rm min}$ decreases with $t$ monotonically, possessing an infinite number of inflection points $t_{n}$ with $dS_{\rm min}(t_{n})/dt_{n}=0$ (Fig.3(a)). They divide the time axis into intervals where the MPEP has different numbers of turning points: as $t$ increases, the transformation of the MPEP with $n-1$ turning points, into one with $n$ points, occurs continuously at $t=t_{n}$.

At $\tau=0$, Eq.(23) coincides with the conventional relaxational equation with a finite friction parameter, $\Gamma{\rm cth}(\Gamma t/2)$. Hence, the closer $q_{aw}$ is to the saddle, the slower the motion near the wall. Thus, $t_{n}\rightarrow\infty$ if $q_{aw}\rightarrow q_{s}$. On the contrary, the time of motion along MPEPs which get to the wall with non-zero velocity (they relate to sections $S_{\rm min}(t)$ with non-zero $dS_{\rm min}/dt$) is less sensitive to the distance $q_{s}-q_{aw}$ and remains finite even if $q_{aw}=q_{s}$. Consequently, as $q_{aw}$ grows, the onset of the fold at $t\approx t_{1}$ (according to numerical calculations) occurs at the critical value $q_{aw}^{(c)}$: $dS_{\rm min}/dt$ is discontinuous at the fold (Fig.3(b)). At $q_{aw}>q_{aw}^{(c)}$, there are intervals of $t$ during which the system (13),(15)-(17) possesses more than one solution footh . It is because, the closer $q_{aw}$ is to $q_{s}$, the larger is the number of such intervals and the maximal possible number of coexisting solutions). This result provides a multi-branch structure for $S(t)$ satisfying (13),(15)-(17) (Fig.3(c)). In order to find the activation energy at a given $t$ one should choose from the solutions of (13),(15)-(17) the minimal one. There are switches between different branches at certain critical times. These can be compared to switching processes, as other parameters vary, in certain escape problems maierstein ; soskin ; see also Sec. III.B below). The switches result in jump-wise changes of the MPEP while the activation energy still remains continuous (Fig.3(c)). At the same time, the switch results in a discontinuity $dS_{\rm min}/dt$: its values on different sides of the fold differ drastically, so that $S_{\rm min}(t)$ and $J(t)$ are still distinctly step-wise (Figs.3(c)).

We have tested some of the above predictions using computer simulations. $S_{\rm min}(t)$ is derived via optimal fitting of $J(t)$ obtained at different $T$. Figs. 2(b) and 3 show reasonable agreement between $S_{\rm min}(t)$ from the theory and from the simulations. The growth of the flux is clearly step-wise (see insets) in both cases.

If the initial state with a given coordinate and velocity, $\{q_{i},\dot{q}_{i}\}$, is shifted from the bottom of the well $\{q_{b},0\}$ then $S_{\rm min}(t)$ changes: cf. Fig.4. Typically, $S_{\rm min}(t)$ becomes non-monotonic: cf. Fig. 4(b) (only if $\dot{q}_{i}=0$ it becomes step-wise i.e. monotonic: cf. Fig. 4(a)). Moreover, as is evident in Fig.4, even a tiny shift of the energy from the bottom results in quite a significant distortion of $S_{\rm min}(t)$: the shift of energy in Fig. 4(a) and Fig. 4(b) is equal to $\Delta U_{p}/100$ and $\Delta U_{p}/200$ respectively. Such strong sensitivity to the initial state can be explained by the singularity in the effective time-dependent damping parameter in equation (23), which describes the MPEP; so, the shift in the activation energy depends non-analytically on the shift of the energy of the initial state.

A non-bottom initial state with a given coordinate and velocity might seem a rather artificial situation but, at the same time, there is always some non-zero initial temperature $T_{0}$ so that various non-bottom states are necessarily involved. The strong sensitivity of the flux $J_{q_{i},\dot{q}_{i}}(t)$ to the shift of $\{q_{i},\dot{q}_{i}\}$ from the bottom, appears to cast doubt on the generality of the stepwise growth $J$ in real situations. However, a rigorous analysis (see below) shows that the flux at short time scales still grows in a stepwise manner for any temperature $T_{0}<T$. Moreover, if $T_{0}/T\ll\Gamma/\omega_{0}$, then the step-wise structure for flux growth is similar to that obtained using the bottom as the initial state.

So, let the distribution of initial coordinates and velocities be quasi-stationary for some low temperature $T_{0}$:

If at the “initial” instant $t=0$ the additional noise source is switched on, so that the effective temperature becomes $T>T_{0}$ footnontherm , the evolution of the flux (6) with the initial distribution (26) becomes relevant. Given the activation-like structure of $J_{q_{0},\dot{q}_{0}}(t)$ (eqs. (10)-(13)), the flux with the thermalized initial state can be presented in the form

There is no room here to provide details but it can be shown that, for any $T_{0}<T$, the function $\tilde{S}_{\rm min}(\frac{T_{0}}{T},t)$ is stepwise in $t$. Analogously to the case of the bottom initial state, $\tilde{S}_{\rm min}$ possesses inflection points with $d\tilde{S}_{\rm min}/dt=0$, provided the wall is not too close to the saddle, and the corresponding MPEPs are described by an equation similar to (23) but with the constant $A$ related to $t$ as

It can be shown that $\tilde{S}_{\rm min}(\frac{T_{0}}{T},t\sim\omega_{0}^{-1})$ is close to $S_{\rm min}(q_{i}=q_{b},\dot{q}_{i}=0,t\sim\omega_{0}^{-1})$ provided

The competition between the two small parameters, $T_{0}/T$ and $\Gamma/\omega_{0}$, is readily interpreted physically. On one hand, the escape flux (on $t\sim\omega_{0}^{-1}$) from the bottom is $\propto\exp(-a\Delta U/(T\Gamma/\omega_{0}))$ where $a\equiv a(t)\sim 1$. On the other hand, if the system starts its motion from an energy $E_{0}$ close to the barrier level, the probability of escape for time $t\sim\omega_{0}^{(-1)}$ will be $\sim 1$, but then the probability to have such starting energy is $\propto\exp(-\Delta U/T_{0})$. It is the competition between these two exponentially weak processes which leads to the relation (30). Fig. 5(a) shows that, for $T_{0}/T=0.01\ll\Gamma/\omega_{0}\approx 0.035$, the MPEP starts close to the bottom while, for $T_{0}/T=0.2\gg\Gamma/\omega_{0}$, the starting energy is $\sim\Delta U$.

It would be interesting to study the following issues.

1. 1.

   Details of the case considered above, including in particular an accurate study of: (i) oscillations of the exit velocity and $dS_{\rm min}/dt$ as the exit time goes, as well as (ii) the transition from a smooth $S_{\rm min}(t)$, with inflection points only, to an $S_{\rm min}(t)$ possessing folds.

2. 2.

   Additional features characteristic of other types of boundary or other types of transitions, in particular inter-well transitions in the symmetric double-well potential - the case particularly relevant in the context of some promising application (see Sec. IV.B.1 below).

3. 3.

   A careful consideration of the case with two absorbing walls while the initial coordinate is close to one of the walls and the initial velocity is directed towards the opposite wall, a case that is relevant e.g. to ionic channels hille ; Zheng ; Igor . The preliminary analysis indicates oscillations of the flux in time.

4. 4.

   Generalization for non-potential systems and/or non-white noise for which, unlike potential systems subject to white noise where switching between different MPEPs gives rise only to folds in $S_{\rm min}(t)$, we anticipate the possibility of jumps in $S_{\rm min}(t)$. Of a particular interest, the cases of various low-frequency noises low-frequency and quasi-resonant noise Huang:19 are since they relate to many real systems.

5. 5.

   Pre-exponential factor.

6. 6.

   Multi-dimensional problems.

1. 1.

   The case with more than two barriers.

2. 2.

   Pre-exponential factor.

3. 3.

   Multi-dimensional problems.

Thermometers or, more generally, meters of noise intensity, which we call further as noisemeters typically can measure temperature or noise intensity respectively in a quite limited range only. In other words, the lower and upper limits of measurements are of the same order of magnitude or, at best, they differ by 1-2 orders of magnitude only. For example, a common room thermometer can measure temperature just in the range 280-320 K.

Generally speaking, an estimate of temperature or noise intensity in case of a non-thermal noise (for the sake of brevity, we shall use one and the same notation $T$ for both cases) can also be based on a measurement of a quasi-stationary noise-induced escape flux. However, to the best of our knowledge, for real routine estimates of $T$ (i.e. for everyday or engineering purposes rather than for just scientific ones) it has not be used. Perhaps, the reason of this is the following: on the one hand, such a measurement is rather time-consuming and, on the other hand, the range of $T$ which can be thus measured is not very large. As for the lower limit, it is about $T_{qs}^{(l)}\approx\Delta U/12$, where $\Delta U\equiv U(q_{aw})-U(q_{b})$ is the value of the “barrier” for a given value $q_{aw}$ (the lowest possible $\Delta U$ is limited with the lowest value of $q_{aw}-q_{b}$ which is possible to measure sufficiently accurately) while the denominator $12$ is explained by that, for larger values of the denominator, the escape probability is so low that it is impossible to measure the escape flux for a realistic time hoban . The main restriction for the estimate of $T$ by means of the measurement of the quasi-stationary escape flux relates to the upper limit of $T$ which can thus be measured: it is inherently limited from above with the value $T_{qs}^{(up)}\approx\Delta U_{max}/3$, where $\Delta U_{max}\equiv U(q_{s})-U(q_{b})$ is the maximal possible value of the potential barrier (cf. Fig. 1). For larger values of $T$, the exponential (activation-like) factor in the dependence of the escape rate $\alpha_{qs}$ (2) on $T$ is not sufficiently sharp and the Kramers-Melnikov formula for $\alpha_{qs}$ Melnikov:84 ; Melnikov_Meshkov:86 ; Melnikov:91 is not valid anymore. The restriction for the upper limit $T_{qs}^{(up)}$ is especially important in case of a non-thermal noise as such a noise may have a very large intensity and, moreover, its variation may be very large (constituting many orders of magnitude).

Our results for the noise-induced escape at time-scales much less than the scale of the formation of the quasi-equilibrium in a single well $t_{f}^{(s)}$ (7) (i.e. those described in Sec. III.A) in case of low friction, which is relevant first of all to nano/micro-mechanical resonators, promise to provide a possibility to measure noise intensity in the range varying by many orders of magnitude while using one and the same devise. Application-wise it may provide a great financial benefit.

There is no room here to provide details unpublished . Rather we just give the main ideas and mention a few difficulties which may be encountered. We see two distinctly different options for an implementation of our ideas.

1. 1.

   The setup with an “absorbing” wall. As compared with the second option described below, the present setup allows us to immediately utilize the results presented in Sec. III.A (in particular, the explicit results for the parabolic approximation of the potential) and, besides, it might be favourable in terms of the duration of the required measurements and of the computation time required for the calculation of $T$ from the measurements. At the same time, the setup might give rise to a serious technical problem: each time when the system reaches the “wall” (being, in fact, just a given coordinate rather than a real wall), it should be somehow returned into the initial state (i.e. in the bottom of the well), and it is desirable for this transition to occur quickly, which may not be easily feasible. Generally speaking, such a return might be fulfilled by means of an interruption of an action of noise on the system (e.g., if noise acts due to an electric connection, then the corresponding connection may be switched off). Then, the time-scale for a single return is $\Gamma^{-1}$. If there is a possibility to strongly increase the diccipation, then the time-scale would further grately decrease. Another possibility is to introduce some additional action on the system which would transfer the system into the vicinity of the bottom of the well. We do realize that the problem of a fast return into the bottom of the well after reaching the coordinate $q_{aw}$ may not be trivial in reality, but meanwhile we assume that it can be resolved somehow.

   Let us discuss the key points of the algorithm of the measurements. Firstly, we should roughly estimate the quasi-stationary escape rate $\alpha_{qs}$. To this end, it would be sufficient to observe just a few (up to $10$) escapes and average them, thus obtaining the rough estimate for $\alpha_{qs}$.

   If the relation $\alpha_{qs}\ll\Gamma$ holds true, then we may conclude that $T\ll\Delta U$ and therefore the Kramers formula for $\alpha_{qs}$ Melnikov:84 ; Melnikov_Meshkov:86 ; Melnikov:91 ; Kramers:40 may be readily used, so that we just need to gain more statistics in order to measure $\alpha_{qs}$ more accurately and then to calculate $T$ from it by means of the Kramers formula.

   Of the main interest in the present context is the complementary case: $\alpha_{qs}\gtrsim\Gamma$. It follows from this relation that $T\gtrsim\Delta U$. In this case, we should measure the flux at relatively small time-scales namely at $t\ll t_{esc}\equiv\alpha_{qs}^{-1}$. In order to obey this inequality while avoiding too poor statistics we should chose a compromise, namely to explore the time $t\approx t^{(N)}\equiv t_{esc}/N$ where $N$ is a moderately large number (about $5-6$). Then we need to compare $t^{(N)}$ with the step width, i.e. with $\pi/\omega_{0}$. If $t^{(N)}\gg\pi/\omega_{0}$, then the step structure is smeared at the time-scale $t^{(N)}$. It is convenient in this case to measure the escape flux at $t=t^{(N)}$ and $t=t^{(N+1)}$ and to compare with each other while T can be shown to obey the following formula:

   	$\displaystyle T=\frac{\Delta U}{\Gamma t_{esc}\ln\left(\frac{J(t^{(N)})}{J(t^{(N+1)})}\right)},\quad\frac{\pi}{\omega_{0}}\ll t^{(N)}\lesssim\frac{1}{\Gamma},$		(40)
   	$\displaystyle t^{(K)}\equiv\frac{t_{esc}}{K},\quad t_{esc}\equiv\alpha_{qs}^{-1},\quad N\approx 6.$

   If $t^{(N)}$ is of the same order as $\pi/\omega_{0}$, then it may be preferable to measure the flux at the centers of the first and second steps rather than at $t^{(N)}$ and $t^{(N+1)}$: the results are much less sensitive to an inaccuracy of a measurement of time. Using formulas in Eq. (19), we obtain:

   $$T=\Delta U\frac{\omega_{0}/(2\pi\Gamma)}{\ln\left(J(t_{2})/J(t_{1})\right)},\quad\frac{t_{esc}}{6}\sim\frac{\pi}{\omega_{0}},$$

   (41)

   where $t_{n}$ is defined in (19).

   Finally, if $t^{(N)}\ll\pi/\omega_{0}$, then one should use the function $S_{min}(t)$ calculated for $t=t^{(N)}$ and $t=t^{(N+1)}$ by methods described in Sec. III.A. Then $T$ can be calculated by means of the formula which is formally valid in a general form for any time-scale and for any position of $q_{aw}$:

   $$T=\frac{S_{min}(t^{(N+1)})-S_{min}(t^{(N)})}{\ln\left(\frac{J(t^{(N)})}{J(t^{(N+1)})}\right)},$$

   (42)

   where $t^{(K)}$ and $N$ are defined in Eq. (40) and we do not restrict the range of its validity to $t^{(N)}\ll\pi/\omega_{0}$ since Eq. (42) is valid in the much broader range: $t^{(N)}\lesssim\Gamma^{-1}$ (Eq. (40) represents a partial case of (42) provided $q_{aw}$ lies sufficienntly close to the bottom of the well, so that the parabolic approximation of $U(q)$ works well)). Of course, various inaccuracies of experimental measurements and theoretical approximations put a limit for the lowest limit of the range of $t^{(N)}$ where Eq. (42) is valid and this determines the upper limit for values of $T$ which we can measure by means of such a method. The limitations will be discussed elsewhere.

   Even if to skip the range of very small times and to restrict ourselves to the range $t_{esc}/6\gtrsim\pi/\omega_{0}$, we can see from Eqs. (40) and (41) that our approach allows one to measure $T$ within the range characterized with the ratio of an upper and lower limits of the order of the quality factor $Q\equiv\omega_{0}/\Gamma$. A few more orders of magnitude may be added for the account of a decrease of an effective $\Delta U$ (by means of shifting $q_{aw}$ closer to the bottom of the well). Quality factors of modern nano/micro-mechanical resonators can rather easily reach values $10^{6}-10^{7}$ Huang:19 ; dykman:22 ; Moser:13 and therefore our method provides a possibility to measure $T$ with one and the same device within a huge range of the order of $10^{8}-10^{10}$.

2. 2.

   The setup with transitions between bottoms of wells of a symmetric double-well potential. If we use a device charecterized with a symmetric double-well potential (for example, it may be a buckled doubly-clamped beam Erbil:20 or a nanoparticle levitating in a bistable one-dimensional opticle trap Flajsmanova:20 ) and consider transitions between close vicinities of bottoms of the potential wells, then the transitions in both directions are equivalent in the context of the transition probability and transition flux. Therefore there is no need to artificialy return the system into the initial state. As compared with the setup with the absorbing wall, this is a big advantage. One of disadvantages consists in that we cannot vary the magnitude of an effective barrier $\Delta U$. Besides, it is necessay to generalize the theory for this case, and the results certainly will not be expressed explicitly. On the other hand, the latter disadvantage (a necessity to use a complicated numerical procudure for a calculation $S_{min}(t)$) is not crucial: for a given potential $U(q)$, one will be able to calculate $S_{min}(t)$ once and forever, so that it will be used just as a known numerical function for any new measurement.