Proceedings TH2002 Supplement (2003) 327 – 350
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/040327-24 $ 1.50+0.20/0
Proceedings TH2002
Kramers Problem on Time-Scales Preceding
Quasi-Stationarity
S.M. Soskin, M. Arrayás, I.Kh. Kaufman, T.L.Linnik, D.G. Luchinsky,
P.V.E. McClintock, R.Mannella and V.I.Sheka
Abstract. The problem of noise-induced escapes/transitions is of great importance
in many applications. For a potential system, it is known as the Kramers problem,
in allusion to the famous paper by Kramers. Until very recently, there remained a
gap in the theory of the Kramers problem: how does the escape flux evolve from
zero, at the initial instant, to the quasi-stationary stage after quasi-equilibrium has
been formed within the metastable state? Intuition suggests the evolution should
be monotonic and without any irregularities, which appears to be confirmed by
the rigorous analysis of some partial regimes. However, as we show, either the flux
itself (multi-well case) or its derivative (single-well case) may evolve in a markedly
non-monotonic fashion. This non-monotonicity, like other irregularities, relates to
characteristic changes in the topology of the most probable escape path. In the
multi-well case, the formation of quasi-equilibrium takes an exponentially long time,
so that our results are relevant to a major portion of the time attainable at small
noise intensities. In the single-well case, our results are relevant to much shorter
time-scales (of the order of a period of eigenoscillation); interest in such short timescales is growing fast, as modern technologies develop.
1 Introduction
In his seminal work [1], Kramers considered a weak noise-induced flux from a single
metastable classical potential well, i.e. he considered a stochastic system
q̈ + Γq̇ + dU/dq = f (t),
f (t) = 0,
(1)
f (t)f (t ) = 2ΓT δ(t − t ),
T ∆U,
which was put initially at the bottom of a metastable potential well U (q) with
a barrier ∆U ; he then calculated the quasi-stationary probability flux across the
barrier. Models of type (1) are relevant to chemical reactions [1], SQUIDs [2],
and many other real systems [3, 4] including the recently designed mechanical
electrometers [5].
There have been many developments and generalizations of the Kramers
problem (see [3, 4] for reviews) but both Kramers and most of those who followed
him considered only the quasi-stationary flux, i.e. the flux established after the
formation of a quasi-equilibrium distribution within the well (up to the barrier).
The quasi-stationary flux is characterized by a slow exponential decay in time t,
an Arrhenius dependence on temperature T , and a relatively weak dependence on
328
Soskin et al.
friction Γ:
Jqs (t) = αqs e−αqs t ,
αqs = P e−
∆U
T
,
Proceedings TH2002
(2)
where P depends on Γ and T in a non-activated way.
But how does the flux evolve from its zero value at initial time to its quasistationary regime (2) at time-scales exceeding the time tf for the formation of
quasi-equilibrium? The answer may obviously depend on initial conditions and a
relevant boundary (i.e. the boundary through which the escape occurs). As for the
boundary, it can be shown that the most general qualitative features of the flux
are valid for any type of boundary (for the sake of simplicity, we shall consider
below only the absorbing wall). As for the initial conditions, their relevance may
vary. The simplest and often relevant initial state is the bottom of the well, since it
is the stable stationary state in the absence of noise: if the noise (not necessarily of
the thermal origin) is switched on at some instant, then the time evolution of the
escape from the bottom becomes relevant. It should be emphasized however that,
if the relevant metastable part of the potential is multi-well, then the flux during
the major part of the relevant time is not sensitive to the initial state provided
it is concentrated just in one well (e.g. it may be thermalized in the well). As for
the single-well case, the flux evolution is more sensitive to the initial state and we
shall consider various cases. But, first, let us discuss the most simple case where
the initial state is just at the bottom of the potential. We shall refer to it as the
bottom initial state.
It may seem natural to assume that the flux evolution from zero to the quasistationary regime is a monotonic function without any “irregularities”. Apart from
the naive argument that “noise smooths everything”, this assumption appears
sound because the probability distribution W is distinctly centered at the bottom
of the well both initially and in the quasi-stationary stage: W (q, q̇, t = 0|q0 =
qb , q̇0 = 0) =√δ(q − qb )δ(q̇) while W (q, q̇, t tf |q0 = qb , q̇0 = 0) is a narrow peak
of width ∝ T around that same state {q = qb , q̇ = 0}. Moreover, it was shown
recently [6] that, both in the underdamped and overdamped limits, the escape flux
J(t) does grow at t ∼ tf in a simple manner.
Despite the above arguments, it can be shown that, generically, J evolves
from J(0) = 0 to Jqs (t tf ) in a quite complicated way.
1. As shown in Sec. II, the flux grows step-wise on time-scales of the order of
a period of eigenoscillation in the bottom of the well. Apart from filling the “gap”
in time-scales in the Kramers problem (cf. the big activity in the 1980s aimed at
filling the gap in the friction ranges [3, 4]), this part of our work is motivated by
the growing interest in the very short time-scales that are now relevant to certain
experiments, such as those studying chemical reactions down to femtosecond timescales [7].
2. As shown below in Sec. III, the evolution of the flux on longer time-scales
in a multi-well metastable potential is also distinctly different from the relatively
simple monotonic function described in [6]: J grows sharply on a logarithmic timescale to a value which is typically very different from Jqs (0) (typically, exponen-
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
329
tially larger) and then evolves to Jqs (t) during the exponentially long time.
It should be emphasized that the qualitative features of J(t) described above
are valid for any reasonable definition of the flux: e.g. the full flux through a boundary or just the first-passage flux, while the boundary may be a given coordinate,
or a boundary of a basin of attraction, or a boundary of the vicinity of another
attractor, etc.
To illustrate our results, we will use the potential
U (q) = q − q 3 /3
(3)
for the single-well case (Fig. 1), and
U (q) = 0.06(q + 1.5)2 − cos(q)
(4)
for the multi-well case (Fig. 6(a)), with an absorbing wall [8] at q = qaw in both
cases.
Experimentally, the flux is measured in the following way. The system is
placed at an initial state, after which it follows the stochastic equation (1) until
either the coordinate of the wall, qaw , or the time limit [9], tl , is reached. It is then
reset to the initial state and everything is repeated. Once the statistics are deemed
adequate, we calculate the flux
J(t) ≡
∆N (t)
Nreset ∆t
1
(5)
where Nreset is the overall number of resets, and ∆N (t) is the number of resets
during the interval [t, t+ ∆t]; ∆t is chosen to be much smaller than a characteristic
time over which the flux (5) may change significantly, but large enough to provide
∆N (t) 1 (roughly, the latter is satisfied provided ∆t tl /Nreset ).
The above experimental definition corresponds to the following theoretical
definition of the flux
J(t) =
Jq0 ,q̇0 (t) =
dq0 dq̇0 Win (q0 , q̇0 )Jq0 ,q̇0 (t),
0
(6)
∞
dq̇ q̇W (q = qaw , q̇, t|q0 , q̇0 ),
where Win (q0 , q̇0 ) is a statistical distribution of the initial coordinate and velocity
and W is the conditional probability density.
The theoretical approach which we use is the method of optimal fluctuation
(cf. e.g. [10, 11]) whose details in application to the present problems are given in
the next two sections. The theoretical results are verified by computer and analog
electronic simulations. Preliminary reports and discussions of some of the central
ideas were given in [12, 13].
330
Soskin et al.
1
(1)
qaw
(c)
qaw
(2)
qaw
Proceedings TH2002
(3)
s
qaw
U
q
−1
b
q
b
−1
q
s
1
Figure 1: The potential U (q) = q − q 3 /3. The bottom and the saddle are marked
as b and s respectively. Triangles indicate four typical positions of the absorbing
wall.
2 Single-well metastable potential
It can be shown directly from the Fokker-Plank equation that the formation of
quasi-equilibrium up to the barrier in the single metastable well typically takes
[14] a time of the order of
(s)
tf ∼
∆U
1
ln(
),
min(Γ, ω02 /Γ)
T
(7)
where ω0 is the frequency of eigenoscillation in the bottom of the well.
In this section, we shall be interested in much smaller time-scales,
(s)
t tf .
(8)
It is convenient to consider first the case of an initial state with a given
coordinate and velocity:
Win (q0 , q̇0 ) = δ(q0 − qi )δ(q̇0 − q̇i ).
(9)
Then the flux is sought as
J(t) ≡ Jqi ,q̇i (t) = P (t)e−
Smin (t)
T
(10)
where the activation energy Smin (t) does not depend on T while the prefactor
P (t) depends on T in a non-activated way. At small T and short t, the factor
exp(−Smin /T ) depends on t much more strongly than P . So, we concentrate on
studying Smin (t), which can be shown [16] to be a minimum of the functional:
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
331
Smin (t) ≡ Smin (qi , q̇i , t) = min[q(τ )],q̇(t) (S),
t
dτ L,
S ≡ Sq̇(t) [q(τ )] =
(11)
L = (q̈ + Γq̇ + dU/dq) /(4Γ),
q(0) = qi , q̇(0) = q̇i , q(t) = qaw .
(12)
(13)
0
2
The minimization is done over an escape path [q(τ )] at a given exit velocity q̇(t),
with a further minimization over this velocity (a minimization similar to the latter
was used in a different context in [17]). Note that the minimization over the exit
velocity (or, more generally, over states on a relevant boundary) was not used
in applications of the method of optimal fluctuation to quasi-stationary escape
rates [16, 18] or related quantities [19] since the exit occured necessarily through
the saddle. The path minimizing S may be called the most probable escape path
(MPEP), in analogy with the quasi-stationary case. The necessary conditions for
the minimum of the functional (11) are as follows.
1. A zero variation, δS = 0: it implies that the MPEP [q(τ )] satisfies the EulerPoisson equation [20, 16]
d ∂L
d2 ∂L
∂L
− (
) + 2(
) = 0,
∂q
dt ∂ q̇
dt ∂ q̈
(14)
which, for the L of the form (12), reads
2
d U
d3 U
d2 U dU
¨q¨+ q̈ 2 2 − Γ2 + q̇ 2 3 + 2
= 0.
dq
dq
dq dq
(15)
2. A zero derivative with respect to the exit velocity, ∂S/∂ q̇(t) = 0, which can
be reduced to ∂L/∂ q̈(t) = 0: for the L of the form (12), it reads
[q̈ + Γq̇ + dU/dq]|τ =t = 0.
(16)
Solutions of Eq.(15) satisfying conditions (13), (16) can be found numerically:
in addition to q(0) and q̇(0) given in (13), one can match q̈(0) and ˙q̈(0) so that the
result of the integration (15) on the interval [0, t] satisfies the last of conditions
(13) and condition (16).
2.1
Bottom initial state
Let us first consider the case of the bottom initial state:
qi = qb ,
q̇i = 0.
(17)
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Soskin et al.
Proceedings TH2002
Before presenting the numerical results, we derive some important general features
of the MPEPs and Smin (t). We show below that, as the boundary moves from the
close vicinity of the bottom towards the saddle, J(t) undergoes qualitative changes
while still being step-wise.
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))
U (q) − U (qb ) ≈
ω02
(q − qb )2 ,
2
(18)
√
where ω0 = 2 and qb = −1, in the case of U (q) (3). Thus (15) reduces to a
linear equation with constant coefficients that can be integrated explicitly. Smin (t)
can be found explicitly too. Rather than presenting some cumbersome formulas we
discuss their most important consequence: if Γ < 2ω0 , then Smin (t) has a step-wise
shape (Fig.2(b)) i.e. possesses inflection points with dSmin /dt = 0 at
nπ
,
ω0 1 − (Γ/2ω0 )2
∆Up
,
S(tn ) =
1 − exp(−Γtn )
t = tn ≡
∆Up ≡ ω02 (qaw − qb )2 /2,
(19)
Γ < 2ω0 ,
n = 1, 2, 3, ...
The flux barely changes near tn whereas it rises sharply beyond this range provided
the corresponding n is not too large [21] (Fig.2(b)). In the underdamped case, the
“length” of each step, tn+1 −tn , is half a period of eigenoscillation and the “height”
Γ→0
of the first steps is large: S(tn ) − S(tn+1 ) ≈ ∆Up ω0 /(πΓn(n + 1)) −→ ∞. As Γ
grows, the length of a step increases while the height decreases and, at Γ = 2ω0 ,
the steps vanish.
The instants tn mark intervals corresponding to different topologies of the
MPEP: for t ≤ t1 , [q(τ )] is monotonic while, for tn < t ≤ tn+1 (n = 1, 2, 3, ...),
[q(τ )] possesses n turning points. As t changes, the MPEP varies continuously for
any t, including t = tn . The exit velocity is non-zero unless t = tn (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 Smin (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 qaw to a distinctly non-parabolic region of
(c)
U (q), but still not too close to the saddle (< qaw ). 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 Γ [16]:
q̈ + Γ q̇ + dU/dq = 0
2
2
(20)
2
[Γ̇ + (Γ − (Γ ) )/2]q̇ = 2ΓẼ,
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
333
0.01
E
∆ Up
(a)
0
−1
−0.9
q
0.02
12
J
0.01
8
S min
____
∆ Up
0
0
5
10
15
t
4
(b)
0
0
2
t
4
6
Figure 2: The case of the bottom initial state. (a) The parabolic approximation
Up (q) ≡ (q +1)2 (thick solid line) of U (q)−U (qb ) near the bottom, and examples of
MPEPs (plotted in the energy-coordinate plane E − q where E ≡ q̇ 2 /2 + Up (q)) at
(1)
Γ = 0.05; the absorbing wall (at qaw = qaw ≡ −0.9) is indicated by triangles; (b)
Smin (t)/∆Up explicitly calculated in the parabolic approximation is shown by the
solid line with markers: circles, squares and triangles indicate regions corresponding to respectively 0, 1 and 2 turning points in the MPEP; Smin (t)/∆U derived
from simulations in U (q) (3) is shown by the jagged line (∆U ≡ U (qaw ) − U (qb )).
Dashed and dotted lines indicate the theoretical 1st and 2nd inflection points
with dSmin /dt = 0, in (b), and the corresponding MPEPs, in (a). The thin solid
line shows the large-time asymptote level (= 1), in (b), and the corresponding
MPEP (which is the time-reversal of the noise-free trajectory from the state
(q = qaw , q̇ = 0)), in (a). The dash-dotted line shows in (a) the MPEP for some
arbitrarily chosen time t = 1.4 (see (b)): it demonstrates that the exit velocity is
typically non-zero. The inset shows J(t) measured at T = ∆U .
334
where
Soskin et al.
∂S
= −L +
Ẽ ≡ −
∂t
∂L
d
−
∂ q̇
dt
∂L
∂ q̈
q̇ +
Proceedings TH2002
∂L
q̈
∂ q̈
(21)
is conserved along the MPEP [20, 16], analogously to energy in mechanics [22].
Given that the initial state is at the bottom, it can be shown that Ẽ ≥ 0 on the
MPEP. Allowing for the fact that ∂S/∂ q̇(t) = 0 on the MPEP,
dSmin
= −Ẽ|MPEP ≤ 0.
dt
(22)
The system (20) has a remarkable feature: if Ẽ = 0, the equation for Γ can
be integrated explicitly [16]. So, the 4th-order equation (15) reduces to a closed
2nd-order equation [23]. Allowing for q̇i = 0, the equation for the time-reversed
trajectory [q̃(τ )] ≡ [q(t − τ )] becomes
d2 q̃
dU (q̃)
1 + AeΓτ dq̃
+
= 0,
+Γ
2
dτ
1 − AeΓτ dτ
dq̃
q̃(0) = qaw .
A = e−Γt
(23)
For the sake of convenience, we have also presented in (23) the initial q̃ which
follows from the third of conditions (13). The derivative dq̃(τ = 0)/dτ must be
chosen such that the condition (16) is satisfied: comparing Eq.(23) at τ = 0 with
Eq.(16), we come to the important conclusion that
dq̃(τ = 0)/dτ = 0,
(24)
i.e. the MPEP has a zero exit velocity if dSmin /dt = 0.
One can show (cf. [16]) that the number of possible finite values of t in
eq.(23), such that q̃(t) = qb , equals the number N of turning points in the noisefree (t = ∞) trajectory. Labelling such times t as tn ≡ tn (qaw ) (n = 1, 2, ..., N ),
one may relate n to the number ntp of turning points in the trajectory (23)-(24):
n = ntp + 1. tn increases with n and, if N = ∞, the trajectory (23)-(24) for t = tn
with n → ∞ coincides with the noise-free trajectory. If
Γ < 2ω0 ,
(25)
then N = ∞ [22, 16] while, if Γ ≥ 2ω0 , then typically N = 0. In rare cases, there
is a finite N = 0 at Γ ≥ 2ω0 [16].
Thus, if Γ < 2ω0 , then Smin decreases with t monotonically, possessing an
infinite number of inflection points tn with dSmin (tn )/dtn = 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 = tn .
At τ = 0, Eq.(23) coincides with the conventional relaxational equation with
a finite friction parameter, Γcth(Γt/2). Hence, the closer qaw is to the saddle, the
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
335
slower the motion near the wall. Thus, tn → ∞ if qaw → qs . On the contrary, the
time of motion along MPEPs which get to the wall with non-zero velocity (they
relate to sections Smin (t) with non-zero dSmin /dt) is less sensitive to the distance
qs − qaw and remains finite even if qaw = qs . Consequently, as qaw grows, the onset
of the fold at t ≈ t1 (according to numerical calculations) occurs at the critical
(c)
(c)
value qaw : dSmin /dt is discontinuous at the fold (Fig.3(b)). At qaw > qaw , there
are intervals of t during which the system (13),(15)-(17) possesses more than one
solution. It is because, the closer qaw is to qs , 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. 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 dSmin /dt: its values on different sides of the fold
differ drastically, so that Smin (t) and J(t) are still distinctly step-wise (Figs.3(c)).
We have tested some of the above predictions using computer simulations.
Smin (t) is derived via optimal fitting of J(t) obtained at different T . Figs. 2(b)
and 3 show a good agreement between Smin (t) from the theory and from the
simulations. The growth of the flux is clearly step-wise (see insets) in both cases.
2.2
Non-bottom initial state with a given coordinate and velocity
If the initial state with a given coordinate and velocity, {qi , q̇i }, is shifted from
the bottom of the well {qb , 0} then Smin (t) changes: cf. Fig.4. Typically, Smin (t)
becomes non-monotonic: cf. Fig. 4(b) (only if q̇i = 0 it becomes stepwise 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 Smin (t): the shift
of energy in Fig. 4(a) and Fig. 4(b) is equal to ∆Up /100 and ∆Up /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.
2.3
Thermalized 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 T0 so that various non-bottom states are necessarily involved.
The strong sensitivity of the flux Jqi ,q̇i (t) to the shift of {qi , q̇i } 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 T0 < T . Moreover, if T0 /T 336
Soskin et al.
Proceedings TH2002
0.01
J
8
/∆ U)/dt
2
4
6
t
4
−d(S
4
0
0
min
S min
____
∆U
0
0
2
4
(a)
6
t
0
0
2
t
4
6
0.01
J
8
/∆ U)/dt
2
4
t
6
4
6
4
−d(S
4
0
0
min
S min
____
∆U
0
0
2
4
(b)
6
t
0
0
2
t
4
6
0.01
J
8
/∆ U)/dt
2
t
4
−d(S
4
0
0
min
S min
____
∆U
0
0
2
4
(c)
6
t
0
0
2
t
4
6
Figure 3: The case of the bottom initial state. The evolution of Smin (t) normalized
by ∆U ≡ U (qaw ) − U (qb ) (thick and jagged lines for the theory and simulations
(2)
(c)
respectively) as qaw increases: (a) qaw = qaw ≡ 0, (b) qaw = 0.371 ≈ qaw , (c)
(3)
qaw = qaw ≡ 0.9. Γ = 0.05. Branches of S(t) corresponding to 0, 1 or 2 turning
points in the escape path are shown by thin lines marked by circles, squares or
triangles respectively: in (a) and (b), only one branch exists at each t while, in
(c), a few branches coexist in some ranges of t (activation energy Smin (t) coincides
with the lowest S(t)). Left and right insets show respectively −d(Smin (t)/∆U )/dt
(theory) and J(t) measured at T = ∆U .
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
337
12
0.02
J
0.01
S min8
____
∆ Up
0
0
5
10
15
t
4
(a)
0
0
5
t
10
15
12
0.02
J
0.01
S min8
____
∆ Up
0
0
5
10
15
t
4
(b)
0
0
5
t
10
15
Figure 4: Comparison between the Smin (t) calculated for an initial state at the
bottom of the potential well (solid line) and for a few other initial states with
given coordinate and velocity, close to those at the bottom (other parameters are
the same as in Fig. 2): (a) q̇0 = 0 while q0 = qb −0.01 (dotted line) or q0 = qb +0.01
(dashed line), (b) q0 = qb while q̇0 = −0.01 (dotted line) or q̇0 = 0.01 (dashed line).
The insets show the corresponding J(t) observed in simulations for T = ∆U .
338
Soskin et al.
Proceedings TH2002
Γ/ω0 , then the stepwise 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 T0 :
Win (q0 , q̇0 ) =
Wqs (q0 , q̇0 ) ≈
Z −1 exp(−E0 /T0 )
0
for
for
E0 <U(qaw ),
E0 >U(qaw ),
(26)
E0 ≡ q̇02 /2 + U (q0 ),
Z=
dq0 dq̇0 exp(−E0 /T0 ).
E0 <U(qaw )
We assume that the probability for the system to leave the well before the relevant
“initial” instant t = 0 is negligible.
If at the “initial” instant t = 0 the additional noise source is switched on,
so that the effective temperature becomes T > T0 [24], the evolution of the flux
(6) with the initial distribution (26) becomes relevant. Given the activation-like
structure of Jq0 ,q̇0 (t) (eqs. (10)-(13)), the flux with the thermalized initial state
can be presented in the form
J(t) ≡ JT0 (t) = P̃ e−
S̃min (t)
T
where P̃ is some prefactor and S̃min is the generalized activation energy:
T
T0
, t = minq0 ,q̇0 Smin (q0 , q̇0 , t) + E0 ,
S̃min ≡ S̃min
T
T0
(27)
(28)
where Smin (q0 , q̇0 , t) is given by (11)-(13) and E0 is defined in (26).
It can be shown that, for any T0 < T , the function S̃min ( TT0 , t) is stepwise
in t. Analogously to the case of the bottom initial state, S̃min possesses inflection
points with dS̃min /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
T0
−Γt
A=e
1−
.
(29)
T
˙
The relevant instants t are determined using the condition q̃(t)
= 0 (unlike the
condition q̃(t) = qb relevant to the bottom initial state).
It can be shown that S̃min ( TT0 , t ∼ ω0−1 ) is close to Smin (qi = qb , q̇i = 0, t ∼
ω0−1 ) provided
Γ
T0
.
T
ω0
(30)
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
339
0.01
E
∆ Up
(a)
0
12
−1
q
2
4
−0.9
0.02
J
0.01
S min8
____
∆ Up
0
0
6
8
10 12 14
t
4
(b)
0
0
2
4
6
t
8
10
12
14
Figure 5: The case of a thermalized initial state. (a) MPEPs for t = 2.222, for
three characteristic values of T0 /T (other parameters are the same as in Fig. 2):
T0 /T = 0 (solid line), 0.01 (dotted line), 0.2 (dashed line); (b) The theoretically
calculated S̃min (t) for T0 /T = 0 (solid line), 0.01 (dotted line), 0.2 (dashed line).
The inset shows the corresponding J(t) measured for the system put at the bottom
of the well at t = −51, after which temperature T0 was valid till t = 0 and then
temperature switched to T = ∆U .
Otherwise S̃min (t ∼ ω0−1 ) is significantly lower and the steps are smeared (Fig.
5(b)).
The competition between the two small parameters, T0 /T and Γ/ω0 , is readily
interpreted physically. On one hand, the escape flux (on t ∼ ω0−1 ) from the bottom
is ∝ exp(−a∆U/(T Γ/ω0 )) where a ≡ a(t) ∼ 1. On the other hand, if the system
starts its motion from an energy E0 close to the barrier level, the probability of
(−1)
escape for time t ∼ ω0
will be ∼ 1, but then the probability to have such starting
energy is ∝ exp(−∆U/T0 ). It is the competition between these two exponentially
weak processes which leads to the relation (30). Fig. 5(a) shows that, for T0 /T =
0.01 Γ/ω0 ≈ 0.035, the MPEP starts close to the bottom while, for T0 /T =
0.2 Γ/ω0 , the starting energy is ∼ ∆U .
3 Multi-well metastable potential
As an example of the multi-well case, we consider the potential (4), which describes
the simplest SQUID [2]. We place an absorbing wall [8] at qaw = 4.5 (Fig.6(a))
while the initial state of the system (1),(4) may be any state within well-1; in
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Proceedings TH2002
simulations, we put it at the bottom of well-1, for the sake of simplicity. We
emphasize also that the type of the boundary is not important either, e.g. our
results are equally valid for the transition rates between non-adjacent wells in the
stable potential with more than two wells [16].
Unlike the single-well case, where the formation time of quasi-equilibrium is
(s)
of the order of tf (7), its formation in the multi-well case proceeds via two distinct
(1)
stages: first, quasi-equilibrium is formed within the initial well which takes tf ∼
(s)
tf : J evolves at this stage quite similarly to the single-well case; secondly, quasiequilibrium between wells becomes established which takes exponentially longer:
(2)
(s)
(1)
tf ∼ tf exp(∆U/T ) tf where ∆U means a minimal internal barrier. During
the latter stage, and during the subsequent quasi-stationary one, the flux J(t) can
be described via a solution of kinetic equations for the well populations, W1 and
W2 , using the concept of constant inter-attractor [25] transition rates αij (cf. [26]):
J(t) ≡ W1 α13 + W2 α23 =
−
α13 e
(2)
tf ≈ α−1
12 ,
T Us1 − U1 ,
t
(2)
t
f
+ αqs
e
t
− tqs
−e
−
t
(2)
t
f
(31)
,
tqs ≈ α−1
qs ≈ α12 /(α12 α23 + α21 α13 ),
(1)
t tf .
The physical meaning of the two terms in (31) is easily understood (cf. Fig. 6).
The first one corresponds to direct escapes, i.e. those that do not go via the bottom of well-2, and it dominates until quasi-equilibrium becomes established. The
second term, corresponding to indirect escapes, i.e. those that involve one or more
intermediate transitions between wells 1 and 2 while the ultimate transition to
3 may occur from either well. It dominates during the ensuing quasi-stationary
stage: it is the asymptotic part of this latter flux, αqs exp(−t/tqs ), that is called
the quasi-stationary flux.
Thus, in order to know the flux dynamics one needs to find the inter-well
transition rates αij . The rates α12 , α21 and the quasi-stationary rate αqs can be
calculated from the Kramers-Melnikov formula [4]. Thus, only one of the four αij
coefficients needs to be found independently. We choose α13 as the independent
coefficient.
The theoretical problem of finding α13 is inherently difficult. Melnikov pointed
out [4] that his method is valid in the multi-well case only if the barriers levels are
equal or at least close to each other (cf. e.g. [4, 27]), a requirement that is often
not satisfied. So, the method of optimal fluctuation (cf. the previous section) was
suggested [16], seeking the escape rate in the form
α13 = P e−
Smin
T
,
(32)
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
341
S
2
U
(a)
S1
1
3
2
q1
q
q
q2
s
1
s
2
qaw
q
−4
x 10
α 13
2
(b)
J
1
t(1)
f
0
0
t(2)
f
200
400
600
800
t
Figure 6: (a) the potential (4) and a sketch of direct (dotted line) and indirect
(dashed line) escape paths 1 → s2 ; thin dashed lines indicate positions of the
local minima (q1 , q2 ) and maxima (qs1 , qs2 ); (b) simulations of the dependence
of the escape flux on time J(t) (thin line) for the model (1),(4) at Γ = 0.15,
T = 0.4. The thick full and dashed lines show the approximation of J(t) by
Eq.(31) in which α12 , α21 , αqs are calculated by the Kramers-Melnikov formula
±1
[4]. For the thick full line, α13,23 = αqs (1 + {Ω1 Ω−1
)/(1 + {m
2 exp[(U1 − U2 )/T ]}
exp[kSmin (s2 → s1 )/T ]}±1 ) where Ω1,2 are the frequencies of eigenoscillation in
the bottom of wells 1,2 respectively, k is equal to 1,-1 for the ranges Γ providing
nf
s2 → 2, 1 respectively, Smin (s2 → s1 ) is calculated from the theory [16] and m is
the only adjustable parameter (m ≈ 1.1 for these parameters); for the dashed line,
α13 = 0 and α23 = αqs (1 + α21 /α12 ).
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Proceedings TH2002
where the action Smin does not depend on T and the dependence of the prefactor
P on T is relatively weak.
One can show that Smin is the minimum of a certain functional [16]
Smin ≡ Smin (1 → s2 ) = min[q(t)],ttr (S),
ttr
1
S ≡ Sttr [q(t)] =
dt(q̈ + Γq̇ + dU/dq)2 ,
4Γ 0
q(0) = q1 , q̇(0) = 0, q(ttr ) = qs2 , q̇(ttr ) = 0,
(33)
where the trajectory [q(t)] does not pass through attractor 2. It can easily be
shown that the ttr yielding Smin is equal to ∞. The [q(t)] yielding Smin is called
[16] the most probable direct transition path (MPDTP). The main features of Smin
and the MPDTP are illustrated in Figs. 7 and 8 for the system (1),(4); see [16] for
a rigorous general treatment [28].
Fig. 7 shows how the excess action
∆S ≡ ∆S(1 → s2 ) = Smin (1 → s2 ) − (Us2 − U1 )
(34)
varies with Γ over the whole range of Γ, from very strong damping to the ultraunderdamped case. One can resolve three distinct regions.
The overdamped region can be defined as Γ ≥ Γ0 = 2Ω2 , where Ω2 is the
frequency of eigenoscillation in the bottom of well–2. Here, there is no MPDTP
1 → s2 at all, so that α13 = 0 in the present approximation.
In the moderate-friction region, [Γ1 , Γ0 ], ∆S(Γ) is monotonic and undergoes
its largest variation: from 0 to Us1 − U2 . The MPDTP (see Fig. 8(a)) is the timeA=A−
nf
reversal of the path s2 → s1 → 1 in which the latter is the noise-free relaxation
from s1 to 1, whereas the former is the solution of (cf. [19] and the previous section)
1 + AeΓt
q̇d + dU (qd )/dqd = 0,
1 − AeΓt
q̇d (0) = 0.
qd (0) = qs2 ,
q̈d + Γ
(35)
Here A = A− is a negative constant providing for the path [qd (t)] to be monotonic
(i.e. without turning points). Note that, in general, there may be an infinite set
of A providing [qd (t)] connecting the saddles: the corresponding trajectories differ
by their number of turning points. But S for the other values of A (their absolute
values are smaller than |A− |) is larger than that for A− .
The underdamped region, Γ ≤ Γ1 , is divided by a number of characteristic
values of the friction Γn≥1 . Each of these Γn provides for a noise-free saddlenf
connection s2 → s1 , which possesses n − 1 turning points. In this region, ∆S(Γ)
undergoes oscillations corresponding to an alternation between two situations.
nf
In the first, [Γ2m , Γ2m−1 ] (m ≥ 1), a noise-free trajectory s2 → 1 exists and
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
2
↑
U −U
1.5
∆ S, ± S
← Γ0
2
1
0.06
1
0.03
min
1
2
(s → s )
s
343
0.5
0
0
−0.03
Γ
0
0.5
1
Γ
2
−0.06
0
−0.5
1
0.1
Γ
1.5
Γ
0.2
2
Figure 7: Theoretical and experimental data on direct escapes/transitions in the
metastable potential (4) (Fig.4(a)). The calculated excess of action over a difference
of energies, ∆S(1 → s2 ) (34), is shown by the full line. It is related to the escape
rate α13 . The calculated ±Smin (s2 → s1 ), related to R (37) by Eq.(38), is shown by
the dotted line. It overlaps the full line in the half-plane of positive ordinates. The
corresponding quantity (39) based on data obtained by electronic and computer
simulations is shown by squares and crosses respectively. Values of Γn≥1 correspond
to noise-free saddle-connections with n − 1 turning points. At Γ = Γ0 = 2Ω2 ≈ 2.1,
nf
nf
the turning points in the noise-free trajectories s2 → 2 and s1 → 2 disappear. The
inset shows the low Γ range enlarged.
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Soskin et al.
Proceedings TH2002
3
(a)
s
2
2
E
1
0
s1
1
−1
2
−8
−4
q
0
4
3
(b)
s
2
2
E
1
0
s1
1
−1
2
−8
−4
q
0
4
3
(c)
s
2
2
E
1
0
s1
1
−1
2
−8
−4
q
0
4
Figure 8: Simulated direct transition paths s2 → 1 (thin full lines) in the energycoordinate plane E − q (where E = q̇ 2 /2 + U (q)) corresponding to (1),(4) at
different Γ: (a) 0.5, (b) 0.05, (c) 0.04 (T = 0.05 for (a) and T = 0.005 for (b), (c)).
nf
nf
The noise-free trajectories s2 → 2 and s1 → 1, 2 are shown by dashed lines. The
MPDTPs s2 → s1 are shown by thick dotted lines.
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
345
the MPDTP is just its time-reversal, with ∆S = 0. In the second situation,
[Γ2m+1 , Γ2m ] (m ≥ 1), the action varies nonmonotonically with Γ, and has cusps.
This is due to a competition between the two paths which are the time-reversals
A−
A+
nf
A±
nf
respectively of s2 → s1 → 1 and s2 → s1 → 1, where s2 → s1 are given by the
solutions of (35) with A+ ≡ A+ (Γ) > 0 and A− ≡ A− (Γ) < 0 respectively: see
Fig.8(b) and Fig.8(c) respectively. As Γ varies, S along one path becomes equal
to S along another, at a certain Γ, leading to switching between the paths and to
the cusp in ∆S(Γ): there are corresponding discontinuities in the non-equilibrium
potential [19] and fluctuational separatrix [29].
Thus, [16] predicts an exponentially strong dependence of the escape rate
α13 on friction, including interesting features such as oscillations and cusps [30],
(1)
for t tf . To establish whether these, and the properties of MPDTPs described
above occur in reality, we have undertaken analogue electronic and computer simulations. A necessary condition is smallness of the temperature: T ∆S, (Us1 −U1 ).
However to obtain reasonable statistics at such a small temperature would require
an unrealistically long time (∝ exp((Us2 − U1 + ∆S)/T )) [31]. We have overcome
this difficulty by exploiting the property of detailed balance [32], which implies
[16] that the MPDTP s2 → 1 is just the time-reversal of the MPDTP 1 → s2 , with
the corresponding actions differing by Us2 − U1 i.e.
∆S(1 → s2 ) = Smin (s2 → 1) = {0
nf
at
s2 →1,
at
s2 →2,
Smin (s2 →s1 )
nf
(36)
so that information about the transition s2 → 1 is equivalent to that for 1 → s2 ,
but the experimental time required is of course much smaller in the former case
(∝ exp(∆S/T )) than in the latter.
Fig.8(a) demonstrates that, for Γ ∈ [Γ1 , Γ0 ], most of the direct paths s2 → 1
A−
nf
do indeed concentrate near s2 → s1 → 1. Figures 8(b) and 8(c) demonstrate
A+
A−
nf
nf
switching of the MPDTP from s2 → s1 → 1 to s2 → s1 → 1 as Γ decreases in the
range [Γ3 , Γ2 ].
In order to study Smin we use the following technique. The system is put at
s2 , and one then follows its stochastic dynamics (1),(4) until either the bottom of
one of the wells is approached or the coordinate qaw is reached. After that, the
system is reset to s2 and the operation is repeated. Once adequate statistics have
been obtained, we calculate the ratio of transitions to wells 1 and 2 respectively:
R ≡ R(T ) =
Ns2 →1
.
Ns2 →2
(37)
It is easy to see that R ∝ exp(±Smin (s2 → s1 )/T ) (where +, − correspond to
nf
ranges of Γ providing s2 → 1, 2 respectively). So, Smin (s2 → s1 ) is related to R
(37) as
±Smin (s2 → s1 ) = lim [T ln(R(T ))],
T →0
(38)
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Proceedings TH2002
nf
where +, − correspond to s2 → 1, 2 respectively.
In practice, however, there is always a lower limit for T in simulations, Tl ,
because the overall simulation time must not become unrealistically long. That is
why the use of (38) may, in practice, introduce significant inaccuracy. To reduce
the influence of the pre-exponential factor we measure R both at Tl and at a
∼
slightly higher temperature, Tl + ∆T (Tl ∆T > Tl2 /Smin (s2 → s1 )), so that:
±Smin (s2 → s1 ) ≈
R(Tl + ∆T )
Tl2
ln(
).
∆T
R(Tl )
(39)
The quantities on the left and right of Eq. (39) are shown in Fig.7 respectively
by the dotted line (theory) and by squares and crosses (electronic and computer
∼
∼
simulations respectively). The agreement is satisfactory, given that 5 < Smin /Tl <
7.
Note that the magnitude of the largest oscillation in action may significantly
exceed Us2 − U1 . This occurs if the initial well–1 is adjacent to an external saddle
s2 while its depth is much less than that of the other well.
Finally, we comment on the experimental consequence of the cutoff of the
(1) ∼
MPDTP s1 → s2 , namely the drastic change of the time evolution of J for tf <
(2)
t tf : at Γ < Γ0 , one may in principle make T small enough that the sharp
∼
(1)
growth of J(t) at t < tf
(1)
tf α13 /(α12 α23 )
(1)
turns into a nearly constant value at tf
t while, at Γ > Γ0 , J(t) ≈ α12 α23 t over the whole relevant timescale: cf. the thin full and dashed lines in Fig.6(b). Another drastic change occurs
with the time evolution of the transition flux s2 → 1: at Γ < Γ0 , it has a high
(1)
narrow peak at t ≈ tf whereas, at Γ > Γ0 , it is a monotonically and very slowly
∼
(2)
growing function of t, at t < tf .
4 Conclusions and open problems
We conclude that: (i) escape from the metastable part of a potential differs
markedly after and before the formation of quasi-equilibrium within the metastable
region; (ii) at time-scales much less than the time of the formation of quasi(s)
equilibrium within a single well tf , the escape flux J grows exponentially strongly
with time and, moreover, if the friction Γ is small, it does so in a step-wise manner
and depends exponentially strongly on Γ; and (iii) if the metastable part of the
potential consists of more than one well, then the formation of quasi-equilibrium
takes an exponentially long period of time. In the latter case, the escape flux during
the most of the period is formed from direct escapes and depends exponentially
strongly on friction; moreover, J(Γ) undergoes oscillations in the underdamped
range and may drop drastically if Γ exceeds a critical value Γ0 ≥ 2Ω2 .
Open problems yet to be addressed include –
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
347
(s)
1. For the range t tf : (a) details of the case considered above, including
an accurate study of oscillations of the exit velocity and dSmin /dt with time, as
well as of the transition from a smooth Smin (t), with inflection points only, to
an Smin (t) possessing folds; (b) additional features characteristic of other types of
absorbing boundary, in particular inter-well transitions in the multi-well case; (c)
a careful consideration of the case with two absorbing walls with the initial state
on one of the walls, a case that is relevant e.g. to ionic channels [33] and the preliminary analysis indicates oscillations of the flux in time; (d) a 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 Smin (t), we anticipate the possibility of jumps in Smin (t);
∼ (2)
(s)
(1)
2. Both for t tf and for tf t < tf (the multi-well case only, for the
latter range): (a) the prefactor; and (b) multi-dimensional problems.
Acknowledgements
The research has been supported by the INTAS and by the EPSRC. We are grateful
to George Pickett for his assistance with some of the figures.
References
[1] H. A. Kramers, Physica 7, 284 (1940).
[2] K.K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon and
Breach, Philadelphia, 1986).
[3] P. Hanggi, P. Talkner and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).
[4] V. I. Mel’nikov, Phys. Rep. 209, 1 (1991).
[5] A.N. Cleland and M.L. Roukes, Nature 392, 160 (1998).
[6] V.A. Shneidman, Phys. Rev. E 56, 5257 (1997).
[7] E. W. G. Diau et al., Science 279, 847 (1998).
[8] The absorbing wall is equivalent to the barrier falling down to −∞, which is
illustrated by Fig. 6(a).
[9] The choice of tl is quite arbitrary: it should exceed the relevant time-scale
and is introduced purely to reduce the overall time of experiment.
[10] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path
Integrals(McGraw-Hill, New York, 1965).
[11] M. I. Dykman, P. V. E. McClintock, V. N. Smelyanski, N. D. Stein and N.
G. Stocks, Phys. Rev. Lett. 68, 2718 (1992).
348
Soskin et al.
Proceedings TH2002
[12] M Arrayás, I Kh Kaufman, D G Luchinsky, P V E McClintock, and S M
Soskin, Phys. Rev. Lett. 84, 2556 (2000).
[13] S.M. Soskin, V.I. Sheka, T.L. Linnik, R. Mannella, Phys. Rev. Lett. 86, 1665
(2001); ibid 87, 059901 (2001).
[14] If the wall is close to the bottom, the estimate (7) can be obtained by consideration of the parabolic potential: the Fokker-Plank equation is then solved
explicitly [15]. It follows directly that, if the initial state corresponds to the
bottom of the well i.e. W (q, q̇,√t = 0) = δ(q − qb )δ(q̇), then W near the energy
level ∆U and the coordinate 2∆U /ω0 evolves close to the equilibrium value
for the time described by Eq.(7). In another characteristic case, when the wall
(3)
is close to the saddle (cf. qaw in Fig.1), the estimate (7) is also typically valid:
it agrees with the explicit results [6] for the underdamped and overdamped
limits provided that the curvatures at the saddle and at the bottom are of
the same order, which in fact is typically the case: cf. U (q) (1).
[15] M.C. Wang, G.E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945).
[16] S.M. Soskin, J. Stat. Phys. 97, 609(1999).
[17] E. Cortes and F. Espinosa, Physica A 267, 414 (1999).
[18] R. S. Maier, D. L. Stein, J. Stat. Phys. 83, 291 (1996).
[19] R. Graham and T. Tel, Phys. Rev. A 33, 1322 (1986).
[20] L. E. Elsgolc Calculus of Variations, (Pergamon Press, London, 1961).
[21] If n → ∞, so that tn → ∞, then S(tn ) reduces to ∆Up while the MPEP
reduces to the time-reversal of the noise-free relaxational trajectory from the
state {q = qaw , q̇ = 0}, thus recovering the conventional Arrhenius activation
energy for the quasi-stationary flux [1] and the corresponding MPEP [19, 16]
respectively. The reduction of the asymptotic (at t → ∞) activation energy
and MPEP to their conventional forms is valid for an arbitrary U (q).
[22] L. D. Landau and E. M. Lifshitz Mechanics, (Pergamon, London, 1976).
[23] A similar equation was obtained in [19] but in a different context and by a
different method.
[24] Note that the additional noise source does not need necessarily be of a thermal
origin. In SQUIDs, for example, it could be an external magnetic flux noise.
[25] For the sake of brevity, we refer to region 3 as an “attractor” too.
[26] Freidlin, M.I. and Wentzell, A.D., Random Perturbations in Dynamical Systems (Springer Verlag, New York, 1984).
Vol. 4, 2003
Kramers Problem on Time-Scales Preceding Quasi-Stationarity
349
[27] R. Ferrando, R. Spadacini, G.E. Tommei, Phyr. Rev. E 48, 2437 (1993).
[28] Some of the results of [16] are closely related to some results of [19] which
were obtained by a different method and in a different context.
[29] M.I. Dykman, M.M. Millonas, V.N. Smelyanskiy, Phys. Lett. A 195, 53 (1994).
[30] In “long” jumps in a multi-barrier potential with equal or nearly equal barriers, a monotonic exponential dependence on friction was found in earlier
works (see e.g. [27] and [4] respectively). However, the obvious reason that
oscillations and cusps were not found in those studies is that they can only
arise in the case of distinctly different barriers levels.
[31] Note however that pre-exponential time-scales in other systems (e.g. SQUIDs)
may be much smaller and the relevant flux may be readily simulated and/or
measured.
[32] H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer-Verlag, Berlin,
1992).
[33] B. Hille, Ionic Channels of Excitable Membranes (Sinauer Associates Inc.,
Sunderland, 1992).
S.M. Soskin, T.L.Linnik and V.I.Sheka
Institute of Semiconductor Physics
National Academy of Sciences of Ukraine
Kiev, Ukraine
S.M. Soskin, M. Arrayás, I.Kh. Kaufman, D.G. Luchinsky and P.V.E. McClintock
Department of Physics
Lancaster University, Lancaster
LA1 4YB, UK
M. Arrayás
Nonlinear Dynamics and Chaos Group
Universidad Rey Juan Carlos
28933 Mostoles, Madrid
Spain
I.Kh. Kaufman and D.G. Luchinsky
Russian Research Institute for Metrological Service
Ozernaya 46, 119361 Moscow
Russia
350
R.Mannella
Dipartimento di Fisica
Università di Pisa and INFM UdR Pisa
56100 Pisa
Italy
Soskin et al.
Proceedings TH2002