5.2 Analytical approximations for the first passage time density
5.2.5 Long time asymptotic
For large T, F(T) decays exponentially, F(T)∝exp(−κT). The decrement of this decay is obtained from a long time asymptotic: κ= limT→∞S0(T) (compare with equations Eq. (4.5) and (5.9)). Thus, in the Hertz approximation Eq. (5.26) one gets
κH = lim
T→∞n1(t) =n0. (5.31)
The behavior in the Stratonovich approximation Eq. (5.30) is determined by limt,t0→∞RT
0 R(t, t0)n1(t0)dt0 =n0τcor, where we have introduced the correlation time τcor = lim
t→∞
Z ∞ 0
R(t, t0)dt0. (5.32)
5.2 Analytical approximations for the first passage time density 87
Figure 5.10: Correlation coefficient R(t,0) of the stationary sequence of upcrossings for a harmonic oscillator driven by white Gaussian noise, theoretical curves. (a) The parameters are γ= 0.01, D= 0.02 (gray line),γ = 0.8, D= 0.1 (dashed line), γ = 10.0, D= 5.5 (black solid line). (b) The same curves on the logarithmictscale. The other parameters are as in Fig. 5.8.
R(t,0) approaches zero at large t with and without oscillations in underdamped and in overdamped regimes, respectively. R(t,0)≈1 on short time scales reveals the nonapproaching character of upcrossings.
Inserting this expression into Eq. (5.30) and expanding the logarithm up to the second term we obtain
κS =n0
1 +1
2n0τcor
(5.33) providing a correction to κH. From the definitions Eqs. (5.28) and (5.32) it immediately follows thatτcor = 0 for a Poisson point process with uncorrelated events. Otherwiseτcor is different from zero, and is not necessarily positive due to the oscillating correlation coefficient.
In the next paragraphs we discuss the meanings ofτcor and of negative correlation times.
Since we are interested in the long time asymptotic, we consider a stationary processx(t), i.e. we shift the initial conditions to t→ −∞. Then, the sequence of upcrossings for x(t) is a stationary point process, whose distribution functions remain unchanged by an arbitrary shift of all their arguments, see Eq. (2.39). Using Eq. (5.28) one obtains the relation between the distribution functionn2(t,0) of a stationary sequence of upcrossings and the correlation coefficient:
n2(t,0) =n20[1−R(t,0)]. (5.34) If R(t,0)≈ 1 for some t, then n2(t,0)≈ 0, i.e. the probability to obtain two upcross-ing separated by the time interval t vanishes. This is the case for small values of t, since the upcrossings produced by a differentiable process x(t) are nonapproaching. In Fig. 5.10 the correlation coefficient is plotted for three parameter sets corresponding to strongly un-derdamped (gray line), slightly unun-derdamped (dashed line), and strongly overdamped (solid black line) situations. The logarithmic scale in Fig. 5.10(b) clearly shows the magnitude of the
Figure 5.11: Typical patterns of upcrossings generated by a harmonic oscillator driven by white Gaussian noise. The parameters are (a) γ = 0.08, D = 0.01, underdamped regime, (b) γ = 10.0, D= 5.5, overdamped regime. The other parameters are as in Fig. 5.8. Black solid line marks the threshold value xb. The upcrossings are clustered; the interval between upcrossings within a cluster is ∼Tp in the underdamped regime, and is extremely small in the overdamped regime.
minimal interval between two successive upcrossingstmin. It approximately equals the period of subthreshold oscillations Tp in the strongly underdamped regime, decreases with increas-ing friction, and becomes extremely small in the overdamped regime. For γ = 10.0, D= 5.5 corresponding toω0/γ= 0.1 the minimal interval between upcrossingstmin∼3·10−3 is much smaller than the mean interval between upcrossings TR= 15.6. In the underdamped regime R(t,0) takes values close to 1 periodically at t ∼ 1.5Tp, 2.5Tp . . .: the trajectory performs almost regular oscillations in this regime, and the appearance of two upcrossings separated by odd multiples of half the period is highly unlikely.
R(t,0) tends to zero fort→ ∞, when the upcrossings become independent, see Eq. (5.34).
R(t,0) approaches its asymptotic value in an oscillatory manner in the underdamped regime and monotonically in the overdamped regime. Negative values ofR(t,0) indicate the increased probability of two upcrossings separated by timet, see Eq. (5.34). In the overdamped regime R(t,0) achieves its minimal values on short time scales (for the example in Fig. 5.10 these are 10−2to 100), indicating the formation of very dense clusters of upcrossings in the overdamped regime. In contrast, R(t,0) has multiple minima at t ∼ Tp, 2Tp . . . in the underdamped regime, indicating the clustering of upcrossings on larger time scales, with the typical interval between successive upcrossings within a cluster being ∼Tp.
Recall that the functionsn2(t2, t1) andR(t2, t1) are symmetric under permutation of their arguments. Using this symmetry and stationarity of the sequence of upcrossings, substitution of Eq. (5.34) into Eq. (5.32 yields:
τcor = lim
t→∞2 Z t
0
1−n2(t0,0) n20
dt0 = lim
t→∞2
t− 1 n20
Z t 0
n2(x,0)dx
. (5.35)
5.2 Analytical approximations for the first passage time density 89
Figure 5.12: Fano factorF(T) for the sequence of upcrossings for a harmonic oscillator driven by white Gaussian noise, theoretical curves. (a) Underdamped regime, γ = 0.01, D = 0.02 (black solid line), γ = 0.08, D= 0.01 (gray solid line), γ = 0.8, D= 0.1 (black dashed line), and γ = 0.8, D = 0.44 (gray dashed line). (b) Overdamped regime, γ = 3.0, D = 0.5 (gray line) andγ = 10.0, D= 5.5 (black line). The other parameters are as in Fig. 5.8. The insets show the same curves on the short time scale.
The correlation time τcor is determined in terms of the first two distribution functions n2(t2, t1) and n1(t).
Now we can derive the relation betweenτcor and the Fano Factor F(T) of the stationary sequence of upcrossings, since the latter is connected by Eqs. (2.44) and (2.45) to the first two distribution functions. Substituting Eq. (2.44) into Eq. (2.45) and using stationarity of the process, we obtain for the Fano factor
F(T) = 1 + 1 n0T 2
Z T 0
dt Z t
0
dx n2(x,0)−n20T2
!
= 1−n0 T
Z T 0
2
t− 1 n20
Z t 0
n2(x,0)dx
dt. (5.36) Hence, in view of Eq. (5.35) the asymptotic value of the Fano factorF∞= limT →∞F(T) is related to the correlation time by
F∞= 1−n0τcor. (5.37)
The correlation timeτcor characterizes the pattern of upcrossings: τcor is negative ifF∞>1, andτcor is positive ifF∞<1.
If the sequence of upcrossings is overdispersed (F∞ > 1), then τcor is negative. The overdispersion results from the existence of two time scales in the point process. In the underdamped regime these are the period of oscillations Tp and the inverse Rice frequency 1/n0=TR. The upcrossings form clusters; the interval between upcrossings within a cluster is
γ D trel n0 τcor F∞ Cvar2
underdamped
0.01 0.02 200.0 0.124 -14.1 2.72 1.42
0.08 0.01 25.0 0.003 -432.0 2.26 2.17
0.8 0.1 2.5 0.003 -2.4 1.007 1.007
0.8 0.44 2.5 0.64 5.1 0.67 0.66
overdamped
3.0 0.5 0.66 0.008 -64.1 1.51 1.48
10.0 5.5 0.2 0.064 -52.4 4.36 4.31
Table 5.1: Time scales and variability coefficients for a sequence of upcrossings generated by a harmonic oscillator driven by a white Gaussian noise. The other parameters are as in Fig. 5.8.
∼Tp(or a multiple ofTp) and the interval between successive clusters is∼TR, see Fig. 5.11(a).
Clustering of upcrossings on the time scale of Tp is pronounced in the Fano factor curves, presented in Fig. 5.12(a) for sequences of upcrossings generated by an underdamped harmonic oscillator for four different parameter sets. On short time scales (inset in Fig. 5.12(a)), F(T) linearly decreases up toTp ≈6: the probability to find more than one upcrossing in an interval shorter than Tp vanishes. On long times scales, the Fano factor saturates at different values, which can be smaller or larger than 1 depending on the parameter values. Qualitatively, if the mean number of upcrossings within a cluster is large (large trel) and the mean interval between clusters is also large (largeTR), then the existence of two time scales manifests itself in F∞ >1 and in the negative correlation time τcor <0. Table 5.1 summarizes the relevant time scales and variability coefficients for all parameter sets used in Figs. 5.4 to 5.9. Note that F∞> Cvar2 indicates positive correlations between upcrossings, see Eq. (2.47).
The Fano factor curves for the sequences of upcrossings in the overdamped regime are de-picted in Fig. 5.12(b). Linear decrease inF(T) on short time scales is practically absent, since the upcrossings form very dense clusters, and two successive upcrossings can follow within an arbitrary small interval, see Fig. 5.11(b). In the overdamped regime two time scales are also present in the sequence of upcrossings. With increasing γ the number of upcrossings within a cluster increases. In contrast to the underdamped situation, F∞ increases with increasing γ. In the overdamped regimeF∞ is always larger than 1 due to the strong separation of time scales corresponding to intervals within and in-between clusters.
Let us return to the long time asymptotic of the Hertz and Stratonovich approximations
5.2 Analytical approximations for the first passage time density 91 Figure 5.13: Mean escape ratesκ(the inverse mean FPT) for a harmonic oscillator driven by white Gaussian noise of moderate inten-sity D and with moderate damping γ = 0.4 (upper curves), γ = 0.8 (middle curves) and γ = 2.6 (lower curves). The other parame-ters are as in Fig. 5.8. The rates κ obtained from numerical simulations are plotted with open circles, from the Hertz approximation with dashed lines and from the Stratonovich approximation with solid lines. Compare the curves with those in Fig. 4.3(a).
Eqs. (5.31) and (5.33). If the pattern of upcrossings is close to the Poisson process, thenF∞
is close to 1, andn0τcor vanishes. In this case, the second order correctionn0τcor/2 provided by Eq. (5.33) toκHis negligibly small and the Hertz approximation is absolutely sufficient. In particular, this is the case forγ = 0.8, D= 0.1, when the Hertz approximation is very accurate in the whole time domain. It is very instructive to compare the quality of approximations plotted in Figs. 5.4 to 5.9 with the corresponding values of variability coefficients in Table 5.1. In the Section 5.4 we use of the condition F∞ ≈ 1 to test the validity of the Hertz approximation.