3.2 Lyapunov exponents
3.2.3 Towards the Fluctuation Theorem
Occurrence of negative local largest Lyapunov exponent can be attributed to a be-haviour similar to the one predicted by the Fluctuation Theorems. Here a simplified and non-rigorous version of the theorem is used. The Fluctuation Theorem states
P(Σ = +a)
P(Σ =−a) = exp(aτΣ)¯ (3.1)
with Σ the entropy production identified by the local largest Lyapunov exponent.
In figure 3.5 the probability distribution for the local largest Lyapunov exponent is assessed for the T21L5 case. The distribution is nearly Gaussian as indicated by the fit, however, especially in the tails a different distribution cannot be ruled out.
The Gaussianity of the distribution is no trivial result as there is no clear reason why the local largest Lyapunov exponent should be Gaussian. However, the fact
−0.2 −0.1 0 0.1 0.2 0.3 10−2
10−1 100
LLLE [1/day]
Relative Frequency
Figure 3.5— The relative frequency of the local largest Lyapunov Exponent (ˆλ) for the T21 resolution with 5 vertical levels. Additionally, a Gaussian fit to the data (solid line) is given.
that it is nearly Gaussian is a good hint towards the validity of equation (3.1) since this relations follows trivially if Σ is Gaussian. Apart from the Gaussianity the next striking observation is the large portion of values less than zero. About 17% of all values are negative so there is a significant occurrence of these events.
The sensitivity of this phenomenon has been investigated and the results are shown in figure 3.6 and 3.7 respectively for a fixed error growth time ofτ = 100 hours.
Figure 3.6 shows the dependence on vertical resolution. In this case the horizontal resolution is kept constant at T21, but the vertical resolution is changed with values of 3,5,10 and 20 levels. For clarity the Gaussian fits are shown in this figure and it can be seen that larger vertical resolutions mostly increase the fluctuation effect.
The distribution is broader, but the mean shifts very little to more positive values.
Furthermore it seems that vertical resolutions larger than 10 levels add nothing further to the dynamics as a saturation effect becomes visible for this low horizontal resolution.
The dependence on the horizontal resolution is shown in figure 3.7 where the vertical resolution has been kept constant at five levels, but the horizontal resolution is changed. T15, T21, T31 and T42 are used and as before only the Gaussian fits are shown. Between T21, T31 and T42 the major difference is a shift of the distribution to the positive side of the graph. The distributions become marginally broader, but this is a minor effect compared to the shift of the mean to much larger values for higher horizontal resolutions. The resolution T is a special case. Its mean is as assumed lower than the means of the other resolutions, but the distribution is much
3.2 Lyapunov exponents 39
sharper than expected by the results from the other resolutions.
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 10−2
10−1 100
LLLE [1/day]
Relative Frequency
L3 L5 L10 L20
Figure 3.6— The relative frequency of the Gaussian fit of the local largest Lyapunov Exponent (ˆλ) for the T21 resolution with different numbers of vertical levels.
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 10−2
10−1 100
LLLE [1/day]
Relative Frequency
T15 T21 T31 T42
Figure 3.7— The relative frequency of the Gaussian fit of the local largest Lyapunov Exponent (ˆλ) for 5 vertical levels and different horizontal resolutions.
0 50 100 150 200
−0.05 0 0.05 0.1 0.15
τ [h]
ˆ λ [1 / d ay ]
0.9 0.99
Figure 3.8— Quantiles of the local largest Lyapunov Exponent (λ) versus the growth timeˆ τfor the T21 resolution with 5 vertical levels.
Equation (3.1) does not solely depend on Σ but on the growth timesτ as well.
To get the respective behaviour for different error growth times τ the local largest Lyapunov exponent is calculated every 6 hours over the course of 10 days or 240 hours. 240 hours are equivalent to 480 time steps in T21 and T31 and 960 time steps in T42. The result is presented in figure 3.8 where for the T21 case with 5 levels the median of the distribution as well as the .9 and .99 quantiles are shown.
The median does not change, however, the distribution slowly sharpens with τ as the quantiles move closer to the median. In the limitτ → ∞the distribution should collapse to the single value of the median which is the global Lyapunov exponent.
With increasingτ the probability to observe negative values does decline due to the sharpening of the distribution.
All the previous observations indicate that the Fluctuation Theorem is applicable here. For this reason the validity of equation (3.1) has been tested for the T21 case with 5 levels. Figure 3.9 shows the results for fixed values of the growth timeτ. For all cases a robust linear relationship between the logarithm of the left hand side of equation (2.16) and the approximated entropy production Σ is evident. It has to be noted that for larger values of Σ this relation becomes increasingly more uncertain due to the data scarcity in this regime. Larger growth times run into this problem earlier since their distribution is already sharper in accordance with the results from figure 3.8. It remains reasonable that the local largest Lyapunov exponent fulfils the Fluctuation Theorem with respect to the entropy production rate Σ.
3.2 Lyapunov exponents 41
0 0.025 0.05 0.075 0.1
0 2 4 6 (a)
logp(+Σ)/p(−Σ)
0 0.025 0.05 0.075 0.1
0 2 4 6 (b)
0 0.025 0.05 0.075 0.1
0 2 4 6 (c)
Σ
logp(+Σ)/p(−Σ)
0 0.025 0.05 0.075 0.1
0 2 4 6 (d)
Σ
Figure 3.9— The right hand side of(2.16)against the values ofΣfor different growth timesτ.
0 50 100 150 200 250
0 0.1 0.2 0.3 0.4 0.5
(e)
τ
Slope
Figure 3.10— The logarithmic slope from figure 3.9 against the growth timeτ. For allτ >160h a least square fit that passes through the origin is calculated.
The relation with the growth time is assessed in figure 3.10. It should be log-linear with τ, however, this relation does not hold for very small τ. The results indicate that such a relationship can be found for values ofτ larger than 160 hours.
For this regime a least square fit that is forced to go though the origin is computed as shown in the figure. Since the data is scattered around this line for the mentioned regime it can furthermore be assumed that equation (2.16) also holds with respect toτ forτ >160 h.