1. Noisy Bistable Systems with Memory
Kramers’ law with time-delayed dynamical systems
Tomasz Piwonski
Physics Department,
University College Cork, Ireland

2. Acknowledgements People who did the real work
John Houlihan, Stephen Hegarty and Thomas Busch
David Goulding, David Curtin, Ian O’Driscoll, Robin Gillen and Sergey Melnik, Tomasz Piwonski
Cristina Masoller

3. Outline What is Kramers’ law Time delayed dynamical systems
A first theoretical calculation
Experiment with a bistable laser
Experiment using an electronic circuit
Asymmetric potential
Future work

4. Trapped particle with noise
A particle is trapped in a potential decays to the minimum of the potential.
If the particle is randomly kicked by noise, then the position of the particle will fluctuate near the minimum.
NOISE

5. Double Potential
For potentials with two minima, the particle can decay to two different positions
Noise can kick the particle from one minimum to the other.
NOISE
x(t)

6. Residence time
The residence time in one depends on the noise and barrier height
From a probabilistic point of view, one can calculate the residence time distribution function.

7. A generic example
We can take a simple potential

8. Two state model
The dynamics is similar to that of a two state particle s(t)=+/-1 where the probability of switching in a time dt is pdt
In this case the Kramers’ time is 1/p

9. Kramers’ formula assumes that the particle has no memory
Each step depends only on the position of the particle and on the noise level
Could we derive a formula for systems with memory
A first step would be to take into account memory at x(t-). For example, the equation below is a simple extension of the generic potential

10. RTD for system with memory
P(T) Or
<p(T)> T ?

11. A potential with memory !
The delay term can be included in the potential
And we could apply Kramers’ law

12. Potential and two state model with memory
The potential barrier will depend on the value of x(t-).
The two state model can be adapted to describe this effect. In this case the switching rate p depends on s(t-).
Tsimring and Pikovsky
PRL 87, (2001)

13. Auto-correlation
This model can be used to calculate the autocorrelation function

14. Power Spectrum

15. In general state x(t- ) is not known !!!
(for 0<t< )
Transition rates
<p(T)> T  p1 p2 ?

16. Analytical prediction

17. The first experiment using lasers
We used a micro-cavity surface emitting laser (VCSEL). This device emits in two possible polarisation.
Near the switching point, the laser spontaneous switch between the two polarizations

18. Kramers’ switching between the two polarisations
The residence time distribution of polarisation switching follows an exponential behavior
Willemsen et al, PRL, 82, 4815 (1999)

19. The experimental set-up

20. Laser experiment
Laser + Delay
Diagnostics

21. The first experimental results
Time trace
RTD for X and Y polarization

22. The first experimental results
Residence time distributions

23. Power Spectra

24. Long delay line

25. Schmitt trigger experiment

26. With delay

27. Comparison between the two experiments
Stronger saturation for the Schmitt trigger
The “well” is deeper
This means that the Schmitt trigger is closer to the two state model and the Vcsel is close to the continuous model
In the continuous model, the residence time distribution is not exactly exponential for t>.

28. Residence time distribution

29. Asymmetric potential

30. Power spectra
* Stochastic resonance Luca Gammaitoni Reviews of Modern Physics, Vol. 70, No. 1, January 1998

31. Conclusion
Presented a short introduction about the behavior of noisy stochastic bistable systems with memory
An experiment using a laser
An experiment using an electrical circuit
Some analytical results
Future direction
Study asymmetric potential
Spatial extension
Coupled systems