1. Diffusion over potential barriers with colored noise

2. Table of contents One dimensional Langevin-equation
Generating colored noise
Diffusion over potential barriers
Inverse harmonic potential
Mexican-hat potential
Conclusion and outlook

3. One dimensional Langevin equation
in the early 20th century Paul Langevin established a phenomenological equation to describe the Brownian motion
: friction coefficient
: external force
: stochastic force
Markovian description

4. Microscopical description- generalized Langevin equation
dissipation fluctuation theorem:
friction is directly related to the stochastic force
autocorrelation leads to different correlation effects
in general a Non-Markovian description

5. Generating colored noise
: correlation function
:random Gaussian
variable
:arbitrary pulse
shape
:mean pulse
rate

6. Diffusion over potential barriers
study on the impact of different potentials and correlation functions to the particle evolution
correlation functions:
D: diffusion coefficient
typical time scales
of the correlation

7. Diffusion over potential barriers
potentials:
inverse harmonic potential
symmetric Mexican hat potential
observables:
average path
transition probability
fluxes
positive and negative

8. Analytical solution system of differential equations:
1) Markovian case:
2) Non-Markovian case:
solution method:
i) Laplace transform of systems 1) & 2)
ii) solving Laplace transform for X(s)
iii) back transform X(s) to receive x(t)
iv) determine and
v) insert solutions of step iv) into equation of transition probability
vi) time derivative of P(t) for total flux

9. Markovian case:
average path:
standard deviation:

10. transition probability:

11. transition probability converges to finite value for
white noise:
colored noise:

12. probability of 50% to cross the potential barrier, if argument of P(t) equals to 0
effective reduction of friction for finite correlation times

13. Inverse harmonic potential
three cases for particle evolution
K<Beff:thermal diffusion
K=Beff : combination of thermal diffusion and kinetic energy of the particle
K>Beff: kinetic energy of particles greater than thermal diffusion
choice of parameters:
m=1, T=0.25, D=1.5, x0=-1, =1

14. Kinetic energy K=0 particles propagate faster in case of colored noise
transition
probability
average
path
particles propagate faster in case of colored noise
ensemble is closer to the potential barrier for white noise
total
flux

15. Kinetic energy K=Beff/2
average
path
Transition
probability
realizations propagate towards the barrier
Smaller correlation effect for increasing correlation times
total
flux

16. Kinetic energy K=Beff
average
path
transition
probability
Average path fluctuates around the maximum of the barrier for small correlation time
total
flux

17. Kinetic energy K=2Beff comparable transition probability
average
path
transition
probability
comparable transition probability
average path crosses the potential barrier
total
flux

18. Comparison to numerics: K=Beff/2
Average
path
Transition
probability
total
flux

19. Mexican hat potential initial distribution
standard Gaussian distributed velocity
choice of parameters:
m=1, T=1.5, x0=-1, D=4
only for the first two correlation functions

20. Mexican hat potential- positive and negative flux

21. Transition probability

22. Average path

23. Conclusion and outlook
autocorrelation leads to different effects:
reduced effective friction
memory effect
smaller amplitude for positive and negative flux in case of Mexican-hat potential
equilibration time rises for increasing correlation times
studies of the Mexican-hat potential can be extended to Kramers' escape rate problem

24. Backup Autocorrelation leads to different correlation effects
In general a non-Markovian description
It can be obtained by a simple model:
Infinitely extended system of harmonic oscillators with masses m
Brownian particles of mass M>>m moved by collisions with the oscillators

25. Generating colored noise
General noise function
Pulse shape for white noise:
: random Gaussian
variable
: arbitrary pulse shape
: diffusion coefficient
: mean pulse rate

26. Characteristic functional of a Gaussian process:
Correlation function of a Gaussian process:
Stationary process:

27. After Wiener-Khinchin theorem spectral density of stationary process is given by Fourier transform of correlations function

28. White noise

31. Colored noise

32. Comparison of analytical with numerical solution- K=0
Average
path
Transition
probability
total
current

33. Kinetic energy K=Beff Average Transition path probability total
current

34. Kinetic energy K=2Beff Transition Average path probability total
current