Diffusion over potential barriers with colored noise

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

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

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

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

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

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

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

Markovian case: average path: standard deviation:

transition probability:

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

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

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

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

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

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

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

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

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

Mexican hat potential- positive and negative flux

Transition probability

Average path

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

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

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

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

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

White noise

Colored noise

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

Kinetic energy K=Beff Average Transition path probability total current

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