The Langevin package provides functions to estimate drift and
diffusion functions from data sets.
Details
This package was developed by the research group
Turbulence, Wind energy and Stochastics (TWiSt) at the Carl von
Ossietzky University of Oldenburg (Germany).
Mathematical Background
A wide range of dynamic systems can be
described by a stochastic differential equation, the Langevin equation. The
time derivative of the system trajectory dot{X}(t) can be expressed as
a sum of a deterministic part D^{(1)} and the product of a stochastic
force Γ(t) and a weight coefficient D^{(2)}. The stochastic
force Γ(t) is δ-correlated Gaussian white noise.
For stationary continuous Markov processes Siegert et al. and Friedrich et
al. developed a method to reconstruct drift D^{(1)} and diffusion
D^{(2)} directly from measured data.
The Langevin equation should be interpreted in the way that for every time
t_i where the system meets an arbitrary but fixed point x in
phase space, X(t_i+τ) is defined by the deterministic function
D^{(1)}(x) and the stochastic function
√{D^{(2)}(x)}Γ(t_i). Both, D^{(1)}(x) and
D^{(2)}(x) are constant for fixed x.
One can integrate drift and diffusion numerically over small intervals. If
the system is at time t in the state x = X(t) the drift can be
calculated for small τ by averaging over the difference of the
system state at t+τ and the state at t. The average has to be
taken over the whole ensemble or in the stationary case over all t =
t_i with X(t_i) = x. Diffusion can be calculated analogously.
Author(s)
Philip Rinn
References
A review of the Langevin method can be found at:
Friedrich, R.; et al. (2011) Approaching Complexity by Stochastic
Methods: From Biological Systems to Turbulence. Physics Reports, 506(5), 87<c3><a2><c2><80><c2><93>162.
For further reading:
Risken, H. (1996) The Fokker-Planck equation. Springer.
Siegert, S.; et al. (1998) Analysis of data sets of stochastic
systems. Phys. Lett. A.
Friedrich, R.; et al. (2000) Extracting model equations from
experimental data. Phys. Lett. A.