Reminder: Exponential Decay of Autocorrelations and Brownian Motion

• AR(1) \[ x_{t+1} = a x_{t} + \xi_t \]
• with autocorrelation function \[ C(\Delta) = e^{-\frac{\Delta}{\tau}} \]

and

• Random walk \[ y_{t+1} = y_{t} + \xi_t \]
• with MSD \[ \langle y_t^2 \rangle = 2Dt \]

Correlated 'Random' Walks

Look at the cummulative process with correlated increments \[ y_t = \sum_{n=1}^{t} x_n \]

• In this case, the MSD for long times scales linearly, but for short times it scales quadratically \[ \langle y^2(t) \rangle = 2D [t-\tau (1-e^{-t/\tau})] \]
• The central limit theorem is still valid for long times

A two Dimensionsional model

In this course, we mostly concentrate on one dimension - generalizing to two dimensions is ofter straight forward. In this case, models and applications are usually in two or three dimensions.

• For a constant velocity with random angle \[ \dot{\vec{r}} = v \left(\begin{array}{c}\cos\phi\\ \sin\phi\end{array}\right) + \sqrt{2D_T}\vec{\xi}, \;\;\;\;\;\; \dot\phi = \sqrt{2D_R}\xi_\phi \;\; [+w] \]
• If w is different from 0, the particle is chiral
• The solution (no chirality case) is \[ \langle r_x(t) \rangle = \frac{v}{D_R}[1-\exp(-D_Rt)] \]
• The MSD goes to a linear scaling for long times, for short times it is steeper, for very short times again linear \[ \langle r^2(t) \rangle = 4D_Tt + 2\frac{v^2}{D_R^2} (D_Rt + e^{-D_Rt} -1) \]

Applications in Biophysics

Many obsevations of self-propelled particles or particles in an active environment can be explained with simple models (see Review: Bechinger et al. Rev. Mod. Phys. 88, 045006 (2016))

e.g. Particles with characteristic time for sticking together build clusters

...as do self-propelled particles with a force towards higher density

Zhang et al. Active phase separation by turning towards regions of higher density. Nat. Phys. 17, 961–967 (2021).

The Joseph-Effect

• ARFIMA(0,d,0) model with d=J-1/2 \[ x_t=\sum_{k=1}^\infty (-1)^{k+1}\frac{\prod_{a=0}^{k-1}(J-1/2-a)}{k!}x_{t-k}+\xi_t \]
• Anomalous diffusive scaling in the cummulative process \[ y_t=\sum_{n=1}^t x_n \;\;\;\;\;\;\;\; \langle y_t^2 \rangle \propto t^{2J} \]
• Continuous time version of this process: fractional Brownian motion
• Power law decay of increment autocorrelations \[ \langle x(t+\Delta)x(t) \rangle = \sigma^2 \frac{\Gamma(2-2J)}{\Gamma(3/2-J)\Gamma(J-1/2)} \frac{\Gamma(\Delta-1/2+J)}{\Gamma(\Delta+3/2-J)} \]

Diverging autocorrelation time

Look at the autocorrelation function \[ \langle x(t+\Delta)x(t) \rangle = \sigma^2 \frac{\Gamma(2-2J)}{\Gamma(3/2-J)\Gamma(J-1/2)} \frac{\Gamma(\Delta-1/2+J)}{\Gamma(\Delta+3/2-J)} \stackrel{\Delta\rightarrow\infty}{\sim} \frac{\Gamma(\Delta)\Delta^{J-1/2}}{\Gamma(\Delta)\Delta^{3/2-J}}=\Delta^{2J-2} \] with \[ \Gamma(z)=\int_0^\infty t^{z-1}e^{-t} \mathrm{d}t \stackrel{z\in\mathbb{N}}{=} (z-1)! \;\;\mbox{with}\;\; \Gamma(\Delta+a) \stackrel{\Delta\rightarrow\infty}{\sim} \Gamma(\Delta)\Delta^{a} \]

The autocorrelation time is the integral \[ \tau=\int_0^\infty \langle x(t+\Delta)x(t) \rangle \mathrm{d}\Delta \]

It can be found to yield long range anipersistence or long memory \[ \tau = 0 \;\;\mbox{for}\;\; 1/2>J>0 \;\;\;\;\;\; \tau \rightarrow\infty \;\;\mbox{for}\;\; 1>J>1/2 \]

Non-Independent Increments and the Central Limit Theorem

• Long range anipersistence or long memory imply non-independence of the increments at all times, so the central limit theorem is not valid, not even in the long time limit
• Gaussian increments lead to Gaussian processes
• The MSD scales with an anomalous exponent
• However, the system is ergodic (time averages equal ensemble averages)

Fractional Brownian Motion as a model for diffusion in crowded environments

e.g. cytoplasms of living cells or artificial solutions; up to 400 mg/ml of macromolecules

See Weiss, PRE 2013

Summary

• In the previous chapters we concentrated on simple forms auf autocorrelations (random walks with white-noise-increments and exponential decay of autocorrelations)
• In this chapter we discussed correlated correlated random walks and showed their relevance in soft matter research
• While increments of correlated random walks still obey the central limit theorem, power-law decay of autocorrelations violates with some exponent 2J-2 the premise of independent increments
• Fractional Brownian Motion (with Fractional Gaussion noise increments (ARFIMA(0,d,0))) is a long-range-correlated random walk - it is also ergodic

How to measure correlations