Mean Squared Displacement

Module 3: Modelling with Randomness

The mean squared displacement (MSD) measures how far a random walker spreads. It is the standard quantity you compute from tracked particles, cells or animals.

Definition

For an ensemble of walkers that start at the origin,

\[ \mathrm{MSD}(t) = \langle |\mathbf{x}(t) - \mathbf{x}(0)|^2 \rangle , \]

where $\langle \cdot \rangle$ is the average over walkers. For the unbiased 1D lattice walk you found $\langle x^2(n) \rangle = n\delta^2$ in The Random Walk.

Diffusive scaling

A walker that takes one step every $\tau$ seconds has made $n = t/\tau$ steps at time $t$. Then $\langle x^2 \rangle = (t/\tau)\,\delta^2$. (berg1993random?, Eqs. 1.9 to 1.11) defines the diffusion coefficient

\[ D = \frac{\delta^2}{2\tau} \tag{1}\]

and obtains

\[ \langle x^2(t) \rangle = 2Dt . \tag{2}\]

Motions along different axes are independent, so each coordinate contributes $2Dt$. In $d$ dimensions

\[ \langle r^2(t) \rangle = 2dDt , \tag{3}\]

which gives $4Dt$ in the plane and $6Dt$ in space (berg1993random?, Eqs. 1.13 and 1.14). The root-mean-square distance grows as $\sqrt{t}$. To go twice as far, a diffusing particle needs four times as long.

A simulated check

The figure uses 2000 planar walkers with independent Gaussian steps of standard deviation $\sigma = 1$ per coordinate and $\tau = 1$. Each coordinate has step variance $\sigma^2$, so $D = \sigma^2 / (2\tau) = 1/2$ and Equation 3 predicts $\mathrm{MSD} = 4Dt = 2t$.

Figure 1: Simulated MSD of 2000 planar Gaussian walkers (dots) and the prediction $4Dt$ with $D = 1/2$ (line). On log-log axes a slope of 1 is the signature of normal diffusion.

Anomalous diffusion

Many experiments find $\mathrm{MSD} \propto t^{\gamma}$ with $\gamma \neq 1$. The case $\gamma < 1$ is subdiffusion and $\gamma > 1$ is superdiffusion (metzler2000random?). Such behaviour requires that one of the random walk assumptions fails: steps are correlated, waiting times have no finite mean, or step lengths have no finite variance. You see the last case in Lévy flights.

Exercise

Estimate $D$ from the simulation by fitting a straight line to $\mathrm{MSD}(t)$ through the origin. How close is your estimate to $1/2$? Then fit the exponent $\gamma$ as the slope of $\log \mathrm{MSD}$ against $\log t$.

Solution

The least-squares slope through the origin is $\sum t\,\mathrm{MSD} / \sum t^2$, and $D$ is that slope divided by $2d = 4$.

slope = np.sum(t * msd) / np.sum(t**2)
gamma = np.polyfit(np.log(t[1:]), np.log(msd[1:]), 1)[0]
print(f"D estimate: {slope / 4:.3f}  (theory 0.5)")
print(f"exponent gamma: {gamma:.3f}  (theory 1)")

D estimate: 0.504  (theory 0.5)
exponent gamma: 1.007  (theory 1)

--- title: "Mean Squared Displacement" subtitle: "Module 3: Modelling with Randomness" format: html --- The mean squared displacement (MSD) measures how far a random walker spreads. It is the standard quantity you compute from tracked particles, cells or animals. ## Definition For an ensemble of walkers that start at the origin, $$ \mathrm{MSD}(t) = \langle |\mathbf{x}(t) - \mathbf{x}(0)|^2 \rangle , $$ where $\langle \cdot \rangle$ is the average over walkers. For the unbiased 1D lattice walk you found $\langle x^2(n) \rangle = n\delta^2$ in [The Random Walk](random-walk.qmd). ## Diffusive scaling A walker that takes one step every $\tau$ seconds has made $n = t/\tau$ steps at time $t$. Then $\langle x^2 \rangle = (t/\tau)\,\delta^2$. @berg1993random [Eqs. 1.9 to 1.11] defines the diffusion coefficient $$ D = \frac{\delta^2}{2\tau} $$ {#eq-msd-D} and obtains $$ \langle x^2(t) \rangle = 2Dt . $$ {#eq-msd-1d} Motions along different axes are independent, so each coordinate contributes $2Dt$. In $d$ dimensions $$ \langle r^2(t) \rangle = 2dDt , $$ {#eq-msd-nd} which gives $4Dt$ in the plane and $6Dt$ in space [@berg1993random, Eqs. 1.13 and 1.14]. The root-mean-square distance grows as $\sqrt{t}$. To go twice as far, a diffusing particle needs four times as long. ## A simulated check The figure uses 2000 planar walkers with independent Gaussian steps of standard deviation $\sigma = 1$ per coordinate and $\tau = 1$. Each coordinate has step variance $\sigma^2$, so $D = \sigma^2 / (2\tau) = 1/2$ and @eq-msd-nd predicts $\mathrm{MSD} = 4Dt = 2t$. ```{python} #| label: fig-msd-check #| fig-cap: "Simulated MSD of 2000 planar Gaussian walkers (dots) and the prediction $4Dt$ with $D = 1/2$ (line). On log-log axes a slope of 1 is the signature of normal diffusion." #| echo: false import matplotlib.pyplot as plt import numpy as np from amlab.random_walks import mean_squared_displacement, simulate_gaussian_walk rng = np.random.default_rng(16) xy = simulate_gaussian_walk(n_walkers=2000, n_steps=500, dim=2, sigma=1.0, rng=rng) msd = mean_squared_displacement(xy) t = np.arange(msd.size) fig, ax = plt.subplots(figsize=(6, 3.5)) sel = np.unique(np.logspace(0, np.log10(t[-1]), 25).astype(int)) ax.loglog(t[sel], msd[sel], "o", label="simulation") ax.loglog(t[1:], 2 * t[1:], "k-", label="$4Dt$") ax.set_xlabel("time $t$") ax.set_ylabel("MSD") ax.legend() plt.tight_layout() plt.show() ``` ## Anomalous diffusion Many experiments find $\mathrm{MSD} \propto t^{\gamma}$ with $\gamma \neq 1$. The case $\gamma < 1$ is subdiffusion and $\gamma > 1$ is superdiffusion [@metzler2000random]. Such behaviour requires that one of the random walk assumptions fails: steps are correlated, waiting times have no finite mean, or step lengths have no finite variance. You see the last case in [Lévy flights](levy-flights.qmd). ::: {.callout-note} ## Exercise Estimate $D$ from the simulation by fitting a straight line to $\mathrm{MSD}(t)$ through the origin. How close is your estimate to $1/2$? Then fit the exponent $\gamma$ as the slope of $\log \mathrm{MSD}$ against $\log t$. ::: ::: {.callout-tip collapse="true"} ## Solution The least-squares slope through the origin is $\sum t\,\mathrm{MSD} / \sum t^2$, and $D$ is that slope divided by $2d = 4$. ```{python} slope = np.sum(t * msd) / np.sum(t**2) gamma = np.polyfit(np.log(t[1:]), np.log(msd[1:]), 1)[0] print(f"D estimate: {slope / 4:.3f} (theory 0.5)") print(f"exponent gamma: {gamma:.3f} (theory 1)") ``` :::