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<title>Mean square displacement - Morphogénie Logiciels</title>
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Mean square displacement
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<i class="fa fa-calendar"></i><time datetime="2014-06-06T12:20:00+02:00"> ven. 06 juin 2014</time>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">IPython.display</span> <span class="k">as</span> <span class="nn">disp</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
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<h1 id="A-short-tutorial-on-the-mean-square-displacement">A short tutorial on the mean square displacement<a class="anchor-link" href="#A-short-tutorial-on-the-mean-square-displacement">¶</a></h1>
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<h2 id="The-raw-mathematical-definition">The raw mathematical definition<a class="anchor-link" href="#The-raw-mathematical-definition">¶</a></h2>
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<p>The mean square displacement for a time difference $\Delta t$ is computed as the squared distance between the position of the particle at time $t$ and its position at time $t + \Delta t$ averaged over each successive time $t$:</p>
<p>$$
\mbox{<span class="caps">MSD</span>}(\Delta t) = \frac{\sum_0^{T - \Delta t} ||\mathbf{r}(t + \Delta t) - \mathbf{r}(t)||^2}{(T - \Delta t) / \delta t} = \frac{d_{t, t+\Delta t}}{(T - \Delta t) / \delta t} $$</p>
<p>Here the vector $\mathbf{r}(t)$ denotes the position of the particle at time $t$.</p>
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<p>So for a given trajectory we’ll mesure the mean square displacement for a delay of 10s ($\Delta t = 10s$) by computing the distance between the particle’s position at time $0 s$ and it’s position at time $9 s$, then between $1 s$ and $10 s$, and so on, square all those distances, and take the mean of those values.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">disp</span><span class="o">.</span><span class="n">SVG</span><span class="p">(</span><span class="s1">'msd_positions.svg'</span><span class="p">)</span>
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<p>The value of the <span class="caps">MSD</span> for a 10 seconds delay is the mean of the squares of the distances reprensented by the arrows above</p>
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<h3 id="Why-bother?">Why bother?<a class="anchor-link" href="#Why-bother?">¶</a></h3>
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<p>The mean square displacement is usefull as it allows to measure the diffusion coefficient of a particle in case the movement is random. When the movement is random, the <span class="caps">MSD</span> grows linearly with the delay, and when it’s linear, it grows like the square of the delay. We’ll see why..</p>
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<h2 id="Let's-create-two-trajectories,-one-random-and-one-linear,-in-2D">Let’s create two trajectories, one random and one linear, in 2D<a class="anchor-link" href="#Let's-create-two-trajectories,-one-random-and-one-linear,-in-2D">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">## Number of points in the trajectories</span>
<span class="n">n_points</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="c1">## Time step between two points</span>
<span class="n">t_step</span> <span class="o">=</span> <span class="mi">1</span>
<span class="c1">## Scale of the random movement (standard diviation)</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mf">1.</span>
<span class="c1">## </span>
<span class="n">xy_random</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="n">scale</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n_points</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">xy_random</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">xy_random</span><span class="p">,</span>
<span class="n">index</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">Index</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_points</span><span class="p">),</span> <span class="n">name</span><span class="o">=</span><span class="s1">'t_stamp'</span><span class="p">),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">'x'</span><span class="p">,</span> <span class="s1">'y'</span><span class="p">])</span>
<span class="n">xy_random</span><span class="p">[</span><span class="s1">'t'</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_points</span><span class="p">)</span> <span class="o">*</span> <span class="n">t_step</span>
<span class="n">xy_linear</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_points</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_points</span><span class="p">)])</span><span class="o">.</span><span class="n">T</span>
<span class="n">xy_linear</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">xy_linear</span><span class="p">,</span>
<span class="n">index</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">Index</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_points</span><span class="p">),</span> <span class="n">name</span><span class="o">=</span><span class="s1">'t_stamp'</span><span class="p">),</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">'x'</span><span class="p">,</span> <span class="s1">'y'</span><span class="p">])</span>
<span class="n">xy_linear</span><span class="p">[</span><span class="s1">'t'</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_points</span><span class="p">)</span> <span class="o">*</span> <span class="n">t_step</span>
</pre></div>
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<h3 id="$x$-vs-$y$-plots-of-the-two-trajectories">$x$ vs $y$ plots of the two trajectories<a class="anchor-link" href="#$x$-vs-$y$-plots-of-the-two-trajectories">¶</a></h3>
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<div class="prompt input_prompt">In [6]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax_rnd</span><span class="p">,</span> <span class="n">ax_lin</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
<span class="n">ax_rnd</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xy_random</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">xy_random</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="s1">'-r+'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
<span class="n">ax_rnd</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">)</span>
<span class="n">ax_rnd</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Random movement'</span><span class="p">)</span>
<span class="n">ax_lin</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xy_linear</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">xy_linear</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="s1">'-k'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
<span class="n">ax_lin</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">)</span>
<span class="n">ax_lin</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Linear motion'</span><span class="p">);</span>
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