Team:Valencia Biocampus/Demonstration/Diffusion3

Proof of a Group Behavior Diffusion Model from a Random Walk Model

For proving that, we will make the assumption that the worm only moves in one dimension ($x$), but it can be extrapolated to $\;n\;$ dimensions.

At each time step $\;q\;$ it either moves a distance $\;h\;$ to the left with probability $\;l\;$, a distance $\;h\;$ to the right with probability $\;r\;$, or stays in the same position with probability $\;1−r−l\;$ (the isotropic random walk has $\;r\;=\;l\;=\;1/2$, so it cannot rest motionless). We also define the probability that a worm is at a position $\;x\;$ at time $\;t\;$ by $\;P(x,t)\;$. One time step earlier, at time $\;t − q\;$, the walker must have been at position $\;x − δ\;$ and then moved to the right, or at position $\;x + δ\;$ and then moved to the left, or at position $\;x\;$ and then not moved at all. Thus:

$$ P(x,t)\;=\;P(x,t-q)\;\left(1 - l - q\right) + P(x-h,t-q)\;r + P(x+h,t-q)\;l $$
Assuming that $\;q\;$ and $\;h\;$ are so small, that are negligible compared to $\;t\;$ and $\;x\;$ respectively, we can expand de function as a Taylor series, around $\;t\;$ and $\;x\;$. Notice that higher terms than $\;q^2\;$ and than $\;h^3\;$ have been included in $\;O(q^2)\;$ and $\;O(h^3)\;$, respectively:

$$ P\;=\;\left(P - q\;\frac{\partial P}{\partial t}\right)\;\left(1 - l - r\right) + \left(P - q\;\frac{\partial P}{\partial t} - h\;\frac{\partial P}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial x^2}\right)\;r + \left(P - q\;\frac{\partial P}{\partial t} + h\;\frac{\partial P}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial x^2}\right)\;l + O(h^3) + O(q^2)$$
Rearranging this gives:

$$ \frac{\partial P}{\partial t}\;=\;\frac{\alpha\;h^2}{2\;q}\;\frac{\partial^2 P}{\partial x^2} - \frac{\beta\;h}{q}\;\frac{\partial P}{\partial x} + O(h^3) + O(q^2)$$
Where $\;\alpha\;=\;r + l\;$ and $\;\beta\;=\;r - l\;$. We now let $\;h,\;q,\;\beta\;\rightarrow\;0\;$ in such a way that the following limits are finite:

$$ D\;=\;\alpha\;\lim\limits_{h,\;q,\;\beta\rightarrow 0}\frac{h^2}{2\;q} $$ $$ v\;=\;\lim\limits_{h,\;q,\;\beta\rightarrow 0}\frac{h\;\beta}{q} $$
So we can neglect $\;O(h^3)\;$ and $\;O(q^2)\;$, resting:

$$ \frac{\partial P}{\partial t}\;=\;D\;\frac{\partial^2 P}{\partial x^2} - v\;\frac{\partial P}{\partial x} $$
Considerations:

• If we set $\;r\;=\;l\;=\;1/2\;$ as in the isotropic random walk, then $\;\beta\;=\;0\;$, so $\;u\;=\;0\;$, giving as a result the non-biased Diffusion Equation:
  
  $$ \frac{\partial P}{\partial t}\;=\;D\;\frac{\partial^2 P}{\partial x^2}$$
  
• In this case, $\;v\;$ is constant for all the space, not as in the case that concerns us, where $\;v\;$ depends on the gradient of the attractant, normally distributed (with Gaussian Distributions) in the space of interest. So, with $\;v\;$ constant, it is possible to obtain an analytical solution, given by Montroll & Shlesinger (1984), with initial condition $\;P(x,0)\;=\;\delta(x)\;$, that is:
  
  $$ P(x,t)\;=\;e^{-\frac{\left(x - v\;t\right)^2}{4\;D\;t}}$$
  

Considerations for the Random Walk:

• Step lenghts ($l_t$) in the order of a pixel in size. That implies, $ \Delta t $ as small as possible.
• Perfect Random Walk, with uniform probabilistic distributions either for $ v_t $, $\dot{\theta_t}$ and $\delta$.
  

Discretizing the whole space into pixels, and assuming, the worm can, either occupy one or not, we can assure that, at each time step, it can only move in four different directions: up, down, right or left from its position. As we considered that each random variable follows an uniform probabilistic distribution, it is equipossible to move in any of these directions, with a probability of $ \frac{1}{4} $ each.

Now, we can compute, the probability that the worm is at position $(x_m,y_m)$ at the iteration $n+1$ as follows:

$$ P_{n+1}(x_m,y_m)\;=\;\frac{1}{4}\;\left(P_{n}(x_m,y_{m+1}) + P_{n}(x_{m+1},y_m) + P_{n}(x_m,y_{m-1}) + P_{n}(x_{m-1},y_m)\right) $$

If we now subtract $ P_{n}(x_m,y_m) $ from both sides:

$$ P_{n+1}(x_m,y_m) - P_{n}(x_m,y_m)\;=\;\frac{1}{4}\;\left(P_{n}(x_m,y_{m+1}) + P_{n}(x_{m+1},y_m) + P_{n}(x_m,y_{m-1}) + P_{n}(x_{m-1},y_m) - 4\;P_{n}(x_m,y_m)\right) $$ $$ P_{n+1}(x_m,y_m) - P_{n}(x_m,y_m)\;=\;\frac{1}{4}\;\left(P_{n}(x_{m+1},y_m) - 2\;P_{n}(x_m,y_m) + P_{n}(x_{m-1},y_m) + P_{n}(x_m,y_{m+1}) - 2\;P_{n}(x_m,y_m) + P_{n}(x_m,y_{m-1})\right) $$
And multiply and divide by $q$ (time step) and $h^2$ (space step squared):

$$ \frac{P_{n+1}(x_m,y_m) - P_{n}(x_m,y_m)}{q}\;=\;\frac{h^2}{4\;q}\;\left(\frac{P_{n}(x_{m+1},y_m) - 2\;P_{n}(x_m,y_m) + P_{n}(x_{m-1},y_m)}{h^2} + \frac{P_{n}(x_m,y_{m+1}) - 2\;P_{n}(x_m,y_m) + P_{n}(x_m,y_{m-1})}{h^2}\right) $$
$$ \frac{P_{n+1}(x_m,y_m) - P_{n}(x_m,y_m)}{q}\;=\;D\;\left(\frac{P_{n}(x_{m+1},y_m) - 2\;P_{n}(x_m,y_m) + P_{n}(x_{m-1},y_m)}{h^2} + \frac{P_{n}(x_m,y_{m+1}) - 2\;P_{n}(x_m,y_m) + P_{n}(x_m,y_{m-1})}{h^2}\right) $$
Where $D\;=\;\frac{h^2}{4\;q}$.

Finally, recalling the definition for the first and second derivates:

$$f^\prime(x)\;=\;\lim\limits_{h\rightarrow 0}\frac{f(x+h) - f(x)}{h}$$ $$f^{\prime\prime}(x)\;=\;\lim\limits_{h\rightarrow 0}\frac{f(x+h) - 2\;f(x) + f(x-h)}{h^2}$$
We reach the basic Diffusion Equation, valid for $q$ and $h$ sufficiently small:

$$\frac{\partial P}{\partial t}\;=\;D \; \left(\frac{\partial^2 P}{\partial x^2} + \frac{\partial^2 P}{\partial y^2}\right) $$

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Okopinska A. (2002) Fokker-Planck equation for bistable potential in the optimized expansion. Physical review E, Volume 65, 062101