Team:Valencia Biocampus/Demonstration/Diffusion3

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

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):

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

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