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2024-03-29T05:08:36Z
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Beta3designs: /* Proof of a Group Behavior Diffusion Model from a Random Walk Model */
2013-10-04T08:53:17Z
<p><span class="autocomment">Proof of a Group Behavior Diffusion Model from a Random Walk Model</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Proof of a Group Behavior Diffusion Model from a Random Walk Model ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Proof of a Group Behavior Diffusion Model from a Random Walk Model ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">For proving </del>that, we will make the assumption that the worm only moves in one dimension ($x$)<del class="diffchange diffchange-inline">, but it can be extrapolated to $\;n\;$ dimensions</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">To prove </ins>that, we will make the assumption that the worm only moves in one dimension ($x$) <ins class="diffchange diffchange-inline">without loss of generality</ins>.</div></td></tr>
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Beta3designs
http://2013.igem.org/wiki/index.php?title=Team:Valencia_Biocampus/Demonstration/Diffusion3&diff=286237&oldid=prev
TonnyESP at 00:07, 4 October 2013
2013-10-04T00:07:44Z
<p></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">== Proof of a Group Behavior Diffusion Model from a Random Walk Model ==</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">In addition to last proof, we will make the assumption that the worm can now move in two dimensions ($x,y$).</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">As before, time step is $\;q\;$, and space step (either in $x$ or in $y$ coordinate) is $\;h\;$. Each time step $\;q\;$, the worm can move a distance $\;h\;$ either up, down, left or right with probabilities $\;u\;$, $\;d\;$, $\;l\;$ and $\;r\;$, respectively. Nevertheless, now we will consider that this probabilities will be dependent on position, so we must rename them as $\;u(x,y)\;$, $\;d(x,y)\;$, $\;l(x,y)\;$ and $\;r(x,y)\;$. The probability of remaining at the same position is, obviously, $\;1 - u(x,y) - d(x,y) - l(x,y) - r(x,y)\;$. The probability of being in position ($x,y$) at time $\;t\;$, $\;P(x,y,t)\;$ is:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ P(x,y,t)\;=\;P(x,y,t-q)\;\left(1 - l(x,y) - r(x,y) - u(x,y) - d(x,y)\right) + P(x-h,y,t-q)\;r(x-h,y) + \\ P(x+h,y,t-q)\;l(x+h,y) + P(x,y-h,t-q)\;u(x,y-h) + P(x,y+h,t-q)\;d(x,y+h)$$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Then, expanding each term as a Taylor series, centered in $\;t\;$, $\;x\;$ and $\;y\;$, including all terms $\;h^3\;$ and $\;q^2\;$ and higher, into $\;O(h^3)\;$ and $\;O(q^2)\;$, respectively:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ P\;=\;\left(P - q\;\frac{\partial P}{\partial t}\right)\;\left(1 - l - r - u - d\right) + \left(P - q\;\frac{\partial P}{\partial t} - h\;\frac{\partial P}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial x^2}\right)\;\left(r - h\;\frac{\partial r}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 r}{\partial x^2}\right) + \\ \left(P - q\;\frac{\partial P}{\partial t} + h\;\frac{\partial P}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial x^2}\right)\;\left(l + h\;\frac{\partial l}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 l}{\partial x^2}\right) + \left(P - q\;\frac{\partial P}{\partial t} - h\;\frac{\partial P}{\partial y} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial y^2}\right)\;\left(u - h\;\frac{\partial u}{\partial y} + \frac{h^2}{2}\;\frac{\partial^2 u}{\partial y^2}\right) + \\ \left(P - q\;\frac{\partial P}{\partial t} + h\;\frac{\partial P}{\partial y} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial y^2}\right)\;\left(d - h\;\frac{\partial d}{\partial y} + \frac{h^2}{2}\;\frac{\partial^2 d}{\partial y^2}\right) + O(h^3) + O(q^2) + O(h\;q)$$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">That, after operating, results:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ q\;\frac{\partial P}{\partial t}\; = \;h\;\left(\frac{\partial l}{\partial x} - \frac{\partial r}{\partial x}\right)\;f + h\;\left(\frac{\partial d}{\partial y} - \frac{\partial u}{\partial y}\right)\;f + \frac{h^2}{2}\left(\frac{\partial^2 r}{\partial x^2} + \frac{\partial^2 l}{\partial x^2}\right)\;f + \frac{h^2}{2}\left(\frac{\partial^2 d}{\partial y^2} + \frac{\partial^2 u}{\partial y^2}\right)\;f + h\;\left(l - r\right)\;\frac{\partial f}{\partial x} + h\;\left(d - u\right)\;\frac{\partial f}{\partial y} + \\ h^2\;\left(\frac{\partial l}{\partial x} + \frac{\partial r}{\partial x}\right)\;\frac{\partial f}{\partial x} + h^2\;\left(\frac{\partial d}{\partial y} + \frac{\partial u}{\partial y}\right)\;\frac{\partial f}{\partial y} + \frac{h^2}{2}\;\left(l + r\right)\;\frac{\partial^2 f}{\partial x^2} + \frac{h^2}{2}\;\left(d + u\right)\;\frac{\partial^2 f}{\partial y^2} + O(h^3) + O(q^2) + O(h\;q)$$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Now we define:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ \nu_x\;=\;\lim\limits_{h,\;q,\;\beta_x\rightarrow 0}\frac{\beta_x\;h}{q} $$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ \nu_y\;=\;\lim\limits_{h,\;q,\;\beta_y\rightarrow 0}\frac{\beta_y\;h}{q} $$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ a_{x,x}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_x\;h^2}{2q} $$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ a_{y,y}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_y\;h^2}{2q} $$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Where $\;\alpha_x \;=\; r + l\;$, $\;\alpha_y \;=\; u + d\;$, $\;\beta_x \;=\; r - l\;$ and $\;\beta_y \;=\; u - d\;$. </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">We now take equation, divide through by $\;q\;$ and take the limit as $\;h,q,\beta_x,\beta_y\rightarrow 0\;$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">such that $\;\beta_x h/q\;$, $\;\beta_y h/q\;$, $\;\alpha_x h^2/2q\;$ and $\;\alpha_y h^2/2q\;$ all tend to a constant, giving:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{x,x}}{\partial x^2} + \frac{\partial^2 f a_{y,y}}{\partial y^2} - \frac{\partial f \nu_x}{\partial x} - \frac{\partial f \nu_y}{\partial y} $$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Or, what is the same:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ \frac{\partial f}{\partial t}\;=\;\nabla·\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \nabla·\left(\underline{\nu} f\right) $$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">With:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ \underline{\underline{D}} \; = \begin{bmatrix}</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">a_{x,x}(x,y) & 0\\</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">0 & a_{y,y}(x,y)\\</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\end{bmatrix}</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\;\;\;\;\;\;\;\;\;\;\;\;</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\underline{\nu} \; = \begin{bmatrix}</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\nu_x(x,y)\\</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\nu_y(x,y)\\</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\end{bmatrix} $$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Nevertheless, in our case of study, we dealed with $\;D\;$ as a constant, so $\;a_{x,x}(x,y)\;=\;a_{y,y}(x,y)\;=\;D\;$, so we can tanke it out from the Laplacian, resting what we were waiting anxiously:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">$$ \frac{\partial f}{\partial t}\;=\;D\;\nabla^2 f - \nabla·\left(\underline{\nu} f\right) $$</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Although, surely it exists an analytical solution of this differential equation, it has not been found yet; that's why, we decided to solve it with numerical tools.</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-----</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-----</div></td></tr>
</table>
TonnyESP
http://2013.igem.org/wiki/index.php?title=Team:Valencia_Biocampus/Demonstration/Diffusion3&diff=286173&oldid=prev
Nightm4re: /* Proof of a Group Behavior Diffusion Model from a Random Walk Model */
2013-10-04T00:01:07Z
<p><span class="autocomment">Proof of a Group Behavior Diffusion Model from a Random Walk Model</span></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 00:01, 4 October 2013</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 136:</td>
<td colspan="2" class="diff-lineno">Line 136:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;D\;\nabla^2 f - \nabla·\left(\underline{\nu} f\right) $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;D\;\nabla^2 f - \nabla·\left(\underline{\nu} f\right) $$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Although, surely it exists an analytical solution of this differential equation, it has not been found yet; that's why, we decided to solve it with numerical tools.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><br/></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-----</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-----</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Okopinska A. (2002) <i>Fokker-Planck equation for bistable potential in the optimized expansion</i>. Physical review E, Volume 65, 062101 <br/> </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Okopinska A. (2002) <i>Fokker-Planck equation for bistable potential in the optimized expansion</i>. Physical review E, Volume 65, 062101 <br/> </div></td></tr>
</table>
Nightm4re
http://2013.igem.org/wiki/index.php?title=Team:Valencia_Biocampus/Demonstration/Diffusion3&diff=286120&oldid=prev
Nightm4re: /* Proof of a Group Behavior Diffusion Model from a Random Walk Model */
2013-10-03T23:54:40Z
<p><span class="autocomment">Proof of a Group Behavior Diffusion Model from a Random Walk Model</span></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 23:54, 3 October 2013</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 100:</td>
<td colspan="2" class="diff-lineno">Line 100:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ <del class="diffchange diffchange-inline">v_x</del>\;=\;\lim\limits_{h,\;q,\;\beta_x\rightarrow 0}\frac{\beta_x\;h}{q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ <ins class="diffchange diffchange-inline">\nu_x</ins>\;=\;\lim\limits_{h,\;q,\;\beta_x\rightarrow 0}\frac{\beta_x\;h}{q} $$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ <del class="diffchange diffchange-inline">v_y</del>\;=\;\lim\limits_{h,\;q,\;\beta_y\rightarrow 0}\frac{\beta_y\;h}{q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ <ins class="diffchange diffchange-inline">\nu_y</ins>\;=\;\lim\limits_{h,\;q,\;\beta_y\rightarrow 0}\frac{\beta_y\;h}{q} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ a_{x,x}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_x\;h^2}{2q} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ a_{x,x}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_x\;h^2}{2q} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ a_{y,y}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_y\;h^2}{2q} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ a_{y,y}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_y\;h^2}{2q} $$</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 112:</td>
<td colspan="2" class="diff-lineno">Line 112:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{x,x}}{\partial x^2} + \frac{\partial^2 f a_{y,y}}{\partial y^2} - \frac{\partial f <del class="diffchange diffchange-inline">v_x</del>}{\partial x} - \frac{\partial f <del class="diffchange diffchange-inline">v_y</del>}{\partial y} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{x,x}}{\partial x^2} + \frac{\partial^2 f a_{y,y}}{\partial y^2} - \frac{\partial f <ins class="diffchange diffchange-inline">\nu_x</ins>}{\partial x} - \frac{\partial f <ins class="diffchange diffchange-inline">\nu_y</ins>}{\partial y} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Or, what is the same:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Or, what is the same:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\nabla·\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \nabla·\left(\underline{<del class="diffchange diffchange-inline">v</del>} f\right) $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\nabla·\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \nabla·\left(\underline{<ins class="diffchange diffchange-inline">\nu</ins>} f\right) $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With:</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 127:</td>
<td colspan="2" class="diff-lineno">Line 127:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\;\;\;\;\;\;\;\;\;\;\;\;</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\;\;\;\;\;\;\;\;\;\;\;\;</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\underline{<del class="diffchange diffchange-inline">v</del>} \; = \begin{bmatrix}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\underline{<ins class="diffchange diffchange-inline">\nu</ins>} \; = \begin{bmatrix}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">v_x</del>(x,y)\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">\nu_x</ins>(x,y)\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">v_y</del>(x,y)\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">\nu_y</ins>(x,y)\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 135:</td>
<td colspan="2" class="diff-lineno">Line 135:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;D\;\nabla^2 f - \nabla·\left(\underline{<del class="diffchange diffchange-inline">v</del>} f\right) $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;D\;\nabla^2 f - \nabla·\left(\underline{<ins class="diffchange diffchange-inline">\nu</ins>} f\right) $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
</table>
Nightm4re
http://2013.igem.org/wiki/index.php?title=Team:Valencia_Biocampus/Demonstration/Diffusion3&diff=286111&oldid=prev
Nightm4re: /* Proof of a Group Behavior Diffusion Model from a Random Walk Model */
2013-10-03T23:52:56Z
<p><span class="autocomment">Proof of a Group Behavior Diffusion Model from a Random Walk Model</span></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 23:52, 3 October 2013</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 117:</td>
<td colspan="2" class="diff-lineno">Line 117:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\<del class="diffchange diffchange-inline">nabla</del>\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \<del class="diffchange diffchange-inline">nabla</del>\left(\underline{v} f\right) $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\<ins class="diffchange diffchange-inline">nabla·</ins>\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \<ins class="diffchange diffchange-inline">nabla·</ins>\left(\underline{v} f\right) $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With:</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 131:</td>
<td colspan="2" class="diff-lineno">Line 131:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>v_y(x,y)\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>v_y(x,y)\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix} $$</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Nevertheless, in our case of study, we dealed with $\;D\;$ as a constant, so $\;a_{x,x}(x,y)\;=\;a_{y,y}(x,y)\;=\;D\;$, so we can tanke it out from the Laplacian, resting what we were waiting anxiously:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">$$ \frac{\partial f}{\partial t}\;=\;D\;\nabla^2 f - \nabla·\left(\underline{v} f\right) $$</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-----</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-----</div></td></tr>
</table>
Nightm4re
http://2013.igem.org/wiki/index.php?title=Team:Valencia_Biocampus/Demonstration/Diffusion3&diff=286068&oldid=prev
Nightm4re: /* Proof of a Group Behavior Diffusion Model from a Random Walk Model */
2013-10-03T23:43:55Z
<p><span class="autocomment">Proof of a Group Behavior Diffusion Model from a Random Walk Model</span></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 23:43, 3 October 2013</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 126:</td>
<td colspan="2" class="diff-lineno">Line 126:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>0 & a_{y,y}(x,y)\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>0 & a_{y,y}(x,y)\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\;\;\;\;</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">\;\;\;\;\;\;\;\;</ins>\;\;\;\;</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\underline{v} \; = \begin{bmatrix}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\underline{v} \; = \begin{bmatrix}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>v_x(x,y)\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>v_x(x,y)\\</div></td></tr>
</table>
Nightm4re
http://2013.igem.org/wiki/index.php?title=Team:Valencia_Biocampus/Demonstration/Diffusion3&diff=286065&oldid=prev
Nightm4re: /* Proof of a Group Behavior Diffusion Model from a Random Walk Model */
2013-10-03T23:43:25Z
<p><span class="autocomment">Proof of a Group Behavior Diffusion Model from a Random Walk Model</span></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 23:43, 3 October 2013</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 100:</td>
<td colspan="2" class="diff-lineno">Line 100:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ <del class="diffchange diffchange-inline">b_x</del>\;=\;\lim\limits_{h,\;q,\;\beta_x\rightarrow 0}\frac{\beta_x\;h}{q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ <ins class="diffchange diffchange-inline">v_x</ins>\;=\;\lim\limits_{h,\;q,\;\beta_x\rightarrow 0}\frac{\beta_x\;h}{q} $$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ <del class="diffchange diffchange-inline">b_y</del>\;=\;\lim\limits_{h,\;q,\;\beta_y\rightarrow 0}\frac{\beta_y\;h}{q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ <ins class="diffchange diffchange-inline">v_y</ins>\;=\;\lim\limits_{h,\;q,\;\beta_y\rightarrow 0}\frac{\beta_y\;h}{q} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ a_{x,x}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_x\;h^2}{2q} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ a_{x,x}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_x\;h^2}{2q} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ a_{y,y}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_y\;h^2}{2q} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ a_{y,y}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_y\;h^2}{2q} $$</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 112:</td>
<td colspan="2" class="diff-lineno">Line 112:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{x,x}}{\partial x^2} + \frac{\partial^2 f a_{y,y}}{\partial y^2} - \frac{\partial f <del class="diffchange diffchange-inline">b_x</del>}{\partial x} - \frac{\partial f <del class="diffchange diffchange-inline">b_y</del>}{\partial y} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{x,x}}{\partial x^2} + \frac{\partial^2 f a_{y,y}}{\partial y^2} - \frac{\partial f <ins class="diffchange diffchange-inline">v_x</ins>}{\partial x} - \frac{\partial f <ins class="diffchange diffchange-inline">v_y</ins>}{\partial y} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Or, what is the same:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Or, what is the same:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\nabla\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \nabla\left(\underline{<del class="diffchange diffchange-inline">b</del>} f\right) $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\nabla\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \nabla\left(\underline{<ins class="diffchange diffchange-inline">v</ins>} f\right) $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With:</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 126:</td>
<td colspan="2" class="diff-lineno">Line 126:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>0 & a_{y,y}(x,y)\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>0 & a_{y,y}(x,y)\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\underline{<del class="diffchange diffchange-inline">b</del>} \; = \begin{bmatrix}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">\;\;\;\;</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">b_x</del>(x,y)\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\underline{<ins class="diffchange diffchange-inline">v</ins>} \; = \begin{bmatrix}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">b_y</del>(x,y)\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">v_x</ins>(x,y)\\</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">v_y</ins>(x,y)\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{bmatrix} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
</table>
Nightm4re
http://2013.igem.org/wiki/index.php?title=Team:Valencia_Biocampus/Demonstration/Diffusion3&diff=286050&oldid=prev
Nightm4re: /* Proof of a Group Behavior Diffusion Model from a Random Walk Model */
2013-10-03T23:41:46Z
<p><span class="autocomment">Proof of a Group Behavior Diffusion Model from a Random Walk Model</span></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 23:41, 3 October 2013</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 100:</td>
<td colspan="2" class="diff-lineno">Line 100:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ <del class="diffchange diffchange-inline">b_1</del>\;=\;\lim\limits_{h,\;q,\;\<del class="diffchange diffchange-inline">beta_1</del>\rightarrow 0}\frac{\<del class="diffchange diffchange-inline">beta_1</del>\;h}{q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ <ins class="diffchange diffchange-inline">b_x</ins>\;=\;\lim\limits_{h,\;q,\;\<ins class="diffchange diffchange-inline">beta_x</ins>\rightarrow 0}\frac{\<ins class="diffchange diffchange-inline">beta_x</ins>\;h}{q} $$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ <del class="diffchange diffchange-inline">b_2</del>\;=\;\lim\limits_{h,\;q,\;\<del class="diffchange diffchange-inline">beta_2</del>\rightarrow 0}\frac{\<del class="diffchange diffchange-inline">beta_2</del>\;h}{q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ <ins class="diffchange diffchange-inline">b_y</ins>\;=\;\lim\limits_{h,\;q,\;\<ins class="diffchange diffchange-inline">beta_y</ins>\rightarrow 0}\frac{\<ins class="diffchange diffchange-inline">beta_y</ins>\;h}{q} $$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ a_{<del class="diffchange diffchange-inline">11</del>}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\<del class="diffchange diffchange-inline">alpha_1</del>\;h^2}{2q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ a_{<ins class="diffchange diffchange-inline">x,x</ins>}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\<ins class="diffchange diffchange-inline">alpha_x</ins>\;h^2}{2q} $$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ a_{<del class="diffchange diffchange-inline">22</del>}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\<del class="diffchange diffchange-inline">alpha_2</del>\;h^2}{2q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ a_{<ins class="diffchange diffchange-inline">y,y</ins>}\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\<ins class="diffchange diffchange-inline">alpha_y</ins>\;h^2}{2q} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Where $\;\<del class="diffchange diffchange-inline">alpha_1 </del>\;=\; r + l\;$, $\;\<del class="diffchange diffchange-inline">alpha_2 </del>\;=\; u + d\;$, $\;\<del class="diffchange diffchange-inline">beta_1 </del>\;=\; r - l\;$ and $\;\<del class="diffchange diffchange-inline">beta_2 </del>\;=\; u - d\;$. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Where $\;\<ins class="diffchange diffchange-inline">alpha_x </ins>\;=\; r + l\;$, $\;\<ins class="diffchange diffchange-inline">alpha_y </ins>\;=\; u + d\;$, $\;\<ins class="diffchange diffchange-inline">beta_x </ins>\;=\; r - l\;$ and $\;\<ins class="diffchange diffchange-inline">beta_y </ins>\;=\; u - d\;$. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>We now take equation, divide through by $\;q\;$ and take the limit as $\;h,q,\<del class="diffchange diffchange-inline">beta_1</del>,\<del class="diffchange diffchange-inline">beta_2</del>\rightarrow 0\;$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>We now take equation, divide through by $\;q\;$ and take the limit as $\;h,q,\<ins class="diffchange diffchange-inline">beta_x</ins>,\<ins class="diffchange diffchange-inline">beta_y</ins>\rightarrow 0\;$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>such that $\;\<del class="diffchange diffchange-inline">beta_1 </del>h/q\;$, $\;\<del class="diffchange diffchange-inline">beta_2 </del>h/q\;$, $\;\<del class="diffchange diffchange-inline">alpha_1 </del>h^2/2q\;$ and $\;\<del class="diffchange diffchange-inline">alpha_2 </del>h^2/2q\;$ all tend to a constant, giving:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>such that $\;\<ins class="diffchange diffchange-inline">beta_x </ins>h/q\;$, $\;\<ins class="diffchange diffchange-inline">beta_y </ins>h/q\;$, $\;\<ins class="diffchange diffchange-inline">alpha_x </ins>h^2/2q\;$ and $\;\<ins class="diffchange diffchange-inline">alpha_y </ins>h^2/2q\;$ all tend to a constant, giving:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{<del class="diffchange diffchange-inline">11</del>}}{\partial x^2} + \frac{\partial^2 f a_{<del class="diffchange diffchange-inline">22</del>}}{\partial y^2} - \frac{\partial f <del class="diffchange diffchange-inline">b_1</del>}{\partial x} - \frac{\partial f <del class="diffchange diffchange-inline">b_2</del>}{\partial y} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{<ins class="diffchange diffchange-inline">x,x</ins>}}{\partial x^2} + \frac{\partial^2 f a_{<ins class="diffchange diffchange-inline">y,y</ins>}}{\partial y^2} - \frac{\partial f <ins class="diffchange diffchange-inline">b_x</ins>}{\partial x} - \frac{\partial f <ins class="diffchange diffchange-inline">b_y</ins>}{\partial y} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Or, what is the same:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Or, what is the same:</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 118:</td>
<td colspan="2" class="diff-lineno">Line 118:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\nabla\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \nabla\left(\underline{b} f\right) $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\nabla\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \nabla\left(\underline{b} f\right) $$</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">With:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">$$ \underline{\underline{D}} \; = \begin{bmatrix}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">a_{x,x}(x,y) & 0\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">0 & a_{y,y}(x,y)\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\end{bmatrix}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\underline{b} \; = \begin{bmatrix}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">b_x(x,y)\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">b_y(x,y)\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\end{bmatrix} $$</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
</table>
Nightm4re
http://2013.igem.org/wiki/index.php?title=Team:Valencia_Biocampus/Demonstration/Diffusion3&diff=286008&oldid=prev
Nightm4re: /* Proof of a Group Behavior Diffusion Model from a Random Walk Model */
2013-10-03T23:34:32Z
<p><span class="autocomment">Proof of a Group Behavior Diffusion Model from a Random Walk Model</span></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 23:34, 3 October 2013</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 113:</td>
<td colspan="2" class="diff-lineno">Line 113:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{11}}{\partial x^2} + \frac{\partial^2 f a_{22}}{\partial y^2} - \frac{\partial f b_1}{\partial x} - \frac{\partial f b_2}{\partial y} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{11}}{\partial x^2} + \frac{\partial^2 f a_{22}}{\partial y^2} - \frac{\partial f b_1}{\partial x} - \frac{\partial f b_2}{\partial y} $$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Or, what is the same:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">$$ \frac{\partial f}{\partial t}\;=\;\nabla\left(\nabla\left(\underline{\underline{D}} f\right)\right) - \nabla\left(\underline{b} f\right) $$</ins></div></td></tr>
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Nightm4re
http://2013.igem.org/wiki/index.php?title=Team:Valencia_Biocampus/Demonstration/Diffusion3&diff=285975&oldid=prev
Nightm4re: /* Proof of a Group Behavior Diffusion Model from a Random Walk Model */
2013-10-03T23:29:04Z
<p><span class="autocomment">Proof of a Group Behavior Diffusion Model from a Random Walk Model</span></p>
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<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 23:29, 3 October 2013</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 102:</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ b_1\;=\;\lim\limits_{h,\;q,\;\beta_1\rightarrow 0}\frac{\beta_1\;h}{q} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ b_1\;=\;\lim\limits_{h,\;q,\;\beta_1\rightarrow 0}\frac{\beta_1\;h}{q} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ b_2\;=\;\lim\limits_{h,\;q,\;\beta_2\rightarrow 0}\frac{\beta_2\;h}{q} $$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>$$ b_2\;=\;\lim\limits_{h,\;q,\;\beta_2\rightarrow 0}\frac{\beta_2\;h}{q} $$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ <del class="diffchange diffchange-inline">a_11</del>\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_1\;h^2}{2q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ <ins class="diffchange diffchange-inline">a_{11}</ins>\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_1\;h^2}{2q} $$</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>$$ <del class="diffchange diffchange-inline">a_22</del>\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_2\;h^2}{2q} $$</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>$$ <ins class="diffchange diffchange-inline">a_{22}</ins>\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_2\;h^2}{2q} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Where $\;\alpha_1 \;=\; r + l\;$, $\;\alpha_2 \;=\; u + d\;$, $\;\beta_1 \;=\; r - l\;$ and $\;\beta_2 \;=\; u - d\;$. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Where $\;\alpha_1 \;=\; r + l\;$, $\;\alpha_2 \;=\; u + d\;$, $\;\beta_1 \;=\; r - l\;$ and $\;\beta_2 \;=\; u - d\;$. </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We now take equation, divide through by $\;q\;$ and take the limit as $\;h,q,\beta_1,\beta_2\rightarrow 0\;$</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We now take equation, divide through by $\;q\;$ and take the limit as $\;h,q,\beta_1,\beta_2\rightarrow 0\;$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>such that $\;\beta_1 h/q\;$, $\;\beta_2 h/q\;$, $\;\alpha_1 h^2/2q\;$ and $\;\alpha_2 h^2/2q\;$ all tend to a constant, giving:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>such that $\;\beta_1 h/q\;$, $\;\beta_2 h/q\;$, $\;\alpha_1 h^2/2q\;$ and $\;\alpha_2 h^2/2q\;$ all tend to a constant, giving:</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">$$ \frac{\partial f}{\partial t}\;=\;\frac{\partial^2 f a_{11}}{\partial x^2} + \frac{\partial^2 f a_{22}}{\partial y^2} - \frac{\partial f b_1}{\partial x} - \frac{\partial f b_2}{\partial y} $$</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-----</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-----</div></td></tr>
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Nightm4re