1v5cells_comparison

Comparison of 1 and 5 cell Simulations

Simulation Set-Up

• Reflective boundary on the left.
• Absorbing boundary on the right.
• For the 5 cell simulation:
  
  • Cells are not connected.
  • Cells may not occupy the same lattice point.
  • Cell 1 is initialized on the reflective boundary
  • Cell 5 is initialized 5 lattice points to the right of cell 1.

Results (Updated 2/9/15)

The results can be found online here.

Passage Time

Update: Plots for the time-averages passage times have been updated. Added a plot of the average first passage time for each cell.

Lets compare the passage times of the 5 cells to that of the single cell diffusion simulation.

The following plots are created by making histograms of the simulation output data. In order to keep the data normalized throughout, a factor dependent on the bin size is added to the probability density. The probability density $P(t_i)$ at certain time is expressed as
$$P(t_i) = \frac{1}{\Delta t} \frac{c(t_i)}{\sum c(t_i)} \\ \Delta t \equiv \text{bin size }$$
The following plots are the histograms of each cell overlayed. One cell refers to the data from the single cell simulation.

Notice that the slope of the one cell and that of the 1st cell in the 5 cell simulation are very similar.

The average first passage times for each respective cell are plotted below.

An interesting ting to note, for the one cell case the first passage time is $\sim 10^4$ and the cell had to diffuse $10^2$ lattice points to get there.

Now lets plot the data while scaling the probability densities and the number of time steps by the mean passage time for each respective cell.
$$\text{let } \tilde{t}_i \equiv \frac{t_i}{\left<t_i\right>} \\ \rightarrow P(\tilde{t}) = \left<t_i\right>P(t)$$
The following plots are with respect to these new quantities, $P(\tilde{t})$ and $\tilde{t}_i$.

Looking more closely at the relationship between the first, last and individual cell:

Position

The following plots compare the probability densities of the position of each cell. The probability of the 5 cells and the one, individual cell are plotted together at different time steps. The probability densities are also plotted with respect to the averaged position.

We would expect that the first cell of the multiple cell simulation to have a normal distribution but it does not appear that way. Boundary conditions tell us that the slope of the first cell’s probability density should be zero at the reflecting boundary. Zooming in on the first cell for small positions shows that the slope approaches zero but the probability density is clearly not flat.

I believe this plot is at the 2500th time step. Looking at different time steps shows very much the same pattern. At small time steps the curve is slightly steeper and as time increases the slope very gradually decreases.

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