Andreas - Projekt

[1] Effect of Clustering

The msd of agents with and without clustering-effect was produced.

The plot can be seen here.
The Header-File used is here.

[2] Density-Dependency of Diffusion-Coefficient

A MC-simulation was made to check the density-dependency of the diffusion-coefficient.
Therefore, the "Pauli-effect" (a patch can only be occupied once), the clustering-effect (if agent has neighbours, the hoppingrate is 0.01 times the "normal" hoppingrate without neighbours) and periodic boundaries were used.
The lattice consists of 100 x 100 patches.

Simulation with different Agent-Densities

A simulation with 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000 agents was made.

The simulation data is here

The result of the density-dependency of the msd is shown in this figure.


Plot of the MSD vs. timelag of different numbers of agents. This plot is logarithmical averaged:


Same plot as above, but "normal" averaged:


The same simulation was done for $\# \text{Agents} > 5000$.

The plots are shown here:


Normally, there should no diffusion if the lattice is completely occupied (100 x 100 Lattice). But one can see a diffusion-constant unequal to zero for #Agents=10000.
This can be explained using this plot of the initial-agent-configuration on the 100 x 100 Lattice:


The lattice is not fully occupied, so diffusion can occur.

[3] Hopping Rate, Diffusion Constant and MSD


I have calculated the relation between the Monte Carlo hopping rate $R$, the size $h$ of a single lattice site and the resulting diffusion constant $D$ for free particle diffusion in 2D. The results can be readily extended to $\eta$-dimensional space:

\begin{equation} D = R h^2 \end{equation}


\begin{align} \overline{\Delta R^2}(t) = 2\; \eta\; D\; t \end{align}

The formula were tested by a simulation. For the case $R=1\;,\;h=1\;,\;\eta=2$ one expects $\overline{\Delta R^2}(t) = 4\; t$. This seems to be indeed the case:


[4] Effective Diffusion constant in 2D


Assume we have NAG particles on a lattice of size XMX * YMX, where each lattice site has a linear size $h$. Then the particle density is

\begin{align} \rho = \mbox{NAG} / (\mbox{XMX}*\mbox{YMX}*h). \end{align}

Let the interaction range of the particles be $r_c = h$, corresponding to nearest neighbor coupling on the lattice. In 2D, the correponding area is

\begin{align} A_c = \pi r_c^2. \end{align}

The probability to have no neighbor within range $r_c$ is

\begin{align} P_{NC} = e^{-A_c \rho} = e^{-\pi r_c^2 \rho}= e^{-\pi h^2 \rho}. \end{align}

Let the normal diffusion constant be $D$ and the slow-down factor be $s$. We then expect a density dependent effective diffusion constant of the form

\begin{align} D_{eff}(\rho) = D \left[ e^{-A_c \rho} + s(1-e^{-A_c \rho}) \right]. \end{align}

This exponential form does not depend on the number of spatial dimensions.

[5] Diffusion-Constant and Agent-Density


The simulated agent-density-dependency of the MSD as shown in [2] is now checked with the analytical result of [4]:

\begin{align} D_{eff}(\rho) = D \left[ e^{-A_c \rho} + s(1-e^{-A_c \rho}) \right]. \end{align}


\begin{align} A_c=\pi h^2 \end{align}

and $h=1$

$s$, the slow downfactor equals $0.01$,


\begin{align} D_{eff}(\rho) = \left[ e^{-\pi \rho} + 0.01\cdot (1-e^{-\pi \rho}) \right] \label{eq:d_eff} \end{align}

$\rho$ is defined as

\begin{align} \rho = \frac{\text{NAG}}{\text{XMX}\cdot\text{YMX}\cdot h} \end{align}

In the case of [2], $\text{NAG}$ is a variable and $\text{XMX}=\text{YMX}=100$.


\begin{align} \rho = 10^{-4}\text{NAG} \label{eq:rho} \end{align}

Combining eq. (9) and (11) yields

\begin{align} D_\text{eff}(\text{NAG})=0.01+0.99\cdot\exp(-10^{-4}\pi\cdot\text{NAG}) \end{align}

Due to $\eta=2$, the MSD can be calculated to

\begin{align} \overline{\Delta R^2}_\text{NAG}(t)=4tD_\text{eff}(\text{NAG}) \end{align}

[2] shows a plot of the MSD at $t=1$ and $t=8$, so the analytical result can be computed an plotted in the plot of the simulated MSD.
The result is shown in this figure:


The assumed poisson-distribution does not fit at all! The first three data-points (1,2,5 agents) could be described by this assumption, but all other agent-densities cannot be explained by poisson-dis.

Alternative Approaches

1. Mean-Field approximation

Clustering takes effect if an agent has at least one neighbour. If $P_\text{NC}$ is the probability to have no neighbour, hence no clustering, the effective diffusion-constant $D_\text{eff}$ can be described as above:

\begin{align} D_\text{eff}=P_\text{NC}D_0+(1-P_\text{NC})s\cdot D_0 \end{align}

Assuming a mean-field approximation,

\begin{align} P_\text{NC}=\left(1-\frac{\text{NAG}}{\text{XMX}\cdot\text{YMX}}\right)^\text{\#Neighbourpatches}=\left(1-\rho \right)^\text{\#Neighbourpatches} \end{align}


\begin{align} D_\text{eff}=D_0\left[P_\text{NC}+(1-P_\text{NC})s\right]=D_0\left[s+P_\text{NC}(1-s)] \end{align}

The simulation defines a neighbour as an agent inside a circle of radius $h$, thus each patch has $8$ neighbours.

The resulting function and the simulation-data is shown in this plot:


The assumed mean-field-approximation yields higher effective diffusion-constanst, because the clustering-effect is not considered correctly:
If clustering occurs, the system does not behave as any equilibrium. Thus, a mean-field approx. is not allowed.

In the next step, the data can be used to fit the distribution, described above, with respect to a critical density $\rho_c$ and a critical maximal agent-number $N_\text{max}$ respectively or an effective number of neighbours $N_\text{NB}$:

\begin{align} D_\text{eff}=D_0\left[s+(1-s)\left(1-\frac{\text{NAG}}{N_\text{max}}}\right)^{N_\text{NB}} \right] \end{align}

The result is shown in this figure:


where $a$ in the figure is $N_\text{max}$

The fit of the effective-number of neighbours is shown here:


2. Another Function

To get a better fit-result, a fit-function

\begin{align} \frac{D_0}{1+\frac{\text{NAG}}{\text{NAG}_0}} \end{align}

The result is shown here:


[6] Simulation of Diffusion only with Pauli-Effect


A simulation without the clustering-effect and with pauli-effect was made to get out if the MSD-Density-Distribution is a result of the pauli effect or of the clustering effect.

The result is shown here:


Thus, the Pauli-Effect alone cannot describe the occured curvature.

[7] Diffusion-Simulation

Due to a problem in the MC-simulation, we decided to simulate the cell-diffusion with an enhanced simulation.
The simulaton is done using a 100 x 100 lattice.
The number of agents varies between 1 and 9999.

The different combinations of diffusion are shown in the next sections:

[7.1] Free-Diffusion

In this simulation, all agents diffuse freely. Thus, one expects that the MSD does not depend on the agent-density. Furthermore, all MSD should look be proportional to $4Dt$.
The plots are shown in this figures:


As expected, the MSD is not a function of the number of agents, because of free diffusion without any special effect. Each agent can diffuse freely.
Furthermore, there are some statistical fluctuations.

[7.2] Diffusion with Pauli-Effect

This simulation uses free-diffusion with the constraint that a patch only can be occupied by one agent.
There should be a small density-dependency of the diffusion-constant.


The MSD vs. time looks as expected: The diffusion is less for a higher number of agents, because an agent cannot diffuse to an already occupied patch.
Furthermore, it looks as if for $\rho=0.5$ the MSD for a fixed time and hence the diffusion-constant $D$ is half the diffusion-constant for one agent.

[7.3] Diffusion with Clustering-Effect

The diffusion is a free diffusion with the constraint that the diffusion-constant of an agent is $D_0$ if the agent has no neighbours, and $D=sD_0$ if the agent has at least one neighbour. Thus, clusters of agents should appear and the MSD should strongly depend on the agent-density.


[7.4] Diffusion with Pauli- and Clustering-Effect

In this simulation, the pauli- and the clustering-effect were used. In the next plots, one can estimate the influence of both effects to the "true" diffusion of agents.

The MSD vs. Time-Plot is shown in this figure:


The MSD vs. #Agents at two different times is shown here:


As one can see, the MSD for an agent-density of $\approx 1$ (#Agents = 9999) the MSD falls dramatically down, as expected. Firstly, the clustering-effect takes effect, so that there is only a diffusion with $D=0.01$, secondly due to the pauli-effect, there is only one patch, where an agent can diffuse. This explains the small diffusion in the MSD vs time-plot and the very low MSD for a high number of agents.

Comparing these plots with the results of clustering only and pauli only suggests that for small agent-densities the clustering effect dominates and the pauli-effect for high agent-densities.

[7.4.1] Fitting the Pauli- and Clustering-Effect Curve

The MSD vs. #Agents-graph was fitted using the function

\begin{align} \text{MSD}(\text{NAG})=\left( \frac{D_0}{1+\frac{\text{NAG}}{a}}\right) \cdot \left(1-\frac{\text{NAG}}{b} \right) \end{align}

$D_0$ is the intial diffusion-constant, thus $D_0=32$ for $t=8.00$ and $D_0=4$ for $t=1.00$.

The fit-results and the resulting plots are shown here:


In this fit, $a$ and $b$ had no initial values. So $b$ does not fit for both fits.


In this fit, $b$ was pre-initalised with $b=9500$ for both cases.

It seems that the fit-routine has some problems fitting these data, so one has to pre-initialise $b$ very carefully.

To get a good $b$, the data for "Pauli-Only" should be fitted using

\begin{align} \text{MSD}(\text{NAG})=\left(1-\frac{\text{NAG}}{b} \right) \end{align}

And the data for "Clustering-Only" using

\begin{align} \text{MSD}(\text{NAG})=\left( \frac{D_0}{1+\frac{\text{NAG}}{a}}\right) \end{align}

to get good initial-conditions for the fit-routine.

This plot:


shows a fit of $a$ only, using $b=10000$, the maximum number of agents on the lattice.

[8] Anti-Clustering


A "Anti-Clustering"-Effect was simulated as well. Therefore, the "slow-down-factor" was chosen to be $1.99$.
The Diffusion of 0-6000 Agents was simulated using an anti-clustering-effect only and the combination of the pauli. and anti-clustering-effect.
The results are shown here:

[8.1] Anti-Clustering Only


This plot shows the MSD at two different times $t=1.00$ and $t=8.00$ as a function of the agent-number.


This plot shows the MSD vs. time for the used agent-numbers.

[8.2] Anti-Clustering and Pauli-Effect


This plot shows the MSD at different times as shown above. In this plot, the Anti-Clustering-Effect and the Pauli-Effect was simuolated.

This plot shows the resulting MSD vs. time graphs:


[9] Simulation of diffusing agents in 3D


[9.1] Free-Diffusion

MSD vs. time:


MSD vs. #Agents


[9.2] Pauli-Only

MSD vs. time:


MSD vs. #Agents


[9.3] Clustering-Only

MSD vs. time


MSD vs. #Agents:


[9.4] Pauli and Clustering

MSD vs. time


MSD vs. #Agents

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