2-D & 3-D Random Walk Simulations of Stochastic Diffusion

Bob Brazzle (Jefferson College)

HexCellEnt: A 2-D Random Walk Game

Game Basics: Place your marker in the center hexagon; roll a die; move one space in the direction indicated by the 6-point rose. Repeat.

Goal: Move your marker off the game board (minimum number is 4 rolls).

Questions: What is the probability of exiting the board in exactly 4 rolls? How could you calculate this probability?

Typical Student Answers (Introductory Lab)

Questions: What is the probability of exiting the board in exactly 4 rolls? How could you calculate this probability?

Most Common Answer: play a bunch of games and count.

Alternate answer: count the “successful” 4-step escape paths.

11

1

1

1162

2 2

2

22

2

2

2

2

2

2 1

1

1

1

1

1

An Elegant Path-Counting Method

After Roll #1 After Roll #2

Setting up the Excel Spreadsheet

A B C D E

1 Roll # Center 1st Ring 2 Main 2 Neighbor

2 2nd 6 2 1 2

3 3rd =6*C2 =2*(C2+E2)+B2+D2 =2*(E2+G2)+C2+F2 =2*(C2+D2+G2)

Setting up the spreadsheet:

Every type of cell gets its own column (it helps to name them, as shown)

Each row is a new roll

For a given cell, set up the equation to add the values from the previous roll in the neighboring cells. (Not shown: Column F is “3rd Ring Main”,column G is “3rd Ring neighbor”,column H is “Number escaping”)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3M

3M 3M

3n3n3n

3n2M

2M2M2n2n

111

111

C

Major Conceptual Shift from path counting {single particle} to relative concentration {infinitely many}

Occupation numbers Occupation probabilities, OR…Relative Concentrations

Example Calculations using the Spreadsheet(most are probably beyond the introductory course)

0 2 4 6 8 10 12 14 16 180

1

2

3

4

5

f(x) = 0.968955765239785 x^0.500707088161058

17-Ring Hexagon Simulation Results

Number of Steps

Dis

tan

ce

Tra

ve

led

We can use the relative concentrations to calculate a weighted average of distance traveled by a particle.

The equation for the best-fit curve is consistent with the well-known result for a random walk – a power law relationship between distance traveled and number of steps taken.

Example Calculation: Flux across a Border

Outbound flux: Notice that all cells in ring #1 have three sides exiting to ring #2. For each cell:Fluxout = (2 * 3/6) ÷ 36

Inbound flux: Some cells have 1 side leading in and some have 2 sides. Total inbound flux is:Fluxin = [(1*6*1/6)+(2*6*2/6)] ÷ 36

Fick’s 1st Law of Diffusion

Compare the net flux across some boundary with the total difference between the relative concentrations across the same boundary. This yields the graph below.

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 00

0.05

0.1

0.15

0.2

0.25

Rings 2 and 3

Radial Concentration (Probability) Gradient

Flux

: Rin

gs 2

to 3

The Continuity Equation…

For any given ring, the rate of change of concentration from one roll to the next {1st time difference}…

Equals the net flux into that ring (must perform calculation across each boundary).

{2nd spatial difference}

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

… becomes the diffusion equation (Fick’s 2nd Law)

-0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

f(x) = 0.166666666666667 x

Rings 1 - 3

2nd Partial Differences (radial)

1st P

artia

l Diff

eren

ces

(tm

e)

Extensions

I have created Excel files to study the following alternatives:• Larger hexagon boards – up to 17 concentric rings• Other plane tiling arrangements (e.g. a “great

rhombitrihexagonal” tiling: dodecagon, hexagon & square)• Soccer-ball-shaped HexCellEnt board• A 3-D space-filling shape – the truncated octahedron