1. 2-D & 3-D Random Walk Simulations of Stochastic Diffusion Bob Brazzle (Jefferson College)

2. 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?

3. Typical Student Answers (Introductory Lab)

4. 11 1 1 1 1 6 2 2 2 2 22 2 2 2 2 2 21 1 1 1 1 1 An Elegant Path-Counting Method After Roll #1 After Roll #2

5. Setting up the Excel Spreadsheet ABCDE 1Roll #Center1 st Ring2 Main2 Neighbor 22 nd 6212 33 rd =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 “3 rd Ring Main”, column G is “3 rd Ring neighbor”, column H is “Number escaping”) 3M 3n 2M 2n 11 1 1 1 1 C

6. Major Conceptual Shift from path counting {single particle} to relative concentration {infinitely many} Occupation numbers Occupation probabilities, OR… Relative Concentrations

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

8. 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: Flux out = (2 * 3/6) ÷ 36 Inbound flux: Some cells have 1 side leading in and some have 2 sides. Total inbound flux is: Flux in = [(1*6*1/6)+(2*6*2/6)] ÷ 36

9. Fick’s 1 st 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.

10. The Continuity Equation…

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

12. 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