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A random walking amoeba

Page history last edited by Joe Redish 5 years, 2 months ago

7.3.1.P4

 

In this problem you will work out a simple model for the entropy associated with an amoeba’s quest for dinner. This amoeba lives in a two-dimensional world, on a surface conveniently divided into grids. For our purposes, the surface is infinite in extent, and everywhere the same. Part of the infinite plane is shown at right. 

 

In a time  the amoeba can move exactly one grid to the right (R), one grid to the left (L), one grid up (U), one grid down (D), or it might stay (S) where it is. You may assume the amoeba’s decisions are completely random, so that each possibility is equally likely. We begin the observing the amoeba at time t=0, with the amoeba at point X.

 

A. After a time of only one Δt, how many different ways (also known as paths, or microstates) could the amoeba have ended up at each the following points?  

  1. A.
  2. B
  3. C
 

B. How many total microstates are associated with one unit of time Δt?

 

C. After two units of time, (so t= 2 Δt ), how many different ways could the amoeba have ended up at each of the following points?

  1. A.
  2. B
  3. C

 

D. How many total microstates are associated with time t=2 Δt ?

 

E. What is the probability P that the amoeba will be found at the point X at time t=2 Δ?

  1. P = 0.25
  2. P = 0.2
  3. P = 0
  4. P = 1
  5. Something else. (What?)

 

F. How many microstates (paths) are there after N units of time (=N Δt )?

 

G. This kind of reasoning naturally leads one to an estimate of the entropy S as a function of N, which accounts for the amount of time since the beginning of the amoeba’s journey. What is that function, S(N)?

 

H. Briefly state what it means to say that a quantity is extensive. Is the entropy in this model an extensive function of N? Why or why not?

 

 

Bill Dorland 3/13/18

 

 

 

 

 

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