Problem 1.3. Counting microstates (a) Calculate the number of possible microstates for each macrostate n for N = 8 parti

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Problem 1 3 Counting Microstates A Calculate The Number Of Possible Microstates For Each Macrostate N For N 8 Parti 1 (158.14 KiB) Viewed 66 times

Problem 1.3. Counting microstates (a) Calculate the number of possible microstates for each macrostate n for N = 8 particles. What is the probability that n = 8? What is the probability that n= 4? It is possible to count the number of microstates for each n by hand if you have enough patience, but because there are a total of 28 = 256 microstates, this counting would be very tedious. An alternative is to obtain an expression for the number of ways that n particles out of N can be in the left half of the box. Motivate such an expression by enumerating the possible microstates for smaller values of N until you see a pattern. (b) The macrostate with n = N/2 is much more probable than the macrostate with n = N. Why? Approach to equilibrium. The macrostates that give us the least amount of information about the associated microstates are the most probable. For example, suppose that we wish to know CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 12 W(n) P(n) 1 microstate L L L L R L L L L R L L L L R L L L L R n 4 3 3 1/16 4 4/16 3 3 2 L R L R R R L L L R L L R R L L L R L NNNNNN 6 6/16 R L R 2 2 2 2 2 R R R R R L R R L R R L R R L R R 1 1 4 4/16 R 1 R R R R 0 1 1/16 Table 1.2: The 24 possible microstates for N = 4 particles which are distributed in the two halves of a box. The quantity W(n) is the number of microstates corresponding to the macroscopic state characterized by n. The probability P(n) of macrostate n is calculated assuming that each microstate is equally likely.