Use a computer to produce a table and graph, like those in this section, for two
ID: 1657326 • Letter: U
Question
Use a computer to produce a table and graph, like those in this section, for two interacting two-state paramagnets, each containing 100 elementary magnetic dipoles. Take a "unit" of energy to be the amount needed to flip a single dipole from the "up" state (parallel to the external field) to the "down" state (antiparallel). Suppose that the total number of units of energy, relative to the state with all dipoles pointing up, is 80; this energy can be shared in any way between the two paramagnets. What is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?
I got the table and graph without any problem. The problem I have is with the probability part. Specifically the total multiplicity. The textbook solution shows that the total multiplicity is 1.6473*10^57 and I just can't figure out how to get that number. I don't need help with any other part of the problem. I know how to get the probability once I have the total multiplicity. I just need a detailed explaination as to how to get the total multiplicity of 1.6473*10^57. I must be missing something simple but I just can't see right now. Please help!
Explanation / Answer
The euation to find multiplicity is
Multiplicity (N, q) = N! / q!(N-q)!
where, N = 100 and q = 80. To find total multiplicity, create two columns. In one table, take N = 100(fixed) and take q as 0 as initial value and go till 80. In other column, take N = 100 (fixed) and take q = 80 as initial value and go till 0. Now multiply the values of two columns to get total multiplicity. I will show you one example.
Let's take, N =100 and q = 0 in column A, the multiplicity will be 100! / 0!(100-0)! = 1
In coulumn B, take N = 100 and q = 80 then the multiplicity will be 100!/80!(100-80)! = 5.36*10^20.
Now multiply values of column A and column B to get total multiplicity, We get 1 * 5.36*10^20 = 5.36*10^20. So this is the procedure. In the same way go till q = 80 (from 0) in coulum A and q = 0(from 80) in column B. After getting total muliplicity of all the values, you can plot the graph to get most probable macrostate.
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