This table was generated by finding all of the numbers that sum (in this case) to 5 which is the macrostate. It shows the number of atoms with a particular energy in the columns headed , the statistical weight of each microstate is in the “weight” column, the probability column next to it shows the probability of randomly selecting this microstate from a given macrostate (in this case 5 atoms and a total energy of 5). The row titled average occupancy shows the expected occupancy of an energy level of type , calculated from the table. Looking at the table there are two equally most likely microstate arrangements. The first of these corresponds to and , both occurring with a probability of 0.238.
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Another possible macrostate is listed below, this time we have 7 atoms and an energy of 7 units. The headings of the table are the same as in the previous example. We can see that the weight of the most probable microstate is 420 and that we have a probability of 0.245 of randomly selecting one of them. The occupancy levels are:
microstate
occupancy of type i
weight
probability
n0
n1
n2
n3
n4
n5
n6
n7
1
6
0
0
0
0
0
0
1
7
0.004
2
5
1
0
0
0
0
1
0
42
0.024
3
5
0
1
0
0
1
0
0
42
0.024
4
5
0
0
1
1
0
0
0
42
0.024
5
4
2
0
0
0
1
0
0
105
0.061
6
4
1
1
0
1
0
0
0
210
0.122
7
4
1
0
2
0
0
0
0
105
0.061
8
4
0
2
1
0
0
0
0
105
0.061
9
3
3
0
0
1
0
0
0
140
0.082
10
3
2
1
1
0
0
0
0
420
0.245
11
3
1
3
0
0
0
0
0
140
0.082
12
2
4
0
1
0
0
0
0
105
0.061
13
2
3
2
0
0
0
0
0
210
0.122
14
1
5
1
0
0
0
0
0
42
0.024
15
0
7
0
0
0
0
0
0
1
0.001
totals
51
30
11
6
3
2
1
1
1716
1
Average occupancy
3.231
1.885
1.028
0.514
0.228
0.086
0.024
0.004
A final example consists of a system of 10 atoms and a total energy of 9. As will be readily seen as the number of atoms and the energy increases the number of microstates corresponding to a given macrostate increases so does the size of the table. It was quite difficult to work out the number of combinations of energy that could occur and I wouldn’t want to do it again for larger tables. In the next part we shall use the method of Lagrange multipliers to massively simplify the calculations for the probabilities and expectations. For the case of 10 atoms and an energy of 9 units.
We see that the most probable microstates have the following occupancy levels:
The most probable microstate has a probability of 0.1555, but there is another microstate that is only slightly less probable (a probability of 0.1300) and this has occupancy levels of:
The two least likely microstates are the following:
Both have a probability of 0.0002 which is very small indeed. Table 3 is below:
d
occupancy of each type i
“weight”
probability
n0
n1
n2
n3
n4
n5
n6
n7
n8
n9
1
9
0
0
0
0
0
0
0
0
1
10
0.000205677
2
8
1
0
0
0
0
0
0
1
0
90
0.00185109
3
8
0
1
0
0
0
0
1
0
0
90
0.00185109
4
8
0
0
1
0
0
1
0
0
0
90
0.00185109
5
8
0
0
0
1
1
0
0
0
0
90
0.00185109
6
7
2
0
0
0
0
0
1
0
0
360
0.00740436
7
7
1
1
0
0
0
1
0
0
0
720
0.014808721
8
7
1
0
1
0
1
0
0
0
0
720
0.014808721
9
7
1
0
0
2
0
0
0
0
0
360
0.00740436
10
7
0
2
0
0
1
0
0
0
0
360
0.00740436
11
7
0
1
1
1
0
0
0
0
0
720
0.014808721
12
7
0
0
3
0
0
0
0
0
0
120
0.00246812
13
6
3
0
0
0
0
1
0
0
0
840
0.017276841
14
6
2
1
0
0
1
0
0
0
0
2520
0.051830522
15
6
2
0
1
1
0
0
0
0
0
2520
0.051830522
16
6
1
2
0
1
0
0
0
0
0
2520
0.051830522
17
6
1
1
2
0
0
0
0
0
0
2520
0.051830522
18
6
0
3
1
0
0
0
0
0
0
840
0.017276841
19
5
4
0
0
0
1
0
0
0
0
1260
0.025915261
20
5
3
1
0
1
0
0
0
0
0
5040
0.103661045
21
5
3
0
2
0
0
0
0
0
0
2520
0.051830522
22
5
2
2
1
0
0
0
0
0
0
7560
0.155491567
23
5
1
4
0
0
0
0
0
0
0
1260
0.025915261
24
4
5
0
0
1
0
0
0
0
0
1260
0.025915261
25
4
4
1
1
0
0
0
0
0
0
6300
0.129576306
26
4
3
3
0
0
0
0
0
0
0
4200
0.086384204
27
3
6
0
1
0
0
0
0
0
0
840
0.017276841
28
3
5
2
0
0
0
0
0
0
0
2520
0.051830522
29
2
7
1
0
0
0
0
0
0
0
360
0.00740436
30