# Information Theory and Thermodynamics

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