Aus: Daniel Schroeder “An - Fakultät für Physik - LMU … Daniel Schroeder “An introduction to...
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N N ! N N e -N ⇧ 2πN Error ln N ! N ln N - N Error 1 1 .922 7.7% 0 -1 ⌅ 10 3628800 3598696 .83% 15.1 13.0 13.8% 100 9 ⇥ 10 157 9 ⇥ 10 157 .083% 364 360 .89% Table 2.3. Comparison of Stirling’s approximation (equations 2.14 and 2.16) to exact values for N = 1, 10, and 100. Copyright c ⇤2000, Addison-Wesley. N , q ≈ few hundred N , q ≈ few thousand q A Multiplicity q A Multiplicity Figure 2.6. Typical multiplicity graphs for two interacting Einstein solids, con- taining a few hundred oscillators and energy units (left) and a few thousand (right). As the size of the system increases, the peak becomes very narrow relative to the full horizontal scale. For N ⌅ q ⌅ 10 20 , the peak is much too sharp to draw. Copyright c ⇤2000, Addison-Wesley. Aus: Daniel Schroeder “An introduction to thermal physics”, Pearson Verlag q A Ω A q B Ω B Ω total 0 1 100 2.8 × 10 81 2.8 × 10 81 1 300 99 9.3 × 10 80 2.8 × 10 83 2 45150 98 3.1 × 10 80 1.4 × 10 85 3 4545100 97 1.0 × 10 80 4.6 × 10 86 4 3.4 × 10 8 96 3.3 × 10 79 1.1 × 10 88 . . . . . . . . . . . . . . . 59 2.2 × 10 68 41 3.1 × 10 46 6.8 × 10 114 60 1.3 × 10 69 40 5.3 × 10 45 6.9 × 10 114 61 7.7 × 10 69 39 8.8 × 10 44 6.8 × 10 114 . . . . . . . . . . . . . . . 100 1.7 × 10 96 0 1 1.7 × 10 96 9.3 × 10 115 1 100 q A Ω total (×10 114 ) 80 60 40 20 0 2 3 4 5 6 7 Figure 2.5. Macrostates and multiplicities of a system of two Einstein solids, with 300 and 200 oscillators respectively, sharing a total of 100 units of energy. Copyright c ⇤2000, Addison-Wesley.
Transcript of Aus: Daniel Schroeder “An - Fakultät für Physik - LMU … Daniel Schroeder “An introduction to...
N N ! NNe�N⇧
2�N Error lnN ! N lnN �N Error1 1 .922 7.7% 0 �1 ⌅
10 3628800 3598696 .83% 15.1 13.0 13.8%100 9⇥ 10157 9⇥ 10157 .083% 364 360 .89%
Table 2.3. Comparison of Stirling’s approximation (equations 2.14 and 2.16) to
exact values for N = 1, 10, and 100. Copyright c⇤2000, Addison-Wesley.
N , q ! few hundred N , q ! few thousand
qA
Mult
iplici
ty
qA
Mult
iplici
ty
Figure 2.6. Typical multiplicity graphs for two interacting Einstein solids, con-
taining a few hundred oscillators and energy units (left) and a few thousand (right).
As the size of the system increases, the peak becomes very narrow relative to the
full horizontal scale. For N ⌅ q ⌅ 1020
, the peak is much too sharp to draw.
Copyright c⇤2000, Addison-Wesley.
Aus: Daniel Schroeder “An introduction to thermal physics”, Pearson Verlag
qA !A qB !B !total
0 1 100 2.8 ! 1081 2.8 ! 1081
1 300 99 9.3 ! 1080 2.8 ! 1083
2 45150 98 3.1 ! 1080 1.4 ! 1085
3 4545100 97 1.0 ! 1080 4.6 ! 1086
4 3.4 ! 108 96 3.3 ! 1079 1.1 ! 1088
......
......
...59 2.2 ! 1068 41 3.1 ! 1046 6.8 ! 10114
60 1.3 ! 1069 40 5.3 ! 1045 6.9 ! 10114
61 7.7 ! 1069 39 8.8 ! 1044 6.8 ! 10114
......
......
...100 1.7 ! 1096 0 1 1.7 ! 1096
9.3 ! 10115
1
100qA
!to
tal(!
10114)
806040200
2
3
4
5
6
7
Figure 2.5. Macrostates and multiplicities of a system of two Einstein solids,
with 300 and 200 oscillators respectively, sharing a total of 100 units of energy.
Copyright c⇤2000, Addison-Wesley.
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