chm 421/621 statistical mechanicshelios.iiserb.ac.in/courses/chm421/lectures/lec22.pdflecture 22...

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CHM 421/621 Statistical Mechanics

Lecture 22

Formalisms of Statistical Mechanics

Lecture Plan

Basic Postulates

Counting microstates in an ideal gas

Boltzmann equation for entropy

Microcanonical Ensemble

Canonical Ensemble

Formalisms of Statistical Mechanics

Boltzmann equation for entropy

System of N 2-level atomsEach atom can be in one of two levels with energy 0 and

Formalisms of Statistical Mechanics

Boltzmann equation for entropy

System of N 2-level atoms

Employing the Stirling formula

For large N

where fraction of excited atoms

Formalisms of Statistical Mechanics

Boltzmann equation for entropy

System of N 2-level atoms

Or

Note that

For=> temperature is negative!!

Population inversion regime. Can occur in isolated nuclear spin systems

Formalisms of Statistical Mechanics

Boltzmann equation for entropy

Simple Harmonic OscillatorConsider a system of N quantum oscillators having a total energy U anda fundamental frequency .

✏n =

✓n+

1

2

◆~!

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Eigenvalues for each oscillator

Total energy: Eµ =NX

i=1

✓nµi +

1

2

◆~!

=

NX

i=1

nµi

!~! +

N

2~!

⌘ Nµ~! + U0/2<latexit sha1_base64="Sdz0p8uL5MkUaL3LyQJcjUj5/wk=">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</latexit><latexit sha1_base64="Sdz0p8uL5MkUaL3LyQJcjUj5/wk=">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</latexit><latexit sha1_base64="Sdz0p8uL5MkUaL3LyQJcjUj5/wk=">AAACr3icbVFda9swFJW9ry77ytbHvYiFjY6N1AqD7aVQVgZ7Ct1YmkKUeLIiJ6KS7ElXhWD89/oD+rZ/M9n1oE17QXA4556jq6usVNJBkvyN4nv3Hzx8tPO49+Tps+cv+i9fnbjCWy4mvFCFPc2YE0oaMQEJSpyWVjCdKTHNzo4afXourJOF+QWbUsw1WxmZS84gUGn/4tuCao/fHWDqvKZKagkureQBqRdjTJXIYc+ksm36gGluGa9IXY1qauVqDe8xXWfMYlposWKY0l6T1LruyuuSOu8N6//wcRveKNcyqfjj5Tket2Ns2SZpsj9K+4NkmLSFbwPSgQHq6jjtX9Jlwb0WBrhizs1IUsK8YhYkV6LuUe9EyfgZW4lZgIZp4eZVu+8avw3MEueFDccAbtnrjopp5zY6C52awdptaw15lzbzkH+ZV9KUHoThVxflXmEocPN5eCmt4KA2ATBuZZgV8zULa4Pwxb2wBLL95NvgZDQkyZD8+DQ4/NqtYwe9Rm/QHiLoMzpE39ExmiAefYx+RrOIxiSexov491VrHHWeXXSjYvkPffDQvA==</latexit><latexit sha1_base64="Sdz0p8uL5MkUaL3LyQJcjUj5/wk=">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</latexit>

Nµ =U

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Taking U0/2 as a reference energy the total number of quanta can be written as

Formalisms of Statistical Mechanics

Boltzmann equation for entropy

Simple Harmonic OscillatorThe number of microstates possible for a given energy U is just the number ofways of partitioning into N parts such that:

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NX

i=1

nµi = Nµ

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This problem is isomorphic to finding ways of making N-1 partitions separating points.

n1 n2 nu

Nµ<latexit sha1_base64="nsQhDdOgbq4rwNY0ngnZ9RPGSo4=">AAAB7HicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bj04kkiuEkgWcPsZDYZMjO7zEMIS77BiwdFvPpB3vwbJ8keNLGgoajqprsrzjjTxve/vZXVtfWNzdJWeXtnd2+/cnDY1KlVhIYk5alqx1hTziQNDTOctjNFsYg5bcWjm6nfeqJKs1Q+mHFGI4EHkiWMYOOk8O6xK2yvUvVr/gxomQQFqUKBRq/y1e2nxAoqDeFY607gZybKsTKMcDopd62mGSYjPKAdRyUWVEf57NgJOnVKHyWpciUNmqm/J3IstB6L2HUKbIZ60ZuK/3kda5KrKGcys4ZKMl+UWI5Miqafoz5TlBg+dgQTxdytiAyxwsS4fMouhGDx5WXSPK8Ffi24v6jWr4s4SnAMJ3AGAVxCHW6hASEQYPAMr/DmSe/Fe/c+5q0rXjFzBH/gff4ArkGOlg==</latexit><latexit sha1_base64="nsQhDdOgbq4rwNY0ngnZ9RPGSo4=">AAAB7HicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bj04kkiuEkgWcPsZDYZMjO7zEMIS77BiwdFvPpB3vwbJ8keNLGgoajqprsrzjjTxve/vZXVtfWNzdJWeXtnd2+/cnDY1KlVhIYk5alqx1hTziQNDTOctjNFsYg5bcWjm6nfeqJKs1Q+mHFGI4EHkiWMYOOk8O6xK2yvUvVr/gxomQQFqUKBRq/y1e2nxAoqDeFY607gZybKsTKMcDopd62mGSYjPKAdRyUWVEf57NgJOnVKHyWpciUNmqm/J3IstB6L2HUKbIZ60ZuK/3kda5KrKGcys4ZKMl+UWI5Miqafoz5TlBg+dgQTxdytiAyxwsS4fMouhGDx5WXSPK8Ffi24v6jWr4s4SnAMJ3AGAVxCHW6hASEQYPAMr/DmSe/Fe/c+5q0rXjFzBH/gff4ArkGOlg==</latexit><latexit sha1_base64="nsQhDdOgbq4rwNY0ngnZ9RPGSo4=">AAAB7HicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bj04kkiuEkgWcPsZDYZMjO7zEMIS77BiwdFvPpB3vwbJ8keNLGgoajqprsrzjjTxve/vZXVtfWNzdJWeXtnd2+/cnDY1KlVhIYk5alqx1hTziQNDTOctjNFsYg5bcWjm6nfeqJKs1Q+mHFGI4EHkiWMYOOk8O6xK2yvUvVr/gxomQQFqUKBRq/y1e2nxAoqDeFY607gZybKsTKMcDopd62mGSYjPKAdRyUWVEf57NgJOnVKHyWpciUNmqm/J3IstB6L2HUKbIZ60ZuK/3kda5KrKGcys4ZKMl+UWI5Miqafoz5TlBg+dgQTxdytiAyxwsS4fMouhGDx5WXSPK8Ffi24v6jWr4s4SnAMJ3AGAVxCHW6hASEQYPAMr/DmSe/Fe/c+5q0rXjFzBH/gff4ArkGOlg==</latexit><latexit sha1_base64="nsQhDdOgbq4rwNY0ngnZ9RPGSo4=">AAAB7HicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bj04kkiuEkgWcPsZDYZMjO7zEMIS77BiwdFvPpB3vwbJ8keNLGgoajqprsrzjjTxve/vZXVtfWNzdJWeXtnd2+/cnDY1KlVhIYk5alqx1hTziQNDTOctjNFsYg5bcWjm6nfeqJKs1Q+mHFGI4EHkiWMYOOk8O6xK2yvUvVr/gxomQQFqUKBRq/y1e2nxAoqDeFY607gZybKsTKMcDopd62mGSYjPKAdRyUWVEf57NgJOnVKHyWpciUNmqm/J3IstB6L2HUKbIZ60ZuK/3kda5KrKGcys4ZKMl+UWI5Miqafoz5TlBg+dgQTxdytiAyxwsS4fMouhGDx5WXSPK8Ffi24v6jWr4s4SnAMJ3AGAVxCHW6hASEQYPAMr/DmSe/Fe/c+5q0rXjFzBH/gff4ArkGOlg==</latexit>

Formalisms of Statistical Mechanics

Boltzmann equation for entropy

Simple Harmonic OscillatorGiven that the partitions are identical and the points are identical we have

⌦ =(Nµ +N � 1)!

Nµ! (N � 1)!⇡ (Nµ +N)!

Nµ!N !<latexit sha1_base64="EOrMiGv3y7Dy2adOFaVxNtL6U7Q=">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</latexit><latexit sha1_base64="EOrMiGv3y7Dy2adOFaVxNtL6U7Q=">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</latexit><latexit sha1_base64="EOrMiGv3y7Dy2adOFaVxNtL6U7Q=">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</latexit><latexit sha1_base64="EOrMiGv3y7Dy2adOFaVxNtL6U7Q=">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</latexit>

(N ! 1)<latexit sha1_base64="5+VWah6XUM/VlG/Miie/UcFbAk4=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoMQL2FXBD0GvXiSCOYB2TXMTmaTIbOzy0yvEkL+w4sHRbz6L978GyePgyYWNBRV3XR3hakUBl3328mtrK6tb+Q3C1vbO7t7xf2DhkkyzXidJTLRrZAaLoXidRQoeSvVnMah5M1wcD3xm49cG5GoexymPIhpT4lIMIpWeijfEh8T4gsV4fC0Uyy5FXcKsky8OSnBHLVO8cvvJiyLuUImqTFtz00xGFGNgkk+LviZ4SllA9rjbUsVjbkJRtOrx+TEKl0SJdqWQjJVf0+MaGzMMA5tZ0yxbxa9ifif184wugxGQqUZcsVmi6JMEvvoJALSFZozlENLKNPC3kpYn2rK0AZVsCF4iy8vk8ZZxXMr3t15qXo1jyMPR3AMZfDgAqpwAzWoAwMNz/AKb86T8+K8Ox+z1pwznzmEP3A+fwA1NJGq</latexit><latexit sha1_base64="5+VWah6XUM/VlG/Miie/UcFbAk4=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoMQL2FXBD0GvXiSCOYB2TXMTmaTIbOzy0yvEkL+w4sHRbz6L978GyePgyYWNBRV3XR3hakUBl3328mtrK6tb+Q3C1vbO7t7xf2DhkkyzXidJTLRrZAaLoXidRQoeSvVnMah5M1wcD3xm49cG5GoexymPIhpT4lIMIpWeijfEh8T4gsV4fC0Uyy5FXcKsky8OSnBHLVO8cvvJiyLuUImqTFtz00xGFGNgkk+LviZ4SllA9rjbUsVjbkJRtOrx+TEKl0SJdqWQjJVf0+MaGzMMA5tZ0yxbxa9ifif184wugxGQqUZcsVmi6JMEvvoJALSFZozlENLKNPC3kpYn2rK0AZVsCF4iy8vk8ZZxXMr3t15qXo1jyMPR3AMZfDgAqpwAzWoAwMNz/AKb86T8+K8Ox+z1pwznzmEP3A+fwA1NJGq</latexit><latexit sha1_base64="5+VWah6XUM/VlG/Miie/UcFbAk4=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoMQL2FXBD0GvXiSCOYB2TXMTmaTIbOzy0yvEkL+w4sHRbz6L978GyePgyYWNBRV3XR3hakUBl3328mtrK6tb+Q3C1vbO7t7xf2DhkkyzXidJTLRrZAaLoXidRQoeSvVnMah5M1wcD3xm49cG5GoexymPIhpT4lIMIpWeijfEh8T4gsV4fC0Uyy5FXcKsky8OSnBHLVO8cvvJiyLuUImqTFtz00xGFGNgkk+LviZ4SllA9rjbUsVjbkJRtOrx+TEKl0SJdqWQjJVf0+MaGzMMA5tZ0yxbxa9ifif184wugxGQqUZcsVmi6JMEvvoJALSFZozlENLKNPC3kpYn2rK0AZVsCF4iy8vk8ZZxXMr3t15qXo1jyMPR3AMZfDgAqpwAzWoAwMNz/AKb86T8+K8Ox+z1pwznzmEP3A+fwA1NJGq</latexit><latexit sha1_base64="5+VWah6XUM/VlG/Miie/UcFbAk4=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoMQL2FXBD0GvXiSCOYB2TXMTmaTIbOzy0yvEkL+w4sHRbz6L978GyePgyYWNBRV3XR3hakUBl3328mtrK6tb+Q3C1vbO7t7xf2DhkkyzXidJTLRrZAaLoXidRQoeSvVnMah5M1wcD3xm49cG5GoexymPIhpT4lIMIpWeijfEh8T4gsV4fC0Uyy5FXcKsky8OSnBHLVO8cvvJiyLuUImqTFtz00xGFGNgkk+LviZ4SllA9rjbUsVjbkJRtOrx+TEKl0SJdqWQjJVf0+MaGzMMA5tZ0yxbxa9ifif184wugxGQqUZcsVmi6JMEvvoJALSFZozlENLKNPC3kpYn2rK0AZVsCF4iy8vk8ZZxXMr3t15qXo1jyMPR3AMZfDgAqpwAzWoAwMNz/AKb86T8+K8Ox+z1pwznzmEP3A+fwA1NJGq</latexit>

S = kB ln⌦(U, V,N)

= kB [(Nµ +N) ln(Nµ +N)� (Nµ +N)� (Nµ lnNµ �Nµ +N lnN �N)]

= kB

Nµ ln

✓1 +

N

◆+N ln

✓1 +

N

◆�

= NkB

U

U0ln

✓1 +

U

U0

◆+ ln

✓1 +

U

U0

◆�

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Formalisms of Statistical Mechanics

Boltzmann equation for entropy

Simple Harmonic OscillatorGiven that the partitions are identical and the points are identical we have

1

T=

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V,N<latexit sha1_base64="uD2NYRxN6PpQCkP/R83tK2IBuUE=">AAACKHicbZDLSgMxFIYzXmu9VV26CRalgpQZEXQjFt24koqdttApJZNm2tDMheSMUIZ5HDe+ihsRRbr1SUzbAbX1h8DHf85Jcn43ElyBaY6MhcWl5ZXV3Fp+fWNza7uws1tXYSwps2koQtl0iWKCB8wGDoI1I8mI7wrWcAc343rjkUnFw6AGw4i1fdILuMcpAW11CleOJwlNrDSppfjoEjuCeVCamk5EJHAi8EP6w3bqSN7rw3EnqZ/cpZ1C0SybE+F5sDIookzVTuHN6YY09lkAVBClWpYZQTsZ304FS/NOrFhE6ID0WEtjQHym2slk0RQfaqeLvVDqEwCeuL8nEuIrNfRd3ekT6KvZ2tj8r9aKwbtoJzyIYmABnT7kxQJDiMep4S6XjIIYaiBUcv1XTPtEhwQ627wOwZpdeR7qp2XLLFv3Z8XKdRZHDu2jA1RCFjpHFXSLqshGFD2hF/SOPoxn49X4NEbT1gUjm9lDf2R8fQP9K6aM</latexit><latexit sha1_base64="uD2NYRxN6PpQCkP/R83tK2IBuUE=">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</latexit><latexit sha1_base64="uD2NYRxN6PpQCkP/R83tK2IBuUE=">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</latexit><latexit sha1_base64="uD2NYRxN6PpQCkP/R83tK2IBuUE=">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</latexit>

=kB~!0

ln

✓1 +

N

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u =~!0

eTET � 1

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Average energy per oscillator is Einstein temperatureTE =

~!0

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Formalisms of Statistical Mechanics

Boltzmann equation for entropy

Simple Harmonic Oscillator

Specific heat

cV = kB

✓TE

T

◆2 eTET

⇣e

TET � 1

⌘2

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0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

f(x)1

cV/kB

T/TE

Such a model was used by Einstein to account for the specific heat of solids.

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