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1-d ideal chain. Link 1. Link 2. N links. Link N. 1-d ideal chain. N links. Part 1. Part 2. Part N. Bath. System. Energy can be exchanged between chain and bath. N links. Part 1. Part 2. Part N. Bath. System. Energy can be moved around bath. N links. Part 1. Part 2. - PowerPoint PPT Presentation
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1-d ideal chain
1
N links
𝑠𝑖=±1
Link 1
Link 2
Link N
. . .
. . .
Part 1
Bath
Part 2 Part N
1-d ideal chain
2
N links
𝑠𝑖=±1
. . .
System
Energy can be exchanged between chain and bath
3
System
N links
𝑠𝑖=±1
Bath
. . .
Part 1 Part 2 Part N. . .
Energy can be moved around bath
4
N links
𝑠𝑖=±1
Bath
Part 1 Part 2 Part N
. . .
. . .
System
Chain can be crinkled in different ways
5
N links
𝑠𝑖=±1
Bath
. . .
Part 1 Part 2 Part N. . .
System
Chain can be crinkled in different ways
6
N links
𝑠𝑖=±1
Bath
. . .
Part 1 Part 2 Part N. . .
System
Chain can be crinkled in different ways
7
N links
𝑠𝑖=±1
Bath
. . .
Part 1 Part 2 Part N. . .
System
Chain can be crinkled in different ways
8
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
𝑠𝑖=±1
Chain can be crinkled in different ways
9
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
𝑠𝑖=±1
Chain can be crinkled in different ways
10
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
𝑠𝑖=±1
Chain can be crinkled in different ways
11
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
𝑠𝑖=±1
Chain can be crinkled in different ways
12
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
𝑠𝑖=±1
Chain can be crinkled in different ways
13
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
𝑠𝑖=±1
Chain can be crinkled in different ways
14
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
𝑠𝑖=±1
Exploring accessible world configurations equally
15
. . .
. . .
. . .
. . .
. . .
Too much total energy
. . .
Too little total energy
X X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
16
Hamiltonian and partition function
𝑍 (𝜏 )= ∑state 1
state 𝑓
𝑒−𝜀 (𝑠1 , 𝑠2 ,⋯ , 𝑠𝑁 )
𝜏
Expectation of elongation
...
X
World
...
...
...
...
... X
Hamiltonian
17
𝜀 (𝑠1 ,𝑠2 ,⋯ ,𝑠𝑁 )=−𝐹∑𝑖=1
𝑁
𝑠𝑖
𝑅
STOP𝑅The animation is oscillating between two states with two values of the system energy e. What are the states and energies?
Full downward extension+1, +1, +1, +1, +1
One upward-directed link+1, +1, +1, -1, +1
e = -5F
e = -3F
R = 5
R = 3
...
X
World
...
...
...
...
... X
Partition function
18
𝜀 (𝑠1 ,𝑠2 ,⋯ ,𝑠𝑁 )=−𝐹∑𝑖=1
𝑁
𝑠𝑖
𝑍 (𝜏 ) := ∑𝜀 𝑖=𝜀𝑀𝐼𝑁
∞
𝑊 𝑆𝑌𝑆 (𝜀𝑖 )𝑒−𝜀𝑖𝜏
¿ ∑state1
state 𝑓
𝑒−𝜀 ( state )
𝜏
𝑠1 ,𝑠2 ,⋯ ,𝑠𝑖 ,⋯ ,𝑠𝑁Particular
-1, +1, +1, +1, +1 -1, +1, +1, -1, +1
...
X
World
...
...
...
...
... X
∑𝑠 1 ,𝑠 2
❑
𝑒−𝜀 (𝑠1 , 𝑠2 )
𝜏 =𝑒−𝜀 (+1 ,+1 )
𝜏 +𝑒−𝜀 (+1 ,− 1)
𝜏
+𝑒−𝜀 (− 1 ,+1)
𝜏 +𝑒−𝜀 (− 1 ,−1 )
𝜏
Partition function
19
𝜀 (𝑠1 ,𝑠2 ,⋯ ,𝑠𝑁 )=−𝐹∑𝑖=1
𝑁
𝑠𝑖
𝑍 (𝜏 )= ∑state 1
state 𝑓
𝑒−𝜀 (𝑠1 , 𝑠2 ,⋯ , 𝑠𝑁 )
𝜏
¿ ∑𝑠1=±1
❑
𝑒−𝜀 (𝑠1 ,+1 )
𝜏 +𝑒−𝜀 (𝑠1 ,− 1)
𝜏
¿ ∑𝑠1=±1
❑
∑𝑠2=± 1
❑
𝑒−𝜀 (𝑠1 , 𝑠2 )
𝜏
...
X
World
...
...
...
...
... X
Partition function
20
𝜀 (𝑠1 ,𝑠2 ,⋯ ,𝑠𝑁 )=−𝐹∑𝑖=1
𝑁
𝑠𝑖
𝑍 (𝜏 )= ∑state 1
state 𝑓
𝑒−𝜀 (𝑠1 , 𝑠2 ,⋯ , 𝑠𝑁 )
𝜏
∑𝑠 1 ,𝑠 2
❑
𝑒−𝜀 (𝑠1 , 𝑠2 )
𝜏 = ∑𝑠1=±1
❑
∑𝑠2=± 1
❑
𝑒−𝜀 (𝑠1 , 𝑠2 )
𝜏
𝑍= ∑𝑠1=±1
❑
⋯ ∑𝑠𝑁− 1=±1
❑
∑𝑠𝑁=± 1
❑
𝑒−𝜀 (𝑠1 ,⋯ , 𝑠𝑁− 1 , 𝑠𝑁 )
𝜏
...
X
World
...
...
...
...
... X
Partition function
21
𝜀 (𝑠1 ,𝑠2 ,⋯ ,𝑠𝑁 )=−𝐹∑𝑖=1
𝑁
𝑠𝑖
𝑍 (𝜏 )= ∑state 1
state 𝑓
𝑒−𝜀 (𝑠1 , 𝑠2 ,⋯ , 𝑠𝑁 )
𝜏
𝑍= ∑𝑠1=±1
❑
⋯ ∑𝑠𝑁− 1=±1
❑
∑𝑠𝑁=± 1
❑
𝑒−𝜀 (𝑠1 ,⋯ , 𝑠𝑁− 1 , 𝑠𝑁 )
𝜏
¿ ∑𝑠1=±1
❑
⋯ ∑𝑠𝑁− 1=±1
❑
∑𝑠𝑁=± 1
❑
𝑒𝐹 (𝑠1+⋯+𝑠𝑁−1+𝑠𝑁 )
𝜏
𝑒𝐹 𝑠1𝜏 ⋯𝑒
𝐹 𝑠𝑁 −1
𝜏 𝑒𝐹 𝑠𝑁𝜏
¿ ∑𝑠1=±1
❑
⋯ ∑𝑠𝑁− 1=±1
❑
𝑒𝐹 𝑠1𝜏 ⋯𝑒
𝐹 𝑠𝑁−1
𝜏 ∑𝑠𝑁=±1
❑
𝑒𝐹𝑠𝑁
𝜏
...
X
World
...
...
...
...
... X
Partition function
22
𝜀 (𝑠1 ,𝑠2 ,⋯ ,𝑠𝑁 )=−𝐹∑𝑖=1
𝑁
𝑠𝑖
𝑍 (𝜏 )= ∑state 1
state 𝑓
𝑒−𝜀 (𝑠1 , 𝑠2 ,⋯ , 𝑠𝑁 )
𝜏
¿ ∑𝑠1=±1
❑
⋯ ∑𝑠𝑁− 1=±1
❑
𝑒𝐹 𝑠1𝜏 ⋯𝑒
𝐹 𝑠𝑁−1
𝜏 ∑𝑠𝑁=±1
❑
𝑒𝐹𝑠𝑁
𝜏
¿ ( ∑𝑠𝑁=±1𝑒
𝐹 𝑠𝑁𝜏 ) ∑𝑠 1=±1
❑
⋯ ∑𝑠𝑁− 1=±1
❑
𝑒𝐹 𝑠1𝜏 ⋯𝑒
𝐹 𝑠𝑁−1
𝜏
¿ ( ∑𝑠 1=±1 𝑒𝐹 𝑠 1𝜏 )⋯( ∑
𝑠𝑁− 1=± 1𝑒𝐹 𝑠𝑁 −1
𝜏 )( ∑𝑠𝑁=±1𝑒
𝐹 𝑠𝑁𝜏 )
...
X
World
...
...
...
...
... X
Partition function
23
𝜀 (𝑠1 ,𝑠2 ,⋯ ,𝑠𝑁 )=−𝐹∑𝑖=1
𝑁
𝑠𝑖
𝑍 (𝜏 )= ∑state 1
state 𝑓
𝑒−𝜀 (𝑠1 , 𝑠2 ,⋯ , 𝑠𝑁 )
𝜏
¿ (∑𝑠=±1
𝑒𝐹 𝑠𝜏 )
𝑁
=(𝑒𝐹𝜏 +𝑒
− 𝐹𝜏 )𝑁
¿ ( ∑𝑠 1=±1 𝑒𝐹 𝑠 1𝜏 )⋯( ∑
𝑠𝑁− 1=± 1𝑒𝐹 𝑠𝑁 −1
𝜏 )( ∑𝑠𝑁=±1𝑒
𝐹 𝑠𝑁𝜏 )
...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
24
Expectation of elongation
Hamiltonian and partition function
𝑍 (𝜏 )= ∑state 1
state 𝑓
𝑒−𝜀 (𝑠1 , 𝑠2 ,⋯ , 𝑠𝑁 )
𝜏...
X
World
...
...
...
...
... X
¿𝑁𝜏2 𝜕𝜕𝜏ln (𝑒
𝐹𝜏 +𝑒
−𝐹𝜏 )𝑁
Expectation of chain energy and downward elongation
25
𝜀 (𝑠1 ,𝑠2 ,⋯ ,𝑠𝑁 )=−𝐹∑𝑖=1
𝑁
𝑠𝑖
𝑍 (𝜏 )=(𝑒𝐹𝜏 +𝑒
−𝐹𝜏 )𝑁
⟨𝜀 ⟩=𝜏 2𝜕 ln 𝑍 (𝜏 )𝜕𝜏
𝑁
¿𝑁𝜏2
𝜕𝜕𝜏 (𝑒
𝐹𝜏 (− 𝐹
𝜏2 )+𝑒− 𝐹𝜏 ( 𝐹𝜏2 ))
𝑒𝐹𝜏+𝑒
−𝐹𝜏
⟨𝜀 ⟩=−𝑁 𝐹𝑒𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏 +𝑒
− 𝐹𝜏
...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
⟨−𝐹∑𝑖=1
𝑁
𝑠𝑖⟩= ⟨𝜀 ⟩=−𝑁 𝐹𝑒𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏 +𝑒
− 𝐹𝜏
Expectation of chain energy and downward elongation
26
𝜀 (𝑠1 ,𝑠2 ,⋯ ,𝑠𝑁 )=−𝐹∑𝑖=1
𝑁
𝑠𝑖
⟨𝜀 ⟩=−𝑁 𝐹𝑒𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏 +𝑒
− 𝐹𝜏
−𝐹 ⟨𝑅 ⟩=−𝑁 𝐹𝑒𝐹𝜏 −𝑒
−𝐹𝜏
𝑒𝐹𝜏 +𝑒
− 𝐹𝜏
𝑅
...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
Expectation of chain energy and downward elongation
27
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
28
𝑦 (𝑥 )=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥=0
0 0
0 001 1
1 1
𝑦 (𝑥 )=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥If x < 0, y(x) < 0
(-)ve (+)ve
If x > 0, y(x) > 0
(-)ve
(+)ve (-)ve
(+)ve
...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
29
(+)ve
(-)ve
𝑑 𝑦𝑑𝑥
= 𝑑𝑑𝑥 (𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥 )¿
(𝑒𝑥−𝑒−𝑥(−1)) (𝑒𝑥+𝑒−𝑥 )− (𝑒𝑥−𝑒−𝑥 ) (𝑒𝑥+𝑒−𝑥 (−1))(𝑒𝑥+𝑒−𝑥 )2
¿(𝑒𝑥+𝑒−𝑥 ) (𝑒𝑥+𝑒−𝑥)− (𝑒𝑥−𝑒−𝑥 ) (𝑒𝑥−𝑒−𝑥 )
(𝑒𝑥+𝑒−𝑥 )2¿
(𝑒𝑥+𝑒−𝑥 )2− (𝑒𝑥−𝑒−𝑥 )2
(𝑒𝑥+𝑒−𝑥 )2
¿1−(𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥 )2
=1− 𝑦2
...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
30
(+)ve
(-)ve
𝑑 𝑦𝑑𝑥
=1−𝑦 2>0
𝑑 𝑦𝑑𝑥
(𝑥 )=1− 𝑦 (𝑥 )2=10 0
𝑦 2=(𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥 )2
=(𝑎−𝑏 )2
(𝑎+𝑏)2=𝑎2−2𝑎𝑏+𝑏2
𝑎2+2𝑎𝑏+𝑏2<1
increasing
increasing
0
denominator
den
numerator
num(<1)
...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
31
(+)ve
(-)ve
𝑑 𝑦𝑑𝑥
=1−𝑦 2
increasing
increasing
𝑑2 𝑦𝑑 𝑥2
= 𝑑𝑑𝑥
(1− 𝑦2 )
𝑑2 𝑦𝑑 𝑥2
=0−2 𝑦 𝑑 𝑦𝑑𝑥(1− 𝑦2 )
(-)ve (+)ve(-), 0, (+)
(+), 0, (-)
x x
...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
32
(+)ve
(-)ve
𝑑 𝑦𝑑𝑥
=1−𝑦 2
increasing
increasing
𝑑2 𝑦𝑑 𝑥2
=−2 𝑦 (1− 𝑦2 ) (+), 0, (-)
lim𝑥→+∞
𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥= lim
𝑥→+∞
1−𝑒−2𝑥
1+𝑒− 2𝑥=+1
lim𝑥→−∞
𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥= lim
𝑥→−∞
𝑒2𝑥−1𝑒2 𝑥+1
=−1
...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
33
increasing
increasing
(+)ve
(-)ve
...
X
World
...
...
...
...
... X
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
34
SaturationUnbiased Partialstretch
PartialstretchSaturation
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥...
X
World
...
...
...
...
... X
𝑦=⟨ 𝑅 ⟩𝑁
=𝑒
𝐹𝜏 −𝑒
− 𝐹𝜏
𝑒𝐹𝜏+𝑒
− 𝐹𝜏
=𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
1-1 0 𝑥=𝐹 /𝜏
1
-1
𝑦=⟨ 𝑅 ⟩𝑁
Expectation of chain energy and downward elongation
35
Expectation of elongation
...
X
World Hamiltonian and partition function
𝑍 (𝜏 )= ∑state 1
state 𝑓
𝑒−𝜀 (𝑠1 , 𝑠2 ,⋯ , 𝑠𝑁 )
𝜏
...
...
...
...
... X
