1-d ideal chain
DESCRIPTION
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 PresentationTRANSCRIPT
1-d ideal chain
1
N links
๐ ๐=ยฑ1
Link 1
Link 2
Link N
. . .
. . .
Part 1
Bath
Part 2 Part N
1-d ideal chain
2
N links
๐ ๐=ยฑ1
. . .
System
Energy can be exchanged between chain and bath
3
System
N links
๐ ๐=ยฑ1
Bath
. . .
Part 1 Part 2 Part N. . .
Energy can be moved around bath
4
N links
๐ ๐=ยฑ1
Bath
Part 1 Part 2 Part N
. . .
. . .
System
Chain can be crinkled in different ways
5
N links
๐ ๐=ยฑ1
Bath
. . .
Part 1 Part 2 Part N. . .
System
Chain can be crinkled in different ways
6
N links
๐ ๐=ยฑ1
Bath
. . .
Part 1 Part 2 Part N. . .
System
Chain can be crinkled in different ways
7
N links
๐ ๐=ยฑ1
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
๐ ๐=ยฑ1
Chain can be crinkled in different ways
9
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
๐ ๐=ยฑ1
Chain can be crinkled in different ways
10
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
๐ ๐=ยฑ1
Chain can be crinkled in different ways
11
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
๐ ๐=ยฑ1
Chain can be crinkled in different ways
12
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
๐ ๐=ยฑ1
Chain can be crinkled in different ways
13
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
๐ ๐=ยฑ1
Chain can be crinkled in different ways
14
N links
Bath
. . .
Part 1 Part 2 Part N. . .
System
๐ ๐=ยฑ1
Exploring accessible world configurations equally
15
. . .
. . .
. . .
. . .
. . .
Too much total energy
. . .
Too little total energy
X X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
16
Hamiltonian and partition function
๐ (๐ )= โstate 1
state ๐
๐โ๐ (๐ 1 , ๐ 2 ,โฏ , ๐ ๐ )
๐
Expectation of elongation
...
X
World
...
...
...
...
... X
Hamiltonian
17
๐ (๐ 1 ,๐ 2 ,โฏ ,๐ ๐ )=โ๐นโ๐=1
๐
๐ ๐
๐
STOP๐ The 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
๐ (๐ 1 ,๐ 2 ,โฏ ,๐ ๐ )=โ๐นโ๐=1
๐
๐ ๐
๐ (๐ ) := โ๐ ๐=๐๐๐ผ๐
โ
๐ ๐๐๐ (๐๐ )๐โ๐๐๐
ยฟ โstate1
state ๐
๐โ๐ ( state )
๐
๐ 1 ,๐ 2 ,โฏ ,๐ ๐ ,โฏ ,๐ ๐Particular
-1, +1, +1, +1, +1 -1, +1, +1, -1, +1
...
X
World
...
...
...
...
... X
โ๐ 1 ,๐ 2
โ
๐โ๐ (๐ 1 , ๐ 2 )
๐ =๐โ๐ (+1 ,+1 )
๐ +๐โ๐ (+1 ,โ 1)
๐
+๐โ๐ (โ 1 ,+1)
๐ +๐โ๐ (โ 1 ,โ1 )
๐
Partition function
19
๐ (๐ 1 ,๐ 2 ,โฏ ,๐ ๐ )=โ๐นโ๐=1
๐
๐ ๐
๐ (๐ )= โstate 1
state ๐
๐โ๐ (๐ 1 , ๐ 2 ,โฏ , ๐ ๐ )
๐
ยฟ โ๐ 1=ยฑ1
โ
๐โ๐ (๐ 1 ,+1 )
๐ +๐โ๐ (๐ 1 ,โ 1)
๐
ยฟ โ๐ 1=ยฑ1
โ
โ๐ 2=ยฑ 1
โ
๐โ๐ (๐ 1 , ๐ 2 )
๐
...
X
World
...
...
...
...
... X
Partition function
20
๐ (๐ 1 ,๐ 2 ,โฏ ,๐ ๐ )=โ๐นโ๐=1
๐
๐ ๐
๐ (๐ )= โstate 1
state ๐
๐โ๐ (๐ 1 , ๐ 2 ,โฏ , ๐ ๐ )
๐
โ๐ 1 ,๐ 2
โ
๐โ๐ (๐ 1 , ๐ 2 )
๐ = โ๐ 1=ยฑ1
โ
โ๐ 2=ยฑ 1
โ
๐โ๐ (๐ 1 , ๐ 2 )
๐
๐= โ๐ 1=ยฑ1
โ
โฏ โ๐ ๐โ 1=ยฑ1
โ
โ๐ ๐=ยฑ 1
โ
๐โ๐ (๐ 1 ,โฏ , ๐ ๐โ 1 , ๐ ๐ )
๐
...
X
World
...
...
...
...
... X
Partition function
21
๐ (๐ 1 ,๐ 2 ,โฏ ,๐ ๐ )=โ๐นโ๐=1
๐
๐ ๐
๐ (๐ )= โstate 1
state ๐
๐โ๐ (๐ 1 , ๐ 2 ,โฏ , ๐ ๐ )
๐
๐= โ๐ 1=ยฑ1
โ
โฏ โ๐ ๐โ 1=ยฑ1
โ
โ๐ ๐=ยฑ 1
โ
๐โ๐ (๐ 1 ,โฏ , ๐ ๐โ 1 , ๐ ๐ )
๐
ยฟ โ๐ 1=ยฑ1
โ
โฏ โ๐ ๐โ 1=ยฑ1
โ
โ๐ ๐=ยฑ 1
โ
๐๐น (๐ 1+โฏ+๐ ๐โ1+๐ ๐ )
๐
๐๐น ๐ 1๐ โฏ๐
๐น ๐ ๐ โ1
๐ ๐๐น ๐ ๐๐
ยฟ โ๐ 1=ยฑ1
โ
โฏ โ๐ ๐โ 1=ยฑ1
โ
๐๐น ๐ 1๐ โฏ๐
๐น ๐ ๐โ1
๐ โ๐ ๐=ยฑ1
โ
๐๐น๐ ๐
๐
...
X
World
...
...
...
...
... X
Partition function
22
๐ (๐ 1 ,๐ 2 ,โฏ ,๐ ๐ )=โ๐นโ๐=1
๐
๐ ๐
๐ (๐ )= โstate 1
state ๐
๐โ๐ (๐ 1 , ๐ 2 ,โฏ , ๐ ๐ )
๐
ยฟ โ๐ 1=ยฑ1
โ
โฏ โ๐ ๐โ 1=ยฑ1
โ
๐๐น ๐ 1๐ โฏ๐
๐น ๐ ๐โ1
๐ โ๐ ๐=ยฑ1
โ
๐๐น๐ ๐
๐
ยฟ ( โ๐ ๐=ยฑ1๐
๐น ๐ ๐๐ ) โ๐ 1=ยฑ1
โ
โฏ โ๐ ๐โ 1=ยฑ1
โ
๐๐น ๐ 1๐ โฏ๐
๐น ๐ ๐โ1
๐
ยฟ ( โ๐ 1=ยฑ1 ๐๐น ๐ 1๐ )โฏ( โ
๐ ๐โ 1=ยฑ 1๐๐น ๐ ๐ โ1
๐ )( โ๐ ๐=ยฑ1๐
๐น ๐ ๐๐ )
...
X
World
...
...
...
...
... X
Partition function
23
๐ (๐ 1 ,๐ 2 ,โฏ ,๐ ๐ )=โ๐นโ๐=1
๐
๐ ๐
๐ (๐ )= โstate 1
state ๐
๐โ๐ (๐ 1 , ๐ 2 ,โฏ , ๐ ๐ )
๐
ยฟ (โ๐ =ยฑ1
๐๐น ๐ ๐ )
๐
=(๐๐น๐ +๐
โ ๐น๐ )๐
ยฟ ( โ๐ 1=ยฑ1 ๐๐น ๐ 1๐ )โฏ( โ
๐ ๐โ 1=ยฑ 1๐๐น ๐ ๐ โ1
๐ )( โ๐ ๐=ยฑ1๐
๐น ๐ ๐๐ )
...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
24
Expectation of elongation
Hamiltonian and partition function
๐ (๐ )= โstate 1
state ๐
๐โ๐ (๐ 1 , ๐ 2 ,โฏ , ๐ ๐ )
๐...
X
World
...
...
...
...
... X
ยฟ๐๐2 ๐๐๐ln (๐
๐น๐ +๐
โ๐น๐ )๐
Expectation of chain energy and downward elongation
25
๐ (๐ 1 ,๐ 2 ,โฏ ,๐ ๐ )=โ๐นโ๐=1
๐
๐ ๐
๐ (๐ )=(๐๐น๐ +๐
โ๐น๐ )๐
โจ๐ โฉ=๐ 2๐ ln ๐ (๐ )๐๐
๐
ยฟ๐๐2
๐๐๐ (๐
๐น๐ (โ ๐น
๐2 )+๐โ ๐น๐ ( ๐น๐2 ))
๐๐น๐+๐
โ๐น๐
โจ๐ โฉ=โ๐ ๐น๐๐น๐ โ๐
โ ๐น๐
๐๐น๐ +๐
โ ๐น๐
...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
โจโ๐นโ๐=1
๐
๐ ๐โฉ= โจ๐ โฉ=โ๐ ๐น๐๐น๐ โ๐
โ ๐น๐
๐๐น๐ +๐
โ ๐น๐
Expectation of chain energy and downward elongation
26
๐ (๐ 1 ,๐ 2 ,โฏ ,๐ ๐ )=โ๐นโ๐=1
๐
๐ ๐
โจ๐ โฉ=โ๐ ๐น๐๐น๐ โ๐
โ ๐น๐
๐๐น๐ +๐
โ ๐น๐
โ๐น โจ๐ โฉ=โ๐ ๐น๐๐น๐ โ๐
โ๐น๐
๐๐น๐ +๐
โ ๐น๐
๐
...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
Expectation of chain energy and downward elongation
27
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
28
๐ฆ (๐ฅ )=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ=0
0 0
0 001 1
1 1
๐ฆ (๐ฅ )=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅIf x < 0, y(x) < 0
(-)ve (+)ve
If x > 0, y(x) > 0
(-)ve
(+)ve (-)ve
(+)ve
...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
29
(+)ve
(-)ve
๐ ๐ฆ๐๐ฅ
= ๐๐๐ฅ (๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ )ยฟ
(๐๐ฅโ๐โ๐ฅ(โ1)) (๐๐ฅ+๐โ๐ฅ )โ (๐๐ฅโ๐โ๐ฅ ) (๐๐ฅ+๐โ๐ฅ (โ1))(๐๐ฅ+๐โ๐ฅ )2
ยฟ(๐๐ฅ+๐โ๐ฅ ) (๐๐ฅ+๐โ๐ฅ)โ (๐๐ฅโ๐โ๐ฅ ) (๐๐ฅโ๐โ๐ฅ )
(๐๐ฅ+๐โ๐ฅ )2ยฟ
(๐๐ฅ+๐โ๐ฅ )2โ (๐๐ฅโ๐โ๐ฅ )2
(๐๐ฅ+๐โ๐ฅ )2
ยฟ1โ(๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ )2
=1โ ๐ฆ2
...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
30
(+)ve
(-)ve
๐ ๐ฆ๐๐ฅ
=1โ๐ฆ 2>0
๐ ๐ฆ๐๐ฅ
(๐ฅ )=1โ ๐ฆ (๐ฅ )2=10 0
๐ฆ 2=(๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ )2
=(๐โ๐ )2
(๐+๐)2=๐2โ2๐๐+๐2
๐2+2๐๐+๐2<1
increasing
increasing
0
denominator
den
numerator
num(<1)
...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
31
(+)ve
(-)ve
๐ ๐ฆ๐๐ฅ
=1โ๐ฆ 2
increasing
increasing
๐2 ๐ฆ๐ ๐ฅ2
= ๐๐๐ฅ
(1โ ๐ฆ2 )
๐2 ๐ฆ๐ ๐ฅ2
=0โ2 ๐ฆ ๐ ๐ฆ๐๐ฅ(1โ ๐ฆ2 )
(-)ve (+)ve(-), 0, (+)
(+), 0, (-)
x x
...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
32
(+)ve
(-)ve
๐ ๐ฆ๐๐ฅ
=1โ๐ฆ 2
increasing
increasing
๐2 ๐ฆ๐ ๐ฅ2
=โ2 ๐ฆ (1โ ๐ฆ2 ) (+), 0, (-)
lim๐ฅโ+โ
๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ= lim
๐ฅโ+โ
1โ๐โ2๐ฅ
1+๐โ 2๐ฅ=+1
lim๐ฅโโโ
๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ= lim
๐ฅโโโ
๐2๐ฅโ1๐2 ๐ฅ+1
=โ1
...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
33
increasing
increasing
(+)ve
(-)ve
...
X
World
...
...
...
...
... X
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
34
SaturationUnbiased Partialstretch
PartialstretchSaturation
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ...
X
World
...
...
...
...
... X
๐ฆ=โจ ๐ โฉ๐
=๐
๐น๐ โ๐
โ ๐น๐
๐๐น๐+๐
โ ๐น๐
=๐๐ฅโ๐โ๐ฅ
๐๐ฅ+๐โ๐ฅ
1-1 0 ๐ฅ=๐น /๐
1
-1
๐ฆ=โจ ๐ โฉ๐
Expectation of chain energy and downward elongation
35
Expectation of elongation
...
X
World Hamiltonian and partition function
๐ (๐ )= โstate 1
state ๐
๐โ๐ (๐ 1 , ๐ 2 ,โฏ , ๐ ๐ )
๐
...
...
...
...
... X