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