How to Calculate Correlation Functions and Persistence Length in a 1D Polymer?

[WarnK]

Homework Statement

1D polymer, fixed segment length a
If the angle between segment j and j+1 is 0, the energy is 0
If the angle is pi the energy is +2J.

Compute the correlation function $$<s_i s_{i+n}>$$, where $$s_j = \pm 1$$ denotes the direction of segment j

Find the persistence length Lp, defined through
$$<s_i s_{i+n}> = e^{-|n|a/Lp}$$

Find an expression for the end-to-end distance $$S(N) = <(x_N - x_0)^2>^{1/2}$$ as a function of temperature and the number of links N

Homework Equations

?

The Attempt at a Solution

$$<s_i s_{i+n}> = \frac{ Tr s_i s_{i+n} e^{-\beta H} }{ Tr e^{-\beta H} }$$

But I don't know any hamiltonian? Or even what sort of trace to do.
The problem sort of reminds me of the 'XY'-modell for spins on a 1d lattice, but I don't really understand how to make any use of that.

 

[nrqed]

Homework Statement

1D polymer, fixed segment length a
If the angle between segment j and j+1 is 0, the energy is 0
If the angle is pi the energy is +2J.

Compute the correlation function $$<s_i s_{i+n}>$$, where $$s_j = \pm 1$$ denotes the direction of segment j

Find the persistence length Lp, defined through
$$<s_i s_{i+n}> = e^{-|n|a/Lp}$$

Find an expression for the end-to-end distance $$S(N) = <(x_N - x_0)^2>^{1/2}$$ as a function of temperature and the number of links N

Homework Equations

?

The Attempt at a Solution

$$<s_i s_{i+n}> = \frac{ Tr s_i s_{i+n} e^{-\beta H} }{ Tr e^{-\beta H} }$$

But I don't know any hamiltonian? Or even what sort of trace to do.
The problem sort of reminds me of the 'XY'-modell for spins on a 1d lattice, but I don't really understand how to make any use of that.

You can write down a Hamiltonian.
It will be the sum of the energy of all the adjacent pairs, taking into account the rule they give. You want an expression that gives zero when $s_j = s_{j+1}$ and which gives 2J when $s_j = -s_{j+1}$. This is simply $J(1-s_j s_{j+1})$.

 

[WarnK]

So, starting with a Hamiltonian like
$$H = J \sum_{j=1}^{N-1} (1-s_j s_{j+1})$$
and proceeding to calculate the partition function much the same way as for the 1d ising chain,
$$Z = Tr e^{-\beta H} = \dots = 2 e^{-\beta J (N-1)} [2 cosh(\beta J)]^{N-1}$$
then noteing I can write
$$<s_j s_{j+n}> = \frac{1}{Z \beta^n} \frac{d^n Z}{d J^n}$$
I end up with
$$<s_j s_{j+n}> = tanh(\beta J)^n$$
is this really right? feels like I missed something

thanks for the help!

 

[physics0]

So, starting with a Hamiltonian like
$$H = J \sum_{j=1}^{N-1} (1-s_j s_{j+1})$$
and proceeding to calculate the partition function much the same way as for the 1d ising chain,
$$Z = Tr e^{-\beta H} = \dots = 2 e^{-\beta J (N-1)} [2 cosh(\beta J)]^{N-1}$$
then noteing I can write
$$<s_j s_{j+n}> = \frac{1}{Z \beta^n} \frac{d^n Z}{d J^n}$$
I end up with
$$<s_j s_{j+n}> = tanh(\beta J)^n$$
is this really right? feels like I missed something

thanks for the help!

the last equation should be (tanh(\beta J))^n

 

[salmana]

the last equation should be (tanh(\beta J))^n

how can we calculate $$S^2(N) = <(x_N-x_0)^2>$$ in above case?

 

[salmana]

How one can calculate

S(N)² = <(x_N-x₀)²>

thanx