Some probabilities

It is instructive to calculate some probabilities occurring in the RIS model. An important probability for computer simulations is the conditional probability  $q_{\xi \eta }$defined below. We shall always assume that the chain is infinitely long, and that end effects may be neglected. In this case $\lambda _{\mathrm{max}}$ is the only eigenvalue we need.

We define the probability that a given bond is in state $\eta$ by

$$p_{\eta} = \langle \delta_{\varphi_i,\eta} \rangle .$$ (1.19)

Explicitly introducing the averaging procedure yields

$${p_{\eta} = \frac{1}{Z} \sum_{\varphi_2} \cdots \sum_{\varphi_{N-... ...1(\varphi_2)} t(\varphi_2,\varphi_3) \cdots t(\varphi_{i-1},\varphi_i) \times }$$

$$\delta_{\varphi_i,\eta} t(\varphi_i,\varphi_{i+1}) \cdots t(\varphi_{N-2},\varphi_{N-1}).$$ (1.20)

Using the same method as in the previous section we find

$$p_{\eta} \frac{1}{Z} \sum_{\varphi_i} (Y^T T^{i-1})_{\varphi_i} \delta_{\varphi_i,\eta} (T^{N-1-i}X)_{\varphi_i}$$ =

$$\frac{1}{Z} (Y^T T^{i-1})_{\eta} (T^{N-1-i}X)_{\eta}$$ =

$$\frac{ \{ \sum_k \lambda_k^{i-1} (Y^T A_k)(B_k^T)_{\eta}\} \{ \su... ...} (A_l)_{\eta} (B_l^T X ) \} } { \sum_k \lambda_k^{N-2} (Y^T A_k) (B_k^T X) } .$$ = (1.21)

Dividing numerator and denominator by $\lambda_{\mathrm{max}}^{N-2}$, and assuming N, i, and N-i are large we get

$$p_{\eta} = (B_{\mathrm{max}}^T)_{\eta} (A_{\mathrm{max}})_{\eta} .$$ (1.22)

Because here and in the remaining part of the chapter we only need $\lambda _{\mathrm{max}}$ and the corresponding vectors A_max and B _max^T, we shall omit the subscript ''max'' and write

$$p_{\eta} = B_{\eta}^T A_{\eta} .$$ (1.23)

Similarly one calculates

 

$$p_{\xi\eta} \langle \delta_{\varphi_{i-1},\xi} \delta_{\varphi_i,\eta} \rangle$$ =

$$B_{\xi}^T \frac{t(\xi,\eta)}{\lambda} A_{\eta} .$$ = (1.24)

An important quantity is $q_{\xi \eta }$, the conditional probability to find bond i in state $\eta$, given that bond i-1 is in state $\xi$

$$q_{\xi\eta} = \frac{p_{\xi\eta}}{p_{\xi}} .$$ (1.25)

Introducing Eqs. (1.23) and (1.24) we get

$$q_{\xi\eta} = \frac{t(\xi,\eta)}{\lambda} \frac{A_{\eta}}{A_{\xi}}.$$ (1.26)

This quantity may be used to generate chain conformations on a computer.