Probability distribution function / propagator

[raintrek]

I have the following equation:

$$\frac{1}{2N^{2}}\int_{s} \int_{p} \left\langle (\textbf{R}_{s} - \textbf{R}_{p} )^{2} \right\rangle$$

which describes the radius of gyration of a polymer. (the term being integrated is the average position between beads p and s)

This is shown to be equivalent to:

$$\frac{1}{2N^{2}}\int_{s} \int_{p} \left| s - p \right| b^{2}$$
where b is the step size.

All my textbook says is that the second equation has been found by using the propagator (probability density function) and two integrals:

$$G(\textbf{R}, N)=\left( \frac{3}{2\pi Nb^{2}}\right)^{3/2}e^{- \frac{3R^{2}}{2Nb^{2}}$$

$$\int^{\infty}_{- \infty} e^{-ax^{2}}=\sqrt{\frac{\pi}{a}}$$
$$\int^{\infty}_{- \infty} x^{2} e^{-ax^{2}}=\frac{1}{2a} \sqrt{\frac{\pi}{a}}$$


I have no idea how this "propagator" is used to reach the second equation. Is anyone able to shed any light on what's happened? Many thanks in advance!

 

[mighty2000]

I can explain how the second equation is derived from the first one. The first equation is a general expression for the radius of gyration of a polymer, where N is the number of beads and \textbf{R}_{s} and \textbf{R}_{p} represent the positions of beads s and p, respectively. The term being integrated, \left\langle (\textbf{R}_{s} - \textbf{R}_{p} )^{2} \right\rangle, is the average squared distance between beads s and p.

To understand how the second equation is derived, we need to use the concept of the propagator. A propagator is a mathematical representation of the probability density function that describes the motion of a particle over time. In this case, the propagator is used to describe the probability of a polymer being at a certain position, \textbf{R}, after N steps.

Using the propagator, we can write the probability of a polymer being at a certain position, \textbf{R}, after N steps as:

P(\textbf{R}, N) = G(\textbf{R}, N) \cdot P(\textbf{R}, N-1)

where G(\textbf{R}, N) is the propagator and P(\textbf{R}, N-1) is the probability of the polymer being at position \textbf{R} after N-1 steps.

We can then use this equation to calculate the average squared distance between beads s and p as:

\left\langle (\textbf{R}_{s} - \textbf{R}_{p} )^{2} \right\rangle = \int_{s} \int_{p} (\textbf{R}_{s} - \textbf{R}_{p} )^{2} \cdot P(\textbf{R}_{s}, N) \cdot P(\textbf{R}_{p}, N) d\textbf{R}_{s} d\textbf{R}_{p}

Using the definition of the propagator and the properties of the Gaussian function, we can simplify this expression to:

\left\langle (\textbf{R}_{s} - \textbf{R}_{p} )^{2} \right\rangle = \left| s - p \right| b^{2}

where b is the step size.