** Next:** Ising Model
**Up:** Lattice simulations. Random Walk.
** Previous:** Lattice simulations. Random Walk.

Following the spirit of the previous lecture, we start with the random walks. They are very important to study diffusion through the non-homogeneous media, and, of course, polymers. Here we will consider the simple example of the random walk on the square lattice in 2D. This can be a model for diffusion, or for non-self avoiding polymer chain.

We do a random walk, and now we are interested in its trajectory, as
this simulates the polymer chain contour. And we are interested in
the properties of this trajectory, as they describe the chain
conformations, not just the probability density as a function of the
coordinates. The important difference now is that we are building the
chain, and we cannot stay at the same place, *we must move at
each step*. But the direction where we move is decided by the MC
procedure.

We consider an example of a polar chain with each bond having a dipole
moment of , parallel to the bond direction,
in the uniform electric field . This model can actually describe
polymer chain in the flux, and liquid-crystalline ordering. The chain
is not self-avoiding, and all segments are independent. Each segment
contributes to the energy.
Probability distribution for conformation of a *N*-segment chain,
described by the set of bond vectors is:

- MC Algorithm
- 1.
- Choose position of the first segment
- 2.
- Loop to generate a conformation
- Pick the direction of the next bond to grow, based on it's weight
*w*_{i}. - update the total energy, and the end-to-end vector.
- go back to the beginning of the loop

- Pick the direction of the next bond to grow, based on it's weight
- 3.
- update current values of the running averages.
- 4.
- go back to step 2

**Fortran code Example 3 :**- Code.

© 1997

Thu Nov 6 22:32:26 EST 1997