Simulation model

Problem setting

Algorithm

Lattice Setting

The simulation of 3-dimentional hopping in positional and energetically disordered media we can realize by two (as minimum) ways :

1. First of all, we have to generate real lattice, which sites are distributed randomly normally with specific width of distribution (SIGMAGAMMA).  As a next step we must give for each site random normal distributed energy (around particular value- SIGMA). And in this lattice hopping probability must be calculated.

, where - constant , Ei - string of random numbers.

But there are some difficulties:

• It is not so simple to generate lattice with sites random distributed.
• And if we have real amorphous lattice it is not so simple to realize boundary condition. For the small lattices boundary conditions are very critical point of the simulation. For example, in the lattice, which has 125 sites (5x5x5), 86 sites are on the boundary.

2. The second way was offered by Bessler[5]. One can generate a cubic lattice, and for each site It have to be assigned normal distributed energy (around particular value- SIGMA) and normal distributed positional displacement (with deviation that is a random number- SIGMAGAMMA(i)).  And in this "virtual" lattice hopping probability must be calculated using next formula:

, where and and Ei - strings of random numbers.

We've chose the second way: initial lattice will be cubic. Here you can download cubic lattices generator (code of the program). 

Boundary conditions

We chose periodical boundary conditions for cubic lattice for boundary effects prevention.

1st site setting

For purpose of averaging we sample all sites with x- coordinate equal to zero as a initial.

Neighbors

We investigate case of "to the nearest neighbor hopping", so, for each site there are 27 neighbors. If site is placed on the boundary we add lacking neighbor according to the boundary rule.

Probability to hop

In our program we compute contemporaneous the case of external field acting and free diffusion, to wit, probability for constrained transport and probability for free diffusion. Therefore after program running you will obtain data about particular random configuration of polymer. 

Sample rule for next site choosing 

See Monte Carlo method

Time of "outstaying"

The computer realization of hopping in the lattice may be done by direct applying sample rule, but if we have polymer with relatively big values of GAMMA (wave function overlapping integral that determines tunneling rate), it is observed only a long staying on the initial site. It is not optimal to compute millions times only one state. This problem can be bypassed by the next algorithm of taking in to account long time sitting on the site till hop.

Let's enter new quantity - time "outstaying" that is:

And from the sequence of probability we exclude the probability to hop to itself and recalculate probabilities values. Then we get  algorithm that do hop for each circle, but also count time of sitting on a site.

Program checking: 

1. Suppose GAMMA=0 and SIGMA=0.

2. For investigation of positional disorder effect we will run program for cubic lattice (GAMMA=constant).

3. Suppose GAMMA=constant and SIGMA=0 for investigation of energetic disorder effects.