# DMPK equations

#### Numerical experiments » DMPK equations

The DMPK equations describe the transport in 1d disordered systems. They provide a full solution for various quantitities, for example the distribution of transmission across a disordered sample. The name comes the intiials of: Dorokhov O. N., Pis’ma Zh. Eksp. Teor. Fiz. 36 (1982) 259; Mello P. A., Pereyra P., Kumar N., Ann. Phys. 181 (1988) 290.

A good - but not easy - reference is C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997) available on this page. DMPK equations are valid for a weakly disordered (localization length larger than microscopic lengths) quasi-1d systems, with several open transverse channels. We will consider only 1d system, with a single channel. The solution is simpler and detailed in A.A. Abrikosov, Solid State Communications 37, 997 (1981) or in C.A. Mueller and D. Delande, arxiv:1005.0915.

The script dmpk.py computes the distribution of transmission (or more precisely, the smoother normalized distribution of -log(transmission)) using the solution of Abrikosov. It involves a simple integral done with scipy. Additionally, it computes some average quantity such as <log(T)>,,<T**2>, etc. Read the script and understand how it works.

Usage:

*dmpk.py t t_max nsteps*

*t*: size of the system in unit of the localization length*t_max*: maximum value of -log(T) for which the distribution is computed

(the minimum value is zero). Hint: of the order of max(4,t+4*sqrt(t))*nsteps*: the number of points where the distribution is comuted. 100 is usually enouhg because the function is very smooth.

Warning: for large t (say above 20), some numerical inaccuracies are present. Could be fixed, but I am too lazy/ignorant.

**Warning:** For some stupid reason, the CMS of this Web site refuses files with the *.py* extension. Thus, you have to manually rename the file, changing the final *_py* to *.py*