README file for Random walks and probability flux in non-equilibrium systems prepared for Spring School 2017

Arvind Ravichandran

[email protected]

Graduate Student in Theoretical Soft Matter and Biophysics Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Juelich, Germany

This contains Python scripts to simulate illustrate concepts of non-equilibrium for the Spring School 2017/2018 course at ICS II. In this course, conducted as a workshop, there are three instances of equilibrium and non-equilibrium systems.

• Task 1 (Unbiased Random Walk): Simulation and analysis of an unbiased random walk using numpy.random.uniform (step length is varied between -0.5 and 0.5)
  Parts of the code are adapted from: http://pages.physics.cornell.edu/~sethna/StatMech/ComputerExercises/PythonSoftware/RandomWalk.py

• Task 2 (Biased Random Walk): Simulation and analysis of a biased random walk using numpy.random.choice (uniform step length). A neat pythonic implementation of Mean Squared Displacement (MSD), and an extemporary approach to compute its standard deviation is given here.

• Task 3 (PFA for Harmonic Oscillators): Probability flux analysis in a coarse grained phase space for a pair of connected harmonic oscillators coupled to independent temperature baths. The flux of energy, and the non-equilibrium signature is seen in the probability flux in phase space. The idea is based on the example from Battle et al. Science. 2016 Apr 29;352(6285):604-7. doi: 10.1126/science.aac8167.

• Problems This contains the problems, with comments for students to complete to get the systems up and running.
• Solutions Contains the solutions to the problems, with the completed code running. There are Cython optimizations, which students are not required to learn. They are implemented for quickly running an ensemble of coupled oscillators serially for Task 3.
• docs Contains the pdf document for the teaching of random walk, diffusion, non-equilibrium, probability flux analysis, and an in-depth perspective on introductory Python.

This project was conducted under the guidance of my PhD advisors, Gerrit Vliegenthart, Thorsten Auth, and with helpful discussions with Ewan Henry. In case of questions, comments, please contact me (Arvind).