Ganesh Swami

In my non-linear physics class, we are covering topics that I find very interesting: Fast-slow dynamics.

Consider the system of coupled differential equations above. If the parameter r is extremely small, the equations exhibit two time-scales. From the perspective of the fast equation, the slow equation is almost constant and from the perspective of the slow equation, the fast equation has already reached steady-state. Thus, by appropriately re-scaling time, you can make approximations about the final behavior of the system. I’m still trying to understand the rigorous mathematical analysis that has gone into these equations. If you want to learn more about this re-scaling trick: search for “adiabatic elimination” and “center manifold.”

These techniques are often used in the field of weather prediction. You have dynamics that can act either locally or globally. By making robust approximations, you can get away with weak assumptions about the local effects. This lets you use a bigger timestep that can give you more relevant results.

From my own experience, I can draw parallels to the simulation of chemical systems. In protein simulations, you have solvent effects that are high frequency actions. This forces you to use a small timestep. Some concrete numbers: water molecules have motions in the picosecond range, while we really are only interested in the motion of the proteins on the nano/micro second range. But the action of the solvent is crucial to the dynamics of the protein, so we really cannot get away with ignoring solvent effects. I wonder if I can use a similar analysis to enable the use of a bigger timestep? Probably not, as somebody would have done this already.

Possibly related:

• Gromacs Workshop
• Avoidable integrator errors
• Anton
• Mersenne Twister
• Fast Molecular Dynamics

This entry was posted on Saturday, February 3rd, 2007 at 1:03 am and is filed under Physics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.