# Time rescaling in overdamped Langevin simulation

I'm simulating a system according to the Langevin equation (with inertia), however my friction coefficient is high enough that I am essentially in the overdamped regime on the timescales of one timestep. I also want the results of a simulation of the same system, but with a higher friction coefficient. I realized that because I am in the overdamped regime, without running a second simulation, I should get the dynamics if I rescale the time and velocities from my first simulation according to the same scale factor I want the friction coefficient to increase by. The scaling factor cancels out in both the overdamped Langevin equation and the Verlet algorithm I am using for my simulation, so it seems like this rescaling is legitimate.

However after rescaling, the effective time I have simulated for becomes longer. This seems too good to be true: what's to stop me from picking a lower friction coefficient (although high enough to still be in the overdamped regime) so that things happen faster, simulating for less time, then rescaling to match the actual parameters I want, without having to do the 'honest work' of simulating those actual parameters, which would take longer?

Edit:

I should have looked more carefully at my equations. The dimensionless overdamped Langevin equation for my system is:

$$x^*(\tau + \Delta\tau) = x^*(\tau) + \frac{\Delta\tau}{6\pi\eta^*R^*}F^*_{det} + \sqrt{\frac{2T^*\Delta\tau}{6\pi\eta^*R^*}}\xi(\tau)$$

therefore increasing $\Delta\tau$ and $\eta^*$ by the same factor will always cancel out (if I increase friction, I should be able to pick a larger timestep), therefore if I simulated using the overdamped equation, my rescaling would not help me gain any simulation time.

However, simulating with the Verlet algorithm, this is not the case. My simulation becomes unstable if I attempt to increase $\Delta \tau$ by the same factor as $\eta^*$. The dimensionless equations are:

$$r_i^*(\tau+\Delta\tau) = r_i^*(\tau) + v_i^*(\tau)\Delta\tau + \frac{f_i^*(\tau)}{2m^*}\Delta\tau^2$$

$$v_i^*(\tau + \Delta\tau) = v_i^* + \frac{f_i^*(\tau+\Delta\tau) + f_i^*(\tau)}{2m^*}\Delta\tau$$

where $f_i^* = -\nabla U(\tau) - 6\pi \eta^*R^*v^*(\tau) + \sqrt{\frac{12\pi T^*\eta^*R^*}{\Delta \tau}}\xi(\tau)$

The force terms are problematic, because there the scaling does not cancel out, but I am in the overdamped regime on the timescales of each timestep, which I thought would mean these force terms would be very close to zero for me, but instead they are blowing up.

• These are not Langevin equations. Did you perform any sanity check simulations with this method for cases with analytic solution? (such as diffusion in harmonic potential) – Alexander Apr 5 '18 at 21:58
• Sorry, I am always a bit confused by this terminology. I meant overdamped Langevin as it is sometimes called Brownian Dynamics (as by this definition. In the Verlet algorithm, my force is given by the equation of motion in Langevin dynamics (I have edited this, sorry it was not clear). – The Hagen Apr 6 '18 at 16:03
• So did you simulate problems with known solutions to check your simulation? – Alexander Apr 6 '18 at 17:01