Langevin dynamics

Simple Langevin dynamics are also provided by NAMD. This consists of adding a random force and subtracting a friction force from each atom during each integration step. The random force is calculated such that the average force is zero and the standard deviation is

$$2 K_{\text{b}} T_0 b_i m_i \Delta t$$

where

• $K_{\text{b}}$ = Boltzman's constant
• T₀ = target temperature specified using the configuration parameter langTemp
• b_i = friction coefficient for atom i
• m_i = mass of atom i
• $\Delta t$ = timestep size

The friction force applied is

$$m_i b_i \frac{{dx}_i}{dt}$$

where

• m_i = mass of atom i
• b_i = friction coefficient for atom i

In order to apply these forces, NAMD uses the same second-order finite difference approximation as X-PLOR. This approximation uses

$$x_i^{n+1} = \left( 1+\frac{b_i \Delta t}{2} \right) \left( 2... ...{\Delta t^2}{m_i} + x_i^{n-1} \frac{b_i \Delta t}{2} \right)$$

to update the position of particle i from step n to step n+1and

$$v_n = \frac{x_{n+1} - x_{n-1}}{2 \Delta t}$$

for the velocity at step n.