Langevin Dynamics (LD) Simulation

The Langevin equation is a stochastic differential equation in which two force terms have been added to Newton's second law to approximate the effects of neglected degrees of freedom. One term represents a frictional force, the other a random force $\vec{R}$. For example, the effects of solvent molecules not explicitly present in the system being simulated would be approximated in terms of a frictional drag on the solute as well as random kicks associated with the thermal motions of the solvent molecules. Since friction opposes motion, the first additional force is proportional to the particle's velocity and oppositely directed. Langevin's equation for the motion of atom i is:

$$\vec{F}_i - \gamma_i \vec{v}_i + \vec{R}_i(t) = m_i \vec{a}_i,$$

where $\vec{F}_i$ is still the sum of all forces exerted on atom i by other atoms explicitly present in the system. This equation is often expressed in terms of the `collision frequency' $\zeta = \gamma/m$.

The friction coefficient is related to the fluctuations of the random force by the fluctuation-dissipation theorem:

$$\langle \vec{R}_i(t) \rangle = 0,$$

$$\int \langle \vec{R}_i(0) \cdot \vec{R}_i(t) \rangle dt = 6 k_BT \gamma_i.$$

In simulations it is often assumed that the random force is completely uncorrelated at different times. That is, the above equation takes the form:

$$\langle \vec{R}_i(t) \cdot \vec{R}_i(t^\prime) \rangle = 6 k_BT \gamma_i \delta(t-t^\prime).$$

The temperature of the system being simulated is maintained via this relationship between $\vec{R}(t)$ and $\gamma$.

The jostling of a solute by solvent can expedite barrier crossing, and hence Langevin dynamics can search conformations better than Newtonian molecular dynamics ($\gamma = 0$).