# Berendsen thermostat

The Berendsen thermostat [1] is a method for controlling the temperature in a molecular dynamics simulation. The Berendsen thermostat uses a weak coupling $(\gamma_i)$ to an external heat bath of temperature $T_0$. This results in the modified equation of motion (Ref. 1 Eq. 8):

$m_i \frac{{\mathrm d} {\mathbf {v}}_i}{{\mathrm d} t} = {\mathbf {F}}_i + m_i \gamma \left( \frac{T_0}{T} -1\right){\mathbf {v}}_i$.

This represents a proportional scaling of the velocities per time step from ${\mathbf {v}}$ to $\lambda {\mathbf {v}}$, where (Ref. 1 Eq. 11)

$\lambda = \left[1 + \frac{\Delta t}{\tau_T} \left( \frac{T_0}{T} -1\right)\right]^{1/2}$

where $\tau_T$ is a time constant associated with the coupling.