Berendsen thermostat

The Berendsen thermostat 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_0} . This results in the modified equation of motion (Ref. 1 Eq. 8):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf {v}}} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau_T} is a time constant associated with the coupling.

References

1. H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and J. R. Haak "Molecular dynamics with coupling to an external bath", Journal of Chemical Physics 81 pp. 3684-3690 (1984)