CSVR thermostat

One popular strategy to control temperature in NVT MD is based on rescaling atomic velocities (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \bold{v}_{i}} ) at a certain predefined frequency by a factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \alpha = \sqrt{\bar{K}/K}} in such a way that the total kinetic energy of the system

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle K= \frac{1}{2} \sum\limits_{i=1}^{N} m_i |\bold{v}_{i}|^2, }

is equal to the average kinetic energy corresponding to given temperature:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \bar{K} = \frac{1}{2}N_f k_B T }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle N_f} is the number of degrees of freedom (e.g., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle N_f = 3N -3 } in the case of 3D periodic systems) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle N } is the number of atoms per the simulation cell. Such a method, however, suffers from several problems. First, the ensemble generated is not strictly canonical. Second, rescaling velocities creates discontinuities in trajectories. As a consequence, the method has no conserved quantity that could be used to guide the choice of simulation parameters, such as the size of the integration step. Also, the rescaling introduces artificial fast fluctuations to velocities, making the evaluation of time correlations is problematic. Finally, the trajectories generated via a naïve rescaling method often suffer from ergodicity issues, such as the flying ice-cube problem, in which kinetic energy of a part of the vibrational degrees of freedom is transferred into translations and/or rotations, violating thus equipartition principle.

The canonical sampling through velocity rescaling (CSVR) proposed by Bussi et al. removes most of the difficulties of the naïve rescaling approach. Here, the term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \bar{K} } is replaced by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle K_{t} } obtained for each time step by propagating in time via auxiliary dynamics

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle dK = (\bar{K} - K) \frac{dt}{\tau} + 2\sqrt{\frac{K\bar{K}}{N_f}} \frac{dW}{\sqrt{\tau}} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): dW is a Wiener noise and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \tau determines the characteristic time scale of the CSVR thermostat. The latter is the only parameter of this thermostat and can be defined via flag CSVR_PERIOD. Importantly, the auxiliary dynamics generates canonical distribution for kinetic energy:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle P(K_t) dK_t \propto K_t^{(N_f/2 - 1)} e^{-K_t/k_B T} dK_t }

The conserved quantity of the CSVR thermostat is the effective energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \tilde{H} } defined as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \tilde{H}(t) = H(t) - \int_0^{t'} (\bar{K}-K)\frac{dt'}{\tau} - 2\int_0^{t} \sqrt{\frac{K{t'}\bar{K}}{N_f}} \frac{dW(t')}{\sqrt{\tau}} }

As shown by Bussi et al., the CSVR thermostat does not significantly affect evaluation of dynamical properties, such as the velocity autocorreletion functions or diffusion coefficients.

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