The Efficiency of a Thermostat and Gentle Thermostats

To understand the reason for this fundamental, qualitative difference between Nos6-Hoover-Langevin and Langevin dynamics, and to compare it with Velocity Rescaling, one may study a quantity that relates the rate of convergence to equilibrium to the rate of growth of the error in the autocorrelation function. In [225], this precise quantity is introduced and termed the efficiency of the thermostat  

The rate of convergence to equilibrium is quite difficult to calculate in general, as it would typically relate to the spectral gap of the diffusion. In light of the observations of Fig. 8.3, one may choose to estimate this in terms of the rate of convergence of the average kinetic energy to the target temperature. 

The rate of accumulation of error due to the thermostat is also difficult to quantify analytically, since it requires calculation of the error in an autocorrelation function over a time interval which may not be short compared to the simulation stepsize, particularly for slow-relaxing systems. Again, an estimate is obtainable by considering the rate of error accumulation at short times (obtained by Taylor expansion of the exact and approximate autocorrelation functions). The result of these two estimates is then an analytical estimate for the efficiency that relates various schemes for various values of the parameters involved. 

In Table 8.1 we show the comparison of the relevant estimates of the above quantities, including the efficiency, for different choices of thermostat. 

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