Giovanni Bussi, Davide Donadio, and Michele Parrinello present a new molecular dynamics algorithm for canonical sampling. The algorithm involves rescaling the velocities of all particles using a random factor, which is formally justified. This method allows for the definition of a quantity that remains constant during the simulation, useful for assessing the accuracy of the sampling. The authors demonstrate the method's effectiveness on Lennard-Jones and TIP4P water models in both solid and liquid phases, showing excellent performance and independence from thermostat parameters. The algorithm is an extension of the Berendsen thermostat, incorporating a random force to enforce the correct kinetic energy distribution. It is shown to be ergodic and can be formulated similarly to Nosé's thermostat. The paper also introduces an effective energy measure to assess the accuracy of numerical integration, which can be used to correct finite time-step errors. The method is easy to implement and outperforms other commonly used thermostats in terms of ergodicity and sampling accuracy.Giovanni Bussi, Davide Donadio, and Michele Parrinello present a new molecular dynamics algorithm for canonical sampling. The algorithm involves rescaling the velocities of all particles using a random factor, which is formally justified. This method allows for the definition of a quantity that remains constant during the simulation, useful for assessing the accuracy of the sampling. The authors demonstrate the method's effectiveness on Lennard-Jones and TIP4P water models in both solid and liquid phases, showing excellent performance and independence from thermostat parameters. The algorithm is an extension of the Berendsen thermostat, incorporating a random force to enforce the correct kinetic energy distribution. It is shown to be ergodic and can be formulated similarly to Nosé's thermostat. The paper also introduces an effective energy measure to assess the accuracy of numerical integration, which can be used to correct finite time-step errors. The method is easy to implement and outperforms other commonly used thermostats in terms of ergodicity and sampling accuracy.