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Most deterministic schemes for controlling temperature by kinetic variables(NH thermsotat), configurational variables (BT thermostat) and all phase spacevariables (PB thermostat) are non-ergodic for systems with a few degrees offreedom. While for the NH thermostat ergodicity has been achieved bycontrolling the higher order moments of kinetic energy, the issues ofnonergodicity of BT and PB thermostats still persist. In this paper, we proposea family of modifications for improving their ergodic characteristics. To doso, we introduce two new measures of configurational temperature (second andthird order) based on the generalized temperature - curvature relationship. Theequations of motion for the existing thermostats are modified by controllingthe relevant higher order temperature variables. The ergodic characteristics ofthe proposed modifications are tested using a single harmonic oscillator. Forthe PB thermostat, the fastest route to ergodicity is by controlling the usualconfigurational and kinetic temperatures along with the $2^{\text{nd}}$ orderconfigurational temperature. For the BT thermostat, controlling the usualconfigurational temperature along with the $2^{\text{nd}}$ orderconfigurational temperature is sufficient to make it ergodic. Our method alsoprovides a new ergodic generalization of NH thermostat that reduces to kineticmoments method for a single harmonic oscillator.

Improving the ergodic characteristics of thermostats using higher order temperatures