Improving the ergodic characteristics of thermostats using higher order   temperatures
  Most deterministic schemes for controlling temperature by kinetic variables (NH thermsotat), configurational variables (BT thermostat) and all phase space variables (PB thermostat) are non-ergodic for systems with a few degrees of freedom. While for the NH thermostat ergodicity has been achieved by controlling the higher order moments of kinetic energy, the issues of nonergodicity of BT and PB thermostats still persist. In this paper, we propose a family of modifications for improving their ergodic characteristics. To do so, we introduce two new measures of configurational temperature (second and third order) based on the generalized temperature - curvature relationship. The equations of motion for the existing thermostats are modified by controlling the relevant higher order temperature variables. The ergodic characteristics of the proposed modifications are tested using a single harmonic oscillator. For the PB thermostat, the fastest route to ergodicity is by controlling the usual configurational and kinetic temperatures along with the $2^{\text{nd}}$ order configurational temperature. For the BT thermostat, controlling the usual configurational temperature along with the $2^{\text{nd}}$ order configurational temperature is sufficient to make it ergodic. Our method also provides a new ergodic generalization of NH thermostat that reduces to kinetic moments method for a single harmonic oscillator. 
Puneet Kumar Patra and Baidurya Bhattacharya
cond-mat.stat-mech