Coupled Nosé-Hoover equations of motion to implement a fluctuating heat-bath temperature

The Nosé-Hoover (NH) equation provides a universal and powerful computer simulation protocol to realize an equilibrium canonical temperature for a target physical system. Here we demonstrate a general formalism to couple such NH equations. We provide a coupled NH equation that is constructed by coupling the NH equation of a target physical system and the NH equation of a temperature system. Thus, in contrast to the conventional single NH equation, the heat-bath temperature is a dynamical variable. The temperature fluctuations are not ad hoc, but instead are generated by the newly defined temperature system, and the statistical distribution of the temperature is completely described with an arbitrarily given probability function. The current equations of motion thus describe the physical system that develops with a predistributed fluctuating temperature, which allows enhanced sampling of the physical system. Since the total system is governed by a prescribed distribution, the equilibrium of the physical system is also reconstructed by reweighting. We have formulated a scheme for specifically setting the distribution of the dynamical inverse temperature and demonstrate the statistical relationship between the dynamical and physical temperatures. The statistical features, dynamical properties, and sampling abilities of the current method are demonstrated via the distributions, trajectories, dynamical correlations, and free energy landscapes for both a model system and a biomolecular system. These results indicated that the current coupled NH scheme works well.