Coupled Nosé–Hoover lattice: A set of the Nosé–Hoover equations with different temperatures

•A simple scheme is presented to couple arbitrary number of the Nosé–Hoover equations with different heat-bath temperatures.•The resulting ordinary differential equation has a smooth invariant measure.•The scheme is numerically validated via a coupled system of two 1-dimensional Nosé–Hoover harmonic oscillators.•The current scheme realizes the coupled differential equations, analogous to the coupled maps.

A simple scheme was presented to couple any number of the Nosé–Hoover equations with different heat-bath temperatures. In general, several practical procedures can be considered to realize such a coupling, where the system is under nonequilibrium. However, the current scheme provides an equilibrium distribution, namely, a smooth invariant measure for the present system. This is attained by a very simple idea, that is, a force scaling. The current scheme realizes coupled differential equations, analogous to coupled maps. Its theoretical possibilities, mathematical framework, and practical utilities are discussed. Numerical validations applying the method to a simple two-oscillator system are provided.