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We study a stochastic perturbation of the Nos\'e-Hoover equation (called theNos\'e-Hoover equation under Brownian heating) and show that the dynamicsconverges at a geometric rate to the augmented Gibbs measure in a weightedtotal variation distance. The joint marginal distribution of the position andmomentum of the particles in turn converges exponentially fast in a similarsense to the canonical Boltzmann-Gibbs distribution. The result applies to ageneral number of particles interacting through wide class of potentialfunctions, including the usual polynomial type as well as the singularLennard-Jones variety.

Exponential relaxation of the Nos\'e-Hoover equation under Brownian heating