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We investigate a novel type of Langevin model that describes thenonequilibrium dynamics of a classical particle interacting with a spatiallyextended environment. In this model, a particle, which interacts with theenvironment through the nonlinear interaction Hamiltonian, is driven by aconstant external force, and subsequently, it reaches a nontrivialnonequilibrium steady state. We derive an effective Langevin equation for theparticle in the nonequilibrium steady states. Using this equation, we calculatethe effective temperature defined as the ratio of the correlation function ofthe velocity fluctuation to the linear response function with respect to asmall perturbation. As a result, it is shown that the effective temperatureassociated with the time scale of the particle is identical to the kinetictemperature if the time scale of the environment and that of the particle arewell separated. Furthermore, a noteworthy expression, which relates the kinetictemperature with the curvature of the driving force-mean velocity curve, isderived.

Nonequilibrium Langevin equation and effective temperature for particle interacting with spatially extended environment