Nanoelectromechanical systems are characterized by an intimate connection between electronic and mechanical degrees of\r\nfreedom. Due to the nanoscopic scale, current flowing through the system noticeably impacts upons the vibrational dynamics of the\r\ndevice, complementing the effect of the vibrational modes on the electronic dynamics. We employ the scattering-matrix approach\r\nto quantum transport in order to develop a unified theory of nanoelectromechanical systems out of equilibrium. For a slow mechanical\r\nmode the current can be obtained from the Landauerââ?¬â??BÃ?¼ttiker formula in the strictly adiabatic limit. The leading correction to\r\nthe adiabatic limit reduces to Brouwerââ?¬â?¢s formula for the current of a quantum pump in the absence of a bias voltage. The principal\r\nresults of the present paper are the scattering-matrix expressions for the current-induced forces acting on the mechanical degrees of\r\nfreedom. These forces control the Langevin dynamics of the mechanical modes. Specifically, we derive expressions for the (typically\r\nnonconservative) mean force, for the (possibly negative) damping force, an effective ââ?¬Å?Lorentzââ?¬Â force that exists even for timereversal-\r\ninvariant systems, and the fluctuating Langevin force originating from Nyquist and shot noise of the current flow. We\r\napply our general formalism to several simple models that illustrate the peculiar nature of the current-induced forces. Specifically,\r\nwe find that in out-of-equilibrium situations the current-induced forces can destabilize the mechanical vibrations and cause limitcycle\r\ndynamics.
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