However, the state with no vibrations (except maybe vibrations at the drive frequency with a very small amplitude) is also stable. For sufficiently strong driving, the mode can have three stable period-3 vibrational states, which all have the same amplitude and differ in phase by 2 π/3. The classical dynamics of such a mode in the absence of noise has been well understood 38. The physical system we consider is a vibrational mode driven close to triple eigenfrequency. The considered model is minimalistic: the scaled equations of motion without noise contain a single parameter. Rather the emergence of the shallow state is a consequence of the symmetry of the system. The dynamics is not controlled by soft modes. We show that such a state exists even in a simple system which has only two dynamical variables. In this paper we consider escape from a shallow metastable state with no detailed balance. Moreover, because there is only one slow variable, the corresponding slow motion has detailed balance. This is a consequence of the onset of a “soft mode” that controls the motion near the relevant bifurcation points 37. For stable states of forced vibrations such scaling has been found in the classical and quantum regimes 34, 35, 36 and has been observed both for the vibrations at the frequency of the driving resonant field 21, 23, 26, 31, 32 and for parametrically excited vibrations at half the drive frequency 22, 24.Ī major feature that underlies the scaling is that, even though the stable states are vibrational, the problem can be mapped on fluctuations of an overdamped particle in a one-dimensional potential well. The rate of switching from a shallow state displays a characteristic scaling with the distance to the bifurcation point in the parameter space. Typically, a stable state becomes “shallow” when one of the parameters of the system approaches the value where the state disappears (a bifurcation point). This significantly simplifies the experiment. Even a comparatively weak noise can lead to escape from a shallow state with an appreciable rate. A simple example is a state at the bottom of a shallow potential well. These are states with a comparatively low barrier for escape. 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33.Ī most detailed comparison of the theory and the experiment can be done for “shallow” metastable states. The examples range from electrons in a Penning trap to cold atoms to nano- and micromechanical systems to Josephson junction based systems, cf. A class of systems that stand out in this respect are resonantly driven mesoscopic vibrational systems where switching occurs between the states of forced vibrations. The experiments require well characterized nonequilibrium systems that remain stable for a time much longer than the relaxation time. However, many problems remain open on the theory side, and much remains to be learned on the experimental side. Much progress has been made over the last few decades on the theory of switching in systems far from thermal equilibrium, see refs. The corresponding theory has been standardly used to characterize Josephson junctions 4, 5, 6, to study magnetic systems 7, 8, 9, 10, 11, and for other applications.
#Noise mapping and symetry free#
Here the major mechanisms of switching are thermal activation over the free energy barrier or, for low temperatures, tunneling. For classical and quantum systems in thermal equilibrium, switching has been well understood 1, 2, 3. Where fluctuations are weak on average, switching is a rare event, with the rate much smaller than the relaxation rate of the system. Fluctuation-induced switching from a metastable state underlies a broad range of phenomena and has been attracting much interest in diverse areas, from statistical physics to chemical kinetics, biophysics, and population dynamics.