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Register a new userStochastic Regimes in the Driven Oscillator with a Step-Like Nonlinearity
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A nonlinear oscillator with an abruptly inhomogeneous restoring force driven by an uniform oscillating force exhibits stochastic properties under specific resonance conditions. This behaviour elucidates the elementary mechanism of the electron energization in the strong electromagnetic wave interaction with thin targets.
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This paper develops further the semi-classical theory of an harmonic oscillator acted on by a Gaussian white noise force discussed in (arXiv:1508.02379). Here I add to that theory the effects of Brownian damping (friction). Albeit semi-classical, the theory can be used to model quantum expectations and probabilities. I consider several examples.
We study a simple run-and-tumble random walk whose switching frequency from run mode to tumble mode and the reverse depend on a stochastic signal. We consider a particularly sharp, step-like dependence, where the run to tumble switching probability jumps from zero to one as the signal crosses a particular value (say y_1 ) from below. Similarly, tumble to run switching probability also shows a jump like this as the signal crosses another value (y_2 < y_1 ) from above. We are interested in characterizing the effect of signaling noise on the long time behavior of the random walker. We consider two different time-evolutions of the stochastic signal. In one case, the signal dynamics is an independent stochastic process and does not depend on the run-and-tumble motion. In this case we can analytically calculate the mean value and the complete distribution function of the run duration and tumble duration. In the second case, we assume that the signal dynamics is influenced by the spatial location of the random walker. For this system, we numerically measure the steady state position distribution of the random walker. We discuss some similarities and differences between our system and E.coli chemotaxis, which is another well-known run-and-tumble motion encountered in nature.
Using the Wigner-Weyl mapping of quantum mechanics to phase space we consider exactly the quantum mechanics of an harmonic oscillator driven by an external white noise force or whose frequency is time dependent, either adiabatically or parametrically. We find several transition probabilities exactly. We also consider the (quantum mechanical) randomizing effects of the external white noise force on the Weyl quantized phase angle and upon other Weyl quantized quantities.
Stochastic resonance (SR) is a coherence enhancement effect due to noise that occurs in periodically-driven nonlinear dynamical systems. A very broad range of physical and biological systems present this effect such as climate change, neurons, neural networks, lasers, SQUIDS, and tunnel diodes, among many others. Early theoretical models of SR dealt only with overdamped bistable oscillators. Here, we propose a simple model that accounts for SR in an underdamped driven Duffing oscillator with added white noise. Furthermore, we develop a theoretical method to predict the effect of white noise on the pump, signal, and idler responses of a Duffing amplifier. We also calculate the power spectral density of the response of the Duffing amplifier. This approach may prove to be useful for assessing the robustness of acoustic, phononic, or mechanical frequency-comb generation to the presence of noise.
The nonlinear dynamics of energetic-particle (EP) driven geodesic acoustic modes (EGAM) is investigated here. A numerical analysis with the global gyrokinetic particle-in-cell code ORB5 is performed, and the results are interpreted with the analytical theory, in close comparison with the theory of the beam-plasma instability. Only axisymmetric modes are considered, with a nonlinear dynamics determined by wave-particle interaction. Quadratic scalings of the saturated electric field with respect to the linear growth rate are found for the case of interest. The EP bounce frequency is calculated as a function of the EGAM frequency, and shown not to depend on the value of the bulk temperature. Near the saturation, we observe a transition from adiabatic to non-adiabatic dynamics, i.e., the frequency chirping rate becomes comparable to the resonant EP bounce frequency. The numerical analysis is performed here with electrostatic simulations with circular flux surfaces, and kinetic effects of the electrons are neglected.
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