Fractal basins and chaotic bifurcations prior to escape from a potential well

The escape of a periodically driven damped oscillator from a potential well is intimately associated with homoclinic tangles, fractal basins, and a variety of chaotic bifurcations. A bifurcation diagram, including an analytical Melnikov solution is presented for a canonical asymmetric cubic potential, and comparisons are made with recent results for the Holmes two-well oscillator.     

Time-dependent escape rate from a potential well

Time-dependent desorption from an interface is studied by obtaining the Green function for the Smoluchowski equation for a one-dimensional model potential barrier and calculating the time-dependence of the number density, particle current, and escape rate. The asymptotic behavior of the system for long times can be described by equations with independent rate constants. For high potential barriers (relative to k_BT) the Kramers expression for the escape rate is recovered, but for low barriers the escape rate can go through a maximum. The steady state Onsager model is related to the transient solution and numerical results are presented for different potential shapes and sizes.     

Noise-activated escape from a sloshing potential well

We treat the noise-activated escape from a one-dimensional potential well of an overdamped particle, to which a periodic force of fixed frequency is applied. Near the well top, the relevant length scales and the boundary layer structure are determined. We show how behavior near the well top generalizes the behavior determined by Kramers, in the case without forcing. Our analysis includes the case when the forcing does not die away in the weak-noise limit. We discuss the relevance of scaling regimes, defined by the relative strengths of the forcing and the noise, to recent optical trap experiments.     

Amplitude modulation control of escape from a potential well

We demonstrate the effectiveness of periodic amplitude modulations in controlling (suppressing and enhancing) escape from a potential well through the universal model of a damped Helmholtz oscillator subjected to an external periodic excitation (the escape-inducing excitation) whose amplitude is periodically modulated (the escape-controlling excitation). Analytical and numerical results show that this multiplicative control works reliably for different subharmonic resonances between the two periodic excitations involved, and that its effectiveness is comparable to those of different methods of additive control. Additionally, we demonstrate the robustness of the multiplicative control against the presence of low-intensity Gaussian noise.     

Diffusion and convection after escape from a potential well

The escape by diffusion of a particle from a potential well in one dimension is strongly influenced by the application of a field in the adjacent half-space. At long times the probability distribution becomes a uniformly moving and steadily broadening gaussian in this half-space. The mean time of escape from the well is given by a simple expression in terms of the mean first passage time and the coefficient of the long-time tail in the occupation probability of the well in the absence of the field. Transient effects in space and time are studied in explicit form for a parabolic potential well.     

Decrease in entropy as a result of measuring the time of escape from a potential well

The decrease in entropy when passing from an equilibrium thermodynamic system to a slightly nonequilibrium system is investigated. A quasi-equilibrium Boltzmann distribution is used to prove the conservation of free energy during this passage. Results are obtained for a Brownian particle in a potential well with a low escape probability. The escape is interpreted as a measurement. It is shown that because of the measurement itself, the distribution function is narrower than that for a system undisturbed by measurement, i.e., an equilibrium system. In this case, the entropy difference between the equilibrium and measurement-disturbed systems is equal to the amount of information entered into the system.     

Incursive fractals: a robust mechanism of basin erosion preceding the optimal escape from a potential well

Comparative cell-to-cell mappings of the basins of driven oscillators with cubic and quartic potential wells show remarkable qualitative and quantitative correlations. We conclude that the recently identified erosion by incursive fractals is a robust phenomenon facilitating the optimal escape from a well.     

Rate of escape from nonattracting chaotic sets

Chaotic transient phenomena occur in the vicinity of nonattracting chaotic sets. The rate of escape measures the average length of the transients. There is a conjecture by Eckmann and Ruelle connecting the rate of escape to the Lyapunov exponents and entropy. We prove an inequality that partially supports the conjecture.     

Calculation of quantum escape rates from a metastable well

The quantum-mechanical decay of a metastable state of a system coupled to a heat bath environment is studied. A functional integral method is presented allowing for the calculation of decay rates at finite temperatures and in the presence of dissipation. Analytical methods for high and intermediate temperatures are combined with an accurate numerical method for low temperatures where the system decays by quantum tunneling. Explicit results are given for a system with a cubic potential and frequency-independent damping.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

Coherence resonance in an excitable potential well

The excitable behaviour is considered as motion of a particle in a potential field in the presence of dissipation. The dynamics of the oscillator proposed in the present paper corresponds to the excitable behaviour in a potential well under condition of positive dissipation. Type-II excitability of the offered system results from intrinsic peculiarities of the potential well, whose shape depends on a system state. Concept of an excitable potential well is introduced. The effect of coherence resonance and self-oscillation excitation in a state-dependent potential well under condition of positive dissipation are explored in numerical experiments.     

Efficient procedure to compute the microcanonical volume of initial conditions that lead to escape trajectories from a multidimensional potential well

A procedure is presented for computing the phase space volume of initial conditions for trajectories that escape or "react" from a multidimensional potential well. The procedure combines a phase space transition state theory, which allows one to construct dividing surfaces that are free of local recrossing and that minimize the directional flux, and a classical spectral theorem. The procedure gives the volume of reactive initial conditions in terms of a sum over each entrance channel of the well of the product of the phase space flux across the dividing surface associated with the channel and the mean residence time in the well of trajectories which enter through the channel. This approach is illustrated for HCN isomerization in three dimensions, for which the method is several orders of magnitude more efficient than standard Monte Carlo sampling.     

Reflection from a composite potential well

Hitherto, the variation of reflectivity as a function of energy for a 1-D potential well has been observed to be either monotonically decreasing or entailing multiple minima. We report that a composite (two-piece) potential well may exhibit an unusual feature, viz., a pronounced single minimum in the reflectivity.     

Hopping of chaotic dynamics in a doubly resonant parametric oscillator

We study a doubly resonant optical parametric oscillator where the pump can feed two pairs of signal-idler modes. We assume the presence of gain at the pump frequency. We investigate the various oscillation states of interest, namely, when only the first pair oscillates with the other pair having null amplitudes and vice versa. We demonstrate the exchange of dynamics between the mode pairs when the relevant parameters of the cavity, namely, the phase mismatch factors or the decay rates switch because of fluctuations. The exchange of dynamics is shown to be independent of the nature of dynamics, i.e. independent of whether the motion isn-periodic or chaotic. We also investigate the case where both the pairs can exhibit chaotic dynamics though these states are difficult to realize because of fluctuations.     

Studies on tunneling escape time of electrons from a quantum well in multilayered heterostructures

By considering the time variation of band-edge profile arising from the decay of injected charge in quantum wells(QWs), we employ a wave packet method to verify that the actual escape time of certain amount of electrons from QWs could be much larger than that for a single electron. The theoretical result is also in agreement with our measurement of escape time, performed by using a newly developed method--transient current response.     

Carrier escape dynamics in a single quantum well waveguide modulator

Picosecond excite-probe studies are performed on a single quantum well waveguide modulator giving a direct measure of the escape of photogenerated carriers from a quantum well. Both the effects of exciton saturation and external field screening are observed in the transient transmission change. The results are consistent with the escape of carriers by thermionic emission.     

On the classical model for microwave induced escape from a Josephson washboard potential

We revisit the interpretation of earlier low temperature experiments on Josephson junctions under the influence of applied microwaves. It was claimed that these experiments unambiguously established a quantum phenomenology with discrete levels in shallow wells of the washboard potential, and macroscopic quantum tunneling. We here apply the previously developed classical theory to a direct comparison with the original experimental observations, and we show that the experimental data can be accurately represented classically. Thus, our analysis questions the necessity of the earlier quantum mechanical interpretation.     

Testing the energy diffusion approximation for the escape of a Brownian particle from a potential pocket

For the first time, the energy diffusion approximation is confronted at the percent level with the exact numerical modeling of thermal decay of a metastable state. This model is useful in many branches of natural sciences: e.g. in biology, nuclear physics, chemistry, etc. The exact (within the statistical errors about 2%) quasistationary decay rates result from the Langevin equations for the coordinate and conjugated momentum. For the energy (or action) diffusion approach, a Langevin-type equation for the action is constructed, validated, and solved numerically. The comparison of these two approaches is performed for four potentials (two of which are anharmonic) in a wide range of two dimensionless scaling parameters: i) the governing parameter G reflecting how high is the barrier with respect to the temperature and ii) the damping parameter φ expressing the friction strength. It turns out that the energy diffusion approach produces the rate which comes into 50%-agreement with the exact one only at φ < 0.02. Thus, we quantify, for the first time by our knowledge, the condition φ ≪ 1 known in the literature.     

Rate of escape of some chaotic Julia sets

We give a formula for the rates of escape for Julia sets with preperiodic critical points and forC endomorphisms of the interval with non-flat pre-periodic critical points outside the basin of attracting periodic points.Research supported by CNPq, Brasil