Super-exponential Decay Research Articles

Exponential decay in a spin bath

We show that the coherence of an electron spin interacting with a bath of nuclear spins can exhibit a well-defined purely exponential decay for special (``narrowed'') bath initial conditions in the presence of a strong applied magnetic field. This is in contrast to the typical case, where spin-bath dynamics have been investigated in the non-Markovian limit, giving superexponential or power-law decay of correlation functions. We calculate the relevant decoherence time ${T}_{2}$ explicitly for free-induction decay and find a simple expression with dependence on bath polarization, magnetic field, shape of the electron wave function, dimensionality, total nuclear spin $I$, and isotopic concentration for experimentally relevant heteronuclear-spin systems.

Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

We show that all eigenfunctions of linear partial differential operators in R n with polynomial coefficients of Shubin type are extended to entire functions in C n of finite exponential type 2 and decay like exp ( − | z | 2 ) for | z | → ∞ in conic neighbourhoods of the form | Im z | ⩽ γ | Re z | . We also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super-exponential decay of the above type, whereas a loss of the holomorphicity occurs, namely we show holomorphic extension into a strip { z ∈ C n | | Im z | ⩽ T } for some T > 0 . The proofs are based on geometrical and perturbative methods in Gelfand–Shilov spaces. The results apply in particular to semilinear Schrödinger equations of the form (∗) − Δ u + | x | 2 u − λ u = F ( x , u , ∇ u ) . Our estimates on homoclinics are sharp. In fact, we exhibit examples of solutions of (∗) with super-exponential decay, which are meromorphic functions, the key point of our argument being the celebrated great Picard theorem in complex analysis.

Force distributions in a triangular lattice of rigid bars

We study the uniformly weighted ensemble of force balanced configurations on a triangular network of nontensile contact forces. For periodic boundary conditions corresponding to isotropic compressive stress, we find that the probability distribution for single-contact forces decays faster than exponentially. This superexponential decay persists in lattices diluted to the rigidity percolation threshold. On the other hand, for anisotropic imposed stresses, a broader tail emerges in the force distribution, becoming a pure exponential in the limit of infinite lattice size and infinitely strong anisotropy.

On pathwise super-exponential decay rates of solutions of scalar nonlinear stochastic differential equations

This paper studies the pathwise asymptotic stability of the zero solution of scalar stochastic differential equation of Itô type. In particular, we provide conditions for solutions to converge to zero at a given rate, which is faster than any exponential rate of decay. The results completely classify the rates of decay of many parameterised families of stochastic differential equations.

Faster than Lyapunov decays of the classical Loschmidt echo.

We show that in the classical interaction picture the echo dynamics, namely, the composition of perturbed forward and unperturbed backward Hamiltonian evolution, can be treated as a time-dependent Hamiltonian system. For strongly chaotic (Anosov) systems we derive a cascade of exponential decays for the classical Loschmidt echo, starting with the leading Lyapunov exponent, followed by a sum of the two largest exponents, etc. In the loxodromic case a decay starts with the rate given as twice the largest Lyapunov exponent. For a class of perturbations of symplectic maps the echo dynamics exhibits a drift resulting in a superexponential decay of the Loschmidt echo.

Analytic continuation and resonance-free regions for Sturm–Liouville potentials with power decay

The analytic continuation of the Jost solution into a sector of the unphysical sheet is established for analytic potentials with power decay. The method also yields resonance-free regions which, in the case of super-exponential decay, are confirmed to be the best possible of their kind by the known asymptotic distribution of resonances.

Pulses in the zero-spacing limit of the GOY model

We study the propagation of localised disturbances in a turbulent, but momentarily quiescent and unforced shell model (an approximation of the Navier–Stokes equations on a set of exponentially spaced momentum shells). These disturbances represent bursts of turbulence travelling down the inertial range, which is thought to be responsible for the intermittency observed in turbulence. Starting from the GOY shell model, we go to the limit where the distance between succeeding shells approaches zero (“the zero spacing limit” ) and helicity conservation is retained. We obtain a discrete field theory which is numerically shown to have pulse solutions travelling with constant speed and with unchanged form. We give numerical evidence that the model might even be exactly integrable, although the continuum limit seems to be singular and the pulses show an unusual super exponential decay to zero as exp(− constant σ n) when n→∞, where σ is the golden mean. For finite momentum shell spacing, we argue that the pulses should accelerate, moving to infinity in a finite time. Finally, we show that the maximal Lyapunov exponent of the GOY model approaches zero in this limit.

Analysis of the bromate-ferroin clock reaction

The bromate-ferroin clock reaction is studied experimentally and the dependence of the clock or induction timetcl on the initial concentration of various reactants determined. Particular attention is paid to the dependence oftclcl on the initial bromide ion concentration [Br−]0. An analytical theory is also derived based on a subset of the Field-Koros-Noyes mechanism. This analysis reveals several features, including exponential decay of [Br−] during the induction period followed by a super-exponential decay in the actual clock event, a linear relationship betweentcl, and ln[Br−] over a wide range of [Br−]0, but departures from this at higher (and lower) concentrations. These features are all confirmed essentially quantitatively by the experimental results. The theory also predicts, and the experiments confirm, that there is a critical bromide ion concentration marking the end of the induction period. This study then provides a firm basis from which to interpret and predict the behaviour of this system in a wider range of experimental situations (such as the reaction-diffusion waves in unstirred media).

Pair correlation function in random sequential adsorption processes

An analytical expression of the pair distribution function is derived for the car-parking problem and for the random sequential adsorption of K-mers onto a one-dimensional lattice. Both on a lattice and in the continuum limit, super-exponential decay of the distribution function is observed. A comparison of the spatial correlations with those at equilibrium demonstrates the influence of the irreversibility of the RSA process.

Superexponential decay of solutions of a semilinear wave equation

Consider a smooth solution of u tt − Δu + q(x) ¦ u ¦ p−1 u = 0 x ϵ R 3 , q ⩾ 0 and is C 1 , and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, q(x) ⩽ a exp (−b ¦ x ¦ c ), a, b > 0, c > 1 , and for simplicity, q r ⩽ 0. Then the local energy decays faster than exponentially.