When a macroscopic system is prepared in an unstable equilibrium state, microscopic random forces trigger its decay, and the time at which a representative observable first reaches a macroscopic threshold fluctuates strongly from realization to realization. Haake, Haus, and Glauber (Phys. Rev. A 23, 3255 (1981)) derived a closed-form asymptotic first-passage-time distribution for this process, proved it exact for the linear isotropic n-component case in the weak-noise large-threshold limit, and extended it to saturating, inertial, and spatially extended (Ginzburg-Landau field) systems, including a nonlinear-field reduction via the Kawasaki-Yalabik-Gunton approximation. The stated validity of these extensions, however, rested on qualitative self-consistency conditions rather than controlled error estimates: a single passage through the threshold, weak sensitivity to wave-vector cutoffs (checked numerically in one case only), an uncontrolled nonlinear mapping, and experimental tests that were proposed but not performed outside quantum optics. Four decades of follow-up work resolved much of the few-variable agenda (colored noise, swept control parameters, non-Markovian baths, exact nonlinear decay times validated against simulations) and produced experimental passage-time statistics in liquid crystals, cold atoms, and electromechanical resonators. What remains open is the core validity question for spatially extended systems: quantifying when multiple threshold passages invalidate the normalization of the asymptotic distribution and computing the corrections; giving cutoff-insensitivity criteria beyond the original single numerical example, including conserved dynamics; and bounding the error of the Kawasaki-Yalabik-Gunton nonlinear mapping; the passage-time experiments in phase-separating systems proposed in the source likewise remain unrealized.

Background

A macroscopic system prepared in an unstable equilibrium state - a maximum of an effective potential, such as an inverted atomic population, the zero-field state of a laser above threshold, or the zero-magnetization state of a ferromagnet below its ordering temperature - does not remain there. Ever-present microscopic random forces eventually push it off the unstable point, and the deterministic amplification that follows magnifies whatever random departure has accumulated. Because the time spent in the noise-dominated neighborhood of the unstable point fluctuates strongly from one realization to the next, such decay processes exhibit very large macroscopic fluctuations, observed experimentally as fluctuating delay times of superfluorescent pulses, laser turn-on statistics, and the onset of convection. (Superfluorescence is the cooperative, spontaneously triggered emission of a short intense light pulse from an initially inverted ensemble of atoms.) A convenient way to characterize these fluctuations is the first-passage time: the time at which a chosen observable first reaches a fixed threshold value M set macroscopically far from the unstable point. Exact first-passage-time distributions were classically known only for simple one-dimensional diffusion-type Markov processes (Siegert 1951; Darling and Siegert 1953), and the scaling theory of transient phenomena of Suzuki (1977) described the anomalous growth of fluctuations in unstable systems without closed-form passage-time distributions. Haake (1978) and Haake, Haus, King, Schröder, and Glauber (1981) analyzed the delay-time statistics of superfluorescent pulses in this context. The source paper (Haake, Haus, and Glauber, Phys. Rev. A 23, 3255 (1981)) gave a general asymptotic construction. For an observable S(t) governed by a Langevin equation - a stochastic differential equation S˙=γS+f(t)\dot S=\gamma S+f(t) with linear amplification rate γ>0\gamma>0 and a Gaussian white-noise force f of strength 2/N (N plays the role of an inverse noise strength, e.g. a number of atoms or microscopic degrees of freedom) - the exact solution admits, for times large compared with 1/γ1/\gamma, the asymptotic form Sas(t,s)=eγtsS_{\rm as}(t,s)=e^{\gamma t}s with a random effective initial value s=S(0)+∫0∞e−γt′f(t′)dt′s=S(0)+\int_0^\infty e^{-\gamma t'}f(t')dt'. The random force is thereby reduced to a Gaussian random initial condition for an otherwise deterministic trajectory, and the first-passage-time density through S=MS=M is approximated by averaging a delta function of Sas(t,s)2−M2S_{\rm as}(t,s)^2-M^2 (with a step function selecting upward crossings) over the distribution of s. The same reduction follows from a transformation of the Fokker-Planck equation (the deterministic partial-differential equation for the probability density of S) to a first-order equation for a convolved density Q. For the linear isotropic amplification of an n-component vector the asymptotic density is closed-form,

W(t)=2Γ(n/2)(NM22e−2t)n/2exp⁡(−NM22e−2t),W(t)=\frac{2}{\Gamma(n/2)}\left(\frac{NM^{2}}{2}e^{-2t}\right)^{n/2}\exp\left(-\frac{NM^{2}}{2}e^{-2t}\right),

where Γ\Gamma is the gamma function; its mean is tˉ=12[ln⁡(NM2/2)−ψ(n/2)]\bar t=\frac12[\ln(NM^{2}/2)-\psi(n/2)] with ψ\psi the digamma function, and its relative variance involves the trigamma function ψ′\psi'. The paper shows that a cubic saturating nonlinearity (van der Pol type) only renormalizes the threshold, M2→M2/(1−M2)M^{2}\to M^{2}/(1-M^{2}), that an inertial (mass) term rescales time by the growing eigenrate γ+\gamma_+ and replaces M2M^2 by M2(1+4m)M^2(1+4m), and - its most ambitious extension - that an n-component order-parameter field in d spatial dimensions obeying a time-dependent Ginzburg-Landau equation (the standard relaxational model for an order parameter, nonconserved or, with an extra Laplacian, conserved as in the Cahn-Hilliard treatment of spinodal decomposition, the coarsening of a mixture quenched into the region where its homogeneous state is unstable) yields a passage-time distribution of the same closed form, with the threshold applied to the field norm and the mode content of the field entering only through two cutoff wave-vector integrals. Here the Fourier modes of the field grow at rates γk=(k2)a(1−k2)\gamma_k=(k^{2})^{a}(1-k^{2}) with a=0 (nonconserved) or a=1 (conserved), so modes with ∣k∣>1|k|>1 decay and an ultraviolet cutoff Λ<1\Lambda<1 (plus an infrared cutoff for the conserved case) must be imposed to keep only growing modes. The authors report a remarkable degree of universality: the form of the distribution is independent of the spatial dimension d, of the conservation law, and - numerically, for one case - of the cutoff. For nonlinear field equations they use the approximation of Kawasaki, Yalabik, and Gunton (1978) - a change of variables S=S0(1+αS02)−1/2S=S_0(1+\alpha S_0^2)^{-1/2} that maps the saturating field to an approximately linear one - which again amounts to a replacement of the threshold. The paper also contains, as a byproduct, an exact result: the Laplace transform of the first-passage-time density for the isotropic n-component linear drift-diffusion process (the n-dimensional Ornstein-Uhlenbeck-type process, the natural multivariate generalization of the classical dissipative and diffusive processes) expressed through Kummer's confluent hypergeometric function, from which the asymptotic formula is recovered in the limit NM2→∞NM^{2}\to\infty. The construction, however, is controlled only in part. The authors state its validity as qualitative conditions: the threshold must lie far from the noise scale (1/N≪M21/N\ll M^2, strengthened to NM2≫nNM^{2}\gg n for the vector case) so that trajectories cross it once and never return; for fields, the passage-time distribution must concentrate on times large compared with the inverse growth rates of all retained modes, and its moments must not depend sensitively on the cutoff wave numbers - a self-consistency demonstrated numerically only for the nonconserved case with n=d=2, M2=1M^2=1, N=104N=10^4, where the cutoff could be varied between 0.7 and 0.999 with about 0.5% variation of the mean passage time (while for N=102N=10^2 the cutoff dependence is visible). For systems with many degrees of freedom the asymptotic distribution is not guaranteed to be normalized to unity, and multiple passages through the threshold can occur, so the normalization integral itself becomes an uncontrolled quantity. The Kawasaki-Yalabik-Gunton mapping is justified only by a qualitative smoothing argument, with the acknowledged distortion of short-wavelength Fourier components, an extension from scalar to multi-component fields without derivation, and corrections claimed only to relative order 1/N without bounds. Finally, the authors note that experimental studies of such decay statistics existed only in quantum optics and express the hope that the paper stimulate experiments in other fields, such as hydrodynamic instabilities and spinodal decomposition. These conditions, limitations, and proposed extensions are not resolved within the paper.

Problem Statement

Establish, with controlled error, the range of validity of the asymptotic first-passage-time distribution of Haake, Haus, and Glauber (1981) for the decay of unstable equilibrium states - completing the validity program that the source left at the level of qualitative self-consistency conditions and a single numerical example. The single scientific objective is to replace the paper's qualitative validity conditions by quantitative, checkable statements for the same models. In terms of the validity conditions and limitations stated in the source, the unresolved parts are: (1) Multiple passages and normalization. For systems with many degrees of freedom (fields) the approximating asymptotic distribution is not guaranteed to be normalized to unity, and multiple passages through the threshold surface are possible. Determine for which field models, thresholds, and noise strengths the normalization integral deviates negligibly from unity, and compute the corrections to W(t)W(t) - or the recrossing contributions to the true first-passage-time density - when it does not. (2) Validity ranges and cutoff sensitivity. The field-theoretic result requires that the passage-time distribution concentrate on times large compared with the inverse growth rates of all retained Fourier modes, and that its moments be insensitive to the ultraviolet cutoff (and to the infrared cutoff in the conserved case). Give quantitative criteria for this insensitivity - beyond the one numerical case reported in the source (n=d=2, nonconserved, N=104N=10^4) - including the conserved dynamics and the transition regime of moderate noise strength, and determine case by case how the cutoff dependence enters. (3) Approximation control for nonlinear field dynamics. The Kawasaki-Yalabik-Gunton change of variables reduces the saturating Ginzburg-Landau field to an effective linear one with a mapped threshold. Bound the overall error of this approximation, including the neglected gradient-squared term, the distortion of short-wavelength modes, the component-wise extension to n>1, and the corrections beyond relative order 1/N, or establish by numerical or refined analytic study when it fails. The linear n-component case is already exactly solved within the source (the Kummer-function Laplace transform) and serves as the benchmark for the controlled theory; the late-stage relaxation near the stable equilibria is a declared scope exclusion of the source, not part of this problem. The requested results are linked by one objective - a controlled asymptotic first-passage-time theory of unstable-state decay; all three parts (1)-(3) are required for a resolution, and a controlled result for a single part is a partial contribution, not a resolution.

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