Abstract
This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein–Uhlenbeck systems (X^varepsilon _t(x))_{tgeqslant 0} with varepsilon -small additive Lévy noise and initial value x. The driving noise processes include Brownian motion, alpha -stable Lévy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp infty /0-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure mu ^varepsilon along a time window centered on a precise varepsilon -dependent time scale mathfrak {t}_varepsilon . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data x we obtain the stronger result mathcal {W}_p(X^varepsilon _{t_varepsilon + r}(x), mu ^varepsilon ) cdot varepsilon ^{-1} rightarrow Kcdot e^{-q r} for any rin mathbb {R} as varepsilon rightarrow 0 for some spectral constants K, q>0 and any pgeqslant 1 whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of mathcal {Q}. Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to varepsilon -small Brownian motion or alpha -stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.
Highlights
The notion of cutoff thermalization has gained growing attention in recent years in the physics literature with applications to quantum Markov chains [72], chemical kinetics [8], quantum information processing [73], the Ising model [79], coagulation-fragmentation equations [83,84], dissipative quantum circuits [67] and open quadratic fermionic systems [104]
The term “cutoff” was originally coined in 1986 by Aldous and Diaconis in their celebrated paper [4] on card shuffling, where they observed and conceptualized the asymptotically abrupt collapse of the total variation distance between the current state of their Markov chain and the uniform limiting distribution at a precise deterministic time scale. At this point we refrain from giving a full account on the mathematical literature on the cutoff phenomenon and refer to the overview article [41] and the introduction of [16]
Due to the mentioned regularity issue concerning the total variation distance the authors state their results under the hypothesis of continuous densities of the marginals, which to date is mathematically not characterized in simple terms
Summary
The notion of cutoff thermalization ( known as the cutoff phenomenon or abrupt thermalization in the literature) has gained growing attention in recent years in the physics literature with applications to quantum Markov chains [72], chemical kinetics [8], quantum information processing [73], the Ising model [79], coagulation-fragmentation equations [83,84], dissipative quantum circuits [67] and open quadratic fermionic systems [104]. The term “cutoff” was originally coined in 1986 by Aldous and Diaconis in their celebrated paper [4] on card shuffling, where they observed and conceptualized the asymptotically abrupt collapse of the total variation distance between the current state of their Markov chain and the uniform limiting distribution at a precise deterministic time scale At this point we refrain from giving a full account on the mathematical literature on the cutoff phenomenon and refer to the overview article [41] and the introduction of [16].
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