Living on the edge of instability

Abstract

Statistical description of stochastic dynamics in highly unstable potentials is strongly affected by properties of divergent trajectories, that quickly leave metastable regions of the potential landscape and never return. Using ideas from theory of Q-processes and quasi-stationary distributions, we analyze position statistics of nondiverging trajectories. We discuss two limit distributions which can be considered as (formal) generalizations of the Gibbs canonical distribution to highly unstable systems. Even though the associated effective potentials differ only slightly, properties of the two distributions are fundamentally different for all highly unstable system. The distribution for trajectories conditioned to diverge in an infinitely distant future is localized and light-tailed. The other distribution, describing trajectories surviving in the meta-stable region at the instant of conditioning, is heavy-tailed. The exponent of the corresponding power-law tail is determined by the leading divergent term of the unstable potential. We discuss different equivalent forms of the two distributions and derive properties of the effective statistical force arising in the ensemble of nondiverging trajectories after the Doob h-transform. The obtained explicit results generically apply to non-linear dynamical models with meta-stable states and fast kinetic transitions.

Infinite ergodic theory meets Boltzmann statistics

“…1f). Our treatment of the weakly-binding potentials below is strongly connected to the work done on periodic potentials in [11,12], and unstable potentials, in [13,14], as well as the work on heterogeneous diffusion in a weakly-binding potential by Wang et al [15]. For additional comments about these works, see the discussion.…”

“…Extensions of our work are certainly worthy, e.g., to higher dimension, interacting systems, and for non-Markovian dynamics (see e.g., [44]). Ryabov et al [13,14] considered a different, though related, setup with an unstable potential that does not allow for the return of the particle to its starting point. Also here the partition function diverges, but again a certain aspect of Boltzmann equilibrium remains.…”

“…e. A periodic potential, with a structure that changes for any x ∈ (−∞, ∞) (also here one finds a non-normalised state, see e.g., [12]). f. An unstable potential (see e.g., [13,14]) . FIG.…”

Uncertainty-induced instantaneous speed and acceleration of a levitated particle

“…Moreover, the initial uncertainty of the levitating particle can be controlled, by postselection 12 , feedback cooling 13 – 16 and ultimately, by coherent scattering to the mechanical ground states 17 – 19 . All these key ingredients encourage broader investigation of the fundamental aspects of statistical mechanics 20 , 21 and accelerate development of applications in mechanical sensing 5 , 22 – 24 and thermodynamical engines 25 , 26 . Recently, the highly unstable motion of a levitating particle in the cubic potential has been analysed 27 , 28 and experimentally verified 29 , 30 in the overdamped regime.…”

Brownian Motion in an External Field Revisited

“…In many interesting physical examples, the partition function is divergent [1] [2] [3] [4]. Thus, the usual toolbox of statistical mechanics becomes unavailable, notwithstanding the well-known fact that the pertinent system may appear to be in a thermal steady state (see, for instance [5] [6] [7] [8] [9]) and references therein]. Our goal here is to deal with a specific divergent partition function, and obtain a finite value for it.…”