Anomalous velocity distributions in slow quantum-tunneling chemical reactions

TL;DR

This paper develops a generalized statistical mechanics theory to explain the non-Maxwellian velocity distributions observed in slow quantum-tunneling chemical reactions involving ions at high densities, aligning well with recent experimental data.

Contribution

It introduces a superstatistical approach deriving a $q$-Maxwellian velocity distribution with a density-dependent entropic index, explaining experimental deviations from classical statistics.

Findings

Velocity distribution is non-Maxwellian, following a $q$-Gaussian form.

The entropic index $q$ depends on the buffer gas density $n$, with $q>1$.

The theory predicts $q(n)$ behavior consistent with experimental data.

Abstract

Recent work [Wild et al., Nature 615, 425 (2023)] has provided an experimental break-through in the realization of a quantum-tunneling reaction involving a proton transfer. The reaction $D^-+H_2 \to H^-+HD$ has an extremely slow reaction rate as it can happen only via quantum tunneling, thus requiring an extremely large density of the reactants in the ion trap. At these high densities strong deviations from Maxwell-Boltzmann statistics are observed. Here we develop a consistent generalized statistical mechanics theory for the above nonequilibrium situation involving quantum effects at high densities. The trapped ions are treated in a superstatistical way and a $q$-Maxwellian velocity distribution with a universal dependence of the entropic index $q$ on the density $n$ of the buffer gas is derived. We show that the velocity distribution of the ions is non-Maxwellian, more precisely $q$-Gaussian, i.e., $p(v) \propto v^2 [1+(q-1)\tilde{\beta} v^2]^{1/(1-q)}$, with entropic index $q>1$ depending on the density $n$ of $H_2$ molecules, in excellent agreement with the experimental observations of Wild et al. Our theory also makes predictions on the statistics of temperature fluctuations in the ion trap which can be tested in future experiments. Through the superstatistical approach, we obtain an analytical expression for $q(n)$ which is consistent with the available experimental data, and which yields $\lim_{n\to 0}q(n)=1$, i.e. recovering the Maxwell-Boltzmann distribution in the ideal gas limit, as well as $\lim_{n\to\infty}q(n)=7/5$.