What is Ballistic Transport?

TL;DR

This survey formalizes ballistic transport in quantum dynamics by defining the Heisenberg position evolution $Q(t)$ and a family of transport exponents $\beta^\pm_\psi(p)$, situating ballistic behavior within a spectrum that also includes diffusion and localization. It builds a general framework of implications among notions (strong ballistic, norm-growth, exponent-based, Abel-averaged) and connects them to scattering, resolvent methods, and spectral types, with Radin–Simon bounds providing universal upper limits. The review then canvasses broad results: general spectral conditions (e.g., Mourre estimates, pure-point absence), periodic/quasi-periodic/limit-periodic settings, and the interplay with other measures of spreading like escape probabilities and dispersion. It contrasts the rigorous mathematical notions with the physics literature, clarifying how macroscopic conductance and bath-induced dissipation alter the interpretation of ballistic transport, and concludes with open questions on AC spectra implying ballistic transport and the rich behavior in Pastur–Tkachenko regimes.

Abstract

In this article, we review some notions of ballistic transport from the mathematics and physics literature, describe their basic interrelations, and contrast them with other commonly studied notions of wave packet spread.

Figures (1)

• Figure 1: An illustration of which notion implies which notion in full generality, for a fixed moment. Dashed arrows represent implications that require additional assumptions, which are written next to the arrow. BT stands for ballistic transport. Note that the horizontal arrow passes from a condition on a state to one on the level of operators; proofs of this condition proceed by proving strong ballistic transport on a subset of states that forms a common core.

Theorems & Definitions (44)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.6
• Proposition 3.1
• Remark 3.2
• proof
• Remark 3.3