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Can quantum dynamics emerge from classical chaos?

Frédéric Faure

TL;DR

This work investigates whether quantum dynamics can arise from classical chaotic motion, focusing on Anosov geodesic flows. It shows that quantum-like evolution is encoded in the discrete Pollicott--Ruelle spectrum, which organizes into vertical bands; when a rightmost band is isolated by a spectral gap, the long-time evolution is effectively governed by a first-band quantum propagator, while the normal directions contract to a rank-one projector. The analysis relies on microlocal methods, anisotropic Sobolev spaces, and a semiclassical bundle, establishing a band-structure theorem and connecting resonances to periodic orbits via trace formulas and dynamical zeta functions, with explicit links to hyperbolic surfaces and the Laplacian spectrum. Together, these results provide a concrete mechanism by which deterministic chaos can mimic quantum dynamics and illuminate deep connections between classical and quantum descriptions in chaotic systems.

Abstract

Anosov geodesic flows are among the simplest mathematical models of deterministic chaos. In this survey we explain how, quite unexpectedly, quantum dynamics emerges from purely classical correlation functions. The underlying mechanism is the discrete Pollicott Ruelle spectrum of the geodesic flow, revealed through microlocal analysis. This spectrum naturally arranges into vertical bands; when the rightmost band is separated from the rest by a gap, it governs an effective dynamics that mirrors quantum evolution.

Can quantum dynamics emerge from classical chaos?

TL;DR

This work investigates whether quantum dynamics can arise from classical chaotic motion, focusing on Anosov geodesic flows. It shows that quantum-like evolution is encoded in the discrete Pollicott--Ruelle spectrum, which organizes into vertical bands; when a rightmost band is isolated by a spectral gap, the long-time evolution is effectively governed by a first-band quantum propagator, while the normal directions contract to a rank-one projector. The analysis relies on microlocal methods, anisotropic Sobolev spaces, and a semiclassical bundle, establishing a band-structure theorem and connecting resonances to periodic orbits via trace formulas and dynamical zeta functions, with explicit links to hyperbolic surfaces and the Laplacian spectrum. Together, these results provide a concrete mechanism by which deterministic chaos can mimic quantum dynamics and illuminate deep connections between classical and quantum descriptions in chaotic systems.

Abstract

Anosov geodesic flows are among the simplest mathematical models of deterministic chaos. In this survey we explain how, quite unexpectedly, quantum dynamics emerges from purely classical correlation functions. The underlying mechanism is the discrete Pollicott Ruelle spectrum of the geodesic flow, revealed through microlocal analysis. This spectrum naturally arranges into vertical bands; when the rightmost band is separated from the rest by a gap, it governs an effective dynamics that mirrors quantum evolution.

Paper Structure

This paper contains 25 sections, 8 theorems, 60 equations, 10 figures.

Table of Contents

  1. Introduction.
  2. Problem of prediction.
  3. Example in arithmetic.
  4. Example in geometry.
  5. Structure of the paper.
  6. Correlation Functions for Anosov Geodesic Flows.
  7. Anosov geodesic flow.
  8. Correlation functions.
  9. Mixing.
  10. Pollicott-Ruelle resonances and eigenvalues.
  11. Resonances and the Pollicott--Ruelle Discrete Spectrum in Anisotropic Sobolev and Banach Spaces.
  12. Band structure of the Pollicott Ruelle spectrum.
  13. A special “ semiclassical” dynamical bundle.
  14. Pollicott-Ruelle spectrum of $X$ for a hyperbolic surface.
  15. Relation with periodic orbits: trace formula and dynamical zeta functions.
  16. ...and 10 more sections

Key Result

Theorem 2.3

anosov_67katok_hasselblattIf $(\mathcal{N},g)$ has strictly negative curvature, then the geodesic vector field $X$ is Anosov. That is, there exists a (Hölder) continuous splitting of the tangent bundle, invariant under the differential of the flow $d\phi^{t}$: Moreover, for $t\gg1$, the action of $d\phi^{t}$ restricted to the unstable direction $E_{\mathrm{u}}$ is expanding, while its action rest

Figures (10)

  • Figure 1.1: Illustration of the geometric sequence (\ref{['eq:suite_geom']}) and the Collatz sequence (\ref{['eq:suite_syracuse']}), starting from different initial values at $t=0$.
  • Figure 1.2: An adhesive tape laid on a vase $\mathcal{N}$ follows the geodesics of the surface. Similarly, a light beam follows geodesics in a medium with spatially varying velocity $c(q)>0$. In this case, the metric is conformal to the Euclidean metric: $g_{q}^{*}\;=\;c(q)\,g_{\mathrm{Eucl}}^{*}.$
  • Figure 1.3: Geodesic trajectories on the https://en.wikipedia.org/wiki/Bolza_surface, represented as a fundamental domain in the Poincaré disk $\mathbb{D}^{2}$: $\mathcal{N}=\Gamma\backslash\mathrm{SL}_{2}\mathbb{R}/\mathrm{SO}_{2}$, with $\Gamma\subset\mathrm{SL}_{2}\mathbb{R}$ cocompact. Color encodes directions. (a) A single trajectory. (b) An ensemble of nearby trajectories spreads exponentially. (c) After some time, a small cloud of points spreads over the whole phase space $(T^{*}\mathcal{N})_{1}$, illustrating exponential mixing towards equilibrium. (d) Exponentially small fluctuations around equilibrium behave like quantum waves.
  • Figure 2.1: Anosov vector field $X$ on $M=(T^{*}\mathcal{N})_{1}$.
  • Figure 2.2: Action of the pullback operator $e^{tX}$ on a smooth function $u\in C^{\infty}(M)$.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 8 more