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Quantitative Hypocoercivity and Lifting of Classical and Quantum Dynamics

Jianfeng Lu

TL;DR

This work addresses quantitative convergence to equilibrium for degenerate-dissipative open systems, unifying classical Langevin and quantum Lindblad dynamics through a space-time Poincaré hypocoercivity framework. It introduces a lifting paradigm that maps a coercive collapsed semigroup to a higher-dimensional lifted semigroup, enabling explicit upper and lower bounds on accelerated convergence and revealing a fundamental quadratic speed-up ceiling. For Langevin dynamics, explicit rates show quadratic acceleration at optimal friction $\gamma\sim\sqrt{m}$; for Lindblad dynamics, a quantum space-time Poincaré inequality yields analogous exponential convergence with rate depending on the spectral gap $\lambda(\mathcal{L}_s)$. The results provide a principled method to design optimal lifts and connect overdamped limits with accelerated, second-order dynamics, with implications for efficient sampling in classical and quantum open systems.

Abstract

We consider quantitative convergence analysis for hypocoercive dynamics such as Langevin and Lindblad equations describing classical and quantum open systems. Our goal is to provide an overview of recent results of hypocoercivity estimates based on space-time Poincare inequality, providing a unified treatment for classical and quantum dynamics. Furthermore, we also present a unified lifting framework for accelerating both classical and quantum Markov semigroups, which leads to upper and lower bounds of convergence rates.

Quantitative Hypocoercivity and Lifting of Classical and Quantum Dynamics

TL;DR

This work addresses quantitative convergence to equilibrium for degenerate-dissipative open systems, unifying classical Langevin and quantum Lindblad dynamics through a space-time Poincaré hypocoercivity framework. It introduces a lifting paradigm that maps a coercive collapsed semigroup to a higher-dimensional lifted semigroup, enabling explicit upper and lower bounds on accelerated convergence and revealing a fundamental quadratic speed-up ceiling. For Langevin dynamics, explicit rates show quadratic acceleration at optimal friction ; for Lindblad dynamics, a quantum space-time Poincaré inequality yields analogous exponential convergence with rate depending on the spectral gap . The results provide a principled method to design optimal lifts and connect overdamped limits with accelerated, second-order dynamics, with implications for efficient sampling in classical and quantum open systems.

Abstract

We consider quantitative convergence analysis for hypocoercive dynamics such as Langevin and Lindblad equations describing classical and quantum open systems. Our goal is to provide an overview of recent results of hypocoercivity estimates based on space-time Poincare inequality, providing a unified treatment for classical and quantum dynamics. Furthermore, we also present a unified lifting framework for accelerating both classical and quantum Markov semigroups, which leads to upper and lower bounds of convergence rates.
Paper Structure (9 sections, 10 theorems, 75 equations)

This paper contains 9 sections, 10 theorems, 75 equations.

Table of Contents

  1. Introduction.
  2. Langevin and Lindblad dynamics
  3. Hypocoercivity and space-time Poincaré inequality
  4. Lifting
  5. Overdamped limit
  6. Semigroups on Hilbert spaces
  7. Lifting in Hilbert spaces
  8. Convergence rate of lifted dynamics
  9. Discussions

Key Result

Theorem 3.2

Assume $U$ is convex, superlinear as $\lvert x\rvert \to \infty$: $\lim_{x\to \infty} \frac{U(x)}{\lvert x\rvert^{\beta}} \to \infty$ for some $\beta > 1$, and satisfy $\lvert\nabla^2 U\rvert \lesssim 1 + \lvert\nabla U\rvert$. Let $f(t, x, v)$ solve the backward Kolmogorov equation eq:kolmogorov co The convergence rate $\nu$ is given explicitly by where $c$ and $C$ are some universal constants.

Theorems & Definitions (13)

  • Theorem 3.2: Hypocoercivity of Langevin dynamics cao2023explicit
  • Theorem 3.3: Space-time Poincaré inequality cao2023explicit
  • Remark 3.4
  • Theorem 3.5: Quantum space-time Poincaré inequality li2024quantum
  • Theorem 3.6: Hypocoercivity of Lindblad dynamics li2024quantum
  • Lemma 4.2
  • Lemma 4.3
  • Definition 4.4: Lifting
  • Theorem 4.5: Upper bound of convergence rate li2025speeding
  • Lemma 4.7
  • ...and 3 more