[Paper Review] Exponential relaxation of the Nosé-Hoover equation under Brownian heating

David P. Herzog|arXiv (Cornell University)|Apr 14, 2018

Stochastic processes and financial applications3 references3 citations

TL;DR

This paper establishes exponential convergence to equilibrium for the Nosé-Hoover thermostat under Brownian heating, using a novel Lyapunov function to prove geometric ergodicity in weighted total variation distance. The method ensures fast sampling from the canonical Boltzmann-Gibbs distribution for systems with general potentials, including polynomial and Lennard-Jones types, resolving a key open problem in molecular dynamics simulation.

ABSTRACT

We study a stochastic perturbation of the Nosé-Hoover equation (called the Nosé-Hoover equation under Brownian heating) and show that the dynamics converges at a geometric rate to the augmented Gibbs measure in a weighted total variation distance. The joint marginal distribution of the position and momentum of the particles in turn converges exponentially fast in a similar sense to the canonical Boltzmann-Gibbs distribution. The result applies to a general number of particles interacting through wide class of potential functions, including the usual polynomial type as well as the singular Lennard-Jones variety.

Motivation & Objective

• To resolve the open problem of convergence rate for the Nosé-Hoover thermostat under Brownian heating in molecular dynamics.
• To establish geometric ergodicity of the stochastic dynamics to the augmented Gibbs measure in a weighted total variation distance.
• To demonstrate exponential convergence of the joint position-momentum distribution to the canonical Boltzmann-Gibbs measure for a broad class of potentials.
• To provide a rigorous analytical foundation for the use of the NHB system in high-dimensional sampling, beyond numerical validation.
• To extend convergence results to singular and polynomial potentials, including Lennard-Jones interactions, via a perturbation-based Lyapunov approach.

Proposed method

• Constructs an explicit Lyapunov function to ensure exponential moments of return times to compact sets.
• Applies perturbation methods from [24, 5, 15] to analyze long-time behavior of the SDE system.
• Uses the weighted total variation distance to measure convergence to stationarity, incorporating a weight function that accounts for energy growth.
• Employs the Chapman-Kolmogorov equation and Fubini-Tonelli theorem to extend discrete-time convergence to continuous time.
• Establishes strong Feller and irreducibility properties of the Markov transition kernel to ensure uniqueness and convergence.
• Integrates out the thermostat variable to recover exponential convergence of the marginal position-momentum distribution to the canonical measure.

Experimental results

Research questions

• RQ1 Does the Nosé-Hoover equation under Brownian heating converge to equilibrium at an exponential rate?
• RQ2 Can the convergence rate be rigorously established for general potential functions, including singular ones like Lennard-Jones?
• RQ3 How does the inclusion of the thermostat variable affect the ergodic properties and convergence speed of the system?
• RQ4 Can a Lyapunov function be explicitly constructed to prove geometric ergodicity in the presence of degenerate noise and a control variable?
• RQ5 Is the convergence of the joint position-momentum distribution to the canonical Boltzmann-Gibbs measure exponential in the weighted total variation norm?

Key findings

• The dynamics of the Nosé-Hoover equation under Brownian heating converges exponentially fast to the augmented Gibbs measure in weighted total variation distance.
• The joint marginal distribution of position and momentum converges exponentially fast to the canonical Boltzmann-Gibbs distribution in the same metric.
• The convergence rate is geometric, with an exponential decay rate determined by a positive constant η > 0.
• The result holds for a wide class of normal potentials, including polynomial and Lennard-Jones interactions, as defined in Definition 2.1.
• The proof relies on constructing a Lyapunov function ensuring exponential moments of return times to compact sets.
• Uniqueness of the invariant measure and strong Feller properties are established, enabling application of ergodicity theorems from [11].