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Harmonic BAOAB thermostat: exact position variance and kinetic bias

Example

Take a one-dimensional harmonic oscillator with mass $m = 2$ , stiffness $k = 8$ , inverse temperature $β = 1$ , friction $γ = 1$ , and BAOAB step $h = 1 / 2$ . The stability threshold is $h < 2 \sqrt{m / k} = 1$ . The example concerns the exact stationary covariance of the discrete BAOAB chain, not Monte Carlo error from a finite trajectory.

Facts & Assumptions

Given: $m = 2$ , $k = 8$ , $β = 1$ , $γ = 1$ , $h = 1 / 2$ , and the harmonic BAOAB theorem.

[F1]

Exact invariant covariance and stability threshold for scalar harmonic BAOAB: when $0 < h < 2 \sqrt{m / k}$ , $Var (Q) = 1 / (β k)$ and $Var (P) = (m / β) (1 - h^{2} k / (4 m))$ .

[A1]

The Axiom of Choice: AC is inherited through [F1]; the parameter substitution and arithmetic use no additional choice.

Verification

Given: $m = 2$ , $k = 8$ , $β = 1$ , $γ = 1$ , $h = 1 / 2$ , and the harmonic BAOAB theorem.

[F1] Exact invariant covariance and stability threshold for scalar harmonic BAOAB: when $0 < h < 2 \sqrt{m / k}$ , $Var (Q) = 1 / (β k)$ and $Var (P) = (m / β) (1 - h^{2} k / (4 m))$ .

[A1] The Axiom of Choice: AC is inherited through [F1]; the parameter substitution and arithmetic use no additional choice.

Given: The parameter values in the Example.

1.1F1givenalgebra

Here $2 \sqrt{m / k} = 2 \sqrt{2 / 8} = 1$ , so $h = 1 / 2$ is in the strict stable range of [F1]. The dimensionless quantity is $r = h^{2} k / (4 m) = (1 / 4) 8 / 8 = 1 / 4$ .

2.1A1F1step 1.1algebra∎

The discrete stationary position variance is $1 / (β k) = 1 / 8$ , exactly the canonical Gibbs position variance. The momentum variance is $(2 / 1) (1 - 1 / 4) = 3 / 2$ , whereas the continuous canonical value is $m / β = 2$ , so the momentum bias is $- 1 / 2 = - h^{2} k / (4 β)$ .

Depends on

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Sources

Benedict Leimkuhler and Charles Matthews, Rational Construction of Stochastic Numerical Methods for Molecular Sampling, §2 and Appendix (standard reference, not scraped)
Jean-Noël Roux, Stéphane Rodts, and Gabriel Stoltz, Introduction à la physique statistique et à la physique quantique, §5.2.2 (standard reference, not scraped)