![Let H = 1
2pT M−1p + V(q) = A + B be a Hamiltonian with the
equations of motion
d
dt
q = M−1
p,
d
dt
p = − qV(q). (1)
The integration schemes under study belong to the
one-parameter family of symplectic two-stage splitting
integrators of the form
ψ∆t = ϕB
b∆t ◦ ϕA
∆t/2 ◦ ϕB
(1−2b)∆t ◦ ϕA
∆t/2 ◦ ϕB
b∆t ,
with b ∈ (0, 1/4].
If b = 1/4, ψ∆t = ψVV
∆t/2 ◦ ψVV
∆t/2 (two velocity Verlet integrator
half steps).
The energy error over an I-steps trajectory is
∆(q, p, ∆t) = H(Ψ∆t,I(q, p)) − H(q, p), where
Ψ∆t,I = ψ∆t ◦ ψ∆t ◦ · · · ◦ ψ∆t
I times
.](https://image.slidesharecdn.com/slidescecam-160317201609/85/Slides-cecam-1-320.jpg)
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- 1. Let H = 1 2pT M−1p + V(q) = A + B be a Hamiltonian with the equations of motion d dt q = M−1 p, d dt p = − qV(q). (1) The integration schemes under study belong to the one-parameter family of symplectic two-stage splitting integrators of the form ψ∆t = ϕB b∆t ◦ ϕA ∆t/2 ◦ ϕB (1−2b)∆t ◦ ϕA ∆t/2 ◦ ϕB b∆t , with b ∈ (0, 1/4]. If b = 1/4, ψ∆t = ψVV ∆t/2 ◦ ψVV ∆t/2 (two velocity Verlet integrator half steps). The energy error over an I-steps trajectory is ∆(q, p, ∆t) = H(Ψ∆t,I(q, p)) − H(q, p), where Ψ∆t,I = ψ∆t ◦ ψ∆t ◦ · · · ◦ ψ∆t I times .
- 2. Given a step size ∆t and the highest frequency ˜ω in a physical system find b(∆t, ˜ω) ≡ system specific integrator that minimizes the expectation of the energy error ∆ 0 ≤ E(∆) ≤ ρ(h, b) = h4 (2b2 (1/2 − b)h2 + 4b2 − 6b + 1)2 8(2 − bh2)(2 − (1/2 − b)h2)(1 − b(1/2 − b)h2) . The bound is found from analysis of the harmonic oscillator and h is a non-dimensional time-step. Then, b∗ (∆t, ˜ω) = arg min max 0<h<˜h ρ(h, b), where ˜h is a function of ∆t and ˜ω.
- 3. • Toxin-bilayer: coarse-grained non-constrained system AIA provides the highest acceptance rates in the Metropolis test of HMC and the most accurate average temperatures in MD, regardless a choice of the time-step. 10 15 20 25 ∆t / number of stages (fs) 0 10 20 30 40 50 60 70 80 90 100 AR(%) VV BCSS AIA Acceptance Rates (HMC) 10 15 20 25 ∆t / number of stages (fs) 308 309 310 311 312 Temperature(K) VV BCSS AIA Target temperature Temperature (MD) AIA is at least as good as the best sampling integrator, i.e., it has the lowest integrated autocorrelation function (IACF) values for the studied observable, regardless a choice of the time-step, for both MD and HMC. 20 30 40 50 60 70 80 90 ∆t x L (ps) 0 10 20 30 40 50 IACFxtime(h) VV BCSS AIA ∆t / nr stages = 10 fs ∆t / nr stages = 15 fs ∆t / nr stages = 20 fs ∆t / nr stages = 22.5 fs IACF equilibration (HMC) 10 15 20 ∆t / number of stages (fs) 0 10 20 30 40 50 60 70 IACFxtime(h) VV BCSS AIA IACF equilibration (MD) Remark: BCSS is a specific two-stage splitting scheme presented in Blanes et al., (2014).
- 4. • Villin protein: atomistic system with hydrogen bonds constrained AIA provides the highest acceptance rates in the Metropolis test of HMC and the most accurate average temperatures in MD, regardless a choice of the time-step. 1 1.5 2 2.5 ∆t / number of stages (fs) 0 10 20 30 40 50 60 70 80 90 100 AR(%) VV BCSS AIA Acceptance Rates (HMC) 1 1.5 2 2.5 ∆t / number of stages (fs) 298 299 300 301 302 Temperature(K) VV BCSS AIA Target temperature Temperature (MD) AIA needs equal or smaller number of force evaluations to start a folding process than the best sampling integrator in both MD and HMC. 2 3 4 5 6 7 8 ∆t x L (ps) 0 10 20 30 40 numberofforceevaluations(x10 5 ) VV BCSS AIA ∆t / nr stages = 1 fs ∆t / nr stages = 1.5 fs ∆t / nr stages = 2 fs Force evaluations folding (HMC) 1 1.5 2 ∆t / number of stages (fs) 0 10 20 30 40numberofforceevaluations(x10 5 ) VV BCSS AIA Force evaluations folding (MD) Remark: BCSS is a specific two-stage splitting scheme presented in Blanes et al., (2014).