Langevin Dynamics

4.3. Langevin Dynamics#

4.3.1. Implicit-Solvent Models#

Sometimes, it’s too expensive to simulate all the solvent around a large solute like a nanoparticle or protein.

  • Treat effects “implicitly”

4.3.2. Langevin Dynamics#

  • Thermodynamics → (effective potential)

  • Dynamics → drag forces, thermal “kicks”

Momentum equation: P = mv = { -rx + ow } T = 2βkT by fluctuation-dissipation theorem

  • Conservative drag force

  • Force: “free drawing” — Conua around a sphere

  • Properly, W is a Wiener process and is not well defined

    • dw = w w(o) = at G

    • Gaussian with zero mean, unit variance, independent for each particle

    • “White noise”: uncorrelated in time (vs. colored noise)

Langevin Equation: mdv = (f - xv) dt + σ dw

To integrate, some people just add drag and random forces to normal Verlet integration. However, more sophisticated schemes (BAOAB, G-JF) are recommended to retain symplectic properties.

Example: BAOAB Scheme

  • Δx(1/2) = p(0) + (Δt/2m) f(0)

  • p(Δt/2) = p(0) + (Δt/2) f

  • p(Δt/2) = exp(-γΔt/2) p(Δt/2) + √(1 - exp(-2γΔt)) √(mkT) G

  • x(Δt) = x(Δt/2) + (Δt/2m) p’(Δt/2)

  • p(Δt) = p’(Δt/2) + (Δt/2) f(Δt)

4.3.3. Brownian Dynamics#

In the “overdamped” limit, friction dominates inertia so p ≈ 0 and:

Equation: dx = f dt + σ dw

It can be shown that the update rule is: x(Δt) = x(0) + (f(0)/γ) Δt + √(2kT/γ) Δt G

By the Einstein relation, we can define a diffusion coefficient: D = kT / γ

If f = 0, ⟨Δx²⟩ = 2DΔt ⟨G_i G_j⟩ = 2DΔt δ_ij

4.3.4. Hydrodynamic Interactions in BD#

Previous equations used free drainage. In real systems, hydrodynamics give rise to drag forces induced by flows from other particles.

From Ermak and McCammon, J. Chem. Phys., 69, 1352 (1978):

Equation: x(Δt) = x(0) + (M·f + kT ∇·M) Δt + B

  • M: mobility tensor, D = kBT M (3N × 3N)

  • B: random displacements

  • ⟨B Bᵀ⟩ = 2kT M Δt

M captures hydrodynamics, e.g., pairwise approximation: M = [M₁₁ M₁₂ … M₁N] [M₂₁ M₂₂ … M₂N] […] [MN₁ MN₂ … MNN]

Free drawing:

  • M_ij = 0

  • Rotne-Prager: M_ij ≠ 0

Both these tensors are divergence-free (i.e., ∇·M = 0), so this term can be ignored. Not true if there are walls, etc.

  • Drawing B is hard! Need to draw with appropriate covariance matrix, which involves a square root. This places some numerical restrictions.

  • Tensors can be long-ranged → need Ewald sums

    • Positively Split Ewald technique [Fiore et al., J. Chem. Phys., 146, 12416 (2019)]

4.3.5. Langevin Thermostat#

By making γ small, can use Langevin Dynamics as a thermostat without significantly perturbing the system.

  • A value γ ≈ 0.1 m/Å is usually considered “weak” coupling and good for thermostating.

  • Beware: Langevin thermostat does not conserve momentum (forces do not sum to zero). Hence, there are no zero degrees of freedom.

  • Hydrodynamics of an explicit solvent with Langevin thermostat may also get screened due to lack of momentum conservation.

4.3.6. Additional Resorces#

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