Skip to content

5. From energy landscape to rates (no Boltzmann inversion, no Kramers)

5. From energy landscape to rates (no Boltzmann inversion, no Kramers)

The PMF is built forward (energy → rates), never inverted from a force profile.

  1. Equilibrium landscape: E_elec(z) = G_pore+part − G_pore − G_part (electrostatics.tex eq:electrostatic_energy), one APBS triple per z.
  2. Tilt by bias (trapping.tex eq:external_energy_model), N_tot = N_body + N_tag + N_eo: E_ext = N_tot·(V_b/14 nm)·(z − 11 nm) for −3 < z < 11 nm; 0 for z ≥ 11 nm; N_tot·V_b for z < −3 nm. Total E_V = E_elec + E_ext.
  3. Escape rate: from a Boltzmann kinetic-energy distribution, energy-independent density of states Z, step transmission c(E)=c₀·θ(E−ΔG) → k = Z c₀ exp(−ΔG/kT) ≡ k₀ exp(−ΔG/kT). Arrhenius, not Kramers — no friction or curvature prefactor anywhere.
  4. Double barrier: 1/τ = k = k_eff^cis·exp(−ΔG^cis/kT) + k_eff^trans·exp(−ΔG^trans/kT), ΔG = ΔG_steric,0 + ΔG_static,0 + N_tag·e·Ψ_tag ∓ (N_net+N_eo)·e·(Δx/L)·V_b (eq:double_barrier_complex).
  5. Translocation probability: P_trans = k^trans/(k^cis + k^trans) (eq:ptrans).
  6. Dwell-time statistics: single bound state → exponential, τ = 1/k, Var = τ². Multi-state → sum of exponentials whose expectation is the arithmetic mean of the level lifetimes (eq:tau_arithmetic_mean) — hence the experimental τ is a plain mean.