Flat-Histogram Reweighting Reference Table
P0 data extract — Table 3.1 from 2007-chipot-free-energy-calculations-book §3.4.1, p. 99 (Shell, Panagiotopoulos, Pohorille). Verbatim. Use to look up the correct microstate probability and reweighting expression when running multicanonical / Wang–Landau / transition-matrix sampling in any flat-histogram macrostate variable (energy U, particle number N, volume V).
Verbatim table
Table 3.1. Common flat-histogram ensembles and their reweighing procedures
| Variable(s) | Microstate probabilities | Reweighting probabilities |
|---|---|---|
| U | ℘(q) ∝ e^{-S(U)} | ρ(U; T) ∝ ℘̃(U) e^{S(U) − βU} |
| U, N | ℘(q, N) ∝ e^{-S(N,U)} | ρ(N, U; μ, T) ∝ ℘̃(N, U) Λ^{-3N} · e^{S(N,U) − βU + βμN} |
| U, V | ℘(q, V) ∝ e^{-S(V,U)} | ρ(V, U; P, T) ∝ ℘̃(V, U) · e^{S(V,U) − βU − βPV} |
| N | ℘(q, N; T₀) ∝ (1/(N! Λ^{3N})) e^{-β₀U − F(N)} | ρ(N; μ, T₀) ∝ ℘̃(N) e^{F(N) + β₀μN} |
| V | ℘(q, V; T₀) ∝ e^{-β₀U − F(V)} | ρ(V; P, T₀) ∝ ℘̃(V) e^{F(V) − β₀PV} |
The first column indicates the flat-histogram variables, the second the prescribed microstate sampling scheme, and the third the appropriate reweighing probabilities. The script variables S and F are the weights to be determined, which converge on ln Ω and ln Q, respectively, in the flat-histogram limit. ℘̃ is the measured distribution from the flat-histogram simulation, frequently dropped if the weights are calculated to high accuracy.
— [Chipot & Pohorille 2007, p. 99, Table 3.1]
How to use this in STRC
- h09 hydrogel self-assembly — when sampling configurational space along an energy or aggregation-number coordinate to build a free-energy profile, look up the corresponding row to construct the right reweighting expression. For temperature-flatness use row 1 (U); for particle-number aggregation use row 2 (U, N).
- h26 dimer dissociation PMF — combine flat-histogram in U with WHAM post-processing if multiple temperature windows are run.
- This table replaces the need to re-derive reweighting expressions from first principles in any future MC/MD methods note.
Connections
[source]2007-chipot-free-energy-calculations-book[part-of]free-energy-methods[see-also]Phase Space Overlap and FEP Sampling