Analysis Tools

The AnalysisTools module post-processes sampling output into thermodynamic quantities: phase-space weights (ωᵢ), the partition function, internal energy, constant-volume heat capacity (cv), grand-canonical reweighting, and — described below — microcanonical inflection-point analysis of phase transitions.

Microcanonical inflection-point analysis

Nested sampling yields the microcanonical entropy $S(E) = \ln g(E)$ almost for free. Along the contiguous cull index $i$ the enclosed prior volume is $X_i = (K/(K+1))^i$, so $\ln X_i = i\,\ln(K/(K+1))$ and, with the recorded energy ladder $E(i)$,

\[S(E) = i\,\ln\!\frac{K}{K+1} - \ln\left|\frac{dE}{di}\right| + \mathrm{const} ,\]

where derivatives are taken against the dense, uniform $i$-axis (local cubic fits) rather than a binned $\ln g$ in energy space. Following Schnabel et al., Phys. Rev. E 84, 011127 (2011) and Qi & Bachmann, Phys. Rev. Lett. 120, 180601 (2018), phase transitions in finite systems appear as inflection points of the inverse caloric temperature $\beta(E) = dS/dE$, and their order follows from the higher derivatives:

transition order	independent signature in $S^{(n)}(E)$
1 (first-order)	$\beta = S'$ has a positive local minimum ($\beta$ backbends; latent heat)
2 (second-order)	$\gamma = S''$ has a negative local maximum
3	$\delta = S'''$ has a positive local minimum
odd $n$	positive local minimum of $S^{(n)}$
even $n$	negative local maximum of $S^{(n)}$

Dependent transitions carry the swapped signatures (odd $n$: negative local maximum; even $n$: positive local minimum) and are reported with kind = :dependent when accompanied by an independent transition of lower order at lower energy. The transition temperature is $T_\mathrm{tr} = 1/(k_B\,\beta(E_\mathrm{tr}))$ — in Kelvin with the default kb in eV/K, or in energy units with kb = 1. This is a purely microcanonical route: it needs no temperature grid, returns transition energies and orders directly, and can resolve transitions that broad canonical cv(T) peaks blur.

Workflow

using FreeBird

df = read_output("ladder.csv")            # canonical NS output (columns :iter, :emax)
K  = 320                                   # number of live walkers used to produce it

E, S = microcanonical_entropy(df, K)               # S(E) = ln g(E)
d    = caloric_derivatives(df, K; max_order = 2)   # d.E, d.β, d.γ
ts   = inflection_transitions(df, K; max_order = 2) # → (E_tr, T_tr, order, kind, strength)

# Recommended: confirm each transition is converged in the walker count K
dfs = [read_output("ladder_K320.csv"),
       read_output("ladder_K640.csv"),
       read_output("ladder_K1280.csv")]
conv = transition_convergence(dfs, [320, 640, 1280])   # flags drifting transitions

Convergence and smoothing

Use enough walkers, and check convergence

The derivative $\gamma = d^2S/dE^2$ amplifies the finite-walker "staircase" granularity of the NS ladder (step $\sim 1/(K\,g)$). This roughness is deterministic in the walker count $K$ — it does not average out over independent runs — so a transition should only be trusted once its temperature and order stop drifting as $K$ grows; use transition_convergence. Near-ground transitions converge last. The light internal smoothing (halfwidth) is a necessary differentiation aid, not a substitute for adequate $K$: smoothing heavily to mask too-few walkers biases the transition temperatures. Energy-window/ground_trim controls remove the $\beta$ divergence as $E \to E_\mathrm{ground}$.

Functions

FreeBird.AnalysisTools — Module

AnalysisTools

Module for analyzing the output of the sampling.