9. Free energy, umbrella sampling, and rare events
Position in the course: Lesson 9 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Reweighting; ensembles; correlated errors
Learning goal: explain the mathematical steps, reproduce the analytical examples, and state the conditions under which the conclusions hold.
1. Why an ordinary energy average is insufficient
Free energy \(F=-\beta^{-1}\ln Z\) depends on the normalization of a distribution, whereas usual Metropolis ratios deliberately cancel it. A small energy error does not imply a small entropy error. For a coordinate \(q(R)\), define its equilibrium marginal density \(p(q)\) and a potential of mean force \(F(q)=-k_BT\ln p(q)+C\) relative to a specified measure. A radial histogram includes a geometric Jacobian; whether \(r^2\) is divided out must be stated. The arbitrary constant \(C\) prevents an ordinary histogram from supplying an absolute free energy.
2. Thermodynamic integration from a derivative
Let \(U_\lambda\) interpolate between two systems, with \(Z_\lambda=\int e^{-\beta U_\lambda}dR\) and fixed measure. Differentiation under the integral gives
Quadrature error in \(\lambda\), sampling uncertainty at each window, and poor equilibration are distinct. If endpoints change support or create singular overlaps, a naive linear path may be problematic; soft-core or staged transformations can improve it.
3. Umbrellas sample otherwise rare coordinates
Add a known bias \(W(q)\) to obtain \(p_b(R)\propto e^{-\beta[U(R)+W(q(R))]}\). The marginal obeys \(p_b(q)\propto p(q)e^{-\beta W(q)}\), hence \(F(q)=-k_BT\ln p_b(q)-W(q)+C_b\). A harmonic umbrella \(W_j=k_j(q-q_j)^2/2\) restrains each window near a selected region. Multiple windows have unknown relative constants; overlap and a consistent histogram or multistate estimator are needed to combine them. Hidden slow coordinates can remain trapped even when the chosen coordinate looks well sampled.
4. Worked harmonic free-energy difference
For \(U_k(x)=kx^2/2\) on the real line, \(Z_k=\sqrt{2\pi/(\beta k)}\). Therefore \(F_{k_1}-F_{k_0}=(k_BT/2)\ln(k_1/k_0)\). With \(k_\lambda=k_0+\lambda(k_1-k_0)\), equipartition gives \(\langle\partial_\lambda U\rangle=(k_1-k_0)/(2\beta k_\lambda)\). Integrating recovers the same logarithm. This exact model checks factors and signs without stochastic execution.
Free-energy perturbation gives \(F_1-F_0=-k_BT\ln\langle e^{-\beta(U_1-U_0)}\rangle_0\) for compatible support and measures. Insert \(e^{-\beta U_1}=e^{-\beta U_0}e^{-\beta(U_1-U_0)}\) into \(Z_1/Z_0\) to derive it. Rare configurations can dominate the exponential average. A precise ordinary mean energy therefore does not establish reliability. Forward and reverse transformations help diagnose overlap problems but agreement alone is not conclusive.
5. Exercises and caveats
Exercise: Does a barrier in \(F(q)\) immediately determine a physical transition rate?
Solution
No. Rates require dynamics, mobility, a suitable reaction coordinate, and possibly recrossing corrections. Ordinary equilibrium Monte Carlo sweep counts do not supply physical seconds.
Exercise: If an umbrella bias is \(W=2k_BT\) at a coordinate, what is its relative unbiasing weight?
Solution
Multiply by \(e^{\beta W}=e^2\), not \(e^{-2}\). Bias suppressed that region, so unbiasing restores it. Normalization still must be determined.
Original teaching schematic of the mathematics or algorithm; it is not simulation or experimental data.
6. Sources and connections
- Kumar et al. (1992), The weighted histogram analysis method
- Shirts and Chodera (2008), Statistically optimal analysis of samples from multiple equilibrium states
Related theory: Molecular dynamics · Quantum Monte Carlo
Software connection: CP2K · Quantum ESPRESSO
These software courses provide related background on energies, orbitals, or convergence management; they do not imply that the Monte Carlo or QMC examples on this page were executed there.
7. Related theory and practice
Molecular dynamics · Quantum Monte Carlo · RASPA3