FR15
Free energy and rare events
A long trajectory can still miss a valley. Follow a bead from Boltzmann populations to biased sampling, learn how to combine its records, then ask why an accurate equilibrium curve can leave the reaction speed unknown. Three experiments keep the landscape, paths and probability estimates close enough to inspect.
Start with occupancy, then ask about time
The first chapter starts with counting configurations. Chapters 2–3 introduce noisy motion and umbrella windows; Chapters 4–5 combine records and examine uncertainty. Chapter 6 develops a second sampling method. Chapters 7–8 use paths and a published paper to test what the resulting curves establish. Each chapter contains a worked example, a prediction task and explained exercises.
Basic probability and logarithms are enough to begin. Derivatives, integrals and boundary-value problems are introduced where needed. Broader thermal and statistical foundations can be read alongside the course. FR25 · Textbook plan
L1 · 1
Why the deepest potential well need not be the most populated
Calculate the populations of wide and narrow wells and see how a projected free energy includes the hidden direction’s entropy.
L2 · 2
How much independent information is in ten thousand frames
Use overdamped Langevin dynamics, detailed balance and autocorrelation to separate dense recording from genuine independent sampling.
L2 · 3
Crossing a barrier with overlapping umbrella windows
Derive static-bias reweighting, estimate window widths and diagnose disconnected overlap or an unequilibrated hidden coordinate.
L3 · 4
How MBAR joins windows into one equilibrium distribution
Derive MBAR from the sampling mixture, calculate target weights in a hand-solvable example and diagnose an unsupported connection.
L3 · 5
Why error bars can miss the largest error
Separate weight concentration, time correlation and sampling bias with block bootstrap, independent seeds and held-out validation.
L3 · 6
How metadynamics gradually fills a visited valley
Derive the well-tempered bias limit, calculate its free-energy scaling and diagnose a collective variable that misses slow motion.
L4 · 7
Why equal free energies can give different reaction speeds
Use friction scaling and committor equations to separate occupancy from speed, then examine hidden coordinates and path ensembles.
L4 · 8
Use the MBAR paper to examine your own calculation
Read Shirts and Chodera’s Figure 1, then compare synthetic two-coordinate data with quadrature and PyMBAR to separate solver and sampling quality.
Three experiments along the route
- Rotate a projection and inspect the configurations it combines
- Move umbrella windows and reconstruct a probability distribution
- Keep equilibrium fixed and stretch the reaction clock
These experiments share a dimensionless two-coordinate potential on a reflecting square. They separate simulated trajectories from quadrature references. The equilibrium workbench demonstrates finite-sample failures as well as successful reweighting.
Two earlier experiments to connect the ideas
Morse molecular bond shows a potential-energy landscape and quantum escape. Here the new question is classical thermal occupancy and sampling. Optical-tweezers calibration connects a harmonic trap to fluctuations. It is useful preparation for umbrella widths; the DNA-hairpin paper remains a separate experimental system.
Sources and reproduced scope
Worked examples and schematic figures are original teaching material. The numerical research slice compares specified estimator equations on our seeded two-coordinate records with independent quadrature and PyMBAR. The final chapter reads Figure 1 of the MBAR paper and states which experimental data remain outside that calculation.
Model definitions, scripts and numerical records
- Sandbox Physics, fixed-length reversible path-ensemble model and validation (2026)
Original discrete endpoint-bridge target, reversible kept-slice shooting derivation, independent path-sum reference and recorded limitations. The computational clock is distinct from the browser Langevin clock.
- Sandbox Physics, reproducible endpoint-bridge benchmark script (2026)
Runs four frozen seeds with accepted and rejected proposals retained, calculates serial-correlation diagnostics and writes path-sampling-summary.json and path-sampling-raw.json.
- PyMBAR 4.0.3, MBAR module documentation
MBAR constructor inputs, reduced-energy matrix, free-energy differences and the independent-sample requirement. Our comparison checks equation solutions on identical records and does not use PyMBAR covariance estimates for correlated frames.
- PLUMED Masterclass 21.2, Statistical errors in MD (documentation v2.9)
Background and Exercises 2, 7–9: marginal probabilities, correlated samples, blocks and bootstrap. The tutorial reports PLUMED 2.7.0 for its original exercises. Our examples use an independently defined two-coordinate model.
- Giovanni Bussi, PLUMED Masterclass 21.3, Umbrella sampling (2021; documentation v2.9)
Exercises on static restraints, combining windows with binless WHAM, reweighting a different coordinate and uncertainty. Its alanine-dipeptide trajectories are not used here.
- Michael R. Shirts and John D. Chodera, Statistically optimal analysis of samples from multiple equilibrium states, J. Chem. Phys. 129, 124105 (2008)
Frozen arXiv v3: Sections II–IV, Eqs. (9), (11)–(14), Section V and Figure 1, Appendices A–C. We implement the equilibrium estimating equations on our synthetic model. Figure 1 is read as an experimental example; its DNA-hairpin traces and plotted values are not reproduced.
- Alessandro Barducci, Giovanni Bussi and Michele Parrinello, Well-Tempered Metadynamics, Phys. Rev. Lett. 100, 020603 (2008)
Frozen arXiv v1: adaptive deposition law and limiting bias, Eqs. (2)–(6). The derivation assumes that microscopic degrees of freedom equilibrate sufficiently fast at a given collective variable. Its alanine-dipeptide numerical experiment is outside our reproduction.
- Max Bonomi, PLUMED Masterclass 21.4, Metadynamics (2021; documentation v2.9)
Theory summary and exercises on reconstruction, block analysis and good or bad collective variables. The long-time bias relation is used with its sampling assumptions; a finite-time bias is not labelled exact free energy.
- Weinan E and Eric Vanden-Eijnden, Towards a theory of transition paths, J. Stat. Phys. 123, 503–523 (2006)
Published transition-path framework and forward-committor definition, checked against the authors’ institutional publication record and the explicit low-dimensional formulation in the next source.
- Mauro Lapelosa and Cameron F. Abrams, Transition-Path Theory Calculations on Non-Uniform Meshes in Two and Three Dimensions using Finite Elements, Comput. Phys. Commun. 184, 2310–2315 (2013)
Transition-Path Theory Calculations on Non-Uniform Meshes in Two and Three Dimensions using Finite Elements (2013), Sections on the backward Kolmogorov problem and reactive currents. Used for the equation and boundary-condition discussion, not for copied mesh data or figures.