10. A reproducible Monte Carlo study
Position in the course: Lesson 10 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: All preceding Monte Carlo lessons
Learning goal: explain the mathematical steps, reproduce the analytical examples, and state the conditions under which the conclusions hold.
1. Turn a physical question into a sampling contract
Ask a concrete question: for a finite square Ising lattice, how does energy depend on temperature, and how reliable is that estimate? Specify lattice size, boundaries, Hamiltonian, units, field, target ensemble, proposal kernel, measurement cadence, and observable definitions. A numerical output without these choices is not a scientific result. Keep input, generator algorithm, seeds, software version, and analysis scripts alongside the output. Report attempted moves as well as sweeps so algorithmic cost can be compared.
2. Verify the algorithm on a state space you can enumerate
For two spins with one bond and no field, \(E=-Js_1s_2\). There are two aligned states of energy \(-J\) and two opposite states of energy \(J\). Thus
The second moment is exactly \(J^2\), so \(\operatorname{Var}(E)=J^2\operatorname{sech}^2(\beta J)\). Construct the four-state transition matrix for uniform single-spin proposals, check each row sums to one, check \(\pi P=\pi\), and confirm every pairwise flow. This deterministic verification tests probability logic before noise complicates interpretation. It is a proposed project specification; the page does not report an executed stochastic run.
3. Separate verification from convergence
Agreement on four states verifies an implementation under that test, not convergence for a large lattice. For larger systems use dispersed starts, energy and magnetization traces, blocking, independent runs, and explicit size dependence. Check that increasing warm-up or chain length does not shift results outside justified uncertainty. Acceptance rates and speed are diagnostics, not universal pass/fail thresholds. Compare local and cluster updates by uncertainty per unit computational effort for the same observable and target.
4. Build an error budget and an honest report
Distinguish numerical mistakes, initial-condition bias, finite sampling uncertainty, finite-size effects, and model inadequacy. Only finite sampling usually falls as the inverse square root of additional independent work. A tiny error bar can coexist with an incorrect Hamiltonian or a trapped chain. Near a transition, finite-size rounding and critical slowing both matter but are different effects. Statistical uncertainty in nonlinear quantities should be based on blockwise recomputation, and data reused for tuning should be identified.
An informative report includes the physical definition, validation model, convergence evidence, uncertainty method, chain and block lengths, independent repetitions, and limitations. Label plots as measurements only if they came from a recorded run. An analytical or schematic plot must say so explicitly. Comparing two energies from independent runs gives \(\operatorname{SE}(E_1-E_2)=\sqrt{s_1^2+s_2^2}\); shared random numbers or reweighting introduce covariance that changes this expression.
Enumerate proposed moves and compare the local energy change with a full energy recomputation. Flipping the same spin twice must restore the state and energy. This tests indexing and double counting separately from acceptance logic. Compare distributions across stochastic implementations rather than expecting identical trajectories from different generators or parallel decompositions. Mathematical, statistical, and bitwise reproducibility are distinct levels.
5. Exercises and solutions
Exercise: At \(\beta J=0\), what energy does the two-spin model predict and why?
Solution
All four configurations are equally likely, so the two \(-J\) states cancel the two \(+J\) states. Mean energy is zero, but its variance remains \(J^2\): zero mean does not imply zero fluctuations.
Exercise: A longer run halves the quoted error but never changes magnetization sign. Is the smaller error enough?
Solution
No. It may estimate conditional fluctuations within one basin. Compare dispersed starts and use a kernel that crosses the relevant bottleneck, or state explicitly that only a restricted sector was sampled.
Original teaching schematic of the mathematics or algorithm; it is not simulation or experimental data.
6. Sources and connections
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