Calculator
Example data table
| m (kg) | ω (rad/s) | T (K) | P | sweeps | Δ (m) | Exact ⟨E⟩ (J) |
|---|---|---|---|---|---|---|
| 9.10938e-31 | 1.00000e14 | 300 | 32 | 5000 | 1.00000e-11 | 6.16984e-21 |
| 1.67262e-27 | 5.00000e12 | 50 | 64 | 8000 | 5.00000e-12 | 7.23565e-22 |
| 9.10938e-31 | 2.00000e13 | 10 | 128 | 12000 | 2.00000e-12 | 1.05457e-21 |
Formula used
The quantum canonical partition function is approximated by a discretized imaginary-time path integral, mapped to a classical “ring polymer” of P beads.
V(x) = ½ m ω² x²
S = Σj=1..P [ m( xj − xj+1 )² / (2τħ²) + τ V(xj) ]
xP+1 = x1
A Metropolis move proposes xj → xj + u, with u ∈ [−Δ, Δ], and accepts it with probability min(1, exp(−ΔS)), where ΔS is the local action change.
For a harmonic oscillator, the virial theorem gives ⟨K⟩ = ⟨V⟩, so the energy estimator used is ⟨E⟩ = 2⟨V⟩. The exact thermal benchmark is ⟨E⟩ = (ħω/2)coth(βħω/2).
How to use this calculator
- Enter the oscillator parameters: m, ω, and temperature T.
- Choose beads P. Use larger P at low temperatures.
- Set sweeps and burn-in. Start with burn-in near 20% of sweeps.
- Pick step size Δ. Enable auto-tune for convenience.
- Submit to run the sampling. Results appear above the form.
- Use CSV or PDF export to store and share your run.
Professional article
1) What this calculator estimates
This tool performs Path Integral Monte Carlo (PIMC) sampling for a one‑dimensional quantum harmonic oscillator at finite temperature. It estimates thermal averages such as ⟨x²⟩, ⟨V⟩, and ⟨E⟩ using a discretized imaginary‑time path. Results include uncertainty and run diagnostics to support trustworthy interpretation.
2) Ring‑polymer discretization and beads
The inverse temperature β = 1/(kBT) is split into P equal slices τ = β/P. The quantum particle is mapped to a classical “ring polymer” of P beads with periodic indexing xP+1 = x1. Increasing P improves accuracy when βħω is large, especially at low T.
3) Kinetic coupling as harmonic springs
The kinetic term becomes a nearest‑neighbor coupling between beads, behaving like springs that penalize sharp jumps along imaginary time. The coefficient scales as m/(2τħ²), so reducing τ (using larger P) strengthens these springs and better resolves short‑time quantum fluctuations. This is the core mechanism behind the polymer analogy.
4) Potential energy for the oscillator
The potential is V(x) = ½ m ω² x² and enters the action through τV(xj) at each bead. Because V is quadratic, the oscillator is a rigorous benchmark: the exact thermal energy and variance are known analytically. That makes it ideal for validating step size, beads, and sweep counts.
5) Metropolis moves and step size Δ
Sampling uses local Metropolis updates that propose xj → xj + u with u in [−Δ, Δ]. The acceptance probability min(1, e−ΔS) depends only on the local action change. Auto‑tuning adjusts Δ during burn‑in to keep acceptance in a practical window (about 0.25–0.55).
6) Burn‑in, stride, and uncertainty
Early sweeps are discarded as burn‑in to remove bias from the initial configuration. After equilibration, the calculator records one sample every “stride” sweeps, reducing serial correlation. Reported standard errors are computed from the sample variance; increasing sweeps and using a moderate stride typically narrows uncertainty.
7) Built‑in exact benchmark
The oscillator’s exact thermal energy is ⟨E⟩ = (ħω/2)coth(βħω/2), with ⟨x²⟩ = (ħ/2mω)coth(βħω/2). The calculator compares Monte Carlo estimates against these values and reports percent energy error. If the error is large, increase P, sweeps, or refine Δ to improve mixing.
8) Practical interpretation for advanced use
For real systems, PIMC generalizes to many dimensions and interacting potentials, but the same controls remain central: beads P set discretization, Δ and acceptance govern mixing, and sweep count sets statistical precision. Use the diagnostics to justify convergence, then export CSV/PDF results for documentation, audits, or further analysis.
FAQs
1) What does “beads P” control?
P sets the number of imaginary‑time slices. Larger P reduces discretization error, especially at low temperature or high ω. Typical starting values are 32–128, increasing when βħω becomes large.
2) Why is acceptance ratio important?
Too low acceptance means moves are rejected and the chain mixes slowly; too high means moves are tiny and exploration is inefficient. A practical range is roughly 0.25–0.55, balancing step size and sampling efficiency.
3) How should I choose burn‑in sweeps?
Use enough burn‑in to forget the initial path. A common choice is 10–30% of total sweeps. If observables drift early, increase burn‑in or enable auto‑tuning for faster stabilization.
4) What does “stride” do?
Stride records one sample every N sweeps, reducing correlation between stored samples. Higher stride often gives more reliable error estimates but fewer samples. If uncertainty looks unstable, increase stride or run more sweeps.
5) Why is energy computed as 2⟨V⟩ here?
For the harmonic oscillator, the virial theorem gives ⟨K⟩ = ⟨V⟩ at equilibrium. Therefore total energy is ⟨E⟩ = ⟨K⟩ + ⟨V⟩ = 2⟨V⟩, a low‑variance estimator.
6) When should I increase P versus sweeps?
Increase P to reduce discretization bias; increase sweeps to reduce statistical noise. If your mean is consistently off from the exact benchmark, raise P. If the mean is close but noisy, raise sweeps or stride.
7) Can I reproduce a run exactly?
Yes. Set an integer random seed and keep all inputs identical. The same seed, P, sweeps, burn‑in, stride, and Δ will generate the same sequence of proposals and results on the same platform.