Theory & Method
Neutron Moderation Simulator · AntraSIM-0100 · version 1.0.0 · build 2026-09-24. This document describes the model as implemented in the application ; the numbers and tables that depend on the model (verification results, regression values, parameter defaults, expected exercise results) were generated at build time from the same core code that the application runs.
How to read this document. Each topic follows the progression Intuition → Physical mechanism → Mathematical formulation → Engineering interpretation → Assumptions and limitations . Section 12 contains laboratory exercises with expected results; each exercise has a Load set-up button that configures the simulator exactly as described.
Model status. Software-verified (Section 7) against analytic lethargy-gain expressions, an exact renewal-theory case and independent quadrature. Not validated against experiments or transport codes. Cross sections are energy-independent, rounded textbook-level values classified as estimate — not verified (Section 8, Section 10). Principle simulator for training; not safety-grade.
1. Physical Background
Intuition A neutron born in fission carries about 2 MeV, but the chain reaction in a water-moderated reactor is sustained mainly by neutrons with about 0.025 eV. Between birth and absorption the neutron must lose more than nine orders of magnitude of energy. It does so by bouncing off nuclei: a collision with a light nucleus takes a large share of the energy, a collision with a heavy nucleus almost none. A moderator is a material made of light nuclei that scatter well and absorb little. Because each collision removes a roughly constant fraction of the energy rather than a constant amount, the neutron energy falls exponentially with the number of collisions, and it is natural to count energy on a logarithmic scale.
Physical mechanism
fission neutron, about 2 MeV → elastic collisions with moderator nuclei → energy loss per collision (random, set by the nuclear mass A) → lethargy gain → number of collisions to reach thermal energy → moderating power and ratio
Neutrons are classified by energy: fast above about 0.1 MeV, intermediate between about 1 eV and 0.1 MeV, slow below about 1 eV, and thermal when they are in approximate equilibrium with the moderator (most probable energy 0.0253 eV at 293 K). The simulator draws these boundaries on the trajectory plot.
Engineering interpretation The number of collisions and the macroscopic scattering cross section together set the distance a neutron travels while slowing down and hence the size and the fuel-to-moderator ratio of a lattice; the absorption in the moderator sets the price of moderation in terms of lost neutrons. Assumptions and limitations The model is a principle model of the collision kinematics only (Section 9); it does not compute space, time, absorption during slowing-down or the thermal spectrum.
2. Assessment of the Source Material and Learning Objectives
Basis The learning objectives are taken from the NUREG-1021 BWR knowledge-and-ability catalog, Reactor Theory, section 292001 (Neutron characteristics) [1]. The catalog was consulted at excerpt level through the project knowledge; no earlier simulator or notebook was the source of this model, so there are no source-model findings to reconcile. Table 2.1 states which objectives the simulator supports and, equally important, which it does not.
Objective (paraphrased) Support in AntraSIM-0100
Define fast, intermediate, slow and thermal neutrons Energy classes marked in the trajectory plot and in the inspection panel. Describe the moderation process and the characteristics of a good moderator Monte Carlo collisions; ξ, moderating power ξΣs and moderating ratio ξΣs /Σa for four materials and a custom nuclide. Explain why light nuclei are efficient moderators Exercises 1–3; α, ξ and N̄ versus mass number. Prompt and delayed neutrons, neutron lifetime, sources Not modelled. Delayed neutrons and kinetics are treated by the existing AntraSIM point-kinetics simulator and by AntraSIM-0130, -0106.Neutron flux distribution and leakage Not modelled here; see AntraSIM-0120 (two-group diffusion).Slowing-down time Not modelled (collision counts only, no velocities or path lengths).
Table 2.1 — Coverage of the catalog objectives (audit finding A-1: the simulator supports the moderation objectives only; the remaining objectives of 292001 are listed for transparency).
ID Finding Resolution
A-1 Objectives of 292001 on lifetime, delayed neutrons and flux shape cannot be met by a collision-kinematics model. Stated in Table 2.1; covered by other simulators of the family. A-2 The frequently quoted “18 collisions” for hydrogen is the leading term L/ξ = 18.2; the expected number of collisions until the energy first falls to Eth is L + 1 = 19.198 because the last collision overshoots the boundary. Both are shown; the difference is explained in Section 3 and used in Exercise 1.
3. Governing Equations
Intuition In an elastic collision the neutron cannot lose more than a limiting fraction of its energy; how much it loses within that limit depends on the scattering angle. If the scattering is equally likely in all directions in the centre-of-mass system (isotropic s-wave scattering, valid for neutrons below about 1 MeV), every outgoing energy between the lower limit and the incoming energy is equally probable.
Mathematical formulation For a target of mass number A at rest the outgoing energy E′ after a collision at energy E is uniformly distributed on [αE, E]:
α = ( A − 1 A + 1 ) 2 , E ′ = E [ α + ( 1 − α ) U ] , U ~ U n i f o r m ( 0 , 1 )
with α the smallest possible ratio E′/E [–] (α = 0 for hydrogen). It is convenient to use lethargy, u = ln(E0 /E), because the gain per collision g = ln(E/E′) then has a distribution independent of energy. Its density and moments are
f ( g ) = e − g 1 − α ( 0 ≤ g ≤ ln ( 1 / α ) ) , ξ = ⟨ g ⟩ = 1 + α ln α 1 − α , ⟨ g 2 ⟩ = 2 − α ln 2 α + 2 α ln α − 2 α 1 − α
where ξ [–] is the mean logarithmic energy decrement per collision (ξ = 1 for hydrogen, ξ ≈ 2/(A + 2/3) for A ≫ 1). For a mixture (for example H2 O) the collision partner j is chosen with probability pj = nj σs,j /Σk nk σs,k and the mixture moments are the pj -weighted sums of the nuclide moments.
Engineering interpretation The neutron has to gain L = ln(E0 /Eth ) = 18.2 lethargy units for the default energies. Since each collision gains ξ on average, the leading estimate of the number of collisions is L/ξ. The gains are independent and identically distributed, so the collision count is a renewal count ; the classical asymptotic renewal results give, including the overshoot of the last collision,
N ¯ ≃ L ξ + ⟨ g 2 ⟩ 2 ξ 2 , V a r ( N ) ≃ ( ⟨ g 2 ⟩ − ξ 2 ) L ξ 3
The first relation is exact for hydrogen, where g is exponentially distributed with mean one and the count is Poisson: N̄ = L + 1. The macroscopic quantities that rank moderators are
Σ s = N m ∑ j n j σ s , j , moderating power = ξ Σ s , moderating ratio = ξ Σ s Σ a
with Nm = ρNA /M the number of molecules per cm³ [cm⁻³], σ [b = 10⁻²⁴ cm²] and Σ [cm⁻¹]. The moderating power says how quickly a material slows neutrons; the moderating ratio says how much slowing-down one obtains per unit of absorption. Assumptions and limitations Elastic, isotropic centre-of-mass scattering; stationary target nuclei; energy-independent cross sections; no absorption or leakage while slowing down.
4. Inputs and Operating Conditions
Inputs Material (light water, heavy water, graphite, beryllium, or a custom single nuclide with its mass number, scattering and absorption cross sections and density); source energy E0 [MeV]; thermalisation energy Eth [eV]; number of histories; random seed; number of displayed trajectories (1–5); inspection collision number. All quantities are entered in the units shown; the inspection collision can be set by clicking the trajectory plot. There is no time-dependent operating input, so no operating-history table is provided.
Cost cap The estimated total number of collisions (renewal estimate × histories) must not exceed 6·10⁶; otherwise the input is refused with the largest admissible number of histories. This keeps every recalculation below a fraction of a second.
Input Configuration (IC) and CSV The Input Configuration (IC) is a JSON file (format “AntraSIM-input-configuration”, schemaVersion 1 ) that stores the complete canonical configuration; every key carries its unit. Loading is all-or-nothing: the file is parsed, identified (format, schema, application id), compared structurally with the reference configuration (same keys, same types, admissible array lengths), range-checked, and only then committed. Older schemas would be migrated by an explicit registered migration or rejected with a message; a migration hook is exercised by the verification suite. Restore defaults returns every input, table and exercise-dependent setting to the reference configuration. CSV exports carry metadata lines (#), the complete IC as one # line and ASCII-only column names.
5. Calculated Quantities
For the selected moderator: the analytic ξ and the Monte Carlo estimate ξMC (total lethargy gain divided by total number of collisions, which is unbiased by Wald’s identity); the Monte Carlo mean number of collisions with its standard error; the renewal-theory prediction and the leading term L/ξ; the standard deviation of N; the histogram of N with a Gaussian of the renewal mean and variance; the histogram of the gain per collision with the analytic density; Σs , ξΣs , Σa and the moderating ratio; and the analytic N̄(A) curve for single nuclides. For the default case (light water, 18.2 lethargy units) the analytic values are ξ = 0.9236, L/ξ = 19.7, renewal N̄ = 20.77, Σs = 1.464 cm⁻¹, ξΣs = 1.352 cm⁻¹ and ξΣs /Σa = 60.91.
Table 5.1 — Moderator comparison for the default energies (generated from the code).
Moderator ξ L/ξ N̄ (renewal) Σs [cm⁻¹] ξΣs [cm⁻¹] ξΣs /Σa Light water (H2O) 0.9236 19.7 20.77 1.464 1.352 60.91 Heavy water (D2O) 0.5083 35.8 36.86 0.3522 0.179 4381 Graphite (C) 0.1578 115.3 116 0.377 0.05949 211.9 Beryllium (Be) 0.2066 88.08 88.77 0.7541 0.1558 137
Table 5.2 — Mass-number dependence of α, ξ and N̄ for single nuclides (generated from the code).
A α ξ 2/(A+2/3) N̄ (renewal) 1 0 1 1 19.2 2 0.1111 0.7253 0.75 25.89 9 0.64 0.2066 0.2069 88.77 12 0.716 0.1578 0.1579 116 16 0.7785 0.1199 0.12 152.4 56 0.9311 0.03529 0.03529 516.3 238 0.9833 0.00838 0.00838 2172
6. Numerical Method
Approach Each history starts at E0 . While E > Eth the collision partner is sampled from the cumulative probabilities pj , the new energy is E[αj + (1 − αj )U], and the gain g = ln(E/E′) is accumulated. The first Ntraj histories store their complete energy sequence. Details Random numbers come from a seeded 32-bit generator (mulberry32) implemented in the core, so the same seed reproduces every result bit for bit. The standard error of the mean is √(Var/H) from the sample variance of N. The histogram uses at most 60 bins of integer width; the gain histogram uses 50 bins on [0, min(12, ln(1/αmin ))]. A history is stopped at 5000 collisions as a safeguard (never reached within the cost cap). There is no time stepping, no tolerance and no iteration. For the default case the calculation takes 23 ms in Node on the reference machine of the build (≈ 10⁷ collisions per second), so results are recomputed automatically after every valid input change.
7. Verification and Numerical Accuracy
Scope Verification shows that the equations are solved correctly and that reference values are reproduced; it is not validation. The suite runs in the application (footer → Verification) and in Node with the same code. A failing test is a finding: the cause is diagnosed, the criterion is never adjusted until it passes. During development three tests initially failed and were traced to test defects, not to the model: the invariant test compared the smallest energy ratio with the wrong nuclide (oxygen instead of the smallest α in the mixture, hydrogen), one exercise set-up exceeded the cost cap, and the regression values had not yet been stored.
Table 7.1 — Release verification suite (13 of 13 passed; generated from an actual run).
ID Test Criterion Result Measured T1 Mean logarithmic decrement ξ and second moment E[X²] vs. Simpson quadrature of the lethargy-gain density (A = 2, 9, 12, 56, 238); ξ(A = 1) = 1 exactly max relative difference ≤ 1e-9; ξ(1) = 1 PASS 2.67e-11 T2 Monte Carlo mean lethargy gain per collision vs. analytic ξ (A = 1, 12, 56) |ξ_MC − ξ| ≤ 4 standard errors PASS max 1.17 σ T3 Hydrogen (A = 1): exact result E[N] = ln(E0/Eth) + 1 (lethargy gains are Exp(1); the number of renewals is Poisson) |N̄ − (L + 1)| ≤ 4 standard errors PASS N̄ = 19.2038 vs 19.1975 (0.65 σ) T4 Light water: Monte Carlo mean collisions vs. renewal-theory prediction L/ξ + E[X²]/(2ξ²) (asymptotic; O(1/L) remainder) |N̄ − N_analytic| ≤ 4 standard errors + 0.05 PASS N̄ = 20.788 vs 20.773 T5 Graphite and beryllium: Monte Carlo mean collisions vs. renewal-theory prediction difference ≤ 4 standard errors + 0.1 PASS excess ≤ 0.000 T6 Determinism: identical seed reproduces every result bit for bit; a different seed changes the sample identical / different PASS seed 12345: 20.80400; seed 999: 20.80330 T7 Invariants: every energy strictly decreases along a history; the last energy is ≤ E_th; no collision loses more than the kinematic limit all histories PASS min E'/E = 4.04e-7 ≥ α_min = 0 T8 Histogram normalisation (collision-number density) and analytic lethargy-gain density integrate to one |∫ − 1| ≤ 1e-9 (histogram); ≤ 0.02 (gain pdf, truncated at 12 lethargy units) PASS 0 ; 5.6e-3 T9 Scale invariance: (E0, E_th) → (2·E0, 2·E_th) gives identical histories for the same seed identical mean, ξ_MC and histogram PASS 20.804000 = 20.804000 T10 IC round trip restores every input exactly; 7 malformed files are rejected; an explicit migration is applied and reported exact; 7 of 7 rejected; migratedFrom = 0 PASS round trip true; rejected 7/7; migrated from 0 T11 All laboratory exercise set-ups are valid and compute 5 of 5 valid PASS slowest 142 ms T12 Default-case regression values (light water, 20000 histories, seed 12345): mean collisions and ξ_MC max relative difference ≤ 1e-9 to the stored values PASS 0 (N̄ = 20.80400, ξ_MC = 0.922365) T13 Invalid inputs are refused: E0 ≤ E_th, cost cap exceeded, non-integer histories, too many trajectories 4 of 4 rejected PASS 4 of 4
Default-case regression values (light water, 20000 histories, seed 12345): N̄ = 20.804, ξMC = 0.9223651; stored in the suite (T12) and printed from the same run. The Monte Carlo results are statistical estimates: the comparison with theory is therefore made in standard errors, and agreement within 4 standard errors is a consistency check of the implementation, not a proof of physical accuracy. Software verification, numerical accuracy and model validation are distinct; no validation against measured data or transport codes exists.
8. Model Parameters and Constants
Evidence classes: externally sourced derived assumption not verified . There is no earlier release, so the “previous value” column of the reasonableness check is empty; the assessment is given in the basis column. The reference case is “2 MeV neutrons in cold light water thermalised at 0.025 eV”.
Parameter Default Unit Class Basis and assessment Editable
E0 2 MeV assumption Typical mean fission-neutron energy about 2 MeV (rounded; the Watt spectrum is not sampled). Reasonable for a training reference; not verified against an evaluated spectrum. yes Eth 0.025 eV externally sourced Most probable energy of a 293 K Maxwellian, 0.0253 eV, rounded to 0.025 eV; a bookkeeping boundary (Section 9). yes Nhist , seed 20000, 12345 – assumption Chosen so that the standard error of the mean collision number is below 0.5 % in about 30 ms; deterministic seed (mulberry32). yes σs (H), σs (O) 20.0, 3.8 b not verified Rounded textbook-level free-atom values, energy independent. Transcribed from memory of standard compilations; not re-checked against ENDF/JEFF in this release. no (material table) σs (D), σs (C), σs (Be) 3.4, 4.7, 6.1 b not verified As above. no (material table) σa (H, D, O, C, Be) 0.332, 0.00052, 0.00019, 0.0035, 0.0092 b not verified 2200 m/s absorption cross sections, rounded textbook-level; used only for the moderating ratio ξΣs /Σa (not for the slowing-down histories). no (material table) ρ (H2 O, D2 O, C, Be) 1.00, 1.105, 1.60, 1.85 g/cm³ assumption Room-temperature values; operating densities (hot water) are lower and shift Σs and the moderating ratio. no (material table) α = ((A−1)/(A+1))² formula – derived Elastic scattering kinematics; exact. no Custom nuclide 1 – assumption Mass number A (also used as molar mass in g/mol); the three further custom parameters set Σs and Σa for the moderating ratio only. yes
Consistency check: with the tabulated values the light-water moderating power is 1.352 cm⁻¹, of the same magnitude as the commonly quoted value near 1.4 cm⁻¹; the moderating ratio is 60.91, which is lower than some textbook values because a molecule-level σa and the rounded H cross section are used. No source value was found to be off by a large factor; none was verified against an evaluated library.
9. Model Scope and Limitations
Represented Elastic scattering kinematics for stationary nuclei with isotropic centre-of-mass scattering; the statistics of the collision count; moderator figures of merit. Not represented Inelastic scattering and (n,2n) reactions (the simulator warns above rough, unverified thresholds: about 6 MeV for oxygen, 4.4 MeV for carbon, 1.9 MeV for beryllium and about 1 MeV for heavy nuclei); anisotropic (p-wave) scattering at high energy; energy dependence of the cross sections, including resonances; chemical binding, thermal motion and up-scattering below about 1 eV (the thermalisation energy is a bookkeeping boundary, not a thermal spectrum); absorption and leakage during slowing-down; the spatial and temporal aspects (slowing-down length, slowing-down time); moderator temperature and void effects (see AntraSIM-0140); the Watt fission spectrum (a single source energy is used).
10. Data Provenance and Open Points
What was found The learning objectives from NUREG-1021 [1] (excerpt level). The kinematic formulas are derived in Section 3 and verified by quadrature and by the exact hydrogen case. What was not found No evaluated-library cross sections (ENDF/B, JEFF) were retrieved for this release; the rounded scattering and absorption cross sections and the densities were entered from standard textbook-level knowledge. Unverified items All cross sections in the material table (class “not verified”); the 2 MeV source energy (assumption); the equivalence of the “moderating ratio” with textbook values. Users should not quote the numerical moderating ratios as design data.
11. References
External standards and reports
[1] U.S. Nuclear Regulatory Commission, Operator Licensing Examination Standards for Power Reactors , NUREG-1021 (BWR knowledge-and-ability catalog, Reactor Theory, section 292001). Consulted at excerpt level (project copy).[2] J. R. Lamarsh and A. J. Baratta, Introduction to Nuclear Engineering , 3rd ed., Prentice Hall, 2001. Further reading — not consulted (source of the terminology and of the textbook comparison values).[3] J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis , Wiley, 1976. Further reading — not consulted (elastic scattering and slowing-down theory).[4] U.S. Department of Energy, DOE Fundamentals Handbook — Nuclear Physics and Reactor Theory , DOE-HDBK-1019/1-93 and -1019/2-93. Further reading — not consulted .[5] D. R. Cox, Renewal Theory , Methuen, 1962. Further reading — not consulted (asymptotic expressions for the renewal count used in Section 3).
Antranova model basis
[A1] Antranova Engineering AB, AntraSIM Production System Prompt v1.0.3 and AntraSIM Template v1.0.0 (software structure, documentation conventions).
12. Exercises and Laboratory Work
The five exercises below are meant for an introductory reactor-physics laboratory (about 20 minutes each). Each begins with a Load set-up button that configures the simulator; the numbers under Expected results were generated from the same code and reproduce exactly with the stated seed. Monte Carlo values fluctuate within their standard error when the seed is changed.
Exercise 1 — Hydrogen, the ideal moderator
Objective: relate the collision kinematics to the exponential lethargy gain and the “+1” of the renewal count. Prerequisite: Section 3.
Learning outcomes. Explain why a hydrogen nucleus can absorb all of the neutron energy in one collision; predict the mean number of collisions for hydrogen from L = ln(E0 /Eth ); distinguish L/ξ from the exact expected count.
Load set-up pure hydrogen (A = 1), 2 MeV → 0.025 eV, 100 000 histories, seed 12345.
Read N̄ (Monte Carlo) and its standard error in the Inspection panel and compare with L/ξ and with renewal N̄. In the “Lethargy gain” plot, look at the analytic density on the logarithmic axis: what shape has a straight line there? Note ξMC and the largest number of collisions observed. Change E0 to 20 MeV and back; how does N̄ change and why by that amount? Expected results. N̄MC = 19.206 ± 0.013; exact L + 1 = 19.198; leading term L/ξ = 18.2. ξMC = 0.9997 (analytic 1). The gain density is a straight line of slope −1 on the logarithmic axis (exponential, unbounded because α = 0). Raising E0 from 2 to 20 MeV adds ln 10 = 2.30 collisions.
Discussion. Why is the 18 collisions quoted in textbooks not the mean of the Monte Carlo sample?
Exercise 2 — Four moderators compared
Objective: rank moderators by collisions, moderating power and moderating ratio.
Learning outcomes. Explain the characteristics of a good moderator; justify light water in a BWR and heavy water where neutron economy is decisive.
Load set-up light water, 50 000 histories; then select D₂O, graphite and beryllium in turn.
For each material record N̄MC , ξ, ξΣs and the moderating ratio (also listed in the Moderator properties panel). Rank the four materials by number of collisions, by moderating power and by moderating ratio. Which material needs the smallest volume to slow neutrons, which the smallest absorption per unit slowing-down? Expected results. Monte Carlo mean collisions: H₂O 20.79 ± 0.021, D₂O 36.86 ± 0.029, graphite 116 ± 0.029, beryllium 88.78 ± 0.026. The analytic comparison is Table 5.1: light water has the highest moderating power, heavy water the highest moderating ratio by a large factor; graphite and beryllium need several times more collisions than water.
Moderator ξ L/ξ N̄ (renewal) Σs [cm⁻¹] ξΣs [cm⁻¹] ξΣs /Σa Light water (H2O) 0.9236 19.7 20.77 1.464 1.352 60.91 Heavy water (D2O) 0.5083 35.8 36.86 0.3522 0.179 4381 Graphite (C) 0.1578 115.3 116 0.377 0.05949 211.9 Beryllium (Be) 0.2066 88.08 88.77 0.7541 0.1558 137
Discussion. Light water is compact and cheap but absorbs neutrons: what does the reactor designer pay for it (enrichment)?
Exercise 3 — Mass-number dependence
Objective: see why heavy nuclei are poor moderators and test the asymptotic formula ξ ≈ 2/(A + 2/3).
Learning outcomes. Explain the trend of α, ξ and N̄ with A; use the log–log N̄(A) plot as a design chart.
Load set-up custom nuclide A = 56 (iron-like), 5000 histories.
Read ξ, L/ξ, the renewal N̄ and the Monte Carlo N̄; look at the trajectory plot: how does the “staircase” differ from hydrogen? Try A = 1, 2, 12, 16 and 238 (for A = 238 reduce the number of histories as the input hint suggests). Locate your material on the N̄(A) plot. Expected results. For A = 56: ξ = 0.03529 (asymptotic formula 0.03529), L/ξ = 515.6, renewal N̄ = 516.3, Monte Carlo N̄ = 516.5 ± 0.19. The trajectories are almost straight lines on the semilogarithmic plot with small steps. Table 5.2 lists all reference nuclides.
Discussion. Why can iron in the core structure hardly moderate but still remove neutrons from the fast region?
Exercise 4 — Energy range and thermal cut-off
Objective: show that only the energy ratio matters for the collision count.
Learning outcomes. Use lethargy instead of energy; predict the effect of a harder source or a higher cut-off.
Load set-up graphite, E0 = 10 MeV, Eth = 1 eV, 20 000 histories.
Record L and N̄ for the loaded case, then restore E0 = 2 MeV and Eth = 0.025 eV (graphite). Now set E0 = 4 MeV and Eth = 0.05 eV. Expected results. Loaded case: L = 16.12, N̄MC = 102.9. Graphite 2 MeV → 0.025 eV: L = 18.2, N̄MC = 116.1. Graphite 4 MeV → 0.05 eV has the same ratio and gives N̄MC = 116.1 — identical to 116.1 because the same seed reproduces the same histories, only scaled in energy.
Discussion. Why does a higher thermalisation cut-off reduce the number of collisions only logarithmically? The simulator shows a warning for the 10 MeV source in graphite: which physical process, neglected here, becomes important at that energy and what would it do to the collision count?
Exercise 5 — Statistics and random seed
Objective: understand Monte Carlo uncertainty and reproducibility.
Learning outcomes. Interpret a standard error; predict the number of histories needed for a target precision.
Load set-up light water, 200 histories, seed 7.
Note N̄ ± SE; change the seed to 8; then use 20 000 histories. Compare the standard errors and the relative sizes; check the 1/√H scaling. Expected results. Seed 7: N̄ = 20.57 ± 0.33; seed 8: N̄ = 21.07 ± 0.35. With 20 000 histories the standard error is 0.034; the ratio of the standard errors (200 versus 20 000 histories) is 9.9, close to √100 = 10. Both 200-history means are compatible with the renewal-theory value 20.77 within about two standard errors.
Discussion. How many histories are needed for a 0.1 % standard error of the mean? (Answer: about 100 times the number that gives 1 %.)