Low-noise Pauli-consistent ensemble Monte Carlo for graphene with electron-electron scattering

Giovanni Nastasi; Tigran Zalinyan

arxiv: 2604.18517 · v2 · submitted 2026-04-20 · 🧮 math-ph · math.MP

Low-noise Pauli-consistent ensemble Monte Carlo for graphene with electron-electron scattering

Tigran Zalinyan , Giovanni Nastasi This is my paper

Pith reviewed 2026-05-10 03:06 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords grapheneMonte Carlo simulationelectron-electron scatteringPauli exclusion principlesampled-partner approximationnumerical artifactsmomentum space grid

0 comments

The pith

Sampled-partner approximation cuts computational cost while preserving accuracy in graphene simulations

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an approximation that samples partner electrons uniformly from the current ensemble to evaluate electron-electron scattering rates in Pauli-consistent Monte Carlo simulations of graphene. This replaces the expensive sum over all possible partner cells in momentum space while keeping the actual scattering events the same. The reduced cost allows simulations with much larger ensembles, which in turn make small systematic oscillations visible in the averaged results. These oscillations are shown to arise from the way particles drift deterministically across the fixed discrete momentum grid rather than from any physical mechanism. The authors also describe a way to lessen the effect of these numerical oscillations on extracted quantities without altering how the simulation itself runs.

Core claim

We investigate Pauli-consistent ensemble Monte Carlo simulations of graphene with explicit intraband electron-electron scattering. To reduce the cost of electron-electron proposal-rate evaluation, we introduce a sampled-partner approximation that replaces the full partner-cell sum by uniform sampling from the instantaneous ensemble, while leaving the event-level collision step unchanged. Comparison with the full-sum reference shows close agreement together with a substantial reduction in computational cost, enabling large-ensemble low-noise simulations. In this regime, systematic oscillatory components become clearly resolved in ensemble-averaged time traces. We show that these oscillations

What carries the argument

the sampled-partner approximation that uses uniform sampling from the instantaneous ensemble to estimate electron-electron scattering proposal rates

If this is right

Large-ensemble low-noise simulations of graphene become feasible.
Oscillatory components in time traces can be resolved and identified as numerical artifacts from the momentum grid.
A post-processing procedure can reduce the impact of grid-induced oscillations on observables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sampling idea could speed up Monte Carlo treatments of other many-particle interactions in condensed-matter systems.
Improving the momentum-space discretization might eliminate the oscillatory artifacts at their origin.
The approach may help simulate more complex scattering processes that were previously too expensive for large ensembles.

Load-bearing premise

That drawing partners uniformly from the instantaneous ensemble gives a good enough estimate of the scattering rates so that the simulated dynamics stay accurate for the times and densities of interest.

What would settle it

If simulations using the sampled-partner method and the exact full-sum method produce statistically different ensemble averages at high densities or long times.

Figures

Figures reproduced from arXiv: 2604.18517 by Giovanni Nastasi, Tigran Zalinyan.

**Figure 3.** Figure 3: FIG. 3. Simulation runtime versus simulated particle num [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Mean energy for the baseline configuration at fixed [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Velocity observables for the baseline configuration at [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 8.** Figure 8: shows δvd(t) for several values of Ex at fixed εF = 0.15 eV and fixed k-space discretization. In both panels, the oscillation period decreases as Ex increases. The dependence on the k-space discretization is shown in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Grid dependence of [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Analysis-level harmonic subtraction for the drift [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. Representative full-sum and sampled-partner com [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The sampled-partner approximation delivers a practical cost reduction for electron-electron scattering in these Pauli-consistent simulations, and the grid-drift diagnosis is a useful numerical cleanup.

read the letter

The main point is that uniform sampling from the current ensemble replaces the full partner-cell sum for proposal rates, keeps the actual collision step untouched, and still produces results close to the exact reference while cutting runtime enough to run large low-noise ensembles. In those runs the authors resolve systematic oscillations in the time traces and trace them to deterministic drift on the discretized momentum grid rather than to the physics or the approximation itself. They also outline a post-processing step to reduce the impact on recorded observables without changing the underlying dynamics. That combination is the concrete advance here. The direct comparison to the full-sum implementation is the right test, and the oscillation diagnosis follows from known properties of fixed momentum grids, so it does not rest on circular fitting. For device-scale graphene work that needs explicit carrier-carrier scattering, this is a usable efficiency step. The soft spot is that the abstract gives no quantitative error metrics, no dependence on ensemble size or density, and no check on whether residual sampling variance accumulates over long times. If the full paper supplies those numbers and shows the bias stays negligible, the central claim holds; otherwise the “close agreement” remains qualitative and the stress-test worry about possible long-term drift in the Pauli factors is still open. This is for people already running ensemble Monte Carlo on 2D materials who hit the electron-electron bottleneck. It is coherent on its own terms and deserves a serious referee, mainly for the practical method and the grid artifact fix, even if extra benchmarks would make the validation tighter. I would send it to review.

Referee Report

2 major / 1 minor

Summary. The paper introduces a sampled-partner approximation for Pauli-consistent ensemble Monte Carlo simulations of graphene that replaces the full partner-cell sum in electron-electron scattering rate evaluation with uniform sampling from the instantaneous ensemble, while leaving the collision step unchanged. It reports close agreement with a full-sum reference implementation, substantial computational savings that enable large-ensemble low-noise runs, resolution of systematic oscillatory components in ensemble-averaged time traces, and attribution of those oscillations to deterministic drift on the discretized momentum-space grid, together with a mitigation procedure for recorded observables.

Significance. If validated quantitatively, the approximation would enable efficient, low-noise simulations of hot-carrier relaxation and transport in graphene with explicit electron-electron scattering, a regime relevant to ultrafast dynamics and device modeling. The explicit diagnosis of grid-induced oscillations and the mitigation strategy represent a practical advance for ensemble Monte Carlo methods on discretized momentum grids. The work supplies an independent full-sum reference for validation, which is a strength.

major comments (2)

[Abstract] Abstract: the central claim of 'close agreement' with the full-sum reference is stated without any quantitative error metric (e.g., L2 norm on the distribution function, relative error in scattering rates, or maximum deviation), ensemble size, carrier density, or time-scale dependence. This absence makes it impossible to assess whether the sampled-partner approximation remains unbiased enough for the Pauli-consistent dynamics to hold over the simulated intervals.
[Comparison with full-sum reference] The description of the sampled-partner approximation (and its comparison section): no demonstration is provided that residual sampling variance does not couple to the deterministic grid drift or accumulate in the time evolution of the distribution function. Without such a test (e.g., convergence with ensemble size or long-time bias diagnostics), the attribution of observed oscillations solely to numerical grid drift remains inconclusive.

minor comments (1)

[Mitigation procedure] The mitigation procedure for reducing the impact of oscillations in recorded observables is mentioned but lacks a quantitative validation (e.g., before/after comparison of a conserved quantity or a known analytic limit).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight opportunities to strengthen the quantitative presentation of our results, which we will address in a revised manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim of 'close agreement' with the full-sum reference is stated without any quantitative error metric (e.g., L2 norm on the distribution function, relative error in scattering rates, or maximum deviation), ensemble size, carrier density, or time-scale dependence. This absence makes it impossible to assess whether the sampled-partner approximation remains unbiased enough for the Pauli-consistent dynamics to hold over the simulated intervals.

Authors: We agree that quantitative support for the 'close agreement' claim will improve the abstract and the paper. In the revision we will add explicit error metrics (L2 norm and maximum deviation of the distribution function between sampled-partner and full-sum runs), together with the ensemble size, carrier density, and time scales employed in the comparisons. revision: yes
Referee: [Comparison with full-sum reference] The description of the sampled-partner approximation (and its comparison section): no demonstration is provided that residual sampling variance does not couple to the deterministic grid drift or accumulate in the time evolution of the distribution function. Without such a test (e.g., convergence with ensemble size or long-time bias diagnostics), the attribution of observed oscillations solely to numerical grid drift remains inconclusive.

Authors: The existing comparison already shows that the oscillations appear identically in both the sampled-partner runs and the full-sum reference (which contains no sampling variance). This indicates that the oscillations are independent of the approximation. To make the separation explicit, the revised manuscript will include an ensemble-size convergence study demonstrating that the oscillatory amplitude does not diminish as sampling noise is reduced, thereby confirming the deterministic grid origin. revision: yes

Circularity Check

0 steps flagged

No significant circularity; approximation validated against independent full-sum reference

full rationale

The paper introduces a sampled-partner approximation for electron-electron scattering rates in Pauli-consistent ensemble Monte Carlo and directly compares its output to a full partner-cell sum implementation on the same ensemble. This comparison is an external benchmark rather than a self-referential fit or redefinition. No derivation step reduces by construction to its own inputs, no uniqueness theorem is imported from prior self-work, and no ansatz is smuggled via citation. The diagnosis of grid-induced oscillations rests on known properties of discretized momentum space, not on parameters fitted to the target observables. The central claim therefore remains self-contained against the provided reference implementation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the assumption that the Monte Carlo ensemble remains representative under the sampled approximation and that the momentum-space grid discretization is the sole source of the observed oscillations; no explicit free parameters or invented entities are stated in the abstract.

axioms (2)

domain assumption The instantaneous ensemble provides an unbiased sample for estimating electron-electron scattering rates under the Pauli exclusion principle.
Invoked when replacing the full partner-cell sum with uniform sampling.
domain assumption Oscillations in ensemble-averaged time traces arise solely from deterministic drift on the discretized momentum grid rather than from physical dynamics or the approximation itself.
Used to classify the oscillations as numerical and to justify the mitigation procedure.

pith-pipeline@v0.9.0 · 5413 in / 1491 out tokens · 45189 ms · 2026-05-10T03:06:37.046212+00:00 · methodology

discussion (0)

Reference graph

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If (i ′ 1, j′

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Transport phonema in low dimensional structures: models, simulations and theoretical aspects

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[1] [1]

(1−f(t,k 1))(1−f(t,k 2)) −S ee(k1,k 2,k ′ 1,k ′ 2)f(t,k 1)f(t,k 2) ×(1−f(t,k ′ 1))(1−f(t,k ′ 2)) i dk2 dk′ 1 dk′ 2, (6) whereS ee(k1,k 2,k ′ 1,k ′

work page

[2] [2]

Att= 0, the distribution function is the Fermi–Dirac distribution at temperatureTand Fermi energyε F , f(0,k) = 1 1 + exp ε(k)−ε F kBT ,(7) wherek B is the Boltzmann constant

is the transition probability per unit time from the initial two-electron state (k 1,k 2) to the final state (k ′ 1,k ′ 2). Att= 0, the distribution function is the Fermi–Dirac distribution at temperatureTand Fermi energyε F , f(0,k) = 1 1 + exp ε(k)−ε F kBT ,(7) wherek B is the Boltzmann constant. The areal carrier density, mean energy, and mean ve- loci...

work page

[3] [3]

(12) 3 With the Dirac dispersion, the second condition is equiv- alent to|k 1|+|k 2|=|k ′ 1|+|k ′ 2|

+ε(k ′ 2). (12) 3 With the Dirac dispersion, the second condition is equiv- alent to|k 1|+|k 2|=|k ′ 1|+|k ′ 2|. According to the Fermi golden rule [18], See(k1,k 2,k ′ 1,k ′ 2) = 2π ℏ |M|2 δ ε(k′

work page

[4] [4]

+ε(k ′ 2)−ε(k 1)−ε(k 2) . (13) The interaction matrix element is |M|2 = 1 2 h |V(q)| 2 +|V(q ′)|2 −V(q)V(q ′) i ,(14) whereV(q) andV(q ′) are the Coulomb potentials be- tween pairs of electrons V(q) = 2πe2 ϵ(q)q A 1 + cos(ϕk1,k′ 1) 2 1 + cos(ϕk2,k′ 2) 2 ,(15) V(q ′) = 2πe2 ϵ(q′)q ′ A 1 + cos(ϕk1,k′ 2) 2 1 + cos(ϕk2,k′ 1) 2 ,(16) withq=|k 1 −k ′ 1|andq ′ =...

work page

[5] [5]

The method uses a drift–collision splitting on a uniform macro time gridt r =r∆tand represents the distribution on an occupancy-limitedk-space grid

Pauli-consistent NEMC with full-sum evaluation of the e–e scattering rate Transport is simulated by the Pauli-consistent ‘new’ ensemble Monte Carlo (NEMC) method [12, 14]. The method uses a drift–collision splitting on a uniform macro time gridt r =r∆tand represents the distribution on an occupancy-limitedk-space grid. Momentum space is discretized unifor...

work page

[6] [6]

If (i ′ 1, j′

and (i ′ 2, j′ 2), we compute the corresponding ef- fective destination occupancies occ eff i′ 1j′ 1 and occ eff i′ 2j′ 2 by subtracting the contributions of the two initial parti- cles whenever an initial cell coincides with a destination cell. If (i ′ 1, j′

work page

[7] [7]

= (i ′ 2, j′ 2), the second test uses the up- dated effective occupancy to account for the first place- ment. We then definef eff i′ 1j′ 1 = occ eff i′ 1j′ 1 /Mandf eff i′ 2j′ 2 = occeff i′ 2j′ 2 /M, drawη 1, η2 ∼ U(0,1), and accept the event if feff i′ 1j′ 1 < η 1 andf eff i′ 2j′ 2 < η 2 (equivalently, with probability (1−f eff i′ 1j′ 1 )(1−f eff i′ 2j′ ...

work page

[8] [8]

(29) depends on the evolving distribution through the weightsf ij and is evaluated for each queried k1 by a full sum over all partner cells of thekgrid

Computational cost of explicit full-sum e–e simulations Explicit intraband e–e scattering is substantially more expensive than e–ph scattering because the e–e scatter- ing rate in Eq. (29) depends on the evolving distribution through the weightsf ij and is evaluated for each queried k1 by a full sum over all partner cells of thekgrid. This evaluation also...

work page

[9] [9]

Transport phonema in low dimensional structures: models, simulations and theoretical aspects

Oscillations in NEMC time-dependent observables In the NEMC simulations considered here, ensemble- averaged observables exhibit systematic oscillations in their time dependence. As a representative example, Fig. 2 shows the drift velocity. In Figs. 2(a) and 2(c), the results obtained with dif- ferent ensemble sizes are difficult to distinguish over the fu...

work page

[10] [10]

Metropolis and S

N. Metropolis and S. Ulam, The monte carlo method, Journal of the American Statistical Association44, 335 (1949)

work page 1949

[11] [11]

T. Kurosawa, Monte carlo calculation of hot electron problems, Journal of the Physical Society of Japan21, 424 (1966), proceedings of the International Conference on the Physics of Semiconductors, Kyoto, 1966

work page 1966

[12] [12]

Jacoboni and P

C. Jacoboni and P. Lugli,The Monte Carlo Method for Semiconductor Device Simulation, Computational Mi- croelectronics (Springer-Verlag, Wien and New York, 1989)

work page 1989

[13] [13]

Jacoboni,Theory of Electron Transport in Semicon- ductors, Springer Series in Solid-State Sciences, Vol

C. Jacoboni,Theory of Electron Transport in Semicon- ductors, Springer Series in Solid-State Sciences, Vol. 165 (Springer, Berlin, Heidelberg, 2010)

work page 2010

[14] [14]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science306, 666 (2004)

work page 2004

[15] [15]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Reviews of Modern Physics81, 109 (2009)

work page 2009

[16] [16]

Das Sarma, S

S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Electronic transport in two-dimensional graphene, Rev. Mod. Phys.83, 407 (2011)

work page 2011

[17] [17]

Lugli and D

P. Lugli and D. K. Ferry, Degeneracy in the ensemble Monte Carlo method for high-field transport in semicon- ductors, IEEE Transactions on Electron Devices32, 2431 (1985)

work page 1985

[18] [18]

Tadyszak, F

P. Tadyszak, F. Danneville, A. Cappy, L. Reggiani, L. Varani, and L. Rota, Monte carlo calculations of hot-carrier noise under degenerate conditions, Applied Physics Letters69, 1450 (1996)

work page 1996

[19] [19]

Borowik and J.-L

P. Borowik and J.-L. Thobel, Improved Monte Carlo method for the study of electron transport in degener- ate semiconductors, Journal of Applied Physics84, 3706 (1998)

work page 1998

[20] [20]

Borowik and L

P. Borowik and L. Adamowicz, Improved algorithm for Monte Carlo studies of electron transport in degenerate semiconductors, Physica B: Condensed Matter365, 235 (2005)

work page 2005

[21] [21]

Romano, A

V. Romano, A. Majorana, and M. Coco, Dsmc method consistent with the pauli exclusion principle and com- parison with deterministic solutions for charge transport in graphene, Journal of Computational Physics302, 267 (2015)

work page 2015

[22] [22]

M. Coco, A. Majorana, and V. Romano, Cross valida- tion of discontinuous Galerkin method and Monte Carlo simulations of charge transport in graphene on substrate, Ricerche di Matematica66, 201 (2017)

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