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Propagation of chaos and approximation error of random batch particle system in the mean field regime

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Pith's one-line read For random batch particle systems, the k-particle law stays within relative entropy O(k^2/N^2 + kτ^2) of the mean-field law, with constants independent of N and k.

desk verdict Sharp RBM relative entropy bound is a real advance, but the key tau-squared lemma needs a repaired proof before the theorem is fully certified. read the letter →

arxiv 2505.12172 v1 pith:2JYC4BSN submitted 2025-05-17 math.NA cs.NAmath.PR

Lei Li , Yuelin Wang , Shi Jin This is my paper · ORCID
classification math.NAcs.NAmath.PR MSC 60H1060K3565C3535Q84
keywords randombatchmethodmean-fieldlimitpropagationofchaosBBGKYhierarchyrelativeentropyinteractingparticlesystemstimediscretizationerror
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Randomly dividing N particles into small batches at every time step instead of computing all pairwise forces does not spoil the mean-field limit: this paper proves that the k-particle marginal of the batch system's law is exponentially close, in relative entropy, to the k-fold tensorized mean-field law. The main theorem gives $H(\mu^k_t\,\|\,\bar{\mu}_t^{\otimes k}) \le C t e^{C t}(k^2/N^2 + k\tau^2)$, with a constant $C$ independent of $N$ and $k$, so the finite-size error scales like $1/N^2$ and the batch-resampling error scales like $\tau^2$. This is a quantitative propagation of chaos for the random batch system and, simultaneously, its approximation error to the classical mean-field dynamics. The $N^{-2}$ rate is sharp and improves earlier entropy estimates that were only $O(1/N)$ for the scaled one-particle entropy, and it is obtained without a logarithmic Sobolev inequality. The standard relative-entropy/total-variation inequality then gives total-variation propagation of chaos at order $\mathcal{O}(k/N + \sqrt{k}\,\tau)$.

What carries the argument

The load-bearing device is a BBGKY-type hierarchy for the time-marginal densities: a chain of equations for $\mu^k_t$ in which the level-$k$ equation involves the $(k+1)$-marginal, obtained here from the continuity equation of the random batch SDE rather than from a path-space representation. Because the random batches are redrawn at each time step, the hierarchy carries an extra residual term $I_k$; the key construction is an auxiliary coupled batch division $\xi_n^i$ satisfying $P(j\in \xi_n(i)\mid \xi_n^i)=(p-1)/(N-1)$, which converts the random-batch interaction into the full mean-field interaction plus a fluctuation. That fluctuation is bounded by Proposition 3.2, a fourth-moment estimate of the difference between two coupled densities, using growth estimates for the logarithmic gradient along one time step. A coarse entropy bound for the full $N$-particle law comes from a large-deviation estimate for sums of centered random variables, and an iterated exponential-integral lemma then converts the hierarchy into the final $k^2/N^2+k\tau^2$ rate.

What would settle it

Take a linear interaction $b(x)=x$ in dimension $1$ with $\sigma>0$, zero external drift, Gaussian initial data, and fixed batch size $p$; both the random batch system and the mean-field SDE are Gaussian, so $H(\mu_t^1\,\|\,\bar{\mu}_t)$ can be computed exactly from the covariance evolution. Let $N$ grow with $\tau$ chosen so small that the $\tau^2$ term is negligible compared with $1/N^2$; if the exact entropy decays slower than $1/N^2$, or if at fixed $N$ it decays as $\tau$ rather than $\tau^2$, then Theorem 2.1's rate is false.

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Extended reading notes

Core claim

The paper's central claim is Theorem 2.1. For the random batch dynamics with constant diffusion $\sigma>0$, a one-sided Lipschitz external drift, a globally Lipschitz interaction kernel with bounded second derivatives, and a sub-Gaussian initial law with finite gradient information, the joint law $\mu^N_t$ of the batch system satisfies $H(\mu^k_t\,\|\,\bar{\mu}_t^{\otimes k})\le C t e^{C t}(k^2/N^2+k\tau^2)$ for every $k\le N$ and $t>0$, where $\mu^k_t$ is the $k$-particle marginal and $\bar{\mu}_t$ solves the mean-field SDE. The proof writes a BBGKY-type hierarchy for time-marginals of the batch process, isolates the extra randomness of batch redrawing in a residual term $I_k$, and controls that residual with a coupled auxiliary division. The constants do not depend on $N$ or $k$; the $1/N^2$ dependence is sharp in the entropy sense.

Load-bearing premise

The force between particles must have a global Lipschitz constant and bounded second derivatives, so the theorem does not cover singular forces such as an electrostatic or gravitational $1/r$ interaction; without that regularity the proof's core estimates do not hold.

Editorial extensions

If this is right

  • For $k=1$, the law of a single batch particle is within relative entropy $O(1/N^2+\tau^2)$ of the mean-field law, so the random batch approximation does not degrade the order of accuracy of the full $N$-particle system.
  • Corollary 2.1 gives $\|\mu^k_t-\bar{\mu}_t^{\otimes k}\|_{\mathrm{TV}}\le C\sqrt{t}e^{Ct}(k/N+\sqrt{k}\tau)$, a total-variation propagation of chaos with the same sharp $N$-dependence.
  • The constants are independent of $N$ and $k$ up to the $t e^{Ct}$ prefactor, so the bound applies uniformly to mesoscopic blocks of $k$ particles, not only to single-particle marginals.
  • Choosing $\tau$ and $N$ together can balance the two error sources: setting $\tau$ of order $1/N$ makes the $k^2/N^2$ and $k\tau^2$ terms comparable, a concrete target for simulation design.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the auxiliary-division coupling is modular and should transfer to other random batch variants (second-order dynamics, nonconstant diffusion) as long as a one-step logarithmic-gradient growth estimate like the paper's Lemma 5.3 can be established.
  • Editorial inference: because the proof avoids logarithmic Sobolev inequalities, the bound is finite-time with an explicit $t e^{Ct}$ prefactor; a uniform-in-time version would require adding a mixing or log-Sobolev condition, a step the paper does not take.
  • Editorial inference: for singular kernels such as $1/r$ forces, a regularized extension would likely add a cutoff-dependent constant to $C$ in Theorem 2.1, and the practical question is how the cutoff, batch size, and time step should be balanced.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the random batch method (RBM) for N-particle interacting diffusions of the form (1.1) and derives a quantitative propagation-of-chaos estimate. The main result, Theorem 2.1, states that under Lipschitz-type assumptions on the interaction kernel, bounded second derivatives, finite initial Fisher information, sub-Gaussian initial data, and sigma > 0, the k-particle marginal mu^k_t of the RBM joint law satisfies H(mu^k_t | bar-mu_t^{otimes k}) <= C t e^{C t} (k^2/N^2 + k tau^2), with C independent of N and k. The proof combines a BBGKY hierarchy for time-marginal distributions, an auxiliary random batch division that couples two batch configurations (Section 3.2), a local discrepancy estimate (Proposition 3.2) that produces the tau^2 rate, a coarse entropy bound via large deviations (Lemma 4.1), and an iteration argument for the entropy hierarchy (Section 4.3). The paper also includes auxiliary lemmas on moment bounds, Fisher-information growth for the RBM density, and iterated exponential integrals.

Significance. If the proof is completed, the result is a significant improvement over [19], which gave O(1/N + tau^2) scaled entropy, by obtaining a sharp O(1/N^2 + tau^2) rate for k=1 and O(k^2/N^2 + k tau^2) for general k, without using log-Sobolev inequalities. The paper is a serious analytic contribution to the mean-field analysis of the RBM. Strengths include a derivation from explicit assumptions with no fitted parameters, an original coupling construction for the random batch divisions, and a main theorem giving explicit and falsifiable rates. The principal limitation is scope: Assumption 2.1.2 requires a globally Lipschitz kernel with bounded second derivatives, so the physically motivated singular kernels mentioned in the introduction are not covered. This is a scope limitation rather than an internal inconsistency.

major comments (3)
  1. [§3.3, Proposition 3.2 (Eqs. (3.8)–(3.9))] The step from the summed quantity E(t) to the per-i bound (3.8) is only justified by the phrase 'by symmetry with respect to i'. Because the N-particle law is exchangeable, each summand has the same expectation, so the argument is valid, but the division by N should be written explicitly: after deriving E(t) <= C N e^{Ct}(t - t_n)^4 from (3.9), one should conclude that each summand, and hence the left-hand side of (3.8), is bounded by C e^{Ct}(t - t_n)^4. Please add this line so that the proof is self-contained.
  2. [§3.3, Eqs. (3.13)–(3.14)] The proof asserts E sum_{j in Lambda_i} int |(p-1) delta b_i|^8 (f + tilde f) ≲ p without any derivation. For the row i itself, (p-1) delta b_i(i) is a sum of p-1 differences of b-values between the original and auxiliary batch, so its 8th moment is typically O(p^4) by Rosenthal-type inequalities, not O(p); the displayed factor p is therefore not immediate. Since the p-dependence can be absorbed into the constant, which is allowed to depend on p, this gap is likely repairable. However, Proposition 3.2 is the unique source of the tau^2 term in Theorem 2.1, so this estimate must be justified for the main theorem to be certified.
  3. [§3.3, Eqs. (3.12)–(3.13)] There is an exponent and notation mismatch. If f_j denotes the density derivative partial_{x_j} f, then int |f_j|^8/(f + tilde f)^7 is not equal to int |grad_{x_j} log f|^8 f; the correct identity is int |partial_{x_j} f|^8 / f^7 = int |grad_{x_j} log f|^8 f. If f_j instead denotes the log-density derivative, then the expansion of grad_x · (...) in (3.12) is not the one obtained from the Liouville equation. The proof should fix the notation so that the estimates match Lemma 5.3 and Lemma 5.4, which are stated in terms of int |grad log f|^q f.
minor comments (5)
  1. [Assumption 2.1] The assumption refers to 'b0 and b1', but b1 is not defined; the interaction kernel is denoted b throughout the paper.
  2. [Lemma 5.4] The proof contains the identity int |grad log g|^q g = int |grad g|^q / g, which is false for q ≠ 2; the correct denominator exponent is q - 1. The statement of Lemma 5.4 itself appears correct, but the displayed identity should be corrected.
  3. [Section 4.3, before Eq. (4.13)] 'For noatational convenience' should read 'For notational convenience'.
  4. [Remark 2.3] The phrase 'the second line of (4.6)' is imprecise because (4.6) is a chain of inequalities; please point to the specific line or renumber the display.
  5. [After Theorem 2.1] Please state explicitly that the constant C may depend on p, T, sigma, and the Lipschitz and Fisher-information constants, and that the estimate is not uniform in time.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is a parameter-free derivation from stated assumptions; the flagged gaps in Proposition 3.2 are proof-completeness issues, not circular reductions.

full rationale

The paper derives Theorem 2.1 from Assumptions 2.1 and 2.2 by a standard entropy/BBGKY argument. No fitted parameter is called a prediction, no input is defined in terms of the target, and no uniqueness claim is imported from the authors' prior work. The τ^2 rate enters only through Proposition 3.2, which is proved via Fisher-information estimates and moment bounds rather than assumed. The authors cite their own earlier work ([26], [10], [19]) for auxiliary moment bounds, Fisher-information estimates, and exponential concentration lemmas, but these are either proved inside the present text (e.g., Lemma 5.3) or are independent published results whose assumptions do not include the target entropy estimate; hence the self-citations are not load-bearing circularity. Two non-circular completeness gaps deserve explicit flagging in Proposition 3.2: (i) the per-i estimate (3.8) is recovered from the summed bound E(t) ≤ C N e^{Ct}(t−t_n)^4 only by an unstated symmetry/exchangeability step, and (ii) the estimate E Σ_{j∈Λ_i} ∫ |(p−1)δb_i|^8(f+f~) ≲ p, used to obtain (3.14), is asserted without proof. These gaps affect the certification of the τ^2 rate but do not make the derivation circular. The restriction to Lipschitz kernels with bounded second derivatives (Assumption 2.1.2) is an explicit scope limitation excluding singular kernels, not a circularity. Overall, the derivation is self-contained in the sense that the central claim is reduced to stated assumptions and independently established lemmas, with no step equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No new physical entities or fitted parameters. The auxiliary random batch division xi^i_n is a proof device, not a claimed physical mechanism.

assumptions (3)
  • domain assumption Interaction kernel b is Lipschitz with bounded second derivatives; external drift b0 is one-sided Lipschitz.
    Assumption 2.1; used throughout to control moment growth and Fisher information.
  • domain assumption Initial law mu0 has finite Fisher information and is sub-Gaussian.
    Assumption 2.2; ensures density regularity and moment bounds (Lemma 5.1, 5.2).
  • standard math Standard tools: Gronwall's inequality, Ito calculus, Hoeffding inequality, Large Deviation bounds, Lemma 5.5 and 5.7.
    These are cited or standard and not proved in the paper.

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Cite this review

Pith. "Pith review of Propagation of chaos and approximation error of random batch particle system in the mean field regime." pith.science (2026). https://pith.science/paper/2JYC4BSN

@misc{pith2026250512172,
  author       = {Pith},
  title        = {Pith review of: Propagation of chaos and approximation error of random batch particle system in the mean field regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JYC4BSN}},
  note         = {Machine review of arXiv:2505.12172}
}
abstract

The random batch method [J. Comput. Phys. 400 (2020) 108877] is not only an efficient algorithm for simulation of classical $N$-particle systems and their mean-field limit, but also a new model for interacting particle system that could be more physical in some applications. In this work, we establish the propagation of chaos for the random batch particle system and at the same time obtain its sharp approximation error to the classical mean field limit of $N$-particle systems. The proof leverages the BBGKY hierarchy and achieves a sharp bound both in the particle number $N$ and the time step $\tau$. In particular, by introducing a coupling of the division of the random batches to resolve the $N$-dependence, we derive an $\mathcal{O}(k^2/N^2 + k\tau^2)$ bound on the $k$-particle relative entropy between the law of the system and the tensorized law of the mean-field limit. This result provides a useful understanding of the convergence properties of the random batch system in the mean field regime.

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Reference graph

Works this paper leans on

46 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [19]

    Mean field error estimate of the random batch method for large interacting particle system

    Zhenyu Huang, Shi Jin, and Lei Li. Mean field error estimate of the random batch method for large interacting particle system. ESAIM: Mathematical Modelling and Numerical Analysis, 59(1):265–289, 2025. 21

    work page 2025

  2. [1]

    Characterization of radially symmetric finite time blowup in multidimensional aggregation equations.SIAM Journal on Mathematical Analysis, 44(2):651–681, 2012

    Andrea L Bertozzi, John B Garnett, and Thomas Laurent. Characterization of radially symmetric finite time blowup in multidimensional aggregation equations.SIAM Journal on Mathematical Analysis, 44(2):651–681, 2012. 20

    work page 2012

  3. [2]

    A duality method for mean-field limits with singular interactions

    Didier Bresch, Mitia Duerinckx, and P-E Jabin. A duality method for mean-field limits with singular interactions. arXiv preprint arXiv:2402.04695 , 2024

    arXiv 2024

  4. [3]

    Biology of synchronous flashing of fireflies, 1966

    John Buck and Elisabeth Buck. Biology of synchronous flashing of fireflies, 1966

    1966

  5. [4]

    Random batch particle methods for the homogeneous Landau equation

    Jos´ e Antonio Carrillo, Shi Jin, and Yijia Tang. Random batch particle methods for the homogeneous Landau equation. arXiv preprint arXiv:2110.06430 , 2021

    work page Pith review arXiv 2021

  6. [5]

    Entropy on the path space and application to singular diffusions and mean-field models

    Patrick Cattiaux. Entropy on the path space and application to singular diffusions and mean-field models. arXiv preprint arXiv:2404.09552 , 2024

    arXiv 2024

  7. [6]

    Propagation of chaos: a review of models, methods and applications

    Louis-Pierre Chaintron and Antoine Diez. Propagation of chaos: a review of models, methods and applications. i. models and methods. arXiv preprint arXiv:2203.00446 , 2022

    arXiv 2022

  8. [7]

    Propagation of chaos: a review of models, methods and applications

    Louis-Pierre Chaintron and Antoine Diez. Propagation of chaos: a review of models, methods and applications. ii. applications. arXiv preprint arXiv:2106.14812 , 2022

    arXiv 2022

Show all 46 references
  1. [8]

    Sanov property, generalized i-projection and a conditional limit theorem

    Imre Csisz´ ar. Sanov property, generalized i-projection and a conditional limit theorem. The Annals of Probability , pages 768–793, 1984

    1984

  2. [9]

    Emergent behavior in flocks

    Felipe Cucker and Steve Smale. Emergent behavior in flocks. IEEE Transactions on automatic control, 52(5):852–862, 2007

    2007

  3. [10]

    A collision-oriented interacting particle system for landau-type equations and the molecular chaos

    Kai Du and Lei Li. A collision-oriented interacting particle system for landau-type equations and the molecular chaos. arXiv preprint arXiv:2408.16252 , 2024

    2024 arXiv

  4. [11]

    Rbmd: A molecular dynamics package enabling to simulate 10 million all-atom particles in a single graphics processing unit

    Weihang Gao, Teng Zhao, Yongfa Guo, Jiuyang Liang, Huan Liu, Maoying Luo, Zedong Luo, Wei Qin, Yichao Wang, Qi Zhou, et al. Rbmd: A molecular dynamics package enabling to simulate 10 million all-atom particles in a single graphics processing unit. arXiv preprint arXiv:2407.093...

    2024

  5. [12]

    On the dynamics of large particle systems in the mean field limit

    Fran¸ cois Golse. On the dynamics of large particle systems in the mean field limit. In Macroscopic and large scale phenomena: coarse graining, mean field limits and ergod- icity, pages 1–144. Springer, 2016

    2016

  6. [13]

    Sharp propagation of chaos for McKean-Vlasov equation with non constant diffusion coefficient

    Jules Grass, Arnaud Guillin, and Christophe Poquet. Sharp propagation of chaos for McKean-Vlasov equation with non constant diffusion coefficient. arXiv preprint arXiv:2410.20874, 2024

    2024 arXiv

  7. [14]

    Uniform-in-time error estimate of the random batch method for the Cucker–Smale model

    Seung-Yeal Ha, Shi Jin, Doheon Kim, and Dongnam Ko. Uniform-in-time error estimate of the random batch method for the Cucker–Smale model. Mathematical Models and Methods in Applied Sciences , 31(06):1099–1135, 2021

    2021

  8. [15]

    A simple proof of the cucker-smale flocking dynamics and mean-field limit

    Seung-Yeal Ha and Jian-Guo Liu. A simple proof of the cucker-smale flocking dynamics and mean-field limit. Communications in Mathematical Sciences , 7(2):297–325, 2009

    2009

  9. [16]

    From particle to kinetic and hydrodynamic descrip- tions of flocking

    Seung-Yeal Ha and Eitan Tadmor. From particle to kinetic and hydrodynamic descrip- tions of flocking. Kinetic and Related Models , 1(3):415–435, 2008

    2008

  10. [17]

    Strong convergence of propagation of chaos for McKean–Vlasov SDEs with singular interactions

    Zimo Hao, Michael R¨ ockner, and Xicheng Zhang. Strong convergence of propagation of chaos for McKean–Vlasov SDEs with singular interactions. SIAM Journal on Math- ematical Analysis, 56(2):2661–2713, 2024

    2024

  11. [18]

    From 1970 until present : the Keller-Segel model in chemotaxis and its consequences I

    Dirk Horstmann. From 1970 until present : the Keller-Segel model in chemotaxis and its consequences I. Jahresbericht der Deutschen Mathematiker-Vereinigung, 105(3):103– 165, 2003

    1970

  12. [20]

    A review of the mean field limits for Vlasov equations

    Pierre-Emmanuel Jabin. A review of the mean field limits for Vlasov equations. Kinetic and Related models, 7(4):661–711, 2014

    2014

  13. [21]

    Mean field limit and propagation of chaos for Vlasov systems with bounded forces

    Pierre-Emmanuel Jabin and Zhenfu Wang. Mean field limit and propagation of chaos for Vlasov systems with bounded forces. Journal of Functional Analysis, 271(12):3588– 3627, 2016

    2016

  14. [22]

    Mean field limit for stochastic particle systems

    Pierre-Emmanuel Jabin and Zhenfu Wang. Mean field limit for stochastic particle systems. Active Particles, Volume 1: Advances in Theory, Models, and Applications , pages 379–402, 2017

    2017

  15. [23]

    Quantitative estimates of propagation of chaos for stochastic systems with W−1,∞ kernels

    Pierre-Emmanuel Jabin and Zhenfu Wang. Quantitative estimates of propagation of chaos for stochastic systems with W−1,∞ kernels. Inventiones mathematicae, 214:523– 591, 2018

    2018

  16. [24]

    Random batch methods for classical and quantum interacting particle systems and statistical samplings

    Shi Jin and Lei Li. Random batch methods for classical and quantum interacting particle systems and statistical samplings. In Active Particles, Volume 3: Advances in Theory, Models, and Applications , pages 153–200. Springer, 2021

    2021

  17. [25]

    On the mean field limit of the Random Batch Method for interacting particle systems

    Shi Jin and Lei Li. On the mean field limit of the Random Batch Method for interacting particle systems. Science China Mathematics , pages 1–34, 2022

    2022

  18. [26]

    Random batch methods (rbm) for interacting particle systems

    Shi Jin, Lei Li, and Jian-Guo Liu. Random batch methods (rbm) for interacting particle systems. Journal of Computational Physics , 400:108877, 2020

    2020

  19. [27]

    On the Random Batch Method for second order interacting particle systems

    Shi Jin, Lei Li, and Yiqun Sun. On the Random Batch Method for second order interacting particle systems. Multiscale Modeling & Simulation , 20(2):741–768, 2022

    2022

  20. [28]

    A random batch Ewald method for par- ticle systems with Coulomb interactions

    Shi Jin, Lei Li, Zhenli Xu, and Yue Zhao. A random batch Ewald method for par- ticle systems with Coulomb interactions. SIAM Journal on Scientific Computing , 43(4):B937–B960, 2021

    2021

  21. [29]

    Ergodicity and long-time behavior of the random batch method for interacting particle systems

    Shi Jin, Lei Li, Xuda Ye, and Zhennan Zhou. Ergodicity and long-time behavior of the random batch method for interacting particle systems. Mathematical Models and Methods in Applied Sciences , 33(01):67–102, 2023

    2023

  22. [30]

    Random batch algorithms for quantum monte carlo simulations

    Shi Jin and Xiantao Li. Random batch algorithms for quantum monte carlo simulations. arXiv preprint arXiv:2008.12990 , 2020

    2008 arXiv

  23. [31]

    Foundations of kinetic theory

    Mark Kac. Foundations of kinetic theory. In Proceedings of The third Berkeley sympo- sium on mathematical statistics and probability , volume 3, pages 171–197, 1956

    1956

  24. [32]

    Self-entrainment of a population of coupled non-linear oscillators

    Yoshiki Kuramoto. Self-entrainment of a population of coupled non-linear oscillators. In International Symposium on Mathematical Problems in Theoretical Physics: January 23–29, 1975, Kyoto University, Kyoto/Japan , pages 420–422. Springer, 1975

    1975

  25. [33]

    Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions

    Daniel Lacker. Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions. Probability and Mathematical Physics , 4(2):377–432, 2023

    2023

  26. [34]

    Sharp uniform-in-time propagation of chaos.Probability Theory and Related Fields , 187(1-2):443–480, 2023

    Daniel Lacker and Luc Le Flem. Sharp uniform-in-time propagation of chaos.Probability Theory and Related Fields , 187(1-2):443–480, 2023

    2023

  27. [35]

    Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations

    Lei Li, Jian-Guo Liu, and Yijia Tang. Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations. Communications in Com- putational Physics, 32(1), 2022

    2022

  28. [36]

    A random-batch Monte Carlo method for many-body systems with singular kernels

    Lei Li, Zhenli Xu, and Yue Zhao. A random-batch Monte Carlo method for many-body systems with singular kernels. SIAM Journal on Scientific Computing , 42(3):A1486– A1509, 2020

    2020

  29. [37]

    Su- perscalability of the random batch Ewald method

    Jiuyang Liang, Pan Tan, Yue Zhao, Lei Li, Shi Jin, Liang Hong, and Zhenli Xu. Su- perscalability of the random batch Ewald method. The Journal of Chemical Physics , 156(1), 2022. 22

    2022

  30. [38]

    Vorticity and incompressible flow

    Andrew J Majda, Andrea L Bertozzi, and A Ogawa. Vorticity and incompressible flow. cambridge texts in applied mathematics. Appl. Mech. Rev., 55(4):B77–B78, 2002

    2002

  31. [39]

    Genealogies and increasing propagation of chaos for feynman-kac and genetic models

    Laurent Miclo and Pierre Del Moral. Genealogies and increasing propagation of chaos for feynman-kac and genetic models. The Annals of Applied Probability , 11(4):1166– 1198, 2001

    2001

  32. [40]

    A new model for self-organized dynamics and its flocking behavior

    Sebastien Motsch and Eitan Tadmor. A new model for self-organized dynamics and its flocking behavior. Journal of Statistical Physics , 144:923–947, 2011

    2011

  33. [41]

    Heterophilious dynamics enhances consensus

    Sebastien Motsch and Eitan Tadmor. Heterophilious dynamics enhances consensus. SIAM review, 56(4):577–621, 2014

    2014

  34. [42]

    Information and information stability of random variables and pro- cesses

    Mark S Pinsker. Information and information stability of random variables and pro- cesses. Holden-Day, 1964

    1964

  35. [43]

    Flocks, herds, and schools: A quantitative theory of flocking

    John Toner and Yuhai Tu. Flocks, herds, and schools: A quantitative theory of flocking. Physical review E, 58(4):4828, 1998

    1998

  36. [44]

    High-dimensional probability: An introduction with applications in data science, volume 47

    Roman Vershynin. High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press, 2018

    2018

  37. [45]

    Springer, 2009

    C´ edric Villani et al.Optimal transport: old and new , volume 338. Springer, 2009

    2009

  38. [46]

    Error analysis of time-discrete random batch method for interacting particle systems and associated mean-field limits.IMA Journal of Numerical Analysis, 44(3):1660–1698, 2024

    Xuda Ye and Zhennan Zhou. Error analysis of time-discrete random batch method for interacting particle systems and associated mean-field limits.IMA Journal of Numerical Analysis, 44(3):1660–1698, 2024. A Computational details in Proposition 3.2. Here we present calculus detail...

    2024

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