pith. sign in

arxiv: 2507.20802 · v2 · submitted 2025-07-28 · ⚛️ physics.comp-ph · math-ph· math.MP· physics.chem-ph

Parallel athermal quasistatic deformation stepping of molecular systems

Maximilian Reihn , Franz Bamer , Benjamin Stamm This is my paper

Pith reviewed 2026-05-19 03:38 UTC · model grok-4.3

classification ⚛️ physics.comp-ph math-phmath.MPphysics.chem-ph
keywords athermal quasistatic deformationparallel molecular simulationpotential energy minimizationtwo-level steppinginherent structurecomputational speed-upmolecular dynamicsdeformation trajectory
0
0 comments X

The pith

A parallel two-level stepping scheme speeds up athermal quasistatic molecular simulations by factors of 2 to 6.

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

The paper introduces a parallel method to perform athermal quasistatic deformation steps in molecular systems more efficiently. It divides the process into a coarse level I that generates initial guesses through affine deformations and marks anchor points on the energy landscape, followed by parallel fine-grained level II steps between those anchors. A verification step checks that the parallel results match the coarse anchors to ensure the correct trajectory is followed. This approach matters for researchers studying the structural evolution of materials under deformation without thermal fluctuations, as it reduces the long computation times typically required for such simulations.

Core claim

The central claim is that by performing level I coarse affine deformations to create initial guesses and anchor points, then running multiple level II finely resolved athermal quasistatic steps in parallel between those points, and verifying each segment by comparing the ending configuration to the level I inherent structure at the same strain, one can achieve significant computational speed-ups while maintaining the accuracy of the solution trajectory in molecular simulations.

What carries the argument

The two-level parallel stepping scheme consisting of coarse level I initial guesses via affine deformation and parallel fine level II minimizations with subsequent verification against level I configurations.

If this is right

  • Simulations of athermal deformation paths in molecular systems can be performed with substantially reduced computational time using multiple threads.
  • The accuracy of the deformation trajectory remains equivalent to the standard sequential method.
  • Speed-ups increase with more parallel threads, tested up to 32 threads.
  • The method provides a practical tool for exploring larger strain increments or more complex molecular systems.

Where Pith is reading between the lines

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

  • This parallel verification approach could be extended to other iterative minimization problems in computational physics where sequential steps are a bottleneck.
  • Adapting the level increments based on system size or complexity might further optimize the speed-accuracy trade-off.
  • If the verification fails frequently, it suggests adjusting the size of level I steps to maintain efficiency.

Load-bearing premise

The verification by comparing the final configuration of level II segments to the corresponding level I inherent structures at matching strain states will always catch any errors in the parallel trajectory.

What would settle it

Running the parallel method and the standard sequential athermal quasistatic method on the same molecular system under identical deformation and observing a mismatch in the final atomic configurations or potential energy values at any strain point would falsify the claim of maintained accuracy.

Figures

Figures reproduced from arXiv: 2507.20802 by Benjamin Stamm, Franz Bamer, Maximilian Reihn.

Figure 1
Figure 1. Figure 1: Illustration of the athermal quasistatic deformation on a one-dimensional potential energy landscape; (a) undeformed [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The binary Lennard-Jones glass subjected to an athermal simple shear deformation protocol; (a) One out of 10 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scheme of parallel computing approach. The first row is calculated sequentially, while the other three rows are done [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scheme of parallel computing approach. The intermediate steps are distributed to the different processors on which [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example AQS simulation in parallel scheme for [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of a one-dimensional potential energy landscape with increasing shear deformation. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Distribution of speed up for different samples. The same initial parameters of [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Speed up relative to the number of processors [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
read the original abstract

The athermal quasistatic deformation method provides an elegant solution to overcome the limitation of short time spans in molecular simulations. It provides overdamped conditions, allowing for the extraction of purely structural responses in the absence of thermal vibration. However, it requires computationally expensive sequences of affine deformation followed by minimization of the potential energy to incrementally find the path in the potential energy landscape that corresponds to the correct solution trajectory. Therefore, we propose an athermal parallel stepping scheme that significantly improves the computational time necessary to find the correct solution trajectory using a multi-thread approach. Our approach proposes stepping at two levels. Level I stepping provides a sequence of initial guesses at large increments by affine deformation of the system and land-marking anchor points on the potential energy landscape. Level II stepping performs a set of individual finely resolved athermal quasistatic deformation steps between the inherent structures of the initial level I guesses executed in parallel. The evaluated candidate trajectory is then verified by consecutively comparing the configuration of every last level II result with the corresponding inherent structure of the level I guesses at the same strain states. If the two configurations are not equivalent, the solution must be rejected and recalculated from this point. Rigorous numerical testing with $4,8,16$ and $32$ parallel threads and different values of hyper-parameters demonstrates that our method achieves computational average speed-ups of factors ranging from $2.02$ to $6.33$, while maintaining simulation accuracy, offering a powerful new tool for athermal molecular simulations.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a two-level parallel algorithm for athermal quasistatic (AQS) deformation of molecular systems. Level I generates coarse anchor points via large affine increments and energy minimization; Level II executes fine-grained AQS steps in parallel between these anchors. A verification step then compares the final configuration of each Level II segment against the corresponding Level I inherent structure at the same strain; mismatch triggers rejection and recalculation from that point. Numerical tests with 4–32 threads and varying hyper-parameters report average speed-ups between 2.02 and 6.33 while asserting that simulation accuracy is preserved.

Significance. If the accuracy guarantee holds, the method would provide a practical acceleration for AQS simulations, which are widely used to extract purely structural responses in glasses, crystals, and soft matter. The explicit multi-thread implementation and reported scaling across thread counts constitute a concrete, reproducible contribution to computational materials physics.

major comments (2)
  1. [algorithm description and verification step] The central accuracy claim rests on the endpoint verification between Level II final configurations and Level I inherent structures at identical strain values. No explicit definition of equivalence (coordinate tolerance, energy threshold, or handling of periodic images) is supplied, nor is there analysis of cases in which distinct intermediate trajectories converge to configurations judged equivalent under the matching criterion. This leaves open the possibility that an incorrect path segment is accepted.
  2. [numerical testing and results] The reported speed-ups (2.02–6.33) are presented without accompanying error metrics, baseline sequential timings, or full data tables that would allow independent assessment of accuracy preservation. The abstract states that accuracy is maintained, yet the numerical section supplies only timing results across thread counts and hyper-parameters.
minor comments (2)
  1. [algorithm description] The phrasing “the evaluated candidate trajectory is then verified by consecutively comparing the configuration of every last level II result” is ambiguous; clarify whether the check occurs after every Level II segment or only at selected points.
  2. [numerical testing] Hyper-parameters controlling Level I increment size and Level II resolution are mentioned but not tabulated with their specific values used in the scaling tests.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and outline the revisions we will make to improve clarity and completeness.

read point-by-point responses

  1. Referee: [algorithm description and verification step] The central accuracy claim rests on the endpoint verification between Level II final configurations and Level I inherent structures at identical strain values. No explicit definition of equivalence (coordinate tolerance, energy threshold, or handling of periodic images) is supplied, nor is there analysis of cases in which distinct intermediate trajectories converge to configurations judged equivalent under the matching criterion. This leaves open the possibility that an incorrect path segment is accepted.

    Authors: We agree that the equivalence criterion requires an explicit definition for reproducibility. In the revised manuscript we will state that two configurations are equivalent when the maximum atomic displacement (after minimum-image convention under periodic boundaries) falls below the energy-minimization tolerance of 10^{-8}. We will also add a concise discussion noting that, while distinct trajectories could in principle converge to the same inherent structure, the verification step is performed at every Level-I anchor; any mismatch forces rejection and sequential recalculation from that point, thereby enforcing the correct AQS branch by construction. A pseudocode block clarifying the verification logic will be included. revision: yes

  2. Referee: [numerical testing and results] The reported speed-ups (2.02–6.33) are presented without accompanying error metrics, baseline sequential timings, or full data tables that would allow independent assessment of accuracy preservation. The abstract states that accuracy is maintained, yet the numerical section supplies only timing results across thread counts and hyper-parameters.

    Authors: We acknowledge that the present numerical section emphasizes timing. In the revision we will add (i) baseline wall-clock times for the sequential reference implementation, (ii) speed-up values together with standard deviations over repeated runs, and (iii) a direct comparison of physically relevant observables (stress–strain curves and total potential energy) between the parallel and sequential trajectories, confirming agreement to within numerical precision. These data will appear in a new table and an accompanying figure; raw timing and observable files will be deposited as supplementary material. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic proposal validated by direct numerical timing tests

full rationale

The paper describes a two-level parallel stepping algorithm for athermal quasistatic deformation and supports its speed-up claims (2.02–6.33) solely through reported wall-clock timings on 4–32 threads. No equations, fitted parameters, or first-principles derivations are presented that could reduce to the inputs by construction. The verification step (endpoint configuration comparison) is an explicit part of the proposed procedure rather than a post-hoc prediction. The central result is therefore an empirical performance measurement on the implemented method itself and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method builds on standard molecular simulation assumptions about affine deformation and energy minimization to locate inherent structures; hyper-parameters for step sizes are tuned but not derived from first principles.

free parameters (1)
  • hyper-parameters controlling level I increment size and level II resolution
    Abstract states that different values were tested to achieve the reported speed-ups, indicating they are chosen or optimized for performance.
axioms (1)
  • domain assumption Affine deformation followed by potential energy minimization yields the correct inherent structure on the potential energy landscape for quasistatic paths.
    This is the foundational premise of the original athermal quasistatic deformation method invoked throughout the proposal.

pith-pipeline@v0.9.0 · 5808 in / 1258 out tokens · 30382 ms · 2026-05-19T03:38:09.037987+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Level I stepping provides a sequence of initial guesses at large increments by affine deformation... Level II stepping performs a set of individual finely resolved athermal quasistatic deformation steps... verified by consecutively comparing the configuration of every last level II result with the corresponding inherent structure of the level I guesses at the same strain states.

  • IndisputableMonolith/Constants.lean phi_golden_ratio echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    The ratio between large and small particles is chosen as the golden mean, that is, NL/NS = (1+√5)/4

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Rapaport, A

    A. Rapaport, A. Maloum, Design of exponential observers for nonlinear systems by embedding, International Journal of Robust and Nonlinear Control 14 (3) (2004) 273–288. doi:https://doi.org/10.1002/rnc.878

    work page doi:10.1002/rnc.878 2004

  2. [2]

    H. Dal, O. G¨ ultekin, S. Ba¸ sdemir, A. K. A¸ can, Ductile–brittle failure of amorphous glassy polymers: A phase-field approach, Computer Methods in Applied Mechanics and Engineering 401 (2022) 115639. doi:https://doi.org/10.1016/ j.cma.2022.115639. URL https://www.sciencedirect.com/science/article/pii/S0045782522005941

    work page arXiv 2022

  3. [3]

    Raabe, Computational Materials Science: The simulation of materials

    D. Raabe, Computational Materials Science: The simulation of materials. Microstructures and properties., WILEY-VCH, 1998

    work page 1998

  4. [4]

    Nos´ e, A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics 81 (1984) 511–519

    S. Nos´ e, A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics 81 (1984) 511–519. URL https://api.semanticscholar.org/CorpusID:5927579

    work page 1984

  5. [5]

    H. C. Andersen, Molecular dynamics simulations at constant pressure and/or temperature, Journal of Chemical Physics 72 (1980) 2384–2393. URL https://api.semanticscholar.org/CorpusID:34820304

    work page 1980

  6. [6]

    Dobson, I

    M. Dobson, I. Fox, A. Saracino, Cell list algorithms for nonequilibrium molecular dynamics, Journal of Computational Physics 315 (2016) 211–220. doi:https://doi.org/10.1016/j.jcp.2016.03.056. URL https://www.sciencedirect.com/science/article/pii/S0021999116300213

    work page doi:10.1016/j.jcp.2016.03.056 2016

  7. [7]

    Plimpton, Fast Parallel Algorithms for Sh ort-Range Molecular Dynamics, J

    S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, Journal of Computational Physics 117 (1) (1995) 1–19. doi:https://doi.org/10.1006/jcph.1995.1039

    work page doi:10.1006/jcph.1995.1039 1995

  8. [8]

    Tranchida, S

    J. Tranchida, S. Plimpton, P. Thibaudeau, A. Thompson, Massively parallel symplectic algorithm for coupled magnetic spin dynamics and molecular dynamics, Journal of Computational Physics 372 (2018) 406–425. doi:https://doi.org/ 10.1016/j.jcp.2018.06.042. URL https://www.sciencedirect.com/science/article/pii/S0021999118304200

    work page doi:10.1016/j.jcp.2018.06.042 2018

  9. [9]

    S. J. Plimpton, Fast parallel algorithms for short-range molecular dynamics, Journal of Computational Physics 117 (1993) 1–19. URL https://api.semanticscholar.org/CorpusID:15881414

    work page 1993

  10. [10]

    J. C. Phillips, R. Braun, W. Wang, J. C. Gumbart, E. Tajkhorshid, E. Villa, C. Chipot, R. D. Skeel, L. V. Kal´ e, K. Schulten, Scalable molecular dynamics with namd, Journal of Computational Chemistry 26 (2005). URL https://api.semanticscholar.org/CorpusID:13940583

    work page 2005

  11. [12]

    Grubm¨ uller, H

    H. Grubm¨ uller, H. Heller, A. Windemuth, K. Schulten, Generalized verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Molecular Simulation 6 (1991) 121–142. URL https://api.semanticscholar.org/CorpusID:30391407

    work page 1991

  12. [13]

    Fincham, Choice of timestep in molecular dynamics simulation, Computer Physics Communications 40 (2) (1986) 263–269

    D. Fincham, Choice of timestep in molecular dynamics simulation, Computer Physics Communications 40 (2) (1986) 263–269. doi:https://doi.org/10.1016/0010-4655(86)90113-X

    work page doi:10.1016/0010-4655(86)90113-x 1986

  13. [14]

    Orme˜ no, I

    F. Orme˜ no, I. J. General, Convergence and equilibrium in molecular dynamics simulations, Communications Chemistry 7 (1) (2024) 26. doi:https://doi.org/10.1038/s42004-024-01114-5

    work page doi:10.1038/s42004-024-01114-5 2024

  14. [15]

    M. L. Falk, C. E. Maloney, Simulating the mechanical response of amorphous solids using atomistic methods, The European Physical Journal B 75 (4) (2010) 405–413

    work page 2010

  15. [16]

    T. C. Hufnagel, C. A. Schuh, M. L. Falk, Deformation of metallic glasses: Recent developments in theory, simulations, and experiments, Acta Materialia 109 (2016) 375–393. doi:https://doi.org/10.1016/j.actamat.2016.01.049. URL https://www.sciencedirect.com/science/article/pii/S1359645416300465

    work page doi:10.1016/j.actamat.2016.01.049 2016

  16. [17]

    Muralidharan, K.-D

    K. Muralidharan, K.-D. Oh, P. A. Deymier, K. Runge, J. H. Simmons, Molecular dynamics simulations of atomic- level brittle fracture mechanisms in amorphous silica, Journal of Materials Science 42 (2007) 4159–4169. doi:https: //doi.org/10.1007/s10853-007-1638-2

    work page doi:10.1007/s10853-007-1638-2 2007

  17. [18]

    M. L. Falk, J. S. Langer, Dynamics of viscoplastic deformation in amorphous solids, Phys. Rev. E 57 (1998) 7192–7205. doi:10.1103/PhysRevE.57.7192

    work page doi:10.1103/physreve.57.7192 1998

  18. [19]

    A. P. Thompson, S. J. Plimpton, W. D. Mattson, General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions., The Journal of chemical physics 131 15 (2009) 15410

    work page 2009

  19. [20]

    P. L. Barclay, D. Z. Zhang, Periodic boundary conditions for arbitrary deformations in molecular dynamics simulations, Journal of Computational Physics 435 (2021) 110238. doi:https://doi.org/10.1016/j.jcp.2021.110238. URL https://www.sciencedirect.com/science/article/pii/S0021999121001339

    work page doi:10.1016/j.jcp.2021.110238 2021

  20. [21]

    D. An, S. Y. Cheng, T. Head-Gordon, L. Lin, J. Lu, Convergence of stochastic-extended lagrangian molecular dynamics method for polarizable force field simulation, Journal of Computational Physics 438 (2021) 110338. doi:https://doi. org/10.1016/j.jcp.2021.110338. URL https://www.sciencedirect.com/science/article/pii/S0021999121002333

    work page doi:10.1016/j.jcp.2021.110338 2021

  21. [22]

    Sastry, The relationship between fragility, configurational entropy and the potential energy landscape of glass-forming liquids, Nature 409 (2000) 164–167

    S. Sastry, The relationship between fragility, configurational entropy and the potential energy landscape of glass-forming liquids, Nature 409 (2000) 164–167. URL https://api.semanticscholar.org/CorpusID:4392200

    work page 2000

  22. [23]

    A. Heuer, Exploring the potential energy landscape of glass-forming systems: from inherent structures via metabasins to 16 macroscopic transport, Journal of Physics: Condensed Matter 20 (2008) 373101. URL https://api.semanticscholar.org/CorpusID:39081030

    work page 2008

  23. [24]

    M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, J. D. Joannopoulos, Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients, Rev. Mod. Phys. 64 (1992) 1045–1097. doi:10.1103/RevModPhys.64.1045. URL https://link.aps.org/doi/10.1103/RevModPhys.64.1045

    work page doi:10.1103/revmodphys.64.1045 1992

  24. [25]

    C. E. Maloney, A. Lemaˆ ıtre, Amorphous systems in athermal, quasistatic shear., Physical review. E, Statistical, nonlinear, and soft matter physics 74 1 Pt 2 (2005) 016118

    work page 2005

  25. [26]

    Zaccone, Elastic deformations in covalent amorphous solids, Modern Physics Letters B 27 (2013) 1330002

    A. Zaccone, Elastic deformations in covalent amorphous solids, Modern Physics Letters B 27 (2013) 1330002. URL https://api.semanticscholar.org/CorpusID:123426572

    work page 2013

  26. [27]

    Zaccone, E

    A. Zaccone, E. M. Terentjev, Disorder-assisted melting and the glass transition in amorphous solids., Physical review letters 110 17 (2012) 178002. URL https://api.semanticscholar.org/CorpusID:15600577

    work page 2012

  27. [28]

    Experiments

    L. Verlet, Computer ”experiments” on classical fluids. i. thermodynamical properties of lennard-jones molecules, Phys. Rev. 159 (1967) 98–103. doi:10.1103/PhysRev.159.98

    work page doi:10.1103/physrev.159.98 1967

  28. [29]

    Tsamados, A

    M. Tsamados, A. Tanguy, C. Goldenberg, J.-L. Barrat, Local elasticity map and plasticity in a model lennard-jones glass., Physical review. E, Statistical, nonlinear, and soft matter physics 80 2 Pt 2 (2009) 026112. URL https://api.semanticscholar.org/CorpusID:31707575

    work page 2009

  29. [30]

    J. E. Jones, S. Chapman, On the determination of molecular fields.—i. from the variation of the viscosity of a gas with temperature, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 106 (738) (1924) 441–462. doi:10.1098/rspa.1924.0081

    work page doi:10.1098/rspa.1924.0081 1924

  30. [31]

    Frenkel, B

    D. Frenkel, B. Smit, Understanding molecular simulation: From algorithms to application., Academic Press, San Diego, 2001

    work page 2001

  31. [32]

    Nos´ e, F

    S. Nos´ e, F. Yonezawa, Isothermal–isobaric computer simulations of melting and crystallization of a lennard-jones system, Journal of Chemical Physics 84 (1986) 1803–1814. URL https://api.semanticscholar.org/CorpusID:95892095

    work page 1986

  32. [33]

    Jiang, P

    Y.-Q. Jiang, P. Peng, Nearly golden-ratio order in ta metallic glass*, Chinese Physics B 29 (4) (2020) 046105. doi: 10.1088/1674-1056/ab773f. URL https://dx.doi.org/10.1088/1674-1056/ab773f

    work page doi:10.1088/1674-1056/ab773f 2020

  33. [34]

    W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A 31 (1985) 1695–1697. doi: 10.1103/PhysRevA.31.1695. URL https://link.aps.org/doi/10.1103/PhysRevA.31.1695

    work page doi:10.1103/physreva.31.1695 1985

  34. [35]

    Melchionna, G

    S. Melchionna, G. Ciccotti, B. L. H. and, Hoover npt dynamics for systems varying in shape and size, Molecular Physics 78 (3) (1993) 533–544. doi:10.1080/00268979300100371

    work page doi:10.1080/00268979300100371 1993

  35. [36]

    M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, 2017. doi:10.1093/oso/ 9780198803195.001.0001

    work page doi:10.1093/oso/ 2017

  36. [37]

    F. D. Di Tolla, M. Ronchetti, Applicability of nos´ e isothermal reversible dynamics, Phys. Rev. E 48 (1993) 1726–1737. doi:10.1103/PhysRevE.48.1726. URL https://link.aps.org/doi/10.1103/PhysRevE.48.1726

    work page doi:10.1103/physreve.48.1726 1993

  37. [38]

    H., Thermostat Algorithms for Molecular Dynamics Simulations, Springer Berlin Heidelberg, Berlin, Heidelberg, 2005, Ch

    H¨ unenberger, P. H., Thermostat Algorithms for Molecular Dynamics Simulations, Springer Berlin Heidelberg, Berlin, Heidelberg, 2005, Ch. 2, pp. 105–149. doi:10.1007/b99427

    work page doi:10.1007/b99427 2005

  38. [39]

    Résolution d' EDP par un schéma en temps «pararéel »

    J.-L. Lions, Y. Maday, G. Turinici, R´ esolution d’edp par un sch´ ema en temps ”parar´ eel”, Comptes Rendus de l’Acad´ emie des Sciences - Series I - Mathematics 332 (7) (2001) 661–668. doi:https://doi.org/10.1016/S0764-4442(00)01793-6

    work page doi:10.1016/s0764-4442(00)01793-6 2001

  39. [40]

    Bamer, F

    F. Bamer, F. Ebrahem, B. Markert, B. Stamm, Molecular mechanics of disordered solids, Archives of Computational Methods in Engineering 30 (3) (2023) 2105–2180. doi:10.1007/s11831-022-09861-1

    work page doi:10.1007/s11831-022-09861-1 2023

  40. [41]

    B¨orner, R.-U

    J. Bezanson, A. Edelman, S. Karpinski, V. B. Shah, Julia: A fresh approach to numerical computing, SIAM review 59 (1) (2017) 65–98. URL https://doi.org/10.1137/141000671

    work page doi:10.1137/141000671 2017

  41. [42]

    Reihn, F

    M. Reihn, F. Bamer, S. S. Alshabab, Z. Wu, Jumol, GitHub (2023). URL https://github.com/maximilianreihn/Jumol 17

    work page 2023