Calculating Macroscopic Consequences of Microscopic Interactions

Last update: 12 Apr 2025 10:32
First version: 19 June 2006

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[A placeholder while I organize my thoughts]

--- I've had that line about "a placeholder" there since 2006; organizing my thoughts on this is clearly the project of several decades, and perhaps my career.

See also:
• Statistical mechanics (especially, perhaps, non-equilibrium statistical mechanics)
• cellular automata (especially their continuum limits)
• evolutionary theory
• economics
• sociology
• simulation modeling
• statistical emulators for simulation models
• large deviations of stochastic processes
• emergence
• Mroi-Zwanzig formalism
• mean-field games and mean-field control
• Agent-Based Modeling

Recommended, big picture:
• Dror Givon, Raz Kupferman and Andrew Stuart, "Extracting macroscopic dynamics: model problems and algorithms", Nonlinearity 17 (2004): R55--R127 [PDF preprint]
• Clark Glymour, "When Is a Brain Like the Planet?", Philosophy of Science 74 (2007): 330--347
• Thomas Schelling, Micromotives and Macrobehavior

Recommended, close-ups:
• David Cai, Louis Tao and David W. McLaughlin, "An embedded network approach for scale-up of fluctuation-driven systems with preservation of spike information", Proceedings of the National Academy of Sciencces (USA) 101 (2004): 14288--14293 [Interesting, but I really needed to read their prior paper on their coarse-graining method (below) to evaluate this.]
• R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner and S. W. Zucker, "Geometric diffusions as a tool for harmonic analysis and structure definition of data"
  1. "Diffusion maps", Proceedings of the National Academy of Sciences 102 (2005): 7426--7431
  2. "Multiscale methods", Proceedings of the National Academy of Sciences 102 (2005): 7432--7437
• Olof Gonerup and Martin Nilsson Jacobi, "A method for inferring hierarchical dynamics in stochastic processes", nlin.AO/0703034
• Navot Israeli and Nigel Goldenfeld, "Coarse-graining of cellular automata, emergence, and the predictability of complex systems", nlin.CG/0508033
• Alberto Montfusco, Dynamic Coarse-Graining via Large-Deviation Theory [Doctoral thesis, ETH Zurich, 2019]
• Martin Nilsson Jacobi, "Quotient Manifold Projections and Hierarchical Dynamics in Smooth Dynamical Systems" [Lie-algebraic techniques for determining when a low-dimensional projection of a high-dimensional dynamical system will itself be a self-contained dynamical system. I heard Martin talk about it, and it's very cool stuff.]
• Martin Nilsson Jacobi, Olof Goernerup, "A dual eigenvector condition for strong lumpability of Markov chains", arxiv:0710.1986

To read:
• David Andrieux, "Bounding the coarse graining error in hidden Markov dynamics", arxiv:1104.1025
• Masano Aoki
  • New Approaches to Macroeconomic Modeling: Evolutionary Stochastic Dynamics, Multiple Equilibria, and Externalities as Field Effects
  • Modeling Aggregate Behavior and Fluctuations in Economics: Stochastic Views of Interacting Agents
• Karen Ball, Tom Kurtz, Lea Popovic and Greg Rempala, "Asymptotic analysis of multiscale approximations to reaction networks", math.PR/0508015
• Robert W. Batterman
  • "The tyranny of scales", phil-sci/8678
  • A Middle Way: A Non-Fundamental Approach to Many-Body Physics
• Sven Banisch, Ricardo Lima, Tanya Araújo, "Aggregation and Emergence in Agent-Based Models: A Markov Chain Approach", arxiv:1207.2255
• David Cai, Louis Tao, Michael Shelley and David McLaughlin, "An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex", Proceedings of the National Academy of Sciences (USA) 101 (2004): 7757--7762
• Nicolas Champagnat, Régis Ferrière, Sylvie Méléard, "Individual-based probabilistic models of adaptive evolution and various scaling approximations", arxiv:math/0510453
• Andreas Degenhard and Javier Rodriguez-Laguna, "Towards the Evaluation of the Relevant Degrees of Freedom in Nonlinear Partial Differential Equations," cond-mat/0106156
• Manh Hong Duong, Mark A. Peletier, Upanshu Sharma, "Coarse-graining and fluctuations: Two birds with one stone", arxiv:1404.1466
• Radek Erban, Ioannis G. Kevrekidis and Hans G. Othmer, "An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal", physics/0505179
• G. Flores Hidalgo and Y. W. Milla, "Dressed (Renormalized) Coordinates in a Nonlinear System", physics/0410238
• Shalev Itzkovitz, Reuven Levitt, Nadav Kashtan, Ron Milo, Michael Itzkovitz and Uri Alon, "Coarse-Graining and Self-Dissimilarity of Complex Networks", q-bio.MN/0405011
• Artemy Kolchinsky, Luis M. Rocha, "Prediction and Modularity in Dynamical Systems", arxiv:1106.3703
• Daniel Korenblum and David Shalloway, "Macrostate data clustering", Physical Review E 67 (2003): 056704
• Peter Kotelenez, Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations [blurb]
• Peter M. Kotelenez and Thomas G. Kurtz, "Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type", Probability Theory and Related Fields 146 (2010): 189--222
• Frederic Legoll and Tony Lelievre, "Some remarks on free energy and coarse-graining", arxiv:1008.3792
• Nadia Loy, Andrea Tosin, "Markov jump processes and collision-like models in the kinetic description of multi-agent systems", Communications in Mathematical Sciences 18 (2020): 1539--1568, arxiv:1905.11343
• Andrew J. Majda, Rafail V. Abramov and Marcus J. Grote, Information Theory and Stochastic for Multiscale Nonlinear Systems [PDF draft?]
• A. Majda, I. Timofeyev and E. Vanden-Eijnden, "Stochastic models for selected slow variables in large deterministic systems", Nonlinearity 19 (2006): 769
• M. Marques-Pita and L. M. Rocha, "Schema Redescription in Cellular Automata: Revisiting Emergence in Complex Systems", arxiv:1102.1691, pp. 233--240 in The 2011 IEEE Symposium on Artificial Life
• Brian A. Maurer, "Statistical mechanics of complex ecological aggregates", Ecological Complexity 2 (2005): 71--85
• Igor Mezic, "Spectral Properties of Dynamical Systems, Model Reduction and Decompositions", Nonlinear Dynamics 41 (2005): 309--325
• Sung Joon Moon and Ioannis G. Kevrekidis, "An equation-free approach to coupled oscillator dynamics: the Kuramoto model example", nlin.AO/0502016
• Sung Joon Moon, R Ghanem, I. G. Kevrekidis, "An equation-free approach to coupled oscillator dynamics", nlin.AO/0509022
• Sung Joon Moon, B. Nabet, Naomi E. Leonard, Simon A. Levin, and I. G. Kevrekidis, "Heterogeneous animal group models and their group-level alignment dynamics; an equation-free approach", q-bio.QM/0606021
• Boaz Nadler, Stephane Lafon, Ronald R. Coifman and Ioannis G. Kevrekidis
  • "Diffusion maps, spectral clustering and reaction coordinates of dynamical systems", math.NA/0503445
  • "Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck operators", math.NA/0506090
• Catherine J. Penington, Barry D. Hughes, and Kerry A. Landman, "Building macroscale models from microscale probabilistic models: A general probabilistic approach for nonlinear diffusion and multispecies phenomena", Physical Review E 84 (2011): 041120
• Liang Qiao, Radek Erban, C. T. Kelley and Ioannis G. Kevrekidis, "Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation", q-bio.QM/0606006
• A. J. Roberts
  • "Normal form transforms separate slow and fast modes in stochastic dynamical systems", arxiv:math.DS/0701623
  • "Resolve the multitude of microscale interactions to model stochastic partial differential equations", math.DS/0506533
• Wolfgang Stadje, "The evolution of aggregated Markov chains", Statistics and Probability Letters 74 (2005): 303--311 ["Given a stationary two-sided Markov chain ... with finite state space ... and a partition ... we consider the aggregated sequence defined by [applying the partition], which is also stationary but in general not Markovian. We present a tractable way to determine the transition probabilities of [the aggregated process], either given a finite part of its past or given its infinite past. These probabilities are linked to the Radon-Nikodym derivative of [the density of an exponentially-decaying sum of aggregated values, conditional on the unaggregated process] with respect to [the unconditional distribution of the exponentially-decaying sum]".]
• Panagiotis Stinis, "A comparative study of two stochastic mode reduction methods", math.NA/0509028
• Thomas N. Thiem, Felix P. Kemeth, Tom Bertalan, Carlo R. Laing and Ioannis G. Kevrekidis, "Global and Local Reduced Models for Interacting, Heterogeneous Agents", Chaos 31 (2021): 073139, arxiv:2105.09398
• Ji-Feng Yang, "Renormalization group equations as 'decoupling' theorems", hep-th/0507024