## Calculating Macroscopic Consequences of Microscopic Interactions

*22 Aug 2015 16:18*

[A placeholder while I organize my thoughts]

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...

- Recommended:
- 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.] - 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 - Olof Gonerup and Martin Nilsson Jacobi, "A method for inferring hierarchical dynamics in stochastic processes", nlin.AO/0703034
- 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
- Thomas Schelling, Micromotives and Macrobehavior

- 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
- 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
- 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" ["We use diffusion semigroups to
generate multiscale geometries in order to organize and represent complex
structures. We show that appropriately selected eigenfunctions or scaling
functions of Markov matrices, which describe local transitions, lead to
macroscopic descriptions at different scales. The process of iterating or
diffusing the Markov matrix is seen as a generalization of some aspects of the
Newtonian paradigm, in which local infinitesimal transitions of a system lead
to global macroscopic descriptions by integration. We provide a unified view of
ideas from data analysis, machine learning, and numerical analysis."]
- "Diffusion maps", Proceedings of the
National Academy of Sciences
**102**(2005): 7426--7431 - "Multiscale methods", Proceedings of the
National Academy of Sciences
**102**(2005): 7432--7437

- "Diffusion maps", Proceedings of the
National Academy of Sciences
- 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
- Navot Israeli and Nigel Goldenfeld, "Coarse-graining of cellular automata, emergence, and the predictability of complex systems", nlin.CG/0508033
- 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
- 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 [An approach to predicting the behavior of macroscopic, coarse-grained variables, which is equation-free in the sense that it "circumvent[s] the derivation of explicit dynamical equations (approximately) governing [their] evolution", substituting rather "short burts of appropriately initialized simulations of the 'fine-scale', detailed" model. This sounds like an interesting idea, applicable e.g. to agent-based models, but I need to read the paper to see if/how it actually works.]
- 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, "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
- Jeroen Wouters, Valerio Lucarini, "Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach",
Journal of Statistical Physics
**151**(2013): 850--860, arxiv:1208.3080 - Ji-Feng Yang, "Renormalization group equations as 'decoupling' theorems", hep-th/0507024