Effective equations in complex systems: from Langevin to machine learning. (English) Zbl 1459.82246
Summary: The problem of effective equations is reviewed and discussed. Starting from the classical Langevin equation, we show how it can be generalized to Hamiltonian systems with non-standard kinetic terms. A numerical method for inferring effective equations from data is discussed; this protocol allows to check the validity of our results. In addition we show that, with a suitable treatment of time series, such protocol can be used to infer effective models from experimental data. We briefly discuss the practical and conceptual difficulties of a pure data-driven approach in the building of models.
MSC:
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 68T05 | Learning and adaptive systems in artificial intelligence |
References:
| [1] | Karplus M and Weaver D L 1976 Protein-folding dynamics Nature260 404-6 · doi:10.1038/260404a0 |
| [2] | Donges J F, Zou Y, Marwan N and Kurths J 2009 Complex networks in climate dynamics Eur. Phys. J. Spec. Top.174 157-79 · doi:10.1140/epjst/e2009-01098-2 |
| [3] | Baldovin M, Cecconi F, Cencini M, Puglisi A and Vulpiani A 2018 The role of data in model building and prediction: a survey through examples Entropy20 807 · doi:10.3390/e20100807 |
| [4] | Dorfman J R and Cohen E G D 1972 Velocity-correlation functions in two and three dimensions: low density Phys. Rev. A 6 776-90 · doi:10.1103/PhysRevA.6.776 |
| [5] | Rubin R J 1960 Statistical dynamics of simple cubic lattices. Model for the study of Brownian motion J. Math. Phys.1 309 · doi:10.1063/1.1703664 |
| [6] | Zwanzig R 1973 Nonlinear generalized Langevin equations J. Stat. Phys.9 215 · doi:10.1007/BF01008729 |
| [7] | Spohn H 1980 Kinetic equations from Hamiltonian dynamics: Markovian limits Rev. Mod. Phys.52 569-615 · doi:10.1103/RevModPhys.52.569 |
| [8] | Mitropolsky I 1967 Averaging method in non-linear mechanics Int. J. Non-Linear Mech.2 69-96 · Zbl 0148.18703 · doi:10.1016/0020-7462(67)90020-0 |
| [9] | Lemons D S and Gythiel A 1997 Paul Langevin’s 1908 ‘Paper on the theory of Brownian Motion’ (1908 ‘Sur la théorie du mouvement Brownien’ C. R. Acad. Sci., Paris 146 530-3) Am. J. Phys.65 1079-81 · doi:10.1119/1.18725 |
| [10] | Pavliotis G and Stuart A 2008 Multiscale Methods and Homogenization (Berlin: Springer) · Zbl 1160.35006 |
| [11] | Marx D and Hutter J 2009 Ab Initio Molecular Dynamics and Advanced Methods (Cambridge: Cambridge University Press) · doi:10.1017/CBO9780511609633 |
| [12] | Car R and Parrinello M 1985 Unified approach for molecular dynamics and density-functional theory Phys. Rev. Lett.55 2471-4 · doi:10.1103/PhysRevLett.55.2471 |
| [13] | Biferale L, Crisanti A, Vergassola M and Vulpiani A 1995 Eddy diffusivities in scalar transport Phys. Fluids7 2725-34 · Zbl 1026.76558 · doi:10.1063/1.868651 |
| [14] | Baldovin M, Puglisi A and Vulpiani A 2018 Langevin equation in systems with also negative temperatures J. Stat. Mech. 043207 · Zbl 1459.82220 · doi:10.1088/1742-5468/aab687 |
| [15] | Givon D, Kupferman R and Stuart A 2004 Extracting macroscopic dynamics: model problems and algorithms Nonlinearity17 R55 · Zbl 1073.82038 · doi:10.1088/0951-7715/17/6/R01 |
| [16] | Kifer Y 2004 Some recent advances in averaging Modern Dynamical Systems and Applications ed M Brin, B Hasselblatt and Y Pesin (Cambridge: Cambridge University Press) · Zbl 1147.37318 |
| [17] | Baldovin M, Vulpiani A, Puglisi A and Prados A 2019 Derivation of a Langevin equation in a system with multiple scales: the case of negative temperatures Phys. Rev. E 99 060101 · doi:10.1103/PhysRevE.99.060101 |
| [18] | Turner R 1960 Motion of a heavy particle in a one dimensional chain Physica26 269-73 · doi:10.1016/0031-8914(60)90022-7 |
| [19] | Ford G W, Kac M and Mazur P 1965 Statistical mechanics of assemblies of coupled oscillators J. Math. Phys.6 504-15 · Zbl 0127.21605 · doi:10.1063/1.1704304 |
| [20] | Gardiner C W 1985 Handbook of Stochastic Methods (Berlin: Springer) |
| [21] | Friedrich R, Peinke J, Sahimi M and Reza Rahimi Tabar M 2011 Approaching complexity by stochastic methods: from biological systems to turbulence Phys. Rep.506 87-162 · doi:10.1016/j.physrep.2011.05.003 |
| [22] | Peinke J, Tabar M and Wächter M 2019 The Fokker-Planck approach to complex spatiotemporal disordered systems Annu. Rev. Condens. Matter Phys.10 107-32 · doi:10.1146/annurev-conmatphys-033117-054252 |
| [23] | Ramsey N F 1956 Thermodynamics and statistical mechanics at negative absolute temperatures Phys. Rev.103 20-8 · Zbl 0074.43007 · doi:10.1103/PhysRev.103.20 |
| [24] | Onsager L 1949 Statistical hydrodynamics Il Nuovo Cimento (1943-1954)6 279-87 · doi:10.1007/BF02780991 |
| [25] | Iubini S, Franzosi R, Livi R, Oppo G-L and Politi A 2013 Discrete breathers and negative-temperature states New J. Phys.15 023032 · doi:10.1088/1367-2630/15/2/023032 |
| [26] | Braun S, Ronzheimer J P, Schreiber M, Hodgman S S, Rom T, Bloch I and Schneider U 2013 Negative absolute temperature for motional degrees of freedom Science339 52-5 · doi:10.1126/science.1227831 |
| [27] | Hilbert S, Hänggi P and Dunkel J 2014 Thermodynamic laws in isolated systems Phys. Rev. E 90 062116 · doi:10.1103/PhysRevE.90.062116 |
| [28] | Cerino L, Puglisi A and Vulpiani A 2015 A consistent description of fluctuations requires negative temperatures J. Stat. Mech. P12002 · Zbl 1456.82409 · doi:10.1088/1742-5468/2015/12/p12002 |
| [29] | Baldovin M, Puglisi A, Sarracino A and Vulpiani A 2017 About thermometers and temperature J. Stat. Mech. 113202 · Zbl 1457.82370 · doi:10.1088/1742-5468/aa933e |
| [30] | Miceli F, Baldovin M and Vulpiani A 2019 Statistical mechanics of systems with long-range interactions and negative absolute temperature Phys. Rev. E 99 042152 · doi:10.1103/PhysRevE.99.042152 |
| [31] | Dunkel J and Hilbert S 2014 Consistent thermostatistics forbids negative absolute temperatures Nat. Phys.10 67-72 · doi:10.1038/nphys2815 |
| [32] | Romero-Rochín V 2013 Nonexistence of equilibrium states at absolute negative temperatures Phys. Rev. E 88 022144 · doi:10.1103/PhysRevE.88.022144 |
| [33] | Campisi M 2015 Construction of microcanonical entropy on thermodynamic pillars Phys. Rev. E 91 052147 · doi:10.1103/PhysRevE.91.052147 |
| [34] | Onsager L and Machlup S 1953 Fluctuations and irreversible processes Phys. Rev.91 1505-12 · Zbl 0053.15106 · doi:10.1103/PhysRev.91.1505 |
| [35] | Scalliet C, Gnoli A, Puglisi A and Vulpiani A 2015 Cages and anomalous diffusion in vibrated dense granular media Phys. Rev. Lett.114 198001 · doi:10.1103/PhysRevLett.114.198001 |
| [36] | Baldovin M, Puglisi A and Vulpiani A 2019 Langevin equations from experimental data: the case of rotational diffusion in granular media PLoS One14 e0212135 · doi:10.1371/journal.pone.0212135 |
| [37] | Cecconi F, Cencini M, Falcioni M and Vulpiani A 2012 Predicting the future from the past: an old problem from a modern perspective Am. J. Phys.80 1001-8 · doi:10.1119/1.4746070 |
| [38] | Hosni H and Vulpiani A 2018 Forecasting in light of big data Phil. Technol.31 557-69 · doi:10.1007/s13347-017-0265-3 |
| [39] | Bradshaw G F, Langley P W and Simon H A 1983 Studying scientific discovery by computer simulation Science222 971-5 · doi:10.1126/science.222.4627.971 |
| [40] | Grabiner J V 1986 Computers and the nature of man: a historian’s perspective on controversies about artificial intelligence Bull. Am. Math. Soc. (N.S.)15 113-26 · Zbl 0624.68073 · doi:10.1090/S0273-0979-1986-15461-3 |
| [41] | Buchanan M 2019 The limits of machine prediction Nat. Phys.15 304 · doi:10.1038/s41567-019-0489-5 |
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