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Thermodynamics and evolutionary biology through optimal control. (English) Zbl 1430.49052

Summary: We consider a particular instance of the lift of controlled systems proposed in the theory of irreversible thermodynamics and show that it is equivalent to a variational principle for an optimal control in the sense of Pontryagin. Then we focus on two important applications: in thermodynamics and in evolutionary biology. In the thermodynamic context, we show that this principle provides a dynamical implementation of the Second Law, which stabilizes the equilibrium states of a system. In the evolutionary context, we show that our principle leads directly to the Optimal Replicator Equation, and we discuss several interesting features: it provides a robust scheme for the coevolution of the population and its fitness landscape; it has a clear formulation in terms of an optimization process; and finally, it extends standard evolutionary dynamics to include phenomena such as the emergence of cooperation.

MSC:

49S05 Variational principles of physics
49K15 Optimality conditions for problems involving ordinary differential equations
80M30 Variational methods applied to problems in thermodynamics and heat transfer
92D15 Problems related to evolution
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[1] Ao, Ping, Laws in Darwinian evolutionary theory, Physics of Life Reviews, 2, 2, 117-156 (2005)
[2] Ao, P., Emerging of stochastic dynamical equalities and steady state thermodynamics from Darwinian dynamics, Communications in Theoretical Physics, 49, 5, 1073 (2008) · Zbl 1392.37093
[3] Arnold, Vladimir Igorevich, Mathematical methods of classical mechanics, vol. 60 (1989), Springer Science & Business Media
[4] Axelrod, Robert M., The evolution of cooperation (2006), Basic books
[5] Baez, John C.; Pollard, Blake S., Relative entropy in biological systems, Entropy, 18, 2, 46 (2016)
[6] Barcelo, Hélene; Capraro, Valerio, Group size effect on cooperation in one-shot social dilemmas, Scientific Reports, 5, 7937 (2015)
[7] Barton, N. H.; Coe, J. B., On the application of statistical physics to evolutionary biology, Journal of Theoretical Biology, 259, 2, 317-324 (2009) · Zbl 1402.92308
[8] Beretta, Gian Paolo, Steepest entropy ascent model for far-nonequilibrium thermodynamics: unified implementation of the maximum entropy production principle, Physical Review E, 90, 4, 042113 (2014)
[9] Bravetti, Alessandro, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19, 10, 535 (2017)
[10] Bravetti, Alessandro, Contact geometry and thermodynamics, International Journal of Geometric Methods in Modern Physics, 1940003 (2018) · Zbl 1421.80002
[11] Bravetti, Alessandro; Cruz, Hans; Tapias, Diego, Contact Hamiltonian mechanics, Annals of Physics, 376, 17-39 (2017) · Zbl 1364.37138
[12] Bravetti, A.; Lopez-Monsalvo, C. S.; Nettel, F., Contact symmetries and Hamiltonian thermodynamics, Annals of Physics, 361, 377-400 (2015) · Zbl 1360.80003
[13] Bravetti, Alessandro; Padilla, Pablo, An optimal strategy to solve the prisoner’s dilemma, Scientific Reports, 8, 1, 1948 (2018)
[14] Bravetti, Alessandro; Tapias, Diego, Liouville’s theorem and the canonical measure for nonconservative systems from contact geometry, Journal of Physics A: Mathematical and Theoretical, 48, 24, 245001 (2015) · Zbl 1321.37067
[15] Bravetti, A.; Tapias, D., Thermostat algorithm for generating target ensembles, Physical Review E, 93, 2, 022139 (2016)
[16] Callen, Herbert B. (1998). Thermodynamics and an Introduction to Thermostatistics.; Callen, Herbert B. (1998). Thermodynamics and an Introduction to Thermostatistics.
[17] Capraro, Valerio, A model of human cooperation in social dilemmas, PLoS One, 8, 8, Article e72427 pp. (2013)
[18] Chakrabarti, Raj; Rabitz, Herschel; Springs, Stacey L.; McLendon, George L., Mutagenic evidence for the optimal control of evolutionary dynamics, Physical Review Letters, 100, 25, 258103 (2008)
[19] Drossel, Barbara, Biological evolution and statistical physics, Advances in Physics, 50, 2, 209-295 (2001)
[20] Eberhard, D.; Maschke, B. M.; Van Der Schaft, A. J., An extension of Hamiltonian systems to the thermodynamic phase space: towards a geometry of nonreversible processes, Reports on Mathematical Physics, 60, 2, 175-198 (2007) · Zbl 1210.80001
[21] England, Jeremy L., Statistical physics of self-replication, Journal of Chemical Physics, 139, 12, 09B623_1 (2013)
[22] Favache, Audrey; Dochain, Denis; Maschke, B., An entropy-based formulation of irreversible processes based on contact structures, Chemical Engineering Science, 65, 18, 5204-5216 (2010)
[23] Favache, Audrey; Martins, Valérie Sylvie Dos Santos; Dochain, Denis; Maschke, Bernhard, Some properties of conservative port contact systems, IEEE Transactions on Automatic Control, 54, 10, 2341-2351 (2009) · Zbl 1367.80002
[24] Fisher, Ronald Aylmer, The genetical theory of natural selection: a complete variorum edition (1999), Oxford University Press
[25] Frank, Steven A., Natural selection maximizes Fisher information, Journal of Evolutionary Biology, 22, 2, 231-244 (2009)
[26] Frank, Steven A., Natural selection. V. How to read the fundamental equations of evolutionary change in terms of information theory, Journal of Evolutionary Biology, 25, 12, 2377-2396 (2012)
[27] Frieden, B. Roy, Physics from Fisher information: a unification (1998), Cambridge University Press · Zbl 0998.81512
[28] Geering, Hans P., Optimal control with engineering applications (2007), Berlin Heidelberg · Zbl 1121.49001
[29] Goto, Shin-itiro, Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics, Journal of Mathematical Physics, 56, 7, 073301 (2015) · Zbl 1327.82072
[30] Goto, Shin-itiro, Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics, Journal of Mathematical Physics, 57, 10, 102702 (2016) · Zbl 1351.37237
[31] Grmela, Miroslav, Contact geometry of mesoscopic thermodynamics and dynamics, Entropy, 16, 3, 1652-1686 (2014)
[32] Grmela, Miroslav; Öttinger, Hans Christian, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Physical Review E, 56, 6, 6620 (1997)
[33] Hudon, Nicolas; Guay, Martin; Dochain, Denis, Control design for thermodynamic systems on contact manifolds, IFAC-Papers OnLine, 50, 1, 588-593 (2017)
[34] Jóźwikowski, Michał; Respondek, Witold, A contact covariant approach to optimal control with applications to sub-riemannian geometry, Mathematics of Control, Signals, and Systems, 28, 3, 27 (2016) · Zbl 1350.49022
[35] Karev, Georgiy P., On mathematical theory of selection: continuous time population dynamics, Journal of Mathematical Biology, 60, 1, 107-129 (2010) · Zbl 1311.92137
[36] Karev, Georgiy P., Replicator equations and the principle of minimal production of information, Bulletin of Mathematical Biology, 72, 5, 1124-1142 (2010) · Zbl 1197.92004
[37] Klimek, Peter; Thurner, Stefan; Hanel, Rudolf, Evolutionary dynamics from a variational principle, Physical Review E, 82, 1, 011901 (2010)
[38] de León, M.; Sardón, C., Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems, Journal of Physics A: Mathematical and Theoretical, 50, 25, 255205 (2017) · Zbl 1425.70031
[39] Marimuthu, Karthikeyan; Chakrabarti, Raj, Dynamics and control of DNA sequence amplification, Journal of Chemical Physics, 141, 16, 10B614_1 (2014)
[40] Markowitz, Harry, Portfolio selection, The Journal of Finance, 7, 1, 77-91 (1952)
[41] Markowitz, Harry M.; Todd, G. P.eter; Sharpe, William F., Mean-variance analysis in portfolio choice and capital markets, vol. 66 (2000), John Wiley & Sons
[42] Maschke, B., About the lift of irreversible thermodynamic systems to the thermodynamic phase space, IFAC-Papers OnLine, 49, 24, 40-45 (2016)
[43] Mrugala, Ryszard; Nulton, James D.; Schön, J. Christian; Salamon, Peter, Contact structure in thermodynamic theory, Reports on Mathematical Physics, 29, 1, 109-121 (1991) · Zbl 0742.58022
[44] Nilsson, Martin; Snoad, Nigel, Error thresholds for quasispecies on dynamic fitness landscapes, Physical Review Letters, 84, 1, 191 (2000)
[45] Nowak, Martin A., Evolutionary dynamics (2006), Harvard University Press · Zbl 1098.92051
[46] Nowak, Martin A., Five rules for the evolution of cooperation, Science, 314, 5805, 1560-1563 (2006)
[47] Ohsawa, Tomoki, Contact geometry of the Pontryagin maximum principle, Automatica, 55, 1-5 (2015) · Zbl 1378.49016
[48] Page, Karen M.; Nowak, Martin A., Unifying evolutionary dynamics, Journal of Theoretical Biology, 219, 1, 93-98 (2002)
[49] Perunov, Nikolay; Marsland, Robert A.; England, Jeremy L., Statistical physics of adaptation, Physical Review X, 6, 2, 021036 (2016)
[50] Ramirez, Hector; Maschke, Bernhard; Sbarbaro, Daniel, Feedback equivalence of input – output contact systems, Systems & Control Letters, 62, 6, 475-481 (2013) · Zbl 1279.93038
[51] Ramirez, Hector; Maschke, Bernhard; Sbarbaro, Daniel, Partial stabilization of input-output contact systems on a legendre submanifold, IEEE Transactions on Automatic Control, 62, 3, 1431-1437 (2017) · Zbl 1366.93525
[52] Smerlak, Matteo, Natural selection as coarsening, Journal of Statistical Physics, 172, 1, 105-113 (2018) · Zbl 1398.82039
[53] Smerlak, Matteo; Youssef, Ahmed, Limiting fitness distributions in evolutionary dynamics, Journal of Theoretical Biology, 416, 68-80 (2017) · Zbl 1368.92125
[54] Traulsen, Arne; Iwasa, Yoh; Nowak, Martin A., The fastest evolutionary trajectory, Journal of Theoretical Biology, 249, 3, 617-623 (2007) · Zbl 1453.92226
[55] Wang, Li; Maschke, B.; van der Schaft, A. J., Stabilization of control contact systems, IFAC-Papers OnLine, 48, 13, 144-149 (2015)
[56] Wang, Kaizhi; Wang, Lin; Yan, Jun, Implicit variational principle for contact Hamiltonian systems, Nonlinearity, 30, 2, 492 (2016) · Zbl 1364.37139
[57] Wang, Kaizhi, Wang, Lin, & Yan, Jun (2018). Aubry-Mather theory for contact Hamiltonian systems, arXiv preprint arXiv:1801.05612; Wang, Kaizhi, Wang, Lin, & Yan, Jun (2018). Aubry-Mather theory for contact Hamiltonian systems, arXiv preprint arXiv:1801.05612
[58] Wilke, Claus O.; Ronnewinkel, Christopher; Martinetz, Thomas, Dynamic fitness landscapes in molecular evolution, Physics Reports, 349, 5, 395-446 (2001)
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