Abstract
The Q2R model is a cellular automaton which is a dynamical variation of the Ising model for ferromagnetism that possesses quite rich and complex dynamics. It has the property of being conservative and reversible but, in practice, it shows irreversible behavior for relatively small system sizes. In this work we review some of its main properties and use it to simulate de behavior of a classical model for irreversible thermodynamical systems: the Ehrenfest’s dog-flea model.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Other dynamical invariants are known in the literature [10].
References
Loschmidt J (1876) Uber das warmegleichgewicht eines systems von korpern mit rucksicht auf die schwere, Sitzungsber. Kais Akad Wiss Wien Math Naturwiss Classe 73:128–142
Zermelo E (1896) Über einen Satz der Dynamik und die mechanische wärmetheorie. Annalen der Physik 293(3):485–494
Zermelo E (1896) Ueber mechanische Erklärungen irreversibler vorgänge. Eine Antwort auf Hrn Boltzmann’s “Entgegnung”, Annalen der Physik 295(12):793–801
Ehrenfest P, Ehrenfest-Afanassjewa T, Über eine Aufgabe aus der Wahrscheinlichkeitsrechnung, die mit der kinetischen Deutung der Entropievermehrung zusammenhängt, Mathematisch Naturwissenschaftliche Blätter 3
Ehrenfest P, Ehrenfest-Afanassjewa T (1907) Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Hirzel
Kac M (1947) Random walk and the theory of brownian motion. Amer Math Month 54(7P1):369–391
Urbina F, Rica S (2016) Master equation approach to reversible and conservative discrete systems. Phys Rev E 94(6):062140
Nicolis G, Nicolis C (1988) Master-equation approach to deterministic chaos. Phys Rev A 38(1):427
Pomeau Y (1984) Invariant in cellular automata. J Phys A: Math Gen 17(8):L415
Takesue S (1995) Staggered invariants in cellular automata. Complex Syst 9(2):149–168
Herrmann H (1986) Fast algorithm for the simulation of ising models. J Stat Phys 45(1–2):145–151
Hermann H, Carmesin H, Stauffer D (1987) Periods and clusters in Ising cellular automata. J Phys A: Math Gen 20(14):4939
Goles E, Rica S (2011) Irreversibility and spontaneous appearance of coherent behavior in reversible systems. Eur Phys J D 62(1):127–137
Onsager L (1944) Crystal statistics. I. A Two-dimensional model with an order-disorder transition. Phys Rev 65:117–149
Grassberger P (1986) Long-range effects in an elementary cellular automaton. J Stat Phys 45(1–2):27–39
Vichniac GY (1984) Simulating physics with cellular automata. Phys D: Nonlinear Phenom 10(1–2):96–116
Montalva-Medel M, Rica S, Urbina F (2020) Phase space classification of an Ising cellular automaton: the Q2R model. Chaos Solitons Fractals 133:109618
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Montalva-Medel, M., Rica, S., Urbina, F. (2022). Thermodynamics of Small Systems Through a Reversible and Conservative Discrete Automaton. In: Adamatzky, A. (eds) Automata and Complexity. Emergence, Complexity and Computation, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-030-92551-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-92551-2_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-92550-5
Online ISBN: 978-3-030-92551-2
eBook Packages: EngineeringEngineering (R0)