ALEX 2018 Workshop: Abstracts

On fluctuations in particle systems and their links to macroscopic models

Rob L. Jack ${}^{(1)}$ , Marcus Kaiser ${}^{(2)}$ , and Johannes Zimmer ${}^{(2)}$

(1) Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge,

Cambridge CB3 0WA, UK

and

Department of Chemistry, University of Cambridge, Lensfield Road,

Cambridge CB2 1EW, UK

(2) Department of Mathematical Sciences, University of Bath,

Bath BA2 7AY, UK

We study particle systems and analyse their fluctuations. These fluctuations can be described by stochastic differential equations or variational formulations related to large deviations. In particular, recently a canonical structure has been introduced [6, 7] to describe dynamical fluctuations in stochastic systems. The resulting theory has several attractive features: Firstly, it applies to a wide range of systems, including finite-state Markov chains and Macroscopic Fluctuation Theory (MFT) [1], see [4]. Secondly, it is based on an action functional which is a relative entropy between probability measures on path spaces — this means that it provides a variational description of the systems under consideration, and the action can be related to large deviation rate functionals. Thirdly, it extends the classical Onsager-Machlup theory [9] in a natural way, by replacing the quadratic functionals that appear in that theory with a pair of convex but non-quadratic Legendre duals $\Psi$ and $\Psi^{\star}$ . We will discuss how this structure can be applied to any finite-state Markov chain and provides a unifying formulation of a wide range of systems [4]. We will discuss large-scale limits of particle systems, closely related to the Energy-Dissipation-Principle, see e.g. [5, 2, 3, 8].

References

1 Lorenzo Bertini, Alberto De Sole, Davide Gabrielli, Giovanni Jona-Lasinio, and Claudio Landim. Macroscopic fluctuation theory. Rev. Modern Phys., 87(2):593–636, 2015.
2 Giovanni A. Bonaschi and Mark A. Peletier. Quadratic and rate-independent limits for a large-deviations functional. Contin. Mech. Thermodyn., 28(4):1191–1219, 2016.
3 Manh Hong Duong, Agnes Lamacz, Mark A. Peletier, and Upanshu Sharma. Variational approach to coarse-graining of generalized gradient flows. Calc. Var. Partial Differential Equations, 56(4):Art. 100, 65, 2017.
4 Marcus Kaiser, Robert L. Jack, and Johannes Zimmer. Canonical structure and orthogonality of forces and currents in irreversible Markov chains. J. Stat. Phys., 170(6):1019–1050, 2018.
5 Matthias Liero, Alexander Mielke, Mark A. Peletier, and D. R. Michiel Renger. On microscopic origins of generalized gradient structures. Discrete Contin. Dyn. Syst. Ser. S, 10(1):1–35, 2017.
6 C. Maes and K. Netočný. Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states. Europhys. Lett. EPL, 82(3):Art. 30003, 6, 2008.
7 C. Maes, K. Netočný, and B. Wynants. On and beyond entropy production: the case of Markov jump processes. Markov Process. Related Fields, 14(3):445–464, 2008.
8 Alexander Mielke. On evolutionary $\varGamma$ -convergence for gradient systems. In Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity, volume 3 of Lect. Notes Appl. Math. Mech., pages 187–249. Springer, [Cham], 2016.
9 L. Onsager and S. Machlup. Fluctuations and irreversible processes. Physical Rev. (2), 91:1505–1512, 1953.

Acknowledgments: MK’s PhD is funded EPSRC (EP/L015684/1, CDT SAMBa). JZ gratefully acknowledges funding by EPSRC (EP/K027743/1), the Leverhulme Trust (RPG-2013-261) and a Royal Society Wolfson Research Merit Award.