Abstract
In this paper a certain type of reaction–diffusion equation—similar to the Allen-Cahn equation—is the starting point for setting up a genuine thermodynamic reduction i.e. involving a finite number of parameters or collective variables of the initial system. We firstly operate a finite Lyapunov–Schmidt reduction of the cited reaction–diffusion equation when reformulated as a variational problem. In this way we gain a finite-dimensional ODE description of the initial system which preserves the gradient structure of the original one and that is exact for the static case and only approximate for the dynamic case. Our main concern is how to deal with this approximate reduced description of the initial PDE. To start with, we note that our approximate reduced ODE is similar to the approximate inertial manifold introduced by Temam and coworkers for Navier–Stokes equations. As a second approach, we take into account the uncertainty (loss of information) introduced with the above mentioned approximate reduction by considering the stochastic version of the ODE. We study this reduced stochastic system using classical tools from large deviations, viscosity solutions and weak KAM Hamilton–Jacobi theory. In the last part we suggest a possible use of a result of our approach in the comprehensive treatment non equilibrium thermodynamics given by Macroscopic Fluctuation Theory.
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Notes
E.g. \(\mathcal {H}\) is the Sobolev space \(W^{2,2}_0(\Omega ;\mathbb {R})\).
By applying twice (2.34), we write \(\left| \Delta ^1 \chi \right| ^2 \ge \Lambda \left| \Delta ^\frac{1}{2} \chi \right| ^2 \ge \Lambda ^2 \left| \Delta ^0 \chi \right| ^2\), from which it follows \( \left| \Delta \chi \right| \ge \Lambda \left| \chi \right| \).
We recall that for the reduced equation the set of critical points is in a one-to-one correspondence with the equilibria of the original reaction–diffusion equation.
Precisely, we consider the Fokker–Planck equation (3.2) with a factor \(\nu /2\) in front of the Laplacian.
When considering lattice gas model, the above introduced continuity equation is the hydrodynamic limit equation for density \(\rho (x,t) \) which in turn is the limit of the empirical density \(\rho _N (z,t)\) of particles in a given point z on the lattice \(\Lambda \subset \mathbb {Z}^d\) with N particles. In the thermodynamic limit \(N \rightarrow \infty \) the empirical density \(\rho _N\) satisfies a large deviations principle, [21].
References
Amann, H., Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7, 539–603 (1980)
Amann, H.: Saddle points and multiple solutions of differential equations. Math. Z. 169, 127–166 (1979)
Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Applied Mathematical Sciences, vol. 125. Springer, New York (1998)
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser Boston Inc, Boston, MA (1997)
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic Fluctuation Theory (2014). arXiv:1404.6466
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Towards a nonequilibrium thermodynamics: a self-contained macroscopic description of driven diffusive systems. J. Stat. Phys. 135(5–6), 857–872 (2009)
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Fluctuations in stationary nonequilibrium states of irreversible processes. Phys. Rev. Lett. 87(4), 040601 (2001)
Boldrighini, C., De Masi, A., Pellegrinotti, A., Presutti, E.: Collective phenomena in interacting particle systems. Stoch. Process. Appl. 25, 137–152 (1987)
Cardin, F., Lovison, A., Putti, M.: Implementation of an exact finite reduction scheme for steady-state reaction-diffusion equations. Int. J. Numer. Methods Eng. 69(9), 1804–1818 (2007)
Cardin, F., Tebaldi, C.: Finite reductions for dissipative systems and viscous fluid-dynamic models on \(\mathbb{T}^2\). J. Math. Anal. Appl. 345, 213–222 (2008)
Cardin, F., Lovison, A.: Finite mechanical proxies for a class of reducible continuum systems. Netw. Heterog. Media 9, 417–432 (2014)
Cardin, F., Masci, L.: A Morse Index invariant reduction of non equilibrium thermodynamics. In: Rendiconti Lincei, Matematica e Applicazioni, vol. IV (2017)
Conley, C.: Isolated Invariant Sets and The Morse Index. In: Regional Conferences Series in Mathematics. Conference Board for the Mathematical Sciences, vol. 38 (1976)
Davini, A., Siconolfi, A.: A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 38(2), 478–502 (2006)
Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics. Cambridge Studies in Advanced Mathematics, vol. 88 (2016)
Foias, C., Manley, O., Temam, R.: Attractors for the Bénard problem: existence and Physical bounds on their fractal dimension. Nonlinear Anal. Theory Model. Appl. (1987)
Foias, C., Manley, O., Temam, R.: Modelling of the interaction of small and large eddies in two dimensional turbulent flows. RAIRO 22, 93–118 (1988)
Freidlin, M.I., Ventzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1998)
Jona-Lasinio, G.: From fluctuations in hydrodynamics to nonequilibrium thermodynamics. Prog. Theor. Phys. Suppl. 184, 262–275 (2010)
Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4), 284–304 (1940)
Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Grundlehren der mathematischen Wissenschaften, vol. 320. Springer, Berlin (1999)
Maragliano, L., Fischer, A., Vanden-Eijnden, E., Ciccotti, G.: String method in collective variables: minimum free energy paths and isocommittor surfaces. J. Chem. Phys. 125, 24106 (2006)
Marion, M.: Approximate inertial manifolds for reaction-diffusion equations in high space dimension. J. Dyn. Differ. Equ. 1, 245–267 (1989)
Moro, G.J.: Kinetic equations for site populations from the Fokker-Planck equation. J. Chem. Phys. 103, 7514–7528 (1995)
Moro, G.J., Cardin, F.: Variational layer expansion for kinetic processes. Phys. Rev. E 55(5), 4918–4934 (1997)
Müller, S., Schlömerkemper, A.: Discrete-to-continuum limit of magnetic forces. C. R. Math. 335, 393–398 (2002)
Temam, R.: Inertial manifolds. Math. Intell. 12(4), 68–74 (1990)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (1997)
Touchette, H.: The large deviation approach to statistical mechanics. Phys. Rep. 478, 1–69 (2009)
Varadhan, S.R.: Large Deviations and Applications. SIAM, Philadelphia (1984)
Viterbo, C.: Recent progress in periodic orbits of autonomous Hamiltonian systems and applications to symplectic geometry. In: Nonlinear Functional Analysis (Newark, NJ, 1987). Lecture Notes in Pure and Applied Mathematics, vol. 121, pp. 227–250. Dekker, New York (1990)
Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17(4), 661–692 (1983)
Yukhnovskiĭ, I.R.: Phase Transitions of the Second Order. World Scientific, Singapore (1987)
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Cardin, F., Favretti, M. & Lovison, A. Inertial Manifold and Large Deviations Approach to Reduced PDE Dynamics. J Stat Phys 168, 1000–1015 (2017). https://doi.org/10.1007/s10955-017-1845-4
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DOI: https://doi.org/10.1007/s10955-017-1845-4
Keywords
- Non-equilibrium thermodynamics
- Lyapunov–Schmidt reduction
- Inertial manifolds
- Collective variables
- Fokker–Planck equation
- Hamilton–Jacobi equation
- Large deviations