Irreversibility in Reversible Systems

We present an overview of recent work together with J. Hoffman, A. Logg and R. Scott developing a new approach towards resolving the classical paradox of irreversibility in formally reversible Hamiltonian systems such as inviscid fluid models and particle models. We base our solution on finite precision computation in the form of General Galerkin G2, instead of statistical mechanics, which is the classical approach developed by Boltzmann. We consider first the Euler equations for an inviscid incompressible or compressible flow fluid and we show that the irreversibility arises because G2 reacts by introducing a dissipative weighted least squares control of the residual if the Euler equations lack solutions with pointwise vanishing residual, which is the general case because of the appearance of turbulence and shocks. We show that the Second Law of Thermodynamics (stating that entropy cannot decrease) is a consequence of the First Law of Thermodynamics (stating conservation of energy) combined with G2 finite precision computation. We thus meet irreversibility in cases of strict decrease of entropy. We also offer an explanation of irreversibility of Euler solutions as a consequence of the stability properties of the Euler equations as expressed in quantitative form by the solution of an associated dual problem. We show that mean value outputs such as drag and lift are stably computable by G2 in forward time, because of considerable cancellation in the corresponding dual solution. We also show that backward in time computation may be unstable because of lack of cancellation in the dual problem. We also consider Hamiltonian particle systems and again explain irreversibility microscopically as a consequence of different stability properties in forward and backward time depending on different effects of cancellation. In particular we show that a forward in time mixing process has stably computable mean value outputs, while the reverse process of unmixing typically is highly unstable. We also explain irreversibility in particle systems macroscopically by noting that mean values of particle solutions generate approximate viscosity solutions to the Euler equations, which satisfy entropy inequalities. We conclude that the stability approach seems to offer a better scientific explanation of irreversibility than approaches based on entropy, because both Nature and G2 computation directly react to stability aspects, but do not seem to have direct sensors for entropy. References: J. Hoffman and C. Johnson, Computational Fulid Dynamics A: Incompressible Flow, Applied Mathematics: Body and Soul Vol V, Springer, 2005. J. Hoffman and C. Johnson, Irreversibility in reversible systems I: The compressible Euler equations in 1d, Preprint Chalmers Finite Element Center, 2005. J. Hoffman and C. Johnson, Irreversibility in reversible systems II: The incompressible Euler equations, Preprint Chalmers Finite Element Center, 2005. J. Hoffman and C. Johnson, Irreversibility in reversible systems III: The compressible Euler equations in 3d, Preprint Chalmers Finite Element Center, 2005. J. Hoffman, C. Johnson, A. Logg and R. Scott, Irreversibility in reversible systems IV: Kinetic gas theory, Preprint Chalmers Finite Element Center, 2005.