Noise and Dissipation on Coadjoint Orbits

We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect product extension. Random attractors are found for this general class of systems when the Lie algebra is semi-simple, provided the top Lyapunov exponent is positive. We study in details two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown.

[1]  Darryl D. Holm,et al.  A geometric theory of selective decay with applications in MHD , 2014 .

[2]  A. Willsky.  Estimation and detection of signals in multiplicative noise , 1974, CDC 1974.

[3]  Petre Birtea,et al.  Stability of Equilibria for the $\mathfrak{so}(4)$ Free Rigid Body , 2008, J. Nonlinear Sci..

[4]  Michael Ghil,et al.  Data-driven non-Markovian closure models , 2014, 1411.4700.

[5]  Shin-Ho Chung,et al.  Algorithm for rigid-body Brownian dynamics. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Minoru Toda,et al.  Springer Series in Solid-State Sciences , 1989 .

[7]  Marc Arnaudon,et al.  Lagrangian Navier–Stokes diffusions on manifolds: Variational principle and stability , 2010 .

[8]  Robert V. Kohn,et al.  Magnetic Elements at Finite Temperature and Large Deviation Theory , 2005, Journal of nonlinear science.

[9]  H. Crauel,et al.  Attractors for random dynamical systems , 1994 .

[10]  Marc Arnaudon,et al.  Stochastic Euler-Poincaré reduction , 2012 .

[11]  Christian Bonatti,et al.  Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective , 2004 .

[12]  T Ratiu.  Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body. , 1981, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Lai-Sang Young,et al.  Strange Attractors for Periodically Forced Parabolic Equations , 2013 .

[14]  T. Ratiu,et al.  Clebsch optimal control formulation in mechanics , 2011 .

[15]  A. Perelomov,et al.  CASIMIR OPERATORS FOR SEMISIMPLE LIE GROUPS , 1968 .

[16]  H. Owhadi,et al.  Stochastic Variational Integrators , 2007, 0708.2187.

[17]  Michael Ghil,et al.  Stochastic climate dynamics: Random attractors and time-dependent invariant measures , 2011 .

[18]  Darryl D. Holm.  Book Review: Geometric Mechanics, Part II: Rotating, Translating and Rolling , 2008 .

[19]  Darryl D. Holm,et al.  Solution Properties of a 3D Stochastic Euler Fluid Equation , 2017, J. Nonlinear Sci..

[20]  M. Rasmussen,et al.  Bifurcation Analysis of a Stochastically Driven Limit Cycle , 2016, Communications in Mathematical Physics.

[21]  Peter Lynch,et al.  The Swinging Spring: A Simple Model of Atmospheric Balance , 2001 .

[22]  Darryl D. Holm,et al.  Selective decay by Casimir dissipation in inviscid fluids , 2013 .

[23]  Lai-Sang Young,et al.  Shear-induced chaos , 2007, 0705.3294.

[24]  Peter E. Kloeden,et al.  Stratonovich and Ito Stochastic Taylor Expansions , 1991 .

[25]  Jerrold E. Marsden,et al.  Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems , 1999 .

[26]  Lai-Sang Young,et al.  Dynamics of periodically kicked oscillators , 2010, 1004.0565.

[27]  A. Iserles,et al.  Geometry, Symmetry and Mechanics, I , 2001 .

[28]  R. Zwanzig.  Nonequilibrium statistical mechanics , 2001, Physics Subject Headings (PhySH).

[29]  Z. Brzeźniak,et al.  Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert Equation , 2008 .

[30]  G. Chirikjian.  Stochastic models, information theory, and lie groups , 2012 .

[31]  J. Marsden,et al.  Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids , 1983 .

[32]  M. Nobili,et al.  Brownian Motion of an Ellipsoid , 2006, Science.

[33]  Darryl D. Holm,et al.  The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories , 1998, chao-dyn/9801015.

[34]  D. Haar,et al.  Statistical Physics , 1971, Nature.

[35]  G. Chirikjian.  Analytic methods and modern applications , 2012 .

[36]  Klaus Reiner Schenk-Hoppé,et al.  Random attractors--general properties, existence and applications to stochastic bifurcation theory , 1997 .

[37]  Lai-Sang Young,et al.  Strange Attractors in Periodically-Kicked Limit Cycles and Hopf Bifurcations , 2003 .

[38]  P. Krishnaprasad,et al.  The Euler-Poincaré equations and double bracket dissipation , 1996 .

[39]  Christian Pötzsche,et al.  Nonautonomous Dynamical Systems , 2010 .

[40]  Classification and Casimir invariants of Lie-Poisson brackets , 1997, math-ph/9904010.

[41]  Darryl D. Holm,et al.  Stepwise Precession of the Resonant Swinging Spring , 2001, SIAM J. Appl. Dyn. Syst..

[42]  Krzysztof Gawedzki,et al.  Universality in turbulence: An exactly soluble model , 1995 .

[43]  Eric Shea-Brown,et al.  Reliability of Coupled Oscillators , 2009, J. Nonlinear Sci..

[44]  Darryl D. Holm,et al.  Poisson brackets and clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity , 1983 .

[45]  B. Rozovskii,et al.  On Equations of Stochastic Fluid Mechanics , 2001 .

[46]  Hans Crauel,et al.  Markov measures for random dynamical systems , 1991 .

[47]  Hans Crauel,et al.  Random attractors , 1997 .

[48]  Darryl D. Holm,et al.  Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics , 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[49]  Gregory S. Chirikjian.  Classical results and geometric methods , 2009 .

[50]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[51]  Darryl D. Holm,et al.  Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows , 2018, J. Nonlinear Sci..

[52]  K. Gawȩdzki,et al.  University in turbulence: An exactly solvable model , 1995, chao-dyn/9504002.

[53]  Lai-Sang Young,et al.  What Are SRB Measures, and Which Dynamical Systems Have Them? , 2002 .

[54]  G. Bellamy.  Lie groups, Lie algebras, and their representations , 2015 .

[55]  Darryl D. Holm.  Variational principles for stochastic fluid dynamics , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[56]  A. Arnaudon.  The stochastic integrable AKNS hierarchy , 2015, 1511.07080.

[57]  Oscar Gonzalez,et al.  On the Stochastic Modeling of Rigid Body Systems with Application to Polymer Dynamics , 2010, Multiscale Model. Simul..

[58]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[59]  Lai-Sang Young,et al.  Entropy formula for random transformations , 1988 .

[60]  V. Araújo.  Random Dynamical Systems , 2006, math/0608162.

[61]  Axel Klar,et al.  Numerical simulation of a moving rigid body in a rarefied gas , 2014, J. Comput. Phys..

[62]  Tudor S. Ratiu,et al.  Stochastic Variational Principles for Dissipative Equations with Advected Quantities , 2015, Journal of Nonlinear Science.

[63]  Z. Brzeźniak,et al.  Large Deviations and Transitions Between Equilibria for Stochastic Landau–Lifshitz–Gilbert Equation , 2012, 1202.0370.

[64]  D. Garanin,et al.  FOKKER-PLANCK AND LANDAU-LIFSHITZ-BLOCH EQUATIONS FOR CLASSICAL FERROMAGNETS , 1997, cond-mat/9805054.

[65]  J. Ortega,et al.  Stochastic hamiltonian dynamical systems , 2007, math/0702787.

[66]  C. Marle,et al.  "Sur une forme nouvelle des ´ equations de la M´ ecanique" , 2013 .

[67]  R. Kraichnan,et al.  Anomalous scaling of a randomly advected passive scalar. , 1994, Physical review letters.

[68]  Hans Crauel,et al.  Additive Noise Destroys a Pitchfork Bifurcation , 1998 .

[69]  B. Goldys,et al.  Large deviations for a stochastic Landau-Lifshitz equation, extended version , 2012 .

[70]  Stefan Bruns,et al.  Measurement of the translational and rotational Brownian motion of individual particles in a rarefied gas. , 2006, Physical review letters.