Abstract
We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schrödinger equation in a random δ-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and using control theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bec J.: Multifractal concentrations of inertial particles in smooth random flows. J. Fluid Mech. 528, 255–277 (2005)
Bec J., Cencini M., Hillerbrand R.: Heavy particles in incompressible flows: the large Stokes number asymptotics. Physica D 226, 11–22 (2007)
Bec J., Cencini M., Hillerbrand R.: Clustering of heavy particles in random self-similar flow. Phys. Rev. E 75, 025301 (2007)
Bec J., Cencini M., Hillerbrand R., Turitsyn K.: Stochastic suspensions of heavy particles. Physica D 237, 2037–2050 (2008)
Duncan K., Mehlig B., Ostlund S., Wilkinson M.: Clustering in mixing flows. Phys. Rev. Lett. 95, 240602 (2005)
Elperin T., Kleeorin N., Rogachevskii I.: Self-Excitation of fluctuations of inertial particle concentration in turbulent fluid flow. Phys. Rev. Lett. 77, 5373–5376 (1996)
Falkovich G., Fouxon A., Stepanov M.G.: Acceleration of rain initiation by cloud turbulence. Nature 419, 151–154 (2002)
Falkovich G., Gawȩdzki K., Vergassola M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975 (2001)
Fouxon I., Horvai P.: Separation of heavy particles in turbulence. Phys. Rev. Lett. 100, 040601 (2008)
Fouxon I., Horvai P.: Fluctuation relation and pairing rule for Lyapunov exponents of inertial particles in turbulence. J. Stat. Mech.: Theor. & Experim. 08, (2007)
Friz, P.K.: An Introduction to Malliavin Calculus, lecture notes, http://www.math.nyu.edu/phd-students/frizpete/malliavin/mall.pdf , 2002
Gawȩdzki, K.: Soluble models of turbulent transport. In: Non-Equilibrium Statistical Mechanics and Turbulence, eds. S. Nazarenko, O. Zaboronski, Cambridge: Cambridge University Press 2008, pp. 44–107
Gradstein, I.S., Rhyzik, I.M.: Table of Integrals, Series, and Products, Vth edition. New York: Academic Press 1994
Halperin B.I.: Green’s functions for a particle in a one-dimensional random potential. Phys. Rev. 139, A104–A117 (1965)
Has’minskii, R.Z.: Stochastic Stability of Differential Equations. alphen aanden Rija, Netherlands: Sijthoff and Noordhoff, 1980
Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. III, Berlin: Springer, 1985
Horvai, P.: Lyapunov exponent for inertial particles in the 2D Kraichnan model as a problem of Anderson localization with complex valued potential, http://arxiv.org/abs/nlin/0511023v1 [nlin.co], 2005
Kraichnan R.H.: Small scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945–953 (1968)
Le Jan Y., Raimond O.: Integration of Brownian vector fields. Ann. Probab. 30, 826–873 (2002)
Le Jan Y., Raimond O.: Flows, coalescence and noise. Ann. Probab. 32, 1247–1315 (2004)
Lifshitz I.M., Gredeskul S., Pastur L.: Introduction to the Theory of Disordered Systems. Wiley, New York (1988)
Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl. 101, 185–232 (2002). doi:10.1016/S0304-4149(02)00150-3 http://dx.doi.org/10.1016/S0304-4149(02)00150-3
Maxey M.R., Riley J.J.: Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889 (1983)
Mehlig B., Wilkinson M.: Coagulation by random velocity fields as a Kramers problem. Phys. Rev. Lett 92, 250602 (2004)
Mehlig B., Wilkinson M., Duncan K., Weber T., Ljunggren M.: On the aggregation of inertial particles in random flows. Phys. Rev. E 72, 051104 (2005)
Meyn S.P., Tweedie R.L.: Markov Chains and Stochastic Stability Communication and Control Engineering Series. Springer-Verlag, London (1993)
Meyn S.P., Tweedie R.L.: Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548 (1993)
Norriss, J.: Simplified Malliavin calculus. In: Séminaire de probabilité XX, Lectures Note in Math. 1204, Berlin: Springer, 1986, pp. 101–130
Nualart, D.: Malliavin Calculus and Related Topics, 2nd edition, Berlin-Heidelberg: Springer, 2006
Piterbarg L.: The top Lyapunov exponent for a stochastic flow modeling the upper ocean turbulence. SIAM J. Appl. Math. 62, 777–800 (2001)
Rey-Bellet, L.: Ergodic properties of Markov processes. In: Open Quantum systems II. The Markovian approach, Lecture notes in Mathematics 1881, Berlin: Springer, 2006, pp. 1–78
Scheutzow M.: Stabilization and destabilization by noise in the plane. Stoch. Anal. Appl. 11, 97–113 (1993)
Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proc. 6-th Berkeley Symp. Math. Stat. Prob., Vol. III, Berkeley: Univ. California Press, 1972, pp. 361368
Wilkinson M., Mehlig B.: The path-coalescence transition and its applications. Phys. Rev. E 68, 040101 (2003)
Wilkinson M., Mehlig B.: Caustics in turbulent aerosols. Europhys. Lett. 71, 186–192 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Aizenman
Rights and permissions
About this article
Cite this article
Gawȩdzki, K., Herzog, D.P. & Wehr, J. Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles. Commun. Math. Phys. 308, 49–80 (2011). https://doi.org/10.1007/s00220-011-1343-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1343-5