# American Institute of Mathematical Sciences

June  2013, 6(2): 429-458. doi: 10.3934/krm.2013.6.429

## Collisionless kinetic theory of rolling molecules

 1 Department of Mathematics, Imperial College London, London SW7 2AZ 2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada 3 Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom

Received  March 2012 Revised  October 2012 Published  February 2013

We derive a collisionless kinetic theory for an ensemble of molecules undergoing nonholonomic rolling dynamics. We demonstrate that the existence of nonholonomic constraints leads to problems in generalizing the standard methods of statistical physics. In particular, we show that even though the energy of the system is conserved, and the system is closed in the thermodynamic sense, some fundamental features of statistical physics such as invariant measure do not hold for such nonholonomic systems. Nevertheless, we are able to construct a consistent kinetic theory using Hamilton's variational principle in Lagrangian variables, by regarding the kinetic solution as being concentrated on the constraint distribution. A cold fluid closure for the kinetic system is also presented, along with a particular class of exact solutions of the kinetic equations.
Citation: Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429
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