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The indefinite integral of the homogenized Ornstein-Uhlenbeck process is awell-known model for physical Brownian motion, modelling the behaviour of anobject subject to random impulses [L. S. Ornstein, G. E. Uhlenbeck: On thetheory of Brownian Motion. In: Physical Review. 36, 1930, 823-841]. One canscale these models by changing the mass of the particle and in the small masslimit one has almost sure uniform convergence in distribution to the standardidealized model of mathematical Brownian motion. This provides one well knownway of realising the Wiener process. However, this result is less robust thanit would appear and important generic functionals of the trajectories of thephysical Brownian motion do not necessarily converge to the same functionals ofBrownian motion when one takes the small mass limit. In presence of a magneticfield the area process associated to the physical process converges - but notto L\'evy's stochastic area. As this area is felt generically in settings wherethe particle interacts through force fields in a nonlinear way, the remark isphysically significant and indicates that classical Brownian motion, with itsusual stochastic calculus, is not an appropriate model for the limitingbehaviour.  We compute explicitly the area correction term and establish convergence, inthe small mass limit, of the physical Brownian motion in the rough path sense.The small mass limit for the motion of a charged particle in the presence of amagnetic field is, in distribution, an easily calculable, but "non-canonical"rough path lift of Brownian motion. Viewing the trajectory of a chargedBrownian particle with small mass as a rough path is informative and allows oneto retain information that would be lost if one only considered it as aclassical trajectory. We comment on the importance of this point of view.

Physical Brownian motion in magnetic field as rough path