Brownian motion of a torus

A torus is one of the few axisymmetric bodies (solids of revolution), which on account of its peculiar shape, shows interesting characteristics during its motion through a fluid. Specifically, it leads to coupling of the translational and rotational degrees of freedom, resulting in non-trivial contributions of the “cross terms” to the diffusion coefficient and relaxation times. The study of Brownian motion of a torus is important on two counts: most stiff or semiflexible polymers like the DNA miniplasmids can be modeled as tori. Secondly, a torus happens to be one of the simplest objects which can explain a self-propelled microogranism. The length scales in both these problems make the study of Brownian motion important. We calculate in this paper, the translational and rotational diffusion coefficient of a torus and show that the coupling contributes to these coefficients, the effect being a function of the slenderness ratio, ϵ . The effect of the coupling is found to be reinforcing, although moderate. The coupling surprisingly has no effect on the autocorrelation function of the twirling degree of freedom ψ , when subject to a harmonic potential. The effect of diffusion on a toroidal swimmer is also calculated and results show that such a swimmer can undergo substantial diffusion before a directional (imposed or self-propelled) motion takes over.