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.