On estimating stress in free-draining Kramers chain simulations using stochastic filtering

For free-draining Kramers chains, several possible algorithms for calculating the stress tensor are examined, when Liu's algorithm is used for the dynamics. Previous methods assumed that Stratonovich calculus was necessary to interpret the stochastic differential equation and calculate the stress, which is incorrect. Here, we consider these previous methods and derive two new ones using Ito calculus. All methods considered give correct averages. However, differences are seen in the statistical error, computational costs, and simplicity. The explicit expression derived by Liu is the most expensive to calculate, and most complicated. A version using 'stochastic filtering', proposed by Grassia and Hinch [P. Grassia, E.J. Hinch, Computer simulations of polymer chain relaxation via Brownian motion, J. Fluid Mech. 308 (1996) 255-288], and independently by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, A.P. Gast, Dynamic simulation of freely draining flexible polymers in steady linear flows, J. Fluid Mech. 334 (1997) 251-291] yields the same statistical error, but requires less computational cost than Liu's expression. A new algorithm is proposed, which yields uncertainties identical to those found by Grassia and Hinch, but is simpler, and requires smaller memory storage. An even simpler expression is proposed which, unlike that using stochastic filtering, requires no additional information from that required for the Brownian dynamics simulations. This last expression yields a larger statistical error than the method proposed here, and is found not to be advantageous for computational costs. Hence, the simplified stress calculator introduced here is the recommended one.