Comment on "Influence of Noise on Force Measurement" [arXiv:1004.0874]

In a recent Letter [arXiv:1004.0874], Volpe et al. describe experiments on a colloidal particle near a wall in the presence of a gravitational field for which they study the influence of noise on the measurement of force. Their central result is a striking discrepancy between the forces derived from experimental drift measurements via their Eq. (1), and from the equilibrium distribution. From this discrepancy they infer the stochastic calculus realised in the system. We comment, however: (a) that Eq. (1) does not hold for space-dependent diffusion, and corrections should be introduced; and (b) that the “force” derived from the drift need not coincide with the “force” obtained from the equilibrium distribution.

💡 Research Summary

In the original Letter, Volpe et al. investigated a colloidal particle sedimenting near a wall under gravity. They measured the particle’s drift velocity v_d(z) from time‑resolved trajectories and also obtained the equilibrium probability distribution P(z). Using the relation F(z)=γ(z)v_d(z) (their Eq. 1) they inferred a “force” from the drift, while from the equilibrium distribution they derived a potential U(z)=−k_BT ln P(z) and defined a second force F_e(z)=−dU/dz. The two forces differed markedly, and the authors concluded that the discrepancy revealed which stochastic calculus (Ito or Stratonovich) governed the underlying Langevin dynamics.

Mannella and McClintock point out that this conclusion rests on an incorrect application of Eq. 1 when the diffusion coefficient D⊥(z) varies with position. Starting from the generic one‑dimensional stochastic differential equation

 dz = f(z) dt + g(z) dW,

with f(z)=F(z)γ(z) and g(z)=√