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 this comment the authors critically examine the recent Letter by Volpe et al. (arXiv:1004.0874) that investigated how thermal noise influences the measurement of forces acting on a colloidal particle near a solid wall in a gravitational field. Volpe et al. reported a striking discrepancy between forces obtained from two independent procedures: (i) a direct measurement of the particle’s drift velocity, interpreted through their Eq. (1) (\langle\dot x\rangle = \mu F), and (ii) an inference from the equilibrium positional distribution, where the force is identified as the gradient of the effective potential (U(x) = -k_{!B}T\ln P_{\rm eq}(x)). From the mismatch they concluded that the underlying stochastic dynamics must realize a particular stochastic calculus (e.g., Stratonovich rather than Itô).

The present comment points out two fundamental flaws in that reasoning. First, Eq. (1) is only valid for a system with a constant diffusion coefficient. In the experimental geometry the diffusion coefficient (D(x)) varies strongly with distance from the wall because of hydrodynamic interactions and confinement. The correct Fokker–Planck description reads
\