# Force Measurement at the Nanoscale

Apart from the technological challenge intrinsic to dealing with forces in the order of femtonewton, it is important to realize that the general concepts we apply in our everyday life cannot be simply scaled down to microscopic objects, mainly due to the presence of thermal noise affecting the motion of small objects (Brownian motion).

We demonstrated how the ineluctable presence of thermal noise alters the measurement of forces acting on microscopic and nanoscopic objects. Our results demonstrated that the force-measurement process is prone to artifacts if the noise is not correctly taken into account. Indeed, the presence of a spatially varying Brownian noise leads to the presence of spurious forces, i.e. forces that exist only due to and in the presence of thermal noise. If overlooked, this leads to erroneous forces, which may severely affect the physical interpretations of experimental data. We quantified this effect exemplarily for a Brownian particle near a wall subjected to gravitational and electrostatic forces.

These results are intimately connected to the long-standing issue of the interpretation of multiplicative noise in stochastic differential equations, i.e. the Ito-Stratonovic dilemma.

The forces acting on a microscopic object immersed in a fluid medium can be assessed either by studying the underlying potential or by studying their effect on the object’s trajectory. The first approach – to which we shall refer as *equilibrium distribution method* – requires sampling of the equilibrium distribution. Accordingly, it can be only applied under conditions where the investigated system is at or close to thermodynamic equilibrium with a heat bath. The second method – to which we shall refer as *non-equilibrium method* or *drift method* – does not require the object to be at or even close to thermal equilibrium. Therefore, it can be applied also to systems which are intrinsically out-of-equilibrium, e.g., molecular machines, transport through pores, DNA stretching. The latter method, however, requires to detect the object trajectory with high sampling rates, which can be technologically more challenging, in particular when combined with a high spatial resolution.