How to Compute Density Fluctuations at the Nanoscale

The standard definition of particle number fluctuations based on point-like particles neglects the excluded volume effect. This leads to a large and systematic finite-size scaling and an unphysical surface term in the isothermal compressibility. We correct these errors by introducing a modified pair distribution function that takes account of the finite size of the particles. For the hard sphere fluid in one-dimension, we show that the compressibility is strictly size-independent, and we reproduce this result from the number fluctuations calculated with the new theory. In general, the present method eliminates the leading finite-size effect, which makes it possible to compute density fluctuations accurately in very small sampling volumes, comparable to a single particle size. These findings open the way for obtaining the local compressibility from fluctuation theory at the nanometer scale.