Correlation functions for restricted Brownian motion from the ballistic through to the diffusive regimes

We calculate the spectra of the correlation functions for fields with arbitrary spatial dependence as seen by Brownian particles in bounded geometries from knowledge of the spectra of the conditional probability density functions in the infinite domain. Our results show a significant difference for the spectra for 1D, 2D and 3D motions. To our knowledge this is the first demonstration of the influence of dimensionality on the form of the correlation functions. Our results also show the different power dependence on frequency for the ballistic and diffusive cases and the treatment of the crossover is unique.

► Spectrum of the correlation function for fields with arbitrary spatial dependence. ► Valid for the diffusive and ballistic motions and the problematic crossover region. ► Calculation of the spectra in 1, 2 and 3 dimensions. ► Application to gradient induced relaxation in nmr, and to the search for new forces.