Testing the Equivalence between Spatial Averaging and Temporal Averaging in Highly Dilute Solutions

Diffusion relates the flux of particles to the local gradient of the particle density in a deterministic way. The question arises as to what happens when the particle density is so low that the local gradient becomes an ill-defined concept. The dilemma was resolved early last century by analyzing the average motion of particles subject to random forces whose magnitude is such that the particles are always in thermal equilibrium with their environment. The diffusion dynamics is now described in terms of the probability density of finding a particle at some position and time and the probabilistic flux density, which is proportional to the gradient of the probability density. In a time average sense, the system thus behaves exactly like the ensemble average. Here, we report on an experimental method and test this fundamental equivalence principle in statistical physics. In the experiment, we study the flux distribution of 20 nm radius polystyrene particles impinging on a circular sink of micrometer dimensions. The particle concentration in the water suspension is approximately 1 particle in a volume element of the dimension of the sink. We demonstrate that the measured flux density is exactly described by the solution of the diffusion equation of an infinite system, and the flux statistics obeys a Poissonian distribution as expected for a Markov process governing the random walk of noninteracting particles. We also rigorously show that a finite system behaves like an infinite system for very long times despite the fact that a finite system converges to a zero flux empty state.