Single-particle and ensemble diffusivities--test of ergodicity.

Diffusion is the omnipresent, random motion of matter, such as atoms and molecules, driven by thermal energy and is the key for innumerable processes in nature and technology. In nearly every chemical reaction diffusion is the key mechanism of bringing the reactants in close proximity, which is an essential prerequisite before any reaction can take place. Additionally many reactions are diffusion-controlled, meaning that the reaction kinetics is limited by the diffusion process. Central to the dynamics of diffusion, and in general matter, is the ergodic theorem, which states that for systems in the equilibrium state the time average taken over a single particle is the same as the ensemble average over many particles. However, while being generally accepted no experimental validation has so far been reported. Here, we present experimental proof of this fundamental theorem by measuring under identical conditions the diffusivities of guest molecules inside a nanostructured porous glass using two conceptually different approaches. The data obtained through the direct observation of dye molecule diffusion by singlemolecule tracking experiments, that is, the time-average, is in perfect agreement with the ensemble value obtained in pulsed-field gradient NMR experiments. After one and a half centuries of diffusion measurements with large ensembles of diffusing particles, the option of single-particle tracking (SPT) with single-molecule sensitivity has recently provided us with a totally new view of diffusion. In this approach, the trajectory of a single, optically labeled molecule can be recorded during a sufficiently long interval of time. The obtained trajectory can thereafter be analyzed to access, for example, the average value of the squared displacement r(t) of a diffusing particle during a time interval t [Eq. (1)].