We present a discussion of positional and velocity correlations of particles in single-file diffusion, based on some earlier work. We consider two physical situations: (a) An open system of N hard-core particles on an infinite line. (b) A large system with a fixed density of hard-core particles at an arbitrary temperature. In the first case (a), moments and correlations show unusual behavior. The average displacement of a particle is nonzero and grows as t1/2. Furthermore it depends on the position of the particle. Particles on the right of center are pushed right and those on the left are pushed left. The mean-square displacement also depends on the position. The diffusion constant is small for particles around the center but grows rapidly toward edges. Certain correlations in particle displacement grow with separation. For the second case (b) we give exact results for velocity-velocity auto-correlator of a tagged particle and establish that with time this correlator becomes negative and approaches zero as a power-law t-3/2 at long times. The mobility of the tagged particle is shown to decrease rapidly with density as has been observed in experiments. [Formula: see text] Special Issue Comments: This article presents mathematical results on the dynamics in expanding files, and constant density files. This article is connected to the Special Issue articles about advanced statistical properties in single file dynamics29 and files with force and advanced formulations.30
Single-file diffusion in a bi-stable potential: Signatures of memory in the barrier-crossing of a tagged-particle
