  The path-integral representation of Smoluchowski equation is exploited to
explore the stochastic dynamics of a tagged Brownian particle within an
interacting system where hydrodynamic effects are neglected. In particular,
this formalism is applied to a particle system confined to a one-dimensional
infinite line aiming to investigate the single-file diffusion phenomenon in
this scenario. In particular, the path-integral method is contrasted against
the standard many-particle Langevin equation for a system of interacting
Brownian particles in a harmonic chain model, exhibiting excellent agreement;
in this case of study a formula defined on the whole time-scale for the
mean-square displacement, in the thermodynamic limit, is found for the tracer
particle in terms of Bessel functions, recovering also the single-file regime.
Additionally, a Brownian particle system with paramagnetic interactions is
considered near crystallization where the total interaction potential is
roughly a harmonic potential. Taking advantage of the path-integral formalism a
simple perturbation treatment is carried out to investigate the single file
diffusion behavior when temperature is increased away from the crystal phase.
