A particle in a one-dimensional channel with excluded volume interaction displays anomalous single-file diffusion with fluctuations scaling as t^(1/4) in the long time limit. This phenomenon has been demonstrated in various experiments involving different types of physical systems such as zeolites, capillary pores, carbon nanotubes or colloids. On a one dimensional lattice, the Symmetric Exclusion Process, particles performing symmetric random walks and interacting by hard-core exclusion, is a pristine model of a single-file diffusion, amenable to quantitative analysis (Spitzer, 1970). At equilibrium, the variance of the position of a tagged particle has been calculated exactly by Arratia in 1983, and this result has been discussed since then in numerous theoretical papers, at various levels of physical intuition or mathematical rigor. However, the full distribution of a tagged particle and its higher cumulants (prone to experimental measurements) are not known. A rigorous work by S. Sethuraman et S. R. S. Varadhan proved the large deviation principle and
asymptotic bounds in 2013.
In a recent Letter, Takashi Imamura (Chiba), Tomohiro Sasamoto (Tokyo) and Kirone Mallick (IPhT) derive an exact and compact formula for the distribution of a tagged particle, calculate the cumulants to all orders as well as the large deviation function, thus answering a problem that has eluded solution for decades. They use the powerful mathematical arsenal of integrable probabilities developed recently to solve the one-dimensional Kardar-Parisi-Zhang equation. Their results can be extended to situations where the system is far from equilibrium, allowing them to prove a Gallavotti-Cohen Fluctuation Relation and to provide a highly nontrivial check of the Macroscopic Fluctuation Theory of G. Jona-Lasinio et al.
Large deviations of a tracer in the symmetric exclusion process, Takashi Imamura, Kirone Mallick and Tomohiro Sasamoto, Phys. Rev. Lett. 118, 160601 (17 April 2017) [Editor's Choice].