Macroscopic fluctuation theory and its application

Non-equilibrium systems are ubiquitous in nature and exhibit a variety
of new phenomena. Classical statistical mechanics does not cover such
systems, and a general theoretical framework is lacking. A recently
proposed macroscopic fluctuation theory offers a promising direction
in building such a framework. It is based on the study of rare
fluctuations of macroscopic variables in non-equilibrium diffusive
systems. I shall present an overview of the theory based on two of my
recent works: single-file diffusion and melting of an Ising quadrant.

The first one is about motion of many particles in narrow channel
where particles can not pass each other. As a consequence of the
forbidden mutual passage the motion of individual particles is
sub-diffusive. I shall apply the macroscopic fluctuation theory to
analyze the probability distribution of the position of a tagged
particle in a large class of single-file system. For the simplest case
of point particles with hard-core repulsion this macroscopic approach
leads to an explicit formula of the large deviation function of the
tagged particle position. I shall compare the results with that
obtained in an exact microscopic analysis. I shall also emphasize the
unusual dependence of the statistics of the tracer position on the
initial state.

In the second example, I shall talk about a non-equilibrium problem of
domain growth in two dimensional Ising ferromagnet endowed with zero
temperature spin-flip dynamics. In one of the simplest scenario,
starting with a quadrant of one phase in the sea of opposite phase I
shall examine how the majority phase invades the minority phase. The
area of the invaded region grows with time and the interface
separating the phases exhibits non-trivial fluctuation. Utilizing a
mapping to exclusion process I cast the problem in the framework of
macroscopic fluctuation theory. This provides a systematic way of
analyzing the statistics of the fluctuating interface and the area of
the invaded region.

I shall conclude with a brief discussion of other problems where
macroscopic fluctuation theory has been successful and also talk about
future directions.

Ref: 
[1] Large deviations in single-file diffusion: P. L. Krapivsky,
Kirone Mallick, Tridib Sadhu, Phys. Rev. Lett. 113, 078101 (2014).
[2] Melting of an Ising quadrant: P. L. Krapivsky, Kirone Mallick,
Tridib Sadhu, J. Phys. A: math theo (under review).