Hydrodynamic noise in one dimension: projected Kubo formula and how it vanishes in integrable models
When many particles interact with each other, the problem of describing their trajectories is extremely difficult even with modern super-computers. But this is not a real problem: after all, we are more interested in what happens on large scales, relevant to us and to most experimental situations. On large scales, we see new, collective behaviours, such as waves in the sea or the weather. These seem to follow their own, somewhat simplified equations, emergent from the more “fundamental” Newton or Schrödinger equations.
The true problem is to understand emergent laws of physics: to go from the very small to the very large. This concept goes beyond particles: in modern parlance, one speaks of agents interacting with each other, such as buyers and sellers in a market, and one looks for emergent global effects. In locally interacting particle systems, it turns out that it is the principles of hydrodynamics that determine the emergent laws. These say that instead of concentrating on all trajectories, one only needs to describe the conserved flows, such as those of mass, energy, or momentum. Conserved flows follow simple equations, where the forgotten microscopic motion of particles re-appears as a small “hydrodynamic noise”.
In this paper, in order to understand this problem in depth, I consider a simple situation: the system is assumed to be constrained to one dimension of space only, as is done in modern experiments on cold atoms or optic fibres. Then I explain how the equations of hydrodynamics come about from basic physical assumptions. Interestingly, I find that the hydrodynamics noise just disappears when there is a large number of conserved flows, a situation that is referred to as integrability and which often happens in one spatial dimension. But I also find that the usual equations of hydrodynamics must be modified at the order where the noise would have been, due to correlations between the flows. This paper introduces the general mathematical formalism that allows us not only to understand these ideas, but also to make very precise observable predictions that come out from them.
Author: Benjamin Doyon