Hydrodynamic gauge fixing and higher order hydrodynamic expansion

Abstract

Hydrodynamics is a powerful emergent theory for the large-scale behaviours in many-body systems, quantum or classical. It is a gradient series expansion, where different orders of spatial derivatives provide an 
effective description on different length scales. We report the first fully general derivation of third-order, or ``dispersive", terms in the hydrodynamic expansion. Our derivation is based on general principles of statistical mechanics, along with the assumption that the complete set of local and quasi-local conserved densities constitutes a good set of emergent degrees of freedom. We obtain fully general Kubo-like expressions for the associated hydrodynamic coefficients (also known as Burnett coefficients), and we determine their exact form in quantum integrable models, introducing in this way purely quantum higher-order terms into generalised hydrodynamics. We emphasise the importance of hydrodynamic gauge fixing at diffusive order, where we claim that it is parity-time-reversal, and not time-reversal, invariance that is at the source of Einstein's relation, Onsager's reciprocal relations, the Kubo formula and entropy production. At higher hydrodynamic orders we introduce a more general, $n$-th order ``symmetric'' gauge, which we show implies the validity of the higher-order hydrodynamic description.

Mesoscopic impurities in generalized hydrodynamics

“…It should be stressed that the results and discussions are only valid in this limit; they do not describe the system at short or long times compared to the Euler-scale time. Any corrections in 1/L (such as diffusive corrections [47][48][49]) are also not considered.…”

“…However, the authors only looked at the λ dependence of E Dr (λ), which works in general for any T (λ, µ). Because the x dependence is also a topic of interest, the situation is more complicated and only gives equation (49) in the case T (λ − µ).…”

“…However, this is not generally true for microscopic impurities. It would be interesting to see whether higher-order gradient expansion [47][48][49] introduces nondeterministic scattering; otherwise, microscopic impurities will be fundamentally different from mesoscopic ones.…”

“…defined using equation (49). Given (∂ λ E) dr and (∂ x E) dr , the solution n(x, λ) can be easily found by computing the characteristics (47).…”

“…Furthermore, following basic ideas from GHD, one can view the regime of mesoscopic impurities as a zeroth-order term of an expansion in 1/L imp , where L imp is the spatial size of the impurity. In thissense, the results can also be a starting point to systematically gain insights into smaller impurities by taking into account diffusion (or further higherorder corrections) to GHD [47][48][49].…”