Periodic Reporting for period 1 - QuasiHydro (Beyond hydrodynamics)
Reporting period: 2021-12-01 to 2023-11-30
For an intuitive model of a fluid one can think of a box filled with many, many randomly moving balls that are constantly colliding with one another. From this perspective it is not hard to see that if one were to reach into the box and flick one of the balls to move it at a higher speed, the perturbed ball will quickly lose its additional speed through collisions with the other balls. However, if we were instead to take all the balls and place them in one corner of the box, because they can only spread out randomly and by colliding with each other, the balls will take time to diffuse into the rest of the box. The end result in both cases will be that the balls are roughly evenly spaced in the box if seen from a distance. The flick is a short-time process, while the diffusion is a slow one. It is these slow processes that are described by hydrodynamics.
Key to hydrodynamics are the concepts of equilibrium (when the balls eventually are uniformly spread in the box) and conservation (the number of balls does not change). Consequently, the mathematics of hydrodynamics describes how the density of conserved charges (e.g. the total number of balls) in space evolves in time until the system reaches equilibrium. In particular hydrodynamics is typically formulated as conservation laws of energy, momentum and particle number in a “gradient expansion” which tells us how the densities of these quantities differ from global equilibrium as we move through the system (box). However, it is not always the case that the systems we wish to describe have exactly conserved energy, momentum and charge. We have now entered the regime of “quasihydrodynamics” or “relaxed hydrodynamics”. The mathematical description of relaxed hydrodynamics is far less developed than its hydrodynamic cousin, and yet it is important to describe exotic states of matter (such as strange metals), traffic flows and active matter.
The main goal of this project was to develop the quasihydrodynamic formalism taking inspiration from observations in gauge-gravity duality.
Key to hydrodynamics are the concepts of equilibrium (when the balls eventually are uniformly spread in the box) and conservation (the number of balls does not change). Consequently, the mathematics of hydrodynamics describes how the density of conserved charges (e.g. the total number of balls) in space evolves in time until the system reaches equilibrium. In particular hydrodynamics is typically formulated as conservation laws of energy, momentum and particle number in a “gradient expansion” which tells us how the densities of these quantities differ from global equilibrium as we move through the system (box). However, it is not always the case that the systems we wish to describe have exactly conserved energy, momentum and charge. We have now entered the regime of “quasihydrodynamics” or “relaxed hydrodynamics”. The mathematical description of relaxed hydrodynamics is far less developed than its hydrodynamic cousin, and yet it is important to describe exotic states of matter (such as strange metals), traffic flows and active matter.
The main goal of this project was to develop the quasihydrodynamic formalism taking inspiration from observations in gauge-gravity duality.
In this project, the structure of and constraints on quasihydrodynamics were investigated. One of the key objectives was to demonstrate that it was possible to write a consistent (quasi-)hydrodynamic theory around stationary states with relaxation. To achieve this, the researcher began by generalising the Drude model which is a phenomenological description for charge flow in a material to a hydrodynamic derivative expansion. The condition relating the velocity of the charge carriers to the phenomenological momentum loss term, a key assumption of the Drude model, became the defining relation for the stationary flow of a fluid. Subsequently, the suite of non-dissipative transport coefficients and DC conductivities were derived allowing us to achieve the initial goal.
Having completed one research objective, the researcher turned to the second – determining the range of validity of quasihydrodynamics. One avenue of research suggested in the original proposal was to consider anomalous hydrodynamics, a particularly constrained quasihydrodynamic theory. Several results for the thermo-electric conductivities of this theory in the presence of a weak, external magnetic field were known in the literature and in mutual contradiction. This naturally represented a challenge to the validity of the quasihydrodynamic framework. However, the researcher managed to resolve this issue by considering the same models with strong magnetic fields and then taking the small field limit, thus allowing us to arrive at a single consistent answer.
In, tackling this inconsistency, the grantee had at one point shown that standard hydrodynamics constraints – namely, Onsager reciprocity – were not satisfied by some of the literature results. This naturally led to questioning how this symmetry (and the related partner constraint of positivity of entropy production) appear in a quasihydrodynamic theory. In a series of works these constraints were examined and seen to restrict the space of possible relaxation terms. Moreover, in achieving this objective, standard kinetic theory was extended to include generic relaxation terms.
The results of this action have been disseminated through six scientific publications in peer-reviewed journals and presentations at five conferences/workshops. One of the papers is an invited review of the grantee's previous work in the field of relaxed hydrodynamics. All the papers are available on the arXiv so that the wider public has access to the results.
Having completed one research objective, the researcher turned to the second – determining the range of validity of quasihydrodynamics. One avenue of research suggested in the original proposal was to consider anomalous hydrodynamics, a particularly constrained quasihydrodynamic theory. Several results for the thermo-electric conductivities of this theory in the presence of a weak, external magnetic field were known in the literature and in mutual contradiction. This naturally represented a challenge to the validity of the quasihydrodynamic framework. However, the researcher managed to resolve this issue by considering the same models with strong magnetic fields and then taking the small field limit, thus allowing us to arrive at a single consistent answer.
In, tackling this inconsistency, the grantee had at one point shown that standard hydrodynamics constraints – namely, Onsager reciprocity – were not satisfied by some of the literature results. This naturally led to questioning how this symmetry (and the related partner constraint of positivity of entropy production) appear in a quasihydrodynamic theory. In a series of works these constraints were examined and seen to restrict the space of possible relaxation terms. Moreover, in achieving this objective, standard kinetic theory was extended to include generic relaxation terms.
The results of this action have been disseminated through six scientific publications in peer-reviewed journals and presentations at five conferences/workshops. One of the papers is an invited review of the grantee's previous work in the field of relaxed hydrodynamics. All the papers are available on the arXiv so that the wider public has access to the results.
In many ways the formalism of relaxed hydrodynamics is now on a much firmer footing thanks to both the results of this project and related other works in the literature. At the beginning of the project period, the state of the art in the holographic literature for relaxed hydrodynamic models was represented by such systems as: Q-lattices where translational symmetry is broken pseudo-spontaneously using scalar fields, probe branes at large charge density and charged hydrodynamics in external magnetic fields. Quasihydrodynamic models of these systems often treated the symmetry breaking parameters as though they were order one in derivatives (and therefore weak). Moreover, there was no reasonably generic formalism which dispensed with having an additional external field sourcing the symmetry breaking. One of the key results of this project has been the development of a formalism that allows one to include strong (order zero in derivative) and reasonably generic relaxation into the (hydrodynamic) stationary state of the system.
Additionally, questions regarding the well-definedness of quasihydrodynamics have been attacked by trying to understand how classic constraints from hydrodynamics, such as Onsager reciprocity and positivity of entropy production, carry over into the quasihydrodynamic regime. It turns out that it is possible to realise relaxed hydrodynamic theories that satisfy both of these constraints. However in certain circumstances, such as for anomalous hydrodynamics, enforcing both of these constraints may be undesirable as they can be incompatible with other desirable constraints such as having finite DC thermo-electric conductivities.
We have only just scratched the surface of the formulation and applications of relaxed hydrodynamics. Much of the formal structure of this effective theory requires further development. Importantly we need to understand how much of the full response of the system can be captured by the effective theory and if quasihydrodynamics predicts its own range of applicability (through hydrodynamic series expansions). Moreover, the grantee plans to continue exploring applications of this framework by examining both theoretical systems (such as probe branes in holography) and more practical models such as active matter.
Additionally, questions regarding the well-definedness of quasihydrodynamics have been attacked by trying to understand how classic constraints from hydrodynamics, such as Onsager reciprocity and positivity of entropy production, carry over into the quasihydrodynamic regime. It turns out that it is possible to realise relaxed hydrodynamic theories that satisfy both of these constraints. However in certain circumstances, such as for anomalous hydrodynamics, enforcing both of these constraints may be undesirable as they can be incompatible with other desirable constraints such as having finite DC thermo-electric conductivities.
We have only just scratched the surface of the formulation and applications of relaxed hydrodynamics. Much of the formal structure of this effective theory requires further development. Importantly we need to understand how much of the full response of the system can be captured by the effective theory and if quasihydrodynamics predicts its own range of applicability (through hydrodynamic series expansions). Moreover, the grantee plans to continue exploring applications of this framework by examining both theoretical systems (such as probe branes in holography) and more practical models such as active matter.