A Model for Hydrodynamics in Kinetic Field Theory

A Model for Hydrodynamics in Kinetic Field Theory

C. Viermann1,2, J.T. Schneider1,3, R. Lilow1,4, F. Fabis1, C. Littek1, E. Kozlikin1, M. Bartelmann1 1 Heidelberg University, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Philosophenweg 12, 69120 Heidelberg, Germany 2 Heidelberg University, Kirchhoff-Institut für Physik, Im Neuenheimer Feld 227, 69120 Heidelberg 3 Heidelberg University, Institut für Theoretische Physik, Philosophenweg 16, 69120 Heidelberg 4 Department of Physics, Technion, Haifa 3200003, Israel
Abstract

In this work, we introduce an effective model for both ideal and viscous fluid dynamics within the framework of kinetic field theory (KFT). The main application we have in mind is cosmic structure formation where gaseous components need to be gravitationally coupled to dark matter. However, we expect that the fluid model is much more widely applicable.

The idea behind the effective model is similar to that of smoothed particle hydrodynamics. By introducing mesoscopic particles equipped with a position, a momentum, and an enthalpy, we construct a free theory for such particles and derive suitable interaction operators. We then show that the model indeed leads to the correct macroscopic evolution equations, namely the continuity, Euler, Navier-Stokes, and energy conservation equations of both ideal and viscous hydrodynamics.

I Introduction

In upcoming years, powerful weak gravitational lensing studies will hopefully deliver detailed data on the matter distribution in the universe. To gain meaningful insight, these measurements must be combined with accurate predictions of the cosmological power spectrum and higher order spectra for all matter.

In past works, we presented an analytic approach to predicting the dark matter power spectrum. This approach is based on a kinetic field theory for classical particles (KFT) originally developed by Das and Mazenko (Das and Mazenko (2013, 2012); Mazenko (2011)) to describe the ergodic to non-ergodic transition in glasses. It was then adapted to cosmic structure formation by Bartelmann et. al (Bartelmann et al. (2016)).

KFT allows the prediction of correlation functions of macroscopic quantities (like density or velocity-density) based only on initial correlations and microscopic particle dynamics. First very promising results were achieved that closely match simulations (Fabis et al. (2018); Bartelmann et al. (2017, 2016); Viermann et al. (2015); Bartelmann (2015); Dombrowski et al. (2018)).

So far, calculations within KFT are limited to ensembles of weakly interacting particles like dark matter particles whose trajectories are only slightly perturbed (which can nonetheless lead to large density perturbations).

For many cosmological applications, the dynamics of dark matter must be coupled to the dynamics of ‘ordinary’ matter. The gaseous ordinary matter behaves fluid-like, i.e. it is in a dynamical regime where particles undergo frequent and strong interactions. As we will argue in more detail below, within KFT this strongly interacting regime is difficult to reach from fundamental principles, although it must be conceptually possible.

As an alternative, we present an effective model for fluid ensembles in the KFT description. The basic idea of this model is similar to the conventional derivation of hydrodynamics and particle-based schemes in numerical fluid codes like smoothed particle hydrodynamics (SPH).

While our main motivation is the cosmological matter power spectrum for mixtures of dark and fluid-like matter, the fluid model in KFT is not limited to this application. It might (in a resummed form) give access to statistical properties of different fluid phenomena, maybe even to turbulent spectra. A combination of the hydrodynamical model with electromagnetic particle interactions might allow the study of magnetohydrodynamics and related instabilities.

Within this work, we introduce the model of hydrodynamics in the framework of KFT, show that the corresponding macroscopic evolution equations match the continuity equation, the Euler or Navier-Stokes equation, respectively, and the energy conservation equation of hydrodynamics. Finally, we present an exemplary calculation of how a localised high-pressure region evolves dynamically within the KFT fluid model.

Before we shift our focus to hydrodynamics, we briefly introduce KFT for readers unfamiliar with the formalism.

Ii Brief summary of KFT

We follow the notation introduced in Bartelmann et al. (2016), to which we refer readers for a more thorough derivation. Here, we only provide a brief summary which is partly taken from Viermann et al. (2015). The latter work also contains an exemplary calculation within KFT which might help in understanding the formalism.

The derivation of this non-equilibrium statistical field theory begins with the equation of motion of an -component classical field in space-time dimensions

(1)

where all terms describing free motion are bundled into and all interaction terms between particles or with external fields are part of .

For a classical theory, the field can only evolve from an initial state into a final state if the transition satisfies the above equation of motion. Starting from this requirement and setting the interaction part of the equation of motion to zero, for the moment, one can derive the centre piece of the theory, i.e. its free generating functional

(2)

where was introduced as a source field conjugate to the degrees of freedom . The free trajectory of each degree of freedom is denoted by an overbar, , and is defined as

(3)

where is the initial field configuration, the retarded Green’s function of the free equation of motion and a second source field that will later allow to introduce interactions. More precisely, it is the source field to an auxiliary field that needs to be introduced in the derivation of the generating functional. In the free functional, the are already integrated out but will reappear once interactions are switched on. Finally, denotes the integration over the initial state space for all degrees of freedom weighted by an initial probability distribution.

So far, the interacting part of the equation of motion has been neglected. If it is included, the generating functional can be brought into the form

(4)

Here we introduced the interaction term whose operator representation is defined by

(5)

with an implied summation over . In Eq. (4), the -derivative contained in the interaction operator acts on the inhomogeneous term in and thus alters the time evolution of the respective degree of freedom.

For explicit calculations, the exponential function containing the interaction operator must be expanded into a perturbation series. In the simplest case, this is just a Taylor expansion in the interaction strength which leads to the generating functional in -th order of canonical perturbation theory

(6)

Note that the expansion in orders of the interaction operator is the only approximation in the derivation of the generating functional from the classical equation of motion.

From the generating functional, expectation values for different degrees of freedom can be extracted by functional derivatives with respect to and . For example, a general field correlator reads

(7)

The derivatives extract the respective time-evolved field components from the generating functional which are then averaged over the initial field configuration, corresponding to an ensemble average.

As the time evolution of degrees of freedom does not generally result in an equilibrium state, the above expression corresponds to a non-equilibrium ensemble average at times specified in the functional derivatives.

While the generating functional is built upon microscopic degrees of freedom and their equations of motion, it is possible to extract information about macroscopic collective fields by means of operators. This is an important advantage of this theory.

An example for such a collective field is the spatial number density , if the degrees of freedom of the ensemble are specified to the phase space coordinates of particles (see for example Bartelmann et al. (2016) or Viermann et al. (2015)). Then, a collective spatial particle density field at position and time is simply the sum over all point-particle contributions at this position and time

(8)

Fourier transforming this expression and replacing the particle positions by derivatives with respect to the conjugate source field, , yields the particle density operator in Fourier space

(9)

We will later need several other collective fields like velocity-density or pressure, which we will introduce as needed.

Iii Incorporating Fluid Dynamics

The full theory of KFT in its exact form of Eq. (4) contains the complete dynamics that can arise in a particle ensemble, including the dynamics of particles in a fluid. However, performing actual calculations is complicated due to the nature of the theory: KFT describes the linear propagation of particles exactly, but in most applications interactions must be included in a perturbation series in the interaction operator. Ensembles dominated by particle propagation with only small disturbances due to interactions can be well represented, as they can be described in a low perturbative order.

In contrast, a fluid is dominated by the interactions of microscopic particles: For ideal hydrodynamics, microscopic particles are assumed to be so tightly coupled that their mean free path is zero. Viscous effects arise when this assumption is relaxed to a finite but still small mean free path. To reach the regime of fluid dynamics in KFT, one would need to go to very high order in perturbation theory. Unfortunately, this is impossible for any practical calculation.

Here, we nonetheless attempt to incorporate fluid dynamics using an idea close to the conventional approach to hydrodynamics: One of the assumptions in the conventional derivation is the existence of the hydrodynamical scale hierarchy. It states that one can choose a scale in such a way that it is much larger than the mean free path of particles and at the same time much smaller than the characteristic scale of the phenomena one is interested in. On this intermediate scale, fluid elements containing many microscopic particles can be introduced. Their properties are then defined as average quantities of all microscopic particles contained.

This has two advantages: First, the frequent collisions of the microscopic particles quickly establish local thermodynamic equilibrium on the scale of each individual fluid element. This, in turn, allows defining local state variables like pressure and energy density. Also, microscopic conservation of particle number, energy, and momentum gives rise to macroscopic conservation laws such that further details of the microscopic interactions are rendered unimportant for the dynamics of the ensemble.

The time evolution of each fluid element is well described by a propagation with the average velocity and perturbations caused by accelerations due to pressure gradients and pressure-volume work. As long as no large gradients in the pressure and velocity field appear, these perturbations stay small. For viscous hydrodynamics, additional effects caused by diffusion of energy and momentum occur, which in most circumstances are only small perturbations as well.

Within the framework of KFT, these simplified dynamics motivate an approach to hydrodynamics based on mesoscopic particles similar to fluid elements. These mesoscopic particles propagate with a locally averaged velocity. The propagation of the mesoscopic particles is described by a Green’s function contained in a free functional. Any necessary interactions can then be included with the help of interaction operators. As long as all interactions remain sufficiently weak, i.e. as long as the fluid does not display large gradients, it is likely that a low-order canonical perturbation theory already captures fluid dynamics.

In the following sections, we set up a free generating functional and interaction operators that model both ideal and viscous hydrodynamics based on mesoscopic particles of equal mass. Each of these particles can be characterised by three properties: the first two are its centre of mass position and velocity (or equivalently momentum ). As this velocity is defined as an average over the velocities of all microscopic particles contained by the mesoscopic particle, it will depend on the scale on which the mesoscopic particles are defined. The average velocity describes only part of the information that is originally contained in the velocities of the microscopic particles. Further information is contained in the velocity dispersion which we include here in the form of a local stress-energy tensor. In summary, we will describe fluid dynamics in KFT by introducing fluid particles, that propagate with the local average velocity field and carry with them a local representation of the stress-energy tensor.

For the specific case of an isotropic fluid, the stress-energy tensor is fully characterised by the pressure . If one assumes in addition, that the pressure is only caused by the random velocities of the particles and that the particles do not have any internal degrees of freedom, the pressure relates to the internal energy-density as .
It is convenient to combine both pressure and internal energy-density into the enthalpy-density

(10)

as this allows to include all necessary information about a fluid particle by assigning the enthalpy as a third property to the mesoscopic particles.

We emphasise again that in principle, it must be possible to derive these three properties as well as all interactions of hydrodynamics from a fully microscopic theory but this remains a task for the future. In an approach similar to smoothed particle hydrodynamics, we start from macroscopic hydrodynamical equations and construct an appropriate free theory and interaction operators to model hydrodynamics based on the mesoscopic particles described above. Thereby, we demand the existence of the hydrodynamical scale hierarchy to assume that the mesoscopic particles have no significant spatial extent in comparison with the fluid phenomena we are interested in and thus can effectively be modelled as point-like.

iii.1 Free theory for mesoscopic particles

In the free theory of hydrodynamics, we include all dynamics that can be described as the propagation of mesoscopic particles. Namely, we set up an ensemble of equal-mass particles which are characterised by three properties: their centre of mass , a centre of mass momentum , and an enthalpy .

The particles change their position according to the propagation with their momentum. Both momentum and enthalpy of each individual particle are conserved and are carried around with the particle. Hence, the momentum-density or enthalpy-density of the fluid might change locally as particles converge or diverge.

The dynamics of both ideal and viscous hydrodynamics, i.e. acceleration due to pressure gradients, pressure-volume work, and diffusive effects will later be included with the help of interaction operators. These effects will change the momentum and enthalpy of individual particles.

To set up the free generating functional with the above dynamics, we first bundle the degrees of freedom of the -th particle into a single vector

(11)

with a corresponding source-field vector defined as

(12)

For the free theory, we set all forces to zero. Hence the position of each particle only changes due to propagation, i.e. linearly with its momentum

(13)

and the particle’s momentum and enthalpy are both conserved

(14)

These equations of motion are solved by the retarded Green’s function

(15)
(16)

where is the three dimensional unit matrix.

Using the above Green’s functions as well as Eq. (4), we can define the free generating functional for mesoscopic particles as

(17)

where the tilde marks the free generating functional specified for mesoscopic particles. For the sake of compact notation we bundle the respective quantities of all particles into a single tensorial object which is denoted by a boldface character. Defining the -dimensional column vector , whose only non-vanishing entry is unity at component , these read

(18)

with a scalar product defined by

(19)

where a sum over is implied. In any other context, the angular brackets indicate the non-equilibrium ensemble average described in the last section.

The time-evolved degrees of freedom of the -th particle read

(20)

with the additional source field

(21)

Finally, the -integration runs over all initial degrees of freedom weighted by their initial probability distribution

(22)

where the probability distribution is chosen such that it samples the smooth macroscopic fields of a fluid.

Note that this choice of dynamics specifies the ensemble to a fluid with isotropic pressure. For the non-isotropic case, the full stress-energy tensor would be needed instead of . However, this would render the notation too involved for the purpose of this paper.

iii.2 Collective fields

To include appropriate interactions between mesoscopic particles, we first need to define collective field operators for number-density, velocity-density, momentum-density, and energy-density as well as pressure.

We can define the operator for the number-density contribution of the -th particle in configuration space as

(23)

the operators for velocity and momentum of the -th particle by

and the operator for the enthalpy of the -th particle by

(24)

As a consequence of the particle-based description, the velocity operator, as well as the momentum and enthalpy operators can never describe the spatial dependence of a collective field in configuration space. This is reflected by the absence of any spatial coordinate in their definition. Therefore, to obtain macroscopic collective information about these quantities in configuration space, they must be paired with a density operator carrying the same particle index (for more details see Viermann et al. (2015)). While we introduce the above ‘naked’ operators for the sake of an easier notation, they must always appear as a velocity-density, momentum-density or enthalpy-density,

(25)
(26)
(27)

Using the enthalpy-density, we can finally define operators for pressure and internal energy

(28)
(29)

Note that the above operators extract properties of a single particle (the -th particle). The corresponding macroscopic fields are found by summing over all particles.

iii.3 Interactions

The free theory set up in the previous section already contains the propagation of mesoscopic particles and the corresponding transport of particle properties.

To arrive at a fluid-like behaviour of the ensemble, we need to incorporate all remaining dynamics with the help of interaction operators. These need to contain accelerations by pressure gradients and pressure-volume work in the case of ideal hydrodynamics and additional effects of momentum and energy diffusion for viscous hydrodynamics.

The main difficulty in the derivation of these interaction operators is the fundamental difference between conventional hydrodynamics and KFT: The hydrodynamical interactions we are looking for depend on derivatives of continuous fields; KFT, however, is a particle-based approach. By adding up particle contributions, it is possible to form fields. However, as particles are point-like, these fields are discontinuous and their local derivatives are ill-defined.

Here, we present two possible derivations that deal with this difficulty and arrive at proper interaction operators. The first is closely linked to smoothed particle hydrodynamics (SPH Monaghan (2005)): With the help of a density kernel, discrete particles and their properties are smoothed into continuous fields. The interactions for these fields are read off the hydrodynamical equations and are rediscretised to describe the dynamics of individual particles.

The second derivation follows heuristic arguments that interpret the physical processes taking place on the scale of the mesoscopic particles.

Both approaches have their merits and lead to interaction terms that only differ in irrelevant details. As the approaches highlight different aspects of the properties and interpretations of the hydrodynamical model, we believe it to be advantageous to present both. The mathematically more rigorous and elegant SPH-derivation is described in the subsequent paragraphs, while the heuristic derivation can be found in Appendix B.

SPH translates the smooth field equations of conventional hydrodynamics to a particle picture. This is done by smoothing out each particle by a three-dimensional normalised kernel centred on the position of the original point-particle .

We adopt this approach and rewrite density, momentum-density, velocity-density, pressure, energy-density, and pressure flux in terms of smoothed particles

(30)
(31)
(32)
(33)
(34)
(35)

where the sums run over all particles. With these fields at hand we can construct the KFT interaction operators.

iii.3.1 Acceleration by pressure gradients

To model acceleration due to pressure gradients, we start with the time evolution of the momentum field as it appears in the Euler equation

(36)

where we inserted the discretised pressure field Eq. (33).

We then pull the position of the -th particle into a Dirac delta distribution that we identify with the density of the -th particle,

(37)

The momentum change at position must be reassigned to particles at this position according to their (spatial) contribution. To this end, we weight the momentum change at with the density contribution of the -th particle and integrate over the entire space,

(38)

iii.3.2 Pressure volume work

To include pressure-volume work in an analogous way, we first determine the time evolution of the enthalpy from the energy conservation equation of ideal hydrodynamics,

(39)

The field cannot be discretised on its own. To circumvent this problem, we can rewrite the derivative as

(40)

Inserting Eq. (33) and Eq. (35) into Eq. (40), we get

(41)

where in the second line we again introduced a Dirac delta distribution which we identify with the density contribution of the -th particle at position .

The enthalpy change is then represented by particles. Picking out the enthalpy change of the -th particle, we get

(42)

Here, the velocity field at position was replaced with the velocity of the th particle. This can be done since velocities are sampled from a smooth field and the appearing specifies the position of the th particle to be .

iii.3.3 Diffusive effects

To find an interaction operator for diffusive effects, we first define diffusive currents of momentum and energy-density and then model their impact on the properties of the mesoscopic particles. Diffusive currents are proportional to gradients in the corresponding densities. For the energy-density and momentum-density currents we get

(43)
(44)

where the indices and denote vector components such that implies the derivative with respect to the -th component of the position vector and is the -th component of the momentum density field at . and are diffusive constants that depend, among other things, on the microscopic particles’ mean free path and collision rates and have the unit . The minus signs ensure that diffusive currents flow from over-dense to under-dense regions and denotes the symmetrization of the stress-energy tensor . This is needed for momentum and angular momentum conservation. More precisely, it ensures that no viscous effects arise for solid body rotation. For a proof see for example Bartelmann (2013).

In the case of isotropic diffusion, only two independent momentum diffusion constants remain

(45)

where describes the diffusion in the direction of the average flow (bulk flow), and the diffusion orthogonal to it (shear flow).

Divergences in the diffusive currents will lead to a local change of enthalpy and momentum. Using the discretization Eq. (31), we get for the momentum change,

(46)

where the indices and indicate vector components and a summation over is implied. In the last line a Dirac delta distribution was used to pull from the kernel and the delta distribution was identified with the contribution of the -th particle.

The momentum change of the field is converted to the momentum change of the -th particle by

(47)

For the enthalpy change, we need to consider an additional phenomenon: the momentum-density current tensor has the unit of pressure and hence performs pressure-volume work. Including this effect in addition to the changes caused by divergences in the energy-density current, the enthalpy change is

(48)

Once again, the velocity field cannot be discretized on its own. We rewrite

(49)

with the discretization

(50)

Inserting Eq. (50) as well as Eq. (43) with the discretization Eq. (34) into Eq. (48) yields

(51)

Introducing Dirac delta distributions and identifying them with the one-particle density contributions yields

(52)

In a last step, we convert the enthalpy change of the field to the enthalpy change of the -th particle and use that velocities are sampled from a smooth velocity field to replace the velocity at position with the velocity of the -th particle,

(53)

iii.4 Interaction operators for ideal and viscous hydrodynamics

As described in the introduction to KFT in section II, the interaction term is defined as

(54)

where and in ideal hydrodynamics are given by Eqs. (38) and Eq. (III.3.2), respectively.

To transform this object into an operator, all particle densities, velocities and enthalpies must be replaced by their respective operators. However, in both and appears an inverse density. This object cannot be expressed by functional derivatives.

We handle this complication by approximating the inverse densities in a Taylor series around the mean density of the ensemble. In this paper we already truncate the approximation at th order, as this is fully sufficient for a proof of concept,

(55)

For further comments on this approximation, we refer to the end of Section IV.

Using this approximation, the interaction for ideal hydrodynamics takes the form

(56)

The notation denotes the sum over and with to exclude self-interactions.

The kernel can be interpreted as the range of the interaction, which is perhaps most evident in the heuristic derivation in Appendix B.

Since conventional hydrodynamics is a local theory we expect to recover it in the limit of vanishing interaction range

(57)

In order to take this limit, we need to remove the derivative from the kernel by replacing

(58)

and integrating by parts. The interaction then turns into

(59)

where we used the abbreviations

(60)

Finally, the expression for the interaction can be turned into an operator by replacing and by their respective functional -derivatives and all particle densities, momenta, and enthalpies by the corresponding operators described in Section III.2.

To add diffusive effects, we take the diffusive momentum and energy transport from Eq. (III.3.3), and Eq. (III.3.3) and proceed analogously: Taking the approximation of the inverse densities Eq. (55), further taking the limit of vanishing interaction ranges and moving the derivative from the kernel to the density contribution of the -th particle, we arrive at

(61)

with the abbreviations from Eq. (60). To arrive at the interaction operators, all appearing positions, momenta and conjugate source fields must again be replaced by their respective operator expressions as described for ideal hydrodynamics.

Adding the interaction operators for ideal hydrodynamics and diffusive effects finally yields the full interaction operator for viscous hydrodynamics

(62)

The interaction operators for ideal and viscous hydrodynamics, together with the free theory described in section III.1, define the dynamics of a fluid within KFT. In the remainder of this paper, we will inspect the properties of this model. To this end, we derive evolution equations for macroscopic fields from mesoscopic dynamics in the following section IV and present the results of an exemplary calculation in section V.

Iv Evolution equations of macroscopic fields

Conventional hydrodynamics is expressed by evolution equations for macroscopic fields, namely the density, velocity and energy-density. In order to compare the behaviour of the KFT fluid model to conventional hydrodynamics, we derive the corresponding macroscopic evolution equations from the dynamics of the mesoscopic particles.

In order to substantially abbreviate future calculations, it is beneficial to derive evolution equations for a more general interaction. The interactions for both ideal and viscous hydrodynamics can be brought into the general form

(63)
(64)

where the hats indicate that all particle positions, momenta and enthalpies as well as the auxiliary fields and are replaced by their respective functional derivatives. The full interacting theory then is