# Poisson-bracket formulation of the dynamics of fluids of deformable
particles

Arthur Hernandez<EMAIL_ADDRESS>Department of Physics,
University of California Santa Barbara, Santa Barbara, CA 93106, USA  M.
Cristina Marchetti  Department of Physics, University of California Santa
Barbara, Santa Barbara, CA 93106, USA<EMAIL_ADDRESS>
###### Abstract

Using the Poisson bracket method, we derive continuum equations for a fluid of
deformable particles in two dimensions. Particle shape is quantified in terms
of two continuum fields: an anisotropy density field that captures the
deformations of individual particles from regular shapes and a shape tensor
density field that quantifies both particle elongation and nematic alignment
of elongated shapes. We explicitly consider the example of a dense biological
tissue as described by the Vertex model energy, where cell shape has been
proposed as a structural order parameter for a liquid-solid transition. The
hydrodynamic model of biological tissue proposed here captures the coupling of
cell shape to flow, and provides a starting point for modeling the rheology of
dense tissue.

##  I Introduction

Many extended systems, such as biological tissue lecuit2007cell , foams
durian1995foam ; cohen2013flow , emulsions mattsson2009soft ;
vlassopoulos2014tunable , and colloidal suspensions mattsson2009soft  can be
described as collections of deformable particles. A variety of mesoscopic
models have been developed to examine the role of particle shape on the
structure and rheology of these soft materials.

Cellular Potts models graner1992simulation ; kabla2012collective  and Vertex
and Voronoi models honda2004three ; hufnagel2007mechanism ;
farhadifar2007influence  have been successfully used to describe dry foams and
confluent layers of biological tissue, where cells completely cover the plane
with no gaps, with extensions to three dimensions hannezo2014theory ;
murisic2015discrete . These models describe cells in confluent tissues as
tightly packed irregular polygons covering the plane and predict a jamming-
unjamming transition tuned by a target cell shape that captures the interplay
of cortex contractility and cell-cell adhesion, with the mean cell shape
serving as a metric for tissue fluidity bi2014energy ; bi2015density ;
bi2016motility . Vertex and Voronoi models do not, however, have a natural
extension to situations where the cell packing fraction is below one, although
gaps between cells have been incorporated in recent work teomy2018confluent ;
kim2020embryonic . In contrast, both particle deformability and density
variations can be incorporated in multi-phase field models and in models of
deformable particles boromand2019role , which have been used to examine solid-
liquid transitions as a functions of both particle shape and density.

Less well developed are continuum descriptions of the rheology of materials
where the constituents can change their shape. An important example is the
classic work by Doi and Ohta that describes the dynamics of the interface
between two immiscible fluids under shear, incorporating formation, rupture
and deformation of droplets doi1991dynamics . Continuum mechanics of confluent
tissue have been constructed phenomenologically and employed to connect
structure and mechanics in _Drosophila_ development sagner2012establishment ;
popovic2017 . Ishihara and collaborators formulated a continuum model that
couples cell shape to mechanical deformations at the tissue scale
ishihara2017cells . Their work, however, only captures _simultaneous_ cell
anisotropy and alignment of elongated cell shapes, without distinguishing
between a tissue where cell shapes are on average isotropic and one where
cells are on average anisotropic, but not aligned, as observed in simulations
of Vertex/Voronoi models bi2016motility ; yang2017correlating . It is in fact
the single-cell anisotropy that provides an order parameter for cell jamming
in Vertex and Voronoi models bi2014energy ; bi2015density ; bi2016motility ,
where fluid states of elongated cells are obtained without nematic order of
elongated cells. The importance of this distinction in a continuum theory of
tissue mechanics was highlighted recently in work by one of us and
collaborators czajkowski2018hydrodynamics .

In this paper we adopt the Poisson-bracket formulation forster1974microscopic
to obtain continuum equations for a fluid of deformable particles in two
dimensions. This method has the advantage of providing a systematic derivation
of the reversible part of the hydrodynamic equations once the continuum fields
have been identified. Our approach is inspired by work by Stark and Lubensky
Stark2003 ; Stark_2005  who used the Poisson-bracket approach to derive the
hydrodynamics of a nematic liquid crystal. As in liquid crystals, we identify
both a continuum scalar field that quantifies fluctuations of individual cell
shape and a cell shape tensor field that captures both cell elongation and
alignment. An important difference is that, while in passive liquid crystals
molecular shape fluctuations decay on fast (non-hydrodynamic) time scales, in
a tissue cell shape is the order parameter for the rigidity transitions, hence
cell-shape fluctuations are long-lived near the transition and must be
incorporated in a hydrodynamic model. The equations derived here provide a
continuum model for collections of interacting deformable “particles” and can
be adapted to describe both confluent and non-confluent systems.

The paper is organized as follows. In Section II we provide the microscopic
definition of the continuum fields used in the hydrodynamic model. In Section
III we briefly summarize the Poisson Bracket (PB) method and the calculation
of the various PBs (with details given in Appendix C), and discuss the
reactive and dissipative contributions to the coarse-grained dynamics. The
final continuum equations are displayed in Section IV. In Section V we discuss
the form of the continuum equations for the specific case of a cellular
tissue, and conclude with a brief discussion in Section VI. Details of the
derivation of the PBs and of the mean-field free-energy of the Vertex model
are given in Appendices.

##  II Continuum fields

We consider a fluid whose constituents are $N$ extended particles of arbitrary
shape. The contour of each particle, referred to below as a ‘cell’, is
described by a polygonal shape joining $n$ vertices located at
$\mathbf{r}_{\mu}^{\alpha}$, where $\mu=1,2,\cdots,n$ labels the vertices and
$\alpha=1,2,\cdots,N$ labels the cells, as shown in Fig. 1. Each cell has a
total mass $m_{c}$, which we assume equally distributed among the $n$
vertices. The shape of each cell is described by a shape tensor defined as

${G}^{\alpha}_{ij}=\frac{1}{n}\sum_{\mu=1}^{n}\Delta x_{i}^{\alpha\mu}\Delta
x_{j}^{\alpha\mu}\;,$ (1)

where
$\Delta\mathbf{r}^{\alpha\mu}=\mathbf{r}^{\alpha\mu}-\mathbf{r}_{\alpha}$,
with $\mathbf{r}_{\alpha}=\frac{1}{n}\sum_{\mu}\mathbf{r}^{\alpha\mu}$, and
Latin indices $i,j$ denote components.

Figure 1: A deformable particle (referred to as a ‘cell’) is described as an
$n$-sided irregular polygon defined by the positions $\mathbf{r}_{\alpha\mu}$
of its vertices, for $\mu=1,\cdots,n$, relative to the location of the
centroid $\mathbf{r}_{\alpha}$ of the polygon.

We define microscopic mass, momentum and cell shape density fields as

$\displaystyle\hat{\rho}(\mathbf{r},t)=\sum_{\alpha\mu}m\leavevmode\nobreak\
\delta(\mathbf{r}-\mathbf{r}^{\alpha\mu}(t))\;,$ (2)
$\displaystyle\hat{\mathbf{g}}(\mathbf{r},t)=\sum_{\alpha\mu}m\dot{\mathbf{r}}^{\alpha\mu}\leavevmode\nobreak\
\delta(\mathbf{r}-\mathbf{r}^{\alpha\mu}(t))\;,$ (3)
$\displaystyle\hat{G}_{ij}(\mathbf{r},t)=\sum_{\alpha}{G}^{\alpha}_{ij}\leavevmode\nobreak\
\delta(\mathbf{r}-\mathbf{r}_{\alpha}(t))\;.$ (4)

with $m=m_{c}/n$. Coarse grained quantities are then defined as
$\rho(\mathbf{r},t)=\left[\hat{\rho}(\mathbf{r},t)\right]_{c}$,
$\mathbf{g}(\mathbf{r},t)=\left[\hat{\mathbf{g}}(\mathbf{r},t)\right]_{c}$ and
$G_{ij}(\mathbf{r},t)=\left[\hat{G}_{ij}(\mathbf{r},t)\right]_{c}$ and
correspond to macroscopic continuum fields describing the system on length
scales large compared to both the size of the particles and their mean
separation. Note that since the microscopic cell-shape tensor
$\hat{\mathbf{G}}^{\alpha}$ has dimensions of length squared, the density of
cellular shape tensor $G_{ij}$ is dimensionless. As we will see below, the
trace of the shape tensor density provides a measure of the density of cell
perimeter, while its traceless part,
$\tilde{G}_{ij}=G_{ij}-\frac{1}{2}\delta_{ij}\text{Tr}[\mathbf{G}]$, captures
both cell anisotropy and local alignment of elongated cells.

The cellular shape tensor can be written in terms of its eigenvalues as

${G}^{\alpha}_{ij}=\frac{1}{2}\left(\lambda_{1}^{\alpha}+\lambda_{2}^{\alpha}\right)\delta_{ij}+\left(\lambda_{1}^{\alpha}-\lambda_{2}^{\alpha}\right)\left(\hat{\nu}_{i}^{\alpha}\hat{\nu}_{j}^{\alpha}-\frac{1}{2}\delta_{ij}\right)\;,$
(5)

where $\lambda_{1}^{\alpha}>\lambda_{2}^{\alpha}$ and
$\hat{\bm{\nu}}^{\alpha}$ is the eigenvector of the largest eigenvalue. For
regular $n-$sided polygons, the shape tensor is diagonal with
$\lambda_{1}^{\alpha}=\lambda_{2}^{\alpha}$. In this case the cell area
$A_{\alpha}^{(n)}$ and perimeter $P_{\alpha}^{(n)}$ can be expressed in terms
of the invariants of the tensor $\mathbf{G}^{\alpha}$ as

$\displaystyle
A_{\alpha}^{(n)}=n\sin\left(\frac{2\pi}{n}\right)\sqrt{\det[\mathbf{G}^{\alpha}]}\;,$
(6) $\displaystyle
P_{\alpha}^{(n)}=2n\sin\left(\frac{\pi}{n}\right)\sqrt{\text{Tr}[\mathbf{G}^{\alpha}]}\;,$
(7)

Single cell anisotropy is measured by
$\Delta_{\alpha}=\lambda_{1}^{\alpha}-\lambda_{2}^{\alpha}$ which vanishes for
regular polygons. To quantify single-cell elongation independently of
alignment of elongated cells, we follow czajkowski2018hydrodynamics , albeit
with a slightly different definition of the shape tensor, and introduce an
anisotropy density field defined as

$\hat{M}(\mathbf{r},t)=\sum_{\alpha}\Delta_{\alpha}\delta(\mathbf{r}-\mathbf{r}_{\alpha}(t))$
(8)

and the associated coarse grained field
$M(\mathbf{r},t)=[\hat{M}(\mathbf{r},t)]_{c}$. Work on Vertex/Voronoi models
of confluent biological tissue, as well as multiphase fields models, has
demonstrated the correlation between tissue fluidity and anisotropy of single
cell shape, as quantified here by $M$. In Vertex models, this anisotropy
provides an order parameter for the solid-liquid transition bi2015density ;
bi2016motility .

In the following, we construct hydrodynamic equations for a fluid of
deformable particles that couple structural changes encoded in cell shape and
alignment of elongated cells to flow. The dynamics of the fluid on scales
large compared to the cell size and mean cell separation is described in terms
of a few continuum fields: the mass density $\rho$, the momentum density
$\mathbf{g}$, the single-cell anisotropy density $M$ and the cell-shape tensor
density $G_{ij}$.

##  III Poisson-Bracket formulation of continuum dynamics

Here we briefly summarize the Poisson-Bracket (PB) formalism. Consider a
system whose microscopic dynamics is determined by canonically conjugate
positions $\mathbf{r}^{\alpha}$ and momenta $\mathbf{p}^{\alpha}$. We describe
the dynamics in terms of a few microscopic density fields
$\hat{\Psi}^{a}(\mathbf{r},t;\\{\mathbf{r}^{\alpha}\\},\\{\mathbf{p}^{\alpha}\\})$,
for $a=1,2,\cdots$. These fields are chosen to be be either hydrodynamic
fields associated with conserved quantities, broken symmetry fields, or quasi-
hydrodynamic fields that decay on times scales large compared to microscopic
ones. In the specific case of interest here
$\\{\hat{\Psi}^{a}\\}=(\hat{\rho},\mathbf{\hat{g}},\hat{G}_{ij},\hat{M})$. The
dynamics of the corresponding coarse-grained fields
$\Psi^{a}(\mathbf{r},t)=[\hat{\Psi}^{a}(\mathbf{r},t;\\{\mathbf{r}^{\alpha}\\},\\{\mathbf{p}^{\alpha}\\})]_{c}$
is governed by the equations

$\partial_{t}\Psi^{a}(\mathbf{r},t)=V^{a}(\mathbf{r},t)+D^{a}(\mathbf{r},t)\;,$
(9)

where $V^{a}$ and $D^{a}$ represent the non-dissipative and dissipative parts
of the dynamics, respectively. The reactive term $V^{a}$ is given by

$V^{a}(\mathbf{r})=-\int_{\mathbf{r}^{\prime}}\\{\Psi^{a}(\mathbf{r}),\Psi^{b}(\mathbf{r}^{\prime})\\}\frac{\delta\mathcal{F}}{\delta\Psi^{b}(\mathbf{r}^{\prime})}\;,$
(10)

where $\mathcal{F}[\\{\Psi^{a}\\}]$ is a phenomenological free energy,

$\\{\Psi^{a}(\mathbf{r}),\Psi^{b}(\mathbf{r}^{\prime})\\}=\left[\\{\hat{\Psi}^{a}(\mathbf{r}),\hat{\Psi}^{b}(\mathbf{r}^{\prime})\\}\right]_{c}\;,$
(11)

and

$\displaystyle\\{\hat{\Psi}^{a}(\mathbf{r}),\hat{\Psi}^{b}(\mathbf{r}^{\prime})\\}=$
$\displaystyle\sum_{\alpha
i}\Big{(}\frac{\partial\hat{\Psi}^{a}(\mathbf{r})}{\partial
p_{i}^{\alpha}}\frac{\partial\hat{\Psi}^{b}(\mathbf{r}^{\prime})}{\partial
r_{i}^{\alpha}}$ (12)
$\displaystyle-\frac{\partial\hat{\Psi}^{a}(\mathbf{r})}{\partial
r_{i}^{\alpha}}\frac{\partial\hat{\Psi}^{b}(\mathbf{r}^{\prime})}{\partial
p_{i}^{\alpha}}\Big{)}\;.$

Finally, the dissipative term in the kinetic equation is controlled by all the
neglected microscopic degrees of freedom and can be written as

$D^{a}(\mathbf{r})=-\Gamma^{ab}\frac{\delta\mathcal{F}}{\delta\Psi^{b}(\mathbf{r})}\;.$
(13)

The dissipation tensor $\Gamma^{ab}$ is in general a functional of the
$\\{\Psi^{a}\\}$ and their gradients. It is a phenomenological quantity
controlled by the requirement that $\partial_{t}\Psi^{a}$ can only couple to
driving forces $\frac{\delta\mathcal{F}}{\delta\Psi^{b}(\mathbf{r})}$ that
have different sign under time reversal, to guarantee that such terms describe
dissipation. In equilibrium it is a symmetric tensor and must obey Onsager’s
principle de1951thermodynamics .

###  III.1 Poisson brackets

The calculation of the PB of mass and momentum density is straightforward and
can be found in the literature Stark2003 , with the result

$\displaystyle\\{\rho(\mathbf{r}),g_{i}(\mathbf{r}^{\prime})\\}=\rho(\mathbf{r}^{\prime})\partial_{i}\delta(\mathbf{r}-\mathbf{r}^{\prime})\;,$
$\displaystyle\\{g_{i}(\mathbf{r}),g_{j}(\mathbf{r}^{\prime})\\}=-\partial_{i}^{\prime}[\delta(\mathbf{r}-\mathbf{r}^{\prime})g_{j}(\mathbf{r}^{\prime})]+\partial_{j}\delta(\mathbf{r}-\mathbf{r}^{\prime})g_{i}(\mathbf{r}^{\prime})\;,$
(14)

The main PBs to be calculated here are those involving the fields describing
cellular shape. The details of the derivation are shown in Appendix C, with
the result

$\displaystyle\\{G_{ij}(\mathbf{r}),g_{k}(\mathbf{r}^{\prime})\\}$
$\displaystyle=\partial_{k}[G_{ij}(\mathbf{r}^{\prime})\delta(\mathbf{r}-\mathbf{r}^{\prime})]$
(15)
$\displaystyle-\left[G_{il}(\mathbf{r})\delta_{jk}+G_{jl}(\mathbf{r})\delta_{ik}\right]\partial_{l}\delta(\mathbf{r}-\mathbf{r}^{\prime})\;,$
$\\{M(\mathbf{r}),g_{i}(\mathbf{r}^{\prime})\\}=[\partial_{i}M(\mathbf{r})]\delta(\mathbf{r}-\mathbf{r}^{\prime})-\frac{2R(\mathbf{r})}{M(\mathbf{r})}\tilde{G}_{ij}(\mathbf{r})\partial_{j}\delta(\mathbf{r}-\mathbf{r}^{\prime})\;.$
(16)

To calculate $\\{M(\mathbf{r}),g_{i}(\mathbf{r}^{\prime})\\}$ we have used the
identity
$\tilde{G}_{ik}^{\alpha}\tilde{G}_{kj}^{\alpha}=\frac{\Delta_{\alpha}^{2}}{4}\delta_{ij}$,
where the tilde denotes the traceless part of ny rank-2 tensor,
$\tilde{G}_{ij}=G_{ij}-\frac{1}{2}\delta_{ij}\text{Tr}[\mathbf{G}]$.
This allows us to write

$\Delta_{\alpha}\\{\Delta_{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha}),g_{i}(\mathbf{r}^{\prime})\\}=2\tilde{G}_{kl}^{\alpha}\\{\tilde{G}_{kl}^{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha}),g_{i}(\mathbf{r}^{\prime})\\}\;.$
(17)

Finally, the other PBs can be obtained using the identity

$\\{\Psi_{n}(\mathbf{r}),\Psi_{m}(\mathbf{r}^{\prime})\\}=-\\{\Psi_{m}(\mathbf{r}^{\prime}),\Psi_{n}(\mathbf{r})\\}\;.$
(18)

###  III.2 Reactive terms

To evaluate the various contributions to the continuum dynamics, we need to
specify the free energy of the system. In general, this has the form

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\mathcal{F}_{K}+\mathcal{F}_{V}$ (19) $\displaystyle=$
$\displaystyle\int_{\mathbf{r}}\left[\frac{\mathbf{g}^{2}}{2\rho}+f(\rho,G_{ij},M)\right]\;,$

where the first term is the kinetic part and the free energy density $f$
depends on the fields and their gradients.

Using the expressions for the Poisson brackets we can then evaluate the
reactive terms $V^{a}$, with the result

$\displaystyle V^{\rho}$ $\displaystyle=$
$\displaystyle-\bm{\nabla}\cdot(\rho\mathbf{v})\;,$ (20) $\displaystyle
V^{g}_{i}$ $\displaystyle=$ $\displaystyle-\partial_{j}(\rho
v_{i}v_{j})-\rho\partial_{i}\frac{\delta\mathcal{F}_{V}}{\delta\rho}+(\partial_{i}M)\frac{\delta\mathcal{F}_{V}}{\delta
M}+\left(\partial_{i}G_{kl}\right)\frac{\delta\mathcal{F}_{V}}{\delta G_{kl}}$
(21)
$\displaystyle+\partial_{j}\left(2G_{jk}\frac{\delta\mathcal{F}_{V}}{\delta
G_{ik}}-\delta_{ij}G_{kl}\frac{\delta\mathcal{F}_{V}}{\delta G_{kl}}\right)$
$\displaystyle+2\partial_{j}\left(\frac{R}{M}\tilde{G}_{ij}\frac{\delta\mathcal{F}_{V}}{\delta
M}\right)\;,$ $\displaystyle V^{G}_{ij}$ $\displaystyle=$
$\displaystyle-\bm{\nabla}\cdot(G_{ij}\mathbf{v})+G_{ik}\partial_{k}v_{j}+G_{jk}\partial_{k}v_{i}\
\;,$ (22) $\displaystyle V^{M}$ $\displaystyle=$
$\displaystyle-\mathbf{v}\cdot\bm{\nabla}M+\frac{2R}{M}\tilde{G}_{ij}\partial_{i}v_{j}\;,$
(23)

where we have defined

$R(\mathbf{r},t)=Tr[\mathbf{G}(\mathbf{r},t)]\;.$ (24)

The field $R$ is essentially a measure of cell perimeter density.

The elastic and density couplings in Eq. (21) can be rewritten in a more
familiar form as gradients of pressure and of an elastic stress. The details
can be found in Appendix B, where it is shown that we can write

$-\rho\partial_{i}\frac{\delta\mathcal{F}_{V}}{\delta\rho}+(\partial_{i}M)\frac{\delta\mathcal{F}_{V}}{\delta
M}+(\partial_{i}G_{kl})\frac{\delta\mathcal{F}_{V}}{\delta
G_{kl}}=-\partial_{i}p+\partial_{j}\sigma_{ij}^{E}$ (25)

where the pressure $p$ and the elastic stress $\sigma_{ij}^{E}$, that plays
the role of the Erickssen stress of nematic liquid crystals, are given by

$\displaystyle p$ $\displaystyle=$ $\displaystyle\rho\frac{\partial
f}{\partial\rho}-f\;,$ (26) $\displaystyle\sigma_{ij}^{E}$ $\displaystyle=$
$\displaystyle-\frac{\partial
f}{\partial\nabla_{j}M}\nabla_{i}M-\frac{\partial
f}{\partial\nabla_{j}G_{kl}}\nabla_{i}G_{kl}\;.$ (27)

The last two terms in Eq.(21) correspond to gradients of a reactive elastic
stress $\sigma^{G}_{ij}$, given by

$\displaystyle\sigma^{G}_{ij}=2\frac{R}{M}\tilde{G}_{ij}\frac{\delta\mathcal{F}_{V}}{\delta
M}+2G_{jk}\frac{\delta\mathcal{F}_{V}}{\delta
G_{ik}}-\delta_{ij}G_{kl}\frac{\delta\mathcal{F}_{V}}{\delta G_{kl}}\;.$ (28)

The reactive term for the momentum density equation can then be written as

$V^{g}_{i}=-\partial_{j}(\rho
v_{i}v_{j})-\partial_{i}p+\partial_{j}\left(\sigma^{G}_{ij}+\sigma^{E}_{ij}\right)\;.$
(29)

###  III.3 Dissipative terms

There is no dissipative term for the mass density $\rho$ if it is conserved.

Dissipative terms in the momentum equation must be odd under time reversal and
hence must couple to gradients of velocity. In general, shape anisotropy and
alignment of elongated cells will entail anisotropic viscosity coefficients,
as in liquid crystals. For simplicity, here we only introduce two viscosities
to account for shear ($\eta$) and bulk ($\eta_{b}$) deformations and write

$D^{g}_{i}=\partial_{j}\sigma_{ij}^{D}\;,$ (30)

with

$\sigma_{ij}^{D}=2\eta
D_{ij}+\eta_{b}\delta_{ij}\bm{\nabla}\cdot\mathbf{v}\;,$ (31)

where $D_{ij}$ is the symmetrized and traceless rate of strain tensor,

$D_{ij}=\frac{1}{2}(\partial_{i}v_{j}+\partial_{j}v_{i}-\delta_{ij}\bm{\nabla}\cdot\mathbf{v})\;.$
(32)

Dissipative couplings in the equations for the shape density tensor $G_{ij}$
and the shape anisotropy field $M$ must be even under time reversal and hence
can couple to $M$, $G_{ij}$ and their gradients. Dissipation will arise from
topological rearrangements, as well as from birth/death events when density
conservation is broken. In general we can write

$\displaystyle D^{G}_{ij}$ $\displaystyle=$
$\displaystyle-\Gamma^{GG}_{ijkl}\frac{\delta\mathcal{F}_{V}}{\delta
G_{kl}}-\Gamma^{GM}_{ij}\frac{\delta\mathcal{F}_{V}}{\delta M}\;,$ (33)
$\displaystyle D^{M}$ $\displaystyle=$
$\displaystyle-\Gamma^{MM}\frac{\delta\mathcal{F}_{V}}{\delta
M}-\Gamma^{MG}_{ij}\frac{\delta\mathcal{F}_{V}}{\delta G_{ij}}\;.$ (34)

The kinetic coefficients $\Gamma^{ab}$ can generally depend on the shape
tensor and anisotropy density field. To linear order in these fields, a
general form is given by

$\displaystyle\Gamma^{GG}_{ijkl}$ $\displaystyle=$
$\displaystyle\frac{M}{2\gamma_{G}}\left(\delta_{ik}\delta_{jl}+\delta_{jk}\delta_{il}\right)$
(35)
$\displaystyle+\frac{1}{\gamma_{1}}\left(\delta_{ik}G_{jl}+\delta_{jk}G_{il}+\delta_{il}G_{jk}+\delta_{jl}G_{ik}\right)\;,$
$\displaystyle\Gamma^{GM}_{ij}$ $\displaystyle=$
$\displaystyle\Gamma^{GM}_{ij}=\frac{G_{ij}}{\gamma_{2}}\;,$ (36)
$\displaystyle\Gamma^{MM}$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma_{M}}+\frac{M}{\gamma_{3}}\;,$ (37)

where the kinetic coefficients $\gamma_{i}$, for $i=G,M,1,2,3$, encode the
characteristic time scales of dissipative processes. For simplicity we have
assumed $\Gamma^{GM}_{ij}=\Gamma^{GM}_{ij}$ although in general the parameters
controlling the relaxation in these terms could differ. Note that the second
term in Eq. (35) has the form introduced in Ref. milner1993dynamical  for the
kinetic coefficient describing the relaxation of the conformation tensor in a
polymer suspension.

##  IV Final equations

Putting it all together, we now write the final form of the equations we have
obtained. It is convenient to write

$\partial_{i}v_{j}=D_{ij}+\omega_{ij}+\frac{1}{2}\delta_{ij}\bm{\nabla}\cdot\mathbf{v}$
(38)

where $D_{ij}$ is the rate of strain tensor given in Eq. (32) and
$\omega_{ij}$ is the vorticity,

$\displaystyle\omega_{ij}=\frac{1}{2}(\partial_{i}v_{j}-\partial_{j}v_{i})\;.$
(39)

The set of continuum equations for our fluid of deformable cells is then given
by

$\displaystyle\partial_{t}\rho=-\bm{\nabla}\cdot\rho\mathbf{v},$ (40)
$\displaystyle\rho\left(\partial_{t}+\mathbf{v}\cdot\bm{\nabla}\right)v_{i}=-\partial_{i}p+\partial_{j}\left(\sigma^{G}_{ij}+\sigma^{E}_{ij}+\sigma^{D}_{ij}\right),$
(41)
$\displaystyle\frac{d}{dt}M=\frac{2R}{M}\tilde{G}_{ij}D_{ij}-\Gamma^{MM}\frac{\delta\mathcal{F}_{V}}{\delta
M}-\Gamma^{MG}_{ij}\frac{\delta\mathcal{F}_{V}}{\delta G_{ij}},$ (42)
$\displaystyle\frac{D}{Dt}G_{ij}=G_{ik}D_{kj}+D_{ik}G_{kj}-\Gamma^{GG}_{ijkl}\frac{\delta\mathcal{F}_{V}}{\delta
G_{kl}}-\Gamma^{GM}_{ij}\frac{\delta\mathcal{F}_{V}}{\delta M},$ (43)

where

$\displaystyle\frac{d}{dt}=\partial_{t}+\mathbf{v}\cdot\bm{\nabla}\;,$
$\displaystyle\frac{D}{Dt}=\frac{d}{dt}+[\bm{\omega},\large\cdot]$ (44)

are the convective derivative, and the convective and corotational derivative.

The equation for the shape tensor $G_{ij}$ contains couplings to flow
vorticity and strain rate which control the tendency of extended and
deformable particles to rotate with flow and align with streamlines. The shape
tensor $G_{ij}$ plays a role similar to that of the conformation tensor in a
polymer suspension beris1994thermodynamics . In fact, if we ignore the
additional anisotropy density field $M$, the equations derived here for a
fluid of deformable particles have the same structure as a one-fluid model of
viscoelastic polymer solutions milner1993dynamical . Unlike in models of
polymer suspensions, however, the coefficient of the coupling to strain rate,
known in that context as the slip parameter beris1994thermodynamics , is found
to be simply equal to $1$ in our PB formulation.

It is also convenient to separate the dynamics of $G_{ij}$ in that of its
trace and deviatoric part. The corresponding equations are given by

$\displaystyle\frac{d}{dt}R$ $\displaystyle=$ $\displaystyle
2\tilde{G}_{ij}D_{ji}-\Gamma^{GG}_{iikl}\frac{\delta\mathcal{F}_{V}}{\delta
G_{kl}}-\Gamma^{GM}_{ii}\frac{\delta\mathcal{F}_{V}}{\delta M},$ (45)
$\displaystyle\frac{D}{Dt}\tilde{G}_{ij}$ $\displaystyle=$ $\displaystyle
RD_{ij}+\tilde{G}_{ik}D_{kj}+D_{ik}\tilde{G}_{kj}-\delta_{ij}\tilde{G}_{kl}D_{kl}$
(46)
$\displaystyle-\left[\bm{\Gamma}^{GG}\mathbin{\vbox{\hbox{$\ldotp$}\hbox{$\ldotp$}}}\frac{\delta\mathcal{F}_{V}}{\delta\mathbf{G}}\right]_{ij}^{ST}-\left[\bm{\Gamma}^{GM}\right]_{ij}^{ST}\frac{\delta\mathcal{F}_{V}}{\delta
M}\;.$

where $[...]^{ST}$ denotes the symmetrized and traceless part of the tensor.
These equations need to be completed by an expression for the free energy
$\mathcal{F}_{V}$ in terms of the shape tensor and anisotropy density field.
Such an expression of course depends on the system of interest. In the next
section we consider the specific case of a model of dense biological tissues.

##  V Cellular tissue

Confluent biological tissue, where cells are tightly packed, with no
intervening gaps, have been modeled extensively using Vertex or Voronoi models
that describe cells as irregular polygons tesselating the plane honda2004three
; hufnagel2007mechanism ; farhadifar2007influence . The behavior of the tissue
is controlled by an energy that describes the tendency of each cell to adjust
its area $A_{a}$ and perimeter $P_{a}$ to target values $A_{0}$ and $P_{0}$,
given by

$E_{V}=\sum_{\alpha}\left[\frac{\kappa_{A}}{2}\left(A_{\alpha}-A_{0}\right)^{2}+\frac{\kappa_{P}}{2}\left(P_{\alpha}-P_{0}\right)^{2}\right]\;,$
(47)

with $\kappa_{A}$ and $\kappa_{P}$ stiffness parameters. The first term arises
from tissue incompressibility in three dimensions and the second captures the
interplay of cell-cell adhesion and cortical contractility. By scaling lengths
with $\sqrt{A_{0}}$ and energies with $\kappa_{A}A_{0}^{2}$, the scaled energy
of each cell is given by

$\epsilon_{a}=\frac{1}{2}\left(a_{a}-1\right)^{2}+\frac{r}{2}\left(p_{a}-p_{0}\right)^{2}\;,$
(48)

with $p_{0}=P_{0}/\sqrt{A_{0}}$ the target shape parameter and
$r=\kappa_{P}/(\kappa_{A}A_{0})$.

Numerical studies of this energy have identified a rigidity transition at a
critical value $p_{0}^{*}$ of the target shape parameter between a rigid,
solid-like state for $p_{0}<p_{0}^{*}$ to a fluid state for $p_{0}>p_{0}^{*}$.
Single-cell anisotropy as quantified by the mean cell-shape index $q=\langle
P_{\alpha}/\sqrt{A_{\alpha}}\rangle$, with the brackets denoting an average
over all cells, provides an order parameter for the transition. Czajkowski _et
al._ czajkowski2018hydrodynamics  derived a mean-field model of this rigidity
transition, albeit using a different definition of the cell shape tensor
$G_{ij}^{\alpha}$ as compared to the one used here. The derivation carried out
with our definition is outlined in Appendix D. The result is a quartic Landau-
type free energy density $f_{M}$ where the the cell shape anisotropy density
$M$ plays the role of an order parameter, given by

$f_{M}=\frac{\alpha(p_{0})}{2}M^{2}+\frac{\beta}{4}M^{4}\;,$ (49)

where $\alpha(p_{0})$ vanishes at $p_{0}=p_{0}^{*}$ and $\beta>0$. The
definition of the shape tensor of individual cells only affects the precise
values of these parameters that also depend on the reference polygonal shape,
but does not change the form of the free energy density nor the value of
$p_{0}^{*}$. The free energy given in Eq. (49) is obtained by assuming small
deformations from regular polygons and constant cell perimeter. It predicts a
mean-field transition at $\alpha=0$ from a state where cells are isotropic
($M=0$) $\alpha<0$ or $p_{0}<p_{0}^{*}$ (the solid state) to a state where
cells are anisotropic ($M=\sqrt{-\alpha/\beta}$) for $\alpha<0$ or
$p_{0}>p_{0}^{*}$ (the liquid state).

This work suggests a phenomenological free energy for a confluent tissue that
captures both fluctuations in the cell anisotropy density $M$ that quantifies
the liquid-solid transition and the shape tensor density $\tilde{G}_{ij}$ that
quantifies alignment of elongated cell as

$\displaystyle\mathcal{F}_{c}=\int_{\mathbf{r}}\bigg{[}$
$\displaystyle\frac{\alpha(p_{0})}{2}M^{2}+\frac{\beta}{4}M^{4}+\frac{K}{2}(\bm{\nabla}M)^{2}$
(50)
$\displaystyle+\frac{\chi}{2}Tr[\tilde{\mathbf{G}}^{2}]+\frac{K_{G}}{2}(\partial_{j}\tilde{G}_{ik})^{2}\bigg{]}\;.$

We do not include terms of order $Tr[\tilde{\mathbf{G}}^{2}]^{2}$ as we do not
expect any nematic order of cellular shapes in the absence of externally
applied or actively generated internal stresses. Also, we have assumed
constant cell perimeter, corresponding to $R=Tr[\mathbf{G}]=$constant. In
general, the various parameters in $\mathcal{F}_{c}$ will depend on $R$.

It is important to stress that $\tilde{G}_{ij}$ and $M$ are not independent.
The traceless tensor $\tilde{G}_{ij}$ can be written as

$\tilde{G}_{ij}=S_{G}\left(n_{i}n_{j}-\frac{1}{2}\delta_{ij}\right)\;,$ (51)

which defines the director field $\mathbf{n}(\mathbf{r},t)$ associated with
alignment of elongated cells and the magnitude $S$ of orientational order.
Cell alignment can only occur if cells are elongated ($M\not=0$), hence
$S_{G}(M)$ must vanish when $M=0$. We assume $S_{G}=MS$, where $S$ plays the
role of a nematic order parameter for orientational order of elongated cells.
Clearly, $S$ is defined only in states where $M$ is finite.

Cell sheets commonly interact with a frictional substrate that eliminates
momentum conservation. Frictional drag with the substrate generally exceeds
inertial forces, and the Navier-Stokes equation for the momentum is replaced
by a Stokes equation quantifying force balance on each fluid element. Within
this overdamped limit, and considering a minimal form for the various
dissipative kinetic coefficients, the tissue dynamics is governed by the
following equations

$\partial_{t}\rho=-\bm{\nabla}\cdot(\rho\mathbf{v}),$ (52) $\xi
v_{i}=-\partial_{i}p+\partial_{j}\left(\sigma^{G}_{ij}+\sigma^{E}_{ij}+\sigma^{D}_{ij}\right),$
(53)
$\frac{d}{dt}M=2\frac{R}{M}\tilde{G}_{ij}D_{ij}-\frac{1}{\gamma_{M}}\frac{\delta\mathcal{F}_{c}}{\delta
M}-\frac{\tilde{G}_{ij}}{\gamma_{2}}\frac{\delta\mathcal{F}_{c}}{\delta\tilde{G}_{ij}}\;,$
(54) $\displaystyle\frac{D}{Dt}\tilde{G}_{ij}$ $\displaystyle=$ $\displaystyle
RD_{ij}+\tilde{G}_{ik}D_{kj}+D_{ik}\tilde{G}_{kj}-\delta_{ij}\tilde{G}_{kl}D_{kl}$
(55) $\displaystyle-$
$\displaystyle\frac{M}{\gamma_{G}}\frac{\delta\mathcal{F}_{c}}{\delta\tilde{G}_{ij}}-\frac{\tilde{G}_{ij}}{\gamma_{2}}\frac{\delta\mathcal{F}_{c}}{\delta
M}\;,$

where $\xi$ is the frictional drag and

$\displaystyle\frac{\delta\mathcal{F}_{c}}{\delta M}$ $\displaystyle=$
$\displaystyle\left[\alpha+\beta M^{2}\right]M-K\nabla^{2}M\;,$ (56)
$\displaystyle\frac{\delta\mathcal{F}_{c}}{\delta\tilde{G}_{ij}}$
$\displaystyle=$
$\displaystyle\chi\tilde{G}_{ij}-K_{G}\nabla^{2}\tilde{G}_{ij}$ (57)

It is useful to consider a simplified form of the equations obtained by
retaining only lowest order terms in fields and gradients. In this case the
Stokes equation and the equations for the shape fields can be written in the
explicit form

$\Gamma\mathbf{v}=-\bm{\nabla}p+\eta\nabla^{2}\mathbf{v}+\eta_{b}\bm{\nabla}\bm{\nabla}\cdot\mathbf{v}+\bm{\nabla}\cdot\bm{\sigma}^{G}\;,$
(58)
$\frac{d}{dt}M=2\frac{R}{M}\tilde{G}_{ij}D_{ij}-\frac{1}{\gamma_{M}}\left[\alpha+\beta
M^{2}\right]M+D\nabla^{2}M\;,$ (59) $\displaystyle\frac{D}{Dt}\tilde{G}_{ij}$
$\displaystyle=$ $\displaystyle
RD_{ij}+\tilde{G}_{ik}D_{kj}+D_{ik}\tilde{G}_{kj}-\delta_{ij}\tilde{G}_{kl}D_{kl}$
(60) $\displaystyle-$ $\displaystyle
rM\tilde{G}_{ij}+D_{G}\nabla^{2}\tilde{G}_{ij}\;,$

where $D=\Gamma_{M}K$, $D_{G}=\Gamma K_{G}$,
$r=\chi/\gamma_{G}+\alpha/\gamma_{2}$ and

$\sigma_{ij}^{G}=2R(\alpha+\beta
M^{2})\tilde{G}_{ij}+\frac{1}{2}\delta_{ij}\chi S_{G}^{2}\;.$ (61)

The single-cell anisotropy field $M$ here plays the role of tissue fluidity.
The first term on the RHS of Eq. (59) captures the fact that shear
deformations, coupled to local cell alignment, can increase cell anisotropy,
driving fluidification. The second term describes relaxation to the ground
state controlled by the tissue free energy, with a cost for spatial variations
in local fluidity controlled by the stiffness $D$. The reactive terms in Eq.
(60) describe flow alignment of elongated cell shape. The term proportional to
$r$ describes changes of cell shape tensor due to dissipative processes, such
as topological rearrangements, at a rate proportional to the tissue fluidity
$M$. The last term in Eq. (60) describes the stiffness against deformations of
local cell alignment.

Finally, in a confluent tissue the cell number density $n=\rho/m_{c}$ is
slaved to the mean cell area $\langle A_{\alpha}\rangle$ with $n=1/\langle
A_{\alpha}\rangle$. For cells that are only slightly deformed from regular
polygons, $\langle
A_{\alpha}\rangle\simeq\sqrt{\det[\mathbf{G}]}\approx\text{Tr}[\mathbf{G}]$,
where we have used Eq. (66). The density equation, Eq (52), can therefore
equivalently be written as an equation for the cell area or for
$|G|\equiv\det[\mathbf{G}]$, given by

$\left(\partial_{t}+\mathbf{v}\cdot\bm{\nabla}\right)|G|=|G|\bm{\nabla}\cdot\mathbf{v}\;.$
(62)

##  VI Conclusion

Using the Poisson bracket formalism, we have derived hydrodynamic equations
for a fluid of deformable particles in two dimensions. Shape fluctuations are
described by two continuum fields: (i) a coarse-grained scalar field that
captures single-particle anisotropy, and (ii) a shape tensor field that
quantifies both particle elongation and nematic alignment of elongated
particles.

We have specifically applied the model to sheets of dense biological tissue,
where single-cell anisotropy was recently identified as the order parameter
for a solid-liquid transition driven by the interplay of cortex contractility
and cell-cell adhesion bi2015density ; bi2016motility . In other words, in
confluent tissue single-cell anisotropy is effectively an experimentally
accessible measure of the rheological properties of the tissue, with isotropic
cell shapes identifying the solid or jammed state and anisotropic shapes
corresponding to a liquid. Previous work has examined the dynamics of a
coarse-grained cell shape tensor and its coupling to mechanical stresses
ishihara2017cells . This work did not, however, distinguish between a tissue
of elongated, but isotropically oriented cells and one were the cells and
elongated and also aligned in a state with nematic liquid crystalline order.
The new ingredient of our work is to distinguish the dynamics of tissue
fluidity, as quantified by the single-cell anisotropy field, from that of cell
alignment, and examine the interplay between flow, which can be either
externally applied or induced by internal active processes, fluidity and
nematic order of cell shapes. Our equations hence provide a starting point for
quantifying the rheology of biological tissue. Future extension needed to
develop a complete framework of tissue rheology include the coupling to the
dynamics of polarized cell motility and the inclusion of structural
rearrangements arising from cell division and death.

Finally, the equations developed here provide a general hydrodynamic model for
any fluid of deformable particles, capable of accounting for both the dynamics
of shape deformations and density changes.

###### Acknowledgements.

MCM thanks to Max Bi, James Cochran, Suzanne Fielding and Holger Stark for
illuminating discussions. This work was supported by the National Science
Foundation through award DMR-1938187.

##  Appendix A Useful identities

The eigenvalues of a $2\times 2$ symmetric matrix $G_{ij}$ are given by

$\lambda_{1,2}=\frac{1}{2}(G_{xx}+G_{yy})\pm\frac{1}{2}\sqrt{(G_{xx}-G_{yy})^{2}+4G_{xy}^{2}}\;,$
(63)

and

$\displaystyle\lambda_{1}-\lambda_{2}=\sqrt{(G_{xx}-G_{yy})^{2}+4G_{xy}^{2}}\;,$
(64) $\displaystyle\lambda_{1}\lambda_{2}=G_{xx}G_{yy}-G_{xy}^{2}\;.$ (65)

We can then show that the following identities apply

$\displaystyle(\lambda_{1}-\lambda_{2})^{2}=[\text{Tr}\mathbf{G}]^{2}-4\det\mathbf{G}\;,$
(66)
$\displaystyle(\lambda_{1}-\lambda_{2})^{2}=2\text{Tr}[\mathbf{G}^{2}]-[\text{Tr}\mathbf{G}]^{2}=2\text{Tr}[\mathbf{\tilde{G}}^{2}]\;.$
(67)

Finally, for a regular polygon, $G_{ij}$ is always diagonal and
$\lambda_{1}=\lambda_{2}=\lambda$. In this case Eq. (66) gives
$\text{Tr}\mathbf{G}=2\sqrt{\det\mathbf{G}}$. For small deformations from a
regular polygon $\text{Tr}\mathbf{G}\sim 2\sqrt{\det\mathbf{G}}$, which
implies that we can think of $\text{Tr}\mathbf{G}$ as either a measure of
square of cell perimeter or a measure of cell area.

##  Appendix B Elastic Stress and Pressure

It is convenient to rewrite some of the term in the reactive part
$\mathbf{V}^{g}$ of the momentum density equation given in Eq. (21) to express
them as gradients of pressure and an elastic stress. The goal is to rewrite
the following terms

$\displaystyle\delta
V_{i}^{g}\equiv-\rho\partial_{i}\frac{\delta\mathcal{F}_{V}}{\delta\rho}+(\partial_{i}M)\frac{\delta\mathcal{F}_{V}}{\delta
M}+\left(\partial_{i}G_{kl}\right)\frac{\delta\mathcal{F}_{V}}{\delta
G_{kl}}\;.$ (68)

By relating functional derivatives of $\mathcal{F}_{V}$ t to derivatives of
the free energy density $f$, which is a function of the hydrodynamic fields
and their gradients, we can write

$\displaystyle-\rho\nabla_{i}\frac{\delta\mathcal{F}_{V}}{\delta\rho}=-\nabla_{i}\left(\rho\frac{\partial
f}{\partial\rho}\right)+\frac{\partial f}{\partial\rho}\nabla_{i}\rho\;,$ (69)
$\displaystyle(\nabla_{i}M)\frac{\delta\mathcal{F}_{V}}{\delta
M}=(\nabla_{i}M)\left(\frac{\partial f}{\partial M}-\nabla_{j}\frac{\partial
f}{\partial\nabla_{j}M}\right)=\frac{\partial f}{\partial M}\nabla_{i}M$
$\displaystyle-\nabla_{j}\left[(\nabla_{i}M)\frac{\partial
f}{\partial\nabla_{j}M}\right]+\frac{\partial
f}{\partial\nabla_{j}M}\nabla_{i}(\nabla_{j}M)\;,$ (70)
$\displaystyle\left(\nabla_{i}G_{kl}\right)\frac{\delta\mathcal{F}_{V}}{\delta
G_{kl}}=\left(\nabla_{i}G_{kl}\right)\left(\frac{\partial f}{\partial
G_{kl}}-\nabla_{j}\frac{\partial f}{\partial\nabla_{j}G_{kl}}\right)$
$\displaystyle=\frac{\partial f}{\partial
G_{kl}}\nabla_{i}G_{kl}-\nabla_{j}\left[(\nabla_{i}G_{kl})\frac{\partial
f}{\partial\nabla_{j}G_{kl}}\right]$ $\displaystyle+\frac{\partial
f}{\partial\nabla_{j}G_{kl}}\nabla_{i}(\nabla_{j}G_{kl})\;.$ (71)

Combining these three terms, and using

$\displaystyle\nabla_{i}f=\frac{\partial
f}{\partial\rho}\nabla_{i}\rho+\frac{\partial f}{\partial
M}\nabla_{i}M+\frac{\partial f}{\partial\nabla_{j}M}\nabla_{i}(\nabla_{j}M)$
$\displaystyle+\frac{\partial f}{\partial
G_{kl}}\nabla_{i}G_{kl}+\frac{\partial
f}{\partial\nabla_{j}G_{kl}}\nabla_{i}(\nabla_{j}G_{kl})$ (72)

we can write

$\delta V_{i}^{g}=-\nabla_{i}p+\nabla_{j}\sigma_{ij}^{E}$ (73)

in terms of the pressure $p$ and an elastic stress $\sigma_{ij}^{E}$, given by

$\displaystyle p$ $\displaystyle=$ $\displaystyle\rho\frac{\partial
f}{\partial\rho}-f\;,$ (74) $\displaystyle\sigma_{ij}^{E}$ $\displaystyle=$
$\displaystyle-\frac{\partial
f}{\partial\nabla_{j}M}\nabla_{i}M-\frac{\partial
f}{\partial\nabla_{j}G_{kl}}\nabla_{i}G_{kl}\;.$ (75)

The stress $\sigma_{ij}^{E}$ plays the role of the Erickssen stress of nematic
liquid crystals.

##  Appendix C Evaluation of Poisson Brackets

First we show the details of the calculation of the fundamental PB
$\\{G_{ij}^{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha}),g_{k}(\mathbf{r}^{\prime})\\}$.
To evaluate the PB we use the following

$\displaystyle\frac{\partial\Delta x_{i}^{\alpha\nu}}{\partial
x_{j}^{\beta\mu}}=\delta^{\alpha\beta}\delta_{ij}\left(\delta^{\mu\nu}-\frac{1}{n}\right)\;,$
(76) $\displaystyle\frac{\partial}{\partial
x_{j}^{\beta\mu}}\delta(\mathbf{r}-\mathbf{r}^{\alpha})=-\frac{\delta^{\alpha\beta}}{n}\partial_{j}\delta(\mathbf{r}-\mathbf{r}^{\alpha})\;,$
(77)
$\displaystyle\delta(\mathbf{r}-\mathbf{r}^{\alpha})=\delta(\mathbf{r}-\mathbf{r}^{\alpha\mu}-\Delta\mathbf{r}^{\alpha\mu})$
$\displaystyle\;\;\;\;\;\;\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
=\delta(\mathbf{r}-\mathbf{r}^{\alpha})-\Delta
x_{i}^{\alpha\mu}\partial_{i}\delta(\mathbf{r}-\mathbf{r}^{\alpha\mu})+\mathcal{O}(\Delta
x^{2}\nabla^{2})\;.$ (78)

We write

$\\{G_{ij}^{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha}),g_{k}(\mathbf{r}^{\prime})\\}=-\sum_{\beta,\nu}\frac{\partial
G_{ij}^{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha})}{\partial
x^{\beta\nu}_{k}}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\beta\nu})\;.$ (79)

Then

$\displaystyle\frac{\partial
G_{ij}^{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha})}{\partial
x^{\beta\nu}_{k}}$
$\displaystyle=-\frac{1}{n}\delta^{\alpha\beta}G_{ij}^{\alpha}\partial_{k}\delta(\mathbf{r}-\mathbf{r}^{\alpha})$
(80)
$\displaystyle+\frac{1}{n}\delta^{\alpha\beta}\delta(\mathbf{r}-\mathbf{r}^{\alpha})\left(\delta_{ik}\Delta
x^{\alpha\nu}_{j}+\delta_{jk}\Delta x^{\alpha\nu}_{i}\right)\;.$

Inserting Eq.(C5) into Eq.(C4) and using that
$\sum_{\mu}\Delta\mathbf{r}^{\alpha\mu}=0$, we obtain

$\displaystyle\\{G_{ij}^{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha}),g_{k}(\mathbf{r}^{\prime})\\}=G_{ij}^{\alpha}\left[\partial_{k}\delta(\mathbf{r}-\mathbf{r}^{\alpha})\right]\frac{1}{n}\sum_{\nu}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha\nu})$
$\displaystyle-\delta(\mathbf{r}-\mathbf{r}^{\alpha})\frac{1}{n}\sum_{\nu}\left[\delta_{ik}\Delta^{\alpha\nu}_{j}+\delta_{jk}\Delta^{\alpha\nu}_{i}\right]\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha\nu})\;.$
(81)

Finally, using

$\displaystyle\delta(\mathbf{r}-\mathbf{r}^{\alpha\nu})$
$\displaystyle=\delta(\mathbf{r}-\mathbf{r}^{\alpha}-\Delta\mathbf{r}^{\alpha\nu})$
(82) $\displaystyle\approx\delta(\mathbf{r}-\mathbf{r}^{\alpha})-\Delta
x^{\alpha\nu}_{k}\partial_{k}\delta(\mathbf{r}-\mathbf{r}^{\alpha})\;,$

we obtain

$\displaystyle\\{G_{ij}^{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha}),g_{k}(\mathbf{r}^{\prime})\\}=G_{ij}^{\alpha}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha})\partial_{k}\delta(\mathbf{r}-\mathbf{r}^{\alpha})$
$\displaystyle+\delta(\mathbf{r}-\mathbf{r}^{\alpha})\left[\delta_{ik}G_{jl}^{\alpha}+\delta_{jk}G_{il}^{\alpha}\right]\partial^{\prime}_{l}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha})\;.$
(83)

From this one can immediately obtain Eq. (15).

To evaluate the PB $\\{M(\mathbf{r}),g_{k}(\mathbf{r}^{\prime})\\}$ we let
$G_{ij}^{\alpha}=\frac{I_{\alpha}}{2}\delta_{ij}+\tilde{G}^{\alpha}_{ij}$,
with
$\tilde{G}^{\alpha}_{ij}=\Delta_{\alpha}\left(\nu^{\alpha}_{i}\nu^{\alpha}_{j}-\frac{1}{2}\delta_{ij}\right)$
and use the following identities

$\displaystyle\tilde{G}_{ik}^{\alpha}\tilde{G}_{kj}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{\alpha}^{2}}{4}\delta_{ij}\;,$ (84)
$\displaystyle\tilde{G}_{ik}^{\alpha}\tilde{G}_{ki}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{\alpha}^{2}}{2}\;.$ (85)

We can then write

$\Delta_{\alpha}\\{\Delta_{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha}),g_{k}(\mathbf{r}^{\prime})\\}=2\tilde{G}^{\alpha}_{ij}\\{\tilde{G}^{\alpha}_{ij}\delta(\mathbf{r}-\mathbf{r}^{\alpha}),g_{k}(\mathbf{r}^{\prime})\\}\;.$
(86)

Using Eq. (83), we find

$\displaystyle\\{\tilde{G}_{ij}^{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha}),g_{k}(\mathbf{r}^{\prime})\\}=\tilde{G}_{ij}^{\alpha}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha})\partial_{k}\delta(\mathbf{r}-\mathbf{r}^{\alpha})$
$\displaystyle+\delta(\mathbf{r}-\mathbf{r}^{\alpha})\left[\delta_{ik}\tilde{G}_{jl}^{\alpha}+\delta_{jk}\tilde{G}_{il}^{\alpha}-\delta_{ij}\tilde{G}_{kl}^{\alpha}\right]\partial^{\prime}_{l}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha})$
$\displaystyle+\frac{I_{\alpha}}{2}\left(\delta_{ik}\delta_{jl}+\delta_{jk}\delta_{il}-\delta_{ij}\delta_{kl}\right)\delta(\mathbf{r}-\mathbf{r}^{\alpha})\partial^{\prime}_{l}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha}).$
(87)

and

$\displaystyle\\{\Delta_{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha})$
$\displaystyle,g_{k}(\mathbf{r}^{\prime})\\}=\Delta_{\alpha}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha})\partial_{k}\delta(\mathbf{r}-\mathbf{r}^{\alpha})$
(88)
$\displaystyle+\Delta_{\alpha}\delta(\mathbf{r}-\mathbf{r}^{\alpha})\partial^{\prime}_{k}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha})$
$\displaystyle+\frac{2I_{\alpha}\tilde{G}_{kl}^{\alpha}}{\Delta_{\alpha}}\delta(\mathbf{r}-\mathbf{r}^{\alpha})\partial^{\prime}_{l}\delta(\mathbf{r}^{\prime}-\mathbf{r}^{\alpha})\;.$

The PB $\\{M(\mathbf{r}),g_{k}(\mathbf{r}^{\prime})\\}$ is then given by

$\displaystyle\\{M(\mathbf{r}),$ $\displaystyle
g_{k}(\mathbf{r}^{\prime})\\}=\delta(\mathbf{r}-\mathbf{r}^{\prime})\partial_{k}M(\mathbf{r})$
(89)
$\displaystyle-2\left[\sum_{\alpha}\frac{I_{\alpha}\tilde{G}_{kl}^{\alpha}}{\Delta_{\alpha}}\delta(\mathbf{r}-\mathbf{r}^{\alpha})\right]\partial_{l}\delta(\mathbf{r}-\mathbf{r}^{\prime})$

and involves a new field

$\sum_{\alpha}\frac{I_{\alpha}\tilde{G}_{kl}^{\alpha}}{\Delta_{\alpha}}\delta(\mathbf{r}-\mathbf{r}^{\alpha})=\sum_{\alpha}I_{\alpha}\left(\nu^{\alpha}_{i}\nu^{\alpha}_{j}-\frac{1}{2}\delta_{ij}\right)\delta(\mathbf{r}-\mathbf{r}^{\alpha})\;.$
(90)

We will need to make approximations to close the equations. We will
approximate as follows

$\sum_{\alpha}\frac{I_{\alpha}\tilde{G}_{kl}^{\alpha}}{\Delta_{\alpha}}\delta(\mathbf{r}-\mathbf{r}^{\alpha})\approx\frac{R(\mathbf{r})\tilde{G}_{ij}(\mathbf{r})}{M(\mathbf{r})}\;.$
(91)

##  Appendix D Mean Field theory of Vertex Model

Following czajkowski2018hydrodynamics , we construct a mean-field free energy
by rewriting the _single-cell_ Vertex model energy in terms of the cell
anisotropy parameter $M_{\alpha}$. Let us define

$\displaystyle M_{\alpha}=\lambda_{1}^{\alpha}-\lambda_{2}^{\alpha}\;,$ (92)
$\displaystyle R_{\alpha}=\lambda_{1}^{\alpha}+\lambda_{2}^{\alpha}\;,$ (93)

which gives $\lambda_{1,2}^{\alpha}=(R_{\alpha}\pm M_{\alpha})/2$. Equations
(6) and (7) are exact for regular polygons, but also hold approximately true
for slightly deformed polygons where the shape tensor remains diagonal and
$M_{\alpha}/R_{\alpha}\ll 1$. We can then write

$\displaystyle P_{\alpha}$ $\displaystyle\approx$ $\displaystyle
2n\sin\left(\frac{\pi}{n}\right)\left(\sqrt{\lambda_{1}^{\alpha}+\lambda_{2}^{\alpha}}\right)\equiv\nu(n)\sqrt{R_{\alpha}}\;,$
(94) $\displaystyle A_{\alpha}$ $\displaystyle=$ $\displaystyle
n\sin\left(\frac{2\pi}{n}\right)\sqrt{\lambda_{1}^{\alpha}\lambda_{2}^{\alpha}}\equiv\frac{\mu(n)}{2}\sqrt{R_{\alpha}^{2}-M_{\alpha}^{2}}\;.$
(95)

The single-cell energy can then be written in terms of $R_{\alpha}$ and
$M_{\alpha}$ as

$\displaystyle
E_{\alpha}=\frac{K_{A}}{2}\left(\frac{\mu}{2}\sqrt{R_{\alpha}^{2}-M_{\alpha}^{2}}-A_{0}\right)^{2}+\frac{K_{P}}{2}(P_{\alpha}-P_{0})^{2}\;.$
(96)

Expanding for $M_{\alpha}/R_{\alpha}\ll 1$, we can write

$\displaystyle E_{\alpha}=$ $\displaystyle\frac{K_{A}}{2}\left[\left(\frac{\mu
A_{0}}{2R_{\alpha}}-\frac{\mu}{2}\right)M_{\alpha}^{2}+\frac{\mu
A_{0}}{8R_{\alpha}^{3}}M_{\alpha}^{4}+\left(\frac{\mu
R_{\alpha}}{2}-A_{0}\right)^{2}\right]$ (97)
$\displaystyle+\frac{K_{P}}{2}\left(\nu\sqrt{R_{\alpha}}-P_{0}\right)^{2}\;.$

We now assume that the cell perimeter is constant, or $P_{\alpha}=P_{0}$,
hence $R_{\alpha}=P_{0}^{2}/\nu^{2}$. Substituting into Eq. (97), we can
rewrite the single-cell energy density $e_{\alpha}=E_{\alpha}/A_{0}$ as

$\displaystyle
e_{\alpha}=e_{0}+\frac{1}{2}\alpha(n,p_{0})\left(\frac{M_{\alpha}}{{A_{0}}}\right)^{2}+\frac{1}{4}\beta(n,p_{0})\left(\frac{M_{\alpha}}{{A_{0}}}\right)^{4}\;,$
(98)

where $e_{0}$ is a constant and

$\displaystyle\alpha(n,p_{0})$ $\displaystyle=$
$\displaystyle\frac{\kappa_{A}A_{0}^{2}\mu^{2}}{4p_{0}^{2}}\left({p_{0}^{*}}^{2}-p_{0}^{2}\right)\;,$
(99) $\displaystyle\beta(n,p_{0})$ $\displaystyle=$
$\displaystyle\frac{\kappa_{a}A_{0}^{4}\mu\nu^{6}}{4p_{0}^{6}}\;,$ (100)

with $p_{0}=P_{0}/\sqrt{A_{0}}$ the shape index, our tuning parameter. Also,
$\alpha(n,p_{0})$ has been written in terms of the critical shape index,

$\displaystyle p_{0}^{*}=\nu\sqrt{\frac{2}{\mu}}=\sqrt{4n\tan(\pi/n)}\;.$
(101)

The value of critical target shape parameter $p_{0}^{*}$ depends on the
specific undeformed polygonal shape, with $p_{0}^{*}=4$ for squares and
$p_{0}^{*}=2\sqrt{2\sqrt{3}}\approx 3.722$ for hexagons. Eq. (99) shows
explicitly that $\alpha$ changes sign at $p_{0}=p_{0}^{*}$, while $\beta>0$.
For $\alpha>0$ the stable ground state has $M=0$ and corresponds to a solid-
like state of isotropic cells. For $\alpha<0$ the stable ground state is a
fluid of anisotropic cells, with $M/A_{0}=\pm\sqrt{-\alpha/\beta}$. At
$\alpha=0$ the system undergoes spontaneous symmetry breaking and fluidizes,
choosing one of two equivalent axial direction along which to elongate. Here
we have defined $M$ as positive by assuming $\lambda_{1}>\lambda_{2}$, hence
breaking from the outside the Ising symmetry of the model. Finally, it was
shown in Ref. czajkowski2018hydrodynamics  that the quartic form given in Eq.
(98) is also obtained by assuming constant cell area, albeit with different
expressions for the coefficients $\alpha$ and $\beta$. In both cases the
coefficient $\alpha$ changes sign at $p_{0}=p_{0}^{*}$ and the behavior near
the transition is unaffected by the approximation used.

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models for English$\leftrightarrow$Tamil, and while a 9-model ensemble was
best for Khmer and for Pashton into English, they found that into Khmer and
Pashto, a single model was best. A second way of combining several models is
to use an additional model to rerank $n$-best hypothesis of an initial model.
libovick-EtAl:2020:WMT attempted right-to-left rescoring (against the normally
produced left-to-right hypothesis), but without seeing any gains for Upper-
Sorbian–German. chen-EtAl:2020:WMT2 test noisy channel reranking for
English–Tamil, but without seeing gains either, although some gains are seen
for Inuktitut$\rightarrow$, presumably because of the high-quality monolingual
news data available to train a good English language model.

##### Knowledge distillation

Knowledge distillation is also a frequently used technique and seems to give
minor gains, although is not as frequently used as backtranslation or
ensembling. The type of knowledge distillation used is that of (kim-
rush-2016-sequence), which consists in using a large or ensemble of models to
translate data on which a new model is trained. A number of teams applies this
iteratively, in combination with backtranslation (xia-EtAl:2019:WMT) or fine-
tuning (li-etal-2019-niutrans), and bei-etal-2019-gtcom mix knowledge
distillation data with genuine and synthetic parallel data to train a new
model to achieve gains in BLEU.

##### Fine-tuning

Mentioned previously in the context of pretraining, fine-tuning was used in
several contexts by a large number of teams. It is inevitably used after
pretraining on language model objectives or on other language pairs (see
above) to adapt the model to the language direction in question. It is also
frequently used on models trained on backtranslated data (fine-tuning to
genuine parallel data) (CITE). A final boost used by a number of top-systems
is achieved through fine-tuning to the development set (shi-etal-2020-oppos;
chen-etal-2020-facebook; zhang-EtAl:2020:WMT; wei-EtAl:2020:WMT1). This was
choice not made by all teams, some of which chose to keep it as a held-out
set, notably to avoid the risk of overfitting.

##### WAT

##### Similar language translation @ WMT ?

###  7.4 Workshops

The Workshop on Technologies for MT of Low Resource Languages (LoResMT) has
run for three years. It aims to close the gap between low-resource translation
systems and their practical use in the real world and there is a focus on
preprocessing tools for languages liu2018proceedings; ws-2019-technologies;
loresmt-2020-technologies. They have published research on a number of
extremely low-resouce languages such as Manipuri, Assamese, and Bambara.

The Workshop on Asian Translation (WAT) has been running for seven years. It
has run a variety of tasks, some of which are low resource. In the last shared
task in 2020 nakazawa-etal-2020-overview there were the following tracks:
Hindi / Thai / Malay / Indonesian $\leftrightarrow$ English IT and Wikinews
tasks; Odia $\leftrightarrow$ English; Bengali / Hindi / Malayalam / Tamil /
Telugu / Marathi / Gujarati $\leftrightarrow$ English Indic multilingual
tasks. They had reasonable participation of 20 teams across different tasks
and ran human evaluation of the top systems.

There have been two African NLP workshops co-hosted with ICLR
171717https://africanlp-workshop.github.io/program.html. Although this
workshop has a broader scope then purely translation, there is a strong focus
on this task with papers looking at translation involving Amharic, Nigerian
Pigin, Wolof and Tigrinya. There will be a similar workshop running for the
first time in 2021 on American languages
181818http://turing.iimas.unam.mx/americasnlp/st.html.

###  7.5 Groups

The Masakhane group nekoto-etal-2020-participatory is a community project that
is designed to promote African language NLP with a focus on resource creation.
They have created a useful resource wiki with recipes for different low-
resource translation scenarios191919https://github.com/masakhane-io/masakhane-
mt/blob/master/MT4LRL.md.

The GoURMET project birch-etal-2019-global is an Horizon 2020 research project
focussed on low-resource machine translation. The project involved the BBC and
Deutsche Welle and provides resources (monolingual and parallel data, models,
software for research and evaluation 202020https://gourmet-project.eu/data-
model-releases/) for 13 low-resource language pairs which are of strategic
intrerest to the media partners. inlineinlinetodo: inlineBarry: It seems a bit
biased to include our own project, there have surely been many LR projects. I
suggest dropping this section

##  8 Conclusion

###  8.1 Research Directions

## Author biographies

##### Alexandra Birch

is a senior research fellow at the University of Edinburgh. She co-ordinates
the H2020 project called GoURMET (2019-2022) and is the PI on the EPSRC
fellowship called MTStretch. Both projects focus on low-resource MT for the
media industry.

##### Rachel Bawden

is a researcher in Machine Translation (MT) at Inria, Paris. She obtained her
PhD in contextual MT in France and has previously worked on the GoURMET low-
resource MT project as a research associate at the University of Edinburgh.

##### Jindřich Helcl

is a PhD student at Charles University in Prague and a research associate in
MT at the University of Edinburgh, working on the GoURMET project focused on
low-resource MT. His other topics of interest include non-autoregressive MT
and multimodal MT.

##### Barry Haddow

is a senior researcher in the School of Informatics at the University of
Edinburgh. He has worked in machine translation for more than 10 years, and
his current interests include low-resource MT, spoken language translation and
evaluation of MT. Barry coordinates the annual WMT conference on machine
translation and associated shared tasks.

##### Antonio Valerio Miceli Barone

is a researcher in Machine Translation at the University of Edinburgh. He has
worked on deep learning methods for multi-lingual natural processing and
machine translation, specifically cross-lingual embedding inductionn, deep
recurrent neural networks and low-resource techniques. He is currently working
on the GoURMET project for translation of low-resource language pairs.