# Kinetics and steady state of polar flock with birth and death

Pratikshya Jena1<EMAIL_ADDRESS>Shradha Mishra1
<EMAIL_ADDRESS>Indian Institute of Technology (BHU)

###### Abstract

We study a collection of polar self-propelled particles or polar flock on a
two dimensional substrate with birth and death. Most of the previous studies
of polar flock with birth and death have assumed the compressible flock, such
that the local density of flock is completely ignored. Effect of birth and
death of particles on the flock with moderate density is focus of our study.
System is modeled using coarse-grained hydrodynamic equations of motion for
local density and velocity of the flock and solved using numerical integration
of the nonlinear coupled partial differential equations of motion and
linearised hydrodynamics about the broken symmetry state. We studied the
ordering kinetics as well as the steady state properties of the immortal flock
and flock with finite birth and death rate. The ordering kinetics of the
velocity field remains unaffected whereas the density field shows a crossover
from asymptotic growth exponent $5/6$ for the immortal flock to diffusive
limit $1/3$ for large birth and death rates. In the steady state, the presence
of birth and death rate leads to the suppression of speed of sound wave and
density fluctuations in the system.

##  I Introduction

Active mater Ramaswamy (2010); Marchetti  _et al._ (2013); Toner  _et al._
(2005); Bechinger  _et al._ (2016); Vicsek  _et al._ (1995a); Semwal  _et al._
(2021) is a novel class of nonequilibrium systems which consist of interacting
units that consume energy individually and generate motion and mechanical
stress collectively. Each individual unit contains an intrinsic asymmetry and
have tendency to self-propelled along the direction of asymmetry. Such systems
span a huge range of length scales, starting from very small scales of few
microns sized molecular motors, bacterial colonies, cells, cytoskeleton
filaments etc. to large scales of the order of few meters and kilometers such
as bird flock, human crowd, animal herd etc. Cichos  _et al._ (2020);
Needleman and Dogic (2017); De Magistris and Marenduzzo (2015); Fodor and
Marchetti (2018); Balasubramaniam  _et al._ (2022); Feder (2007); Bottinelli
_et al._ (2017). The collection of such particles show interesting features
such as pattern formation Liebchen and Levis (2017), nonequilibrium disorder
to order transitions Mahault  _et al._ (2018); Bhattacherjee  _et al._ (2015);
Chaté  _et al._ (2008); Mishra and Pattanayak (2017); Singh  _et al._ (2021);
Dikshit and Mishra (2022), anomalous fluctuations Liverpool (2003) etc., those
are not present in corresponding analogous equilibrium systems.
Flocking Toner  _et al._ (2005) or collective and coherent motion of large
number of organisms is the most interesting and ubiquitous phenomenon in many
biological systems. Unlike the corresponding equilibrium system Mermin and
Wagner (1966), flock shows a true long ranged, long time behavior in two
dimensions. The appearance of dense ordered clusters of particles, propagating
sound modes and the giant number fluctuation are observed Tu  _et al._ (1998)
in flock in the steady state.
Most of the previous studies of polar flock focused on the immortal flock,
where number of particles remain constant Toner and Tu (1995, 1998); Chaté
_et al._ (2008); Vicsek  _et al._ (1995b). In a recent study Toner (2012), the
author studied a system of polar flock without number conservation and showed
that the system behaves differently from the immortal flock i.e, the
longitudinal velocity fluctuations in the direction of flock motion replace
the sound modes of number conserving flock. The study of Toner (2012) focused
on the high density limit of polar flock with birth and death, such that the
density field is completely ignored. The properties of immortal flock with
finite density is not explored, and it is the focus of our study.
In this letter, we study the polar flock without number conservation
(Malthusian flock) Malthus  _et al._ (1992). We model the collection of polar
self-propelled particles with birth and death. We write the coarse-grained
hydrodynamic equations of motion for local density and velocity and study it
using numerical solution of nonlinear coupled partial differential equations
of motion and linearised hyrodynamics about the broken symmetry state.
Starting from random homogeneous ordered state of density and velocity, we
study the ordering kinetics of both the fields and find that the ordering of
velocity field remains unaffected due to birth and death, whereas the density
field shows a strong dependence on the birth and death rate. Density shows an
asymptotic growth law of $5/6$ Toner (2012) for immortal flock and low birth
and death rate. For large birth and death rate the approach to asymptotic
growth is faster and towards the diffusive growth law $1/3$ Bray (2002). In
the steady state, the presence of birth and death rate suppresses the speed of
traversing sound mode and density fluctuations in the system.

Figure 1: (color online) Real space snapshots of local velocity $v({\bf r},t)$
(upper panel) $(a-c)$ and density $\rho({\bf r},t)$ (lower panel) $(f-h)$ for
the IF and at times $t=0,900$ and $200$ respectively. $(d-e)$ and $(i-j)$ are
the snapshots for local velocity and density respectively at fixed time
$t=900$ and for different $\lambda=0.05$, $0.1$.

##  II Model

We model a collection of polar self-propelled particles with birth and death.
We approach the problem using coarse-grained hydrodynamic equations of motion
for two slow variables, viz .(i) local density $\rho({\bf r},t)$ and (ii)
velocity ${\it{\bf v}}({\bf r},t)$ of particles at position ${\bf r}$ and at
time $t$. The hydrodynamic equation for the density $\rho({\bf r},t)$ is:

$\partial{{}_{t}}\rho+v_{0}\nabla\cdot({\it{\bf
v}}\rho)=D_{\rho}\nabla^{2}\rho+g(\rho)$ (1)

and for the local velocity ${\it{\bf v}}({\bf r},t)$ is:

$\displaystyle\partial{{}_{t}}{\bf v}+{}$
$\displaystyle\lambda{{}_{1}}({\it{\bf v}}.\nabla){\it{\bf
v}}+\lambda{{}_{2}}(\nabla\cdot{\it{\bf v}}){\it{\bf v}}=\
\alpha(\rho){\it{\bf v}}$ (2) $\displaystyle\ -\beta\mid{\it{\bf
v}}\mid^{2}{\it{\bf v}}-\sigma_{1}\nabla\rho+D{{}_{T}}\nabla^{2}{\it{\bf
v}}+{\it{\bf f}}$

The density equation is continuity equation with addition to birth and death.
The term on the left hand side is the current due to self-propulsion. The
first term on the right side is the diffusion current and the last term is an
additional birth and death term of the form
$g(\rho)$=$\lambda\rho(1-\frac{\rho}{\rho_{0}})$. The parameter $\lambda$
controls the rate of birth and death. For $\lambda=0$, the model reduces to
the immortal flock (IF) Toner and Tu (1998) and for finite $\lambda$ we have
flock with birth and death (MF) Malthus and Malthus (1996); Mishra and Mishra
(2022). Here, $\rho_{0}$ is constant that controls the mean density. The
velocity Eq.2 is the same as introduced by Toner and Tu for the immortal flock
Toner and Tu (1998). In Eq.2, the right hand side, the terms,
$\alpha(\rho)\bf{v}$, $\beta\mid{\it{\bf v}}\mid^{2}{\it{\bf v}}$ and
$D_{T}\nabla^{2}{\it{\bf v}}$ arise in analogy with the time dependent
Ginzburg-Landau equation (TDGL) Goldenfeld (2018) which is responsible for
ordering towards the broken symmetry phase. $\lambda_{1}({\bf
v}\cdot{\nabla}){\bf v}$, $\sigma_{1}\nabla\rho$ terms are analogous to the
convection and pressure due to gradient in density respectively and similar to
the terms present in Navier-Stoke’s equation for fluid dynamics.
$\lambda_{2}(\nabla\cdot{\bf v}){\bf v}$ is another convective non linearity
appears due to absence of Galilean invariance in the system. The last term in
Eq.2 ${\bf f}({\bf r},t)=(f_{1},f_{2})$ is a random driving force representing
noise and it is Gaussian with white-noise correlation and strength $\Delta$;

$\langle f{{}_{i}}({\bf r},t)f{{}_{j}}({\bf
r^{{}^{\prime}}},t{{}^{\prime}})\rangle=\Delta\delta_{ij}\delta({\bf r}-{\bf
r}^{{}^{\prime}})\delta(t-t^{{}^{\prime}})$ (3)

Numerical integration of Eq.1 and 2 is performed with homogeneous initial
density (with mean $\rho_{0}=0.75$) and random velocity in a box of size
$K\times K$ with periodic boundary condition in the both directions. The
integration is performed using Euler’s scheme Bally and Talay (1996) with
$\Delta x=1.0$ and $\Delta t=0.1$. The other parameters of the system are
fixed to $\beta=1.0$, $\lambda_{1}=0.5$, $\lambda_{2}=1.0$,
$\alpha(\rho)=(1-\rho/\rho_{c})$ ($\rho_{c}=0.5$), $D_{\rho}=1.0$,
$\sigma_{1}=v_{0}/\rho_{0}$, $D_{T}=1.0$, $v_{0}=0.5$ and varied the birth and
death rate $\lambda$ from $0$ to $1.0$. We checked that our numerical
integration is stable for the above choice of parameters. The simulation is
carried out by varying the system size i.e $K=256$ to $2048$. Starting with
random homogeneous state, system is quenched to the above parameters such that
steady state is an ordered state. We studied the ordering kinetics of the
velocity and the density field and properties of the steady state for
different birth and death rate $\lambda$. Steady state results are obtained
for $K=256$ to $1000$ for time $t=2\times 10^{5}$ whereas coarsening studies
are performed on relatively large system size $K=2048$ with final time
$t=7\times 10^{3}$. For better statistical averaging results are averaged over
$100$ independent realisations.

##  III Result

We first plot the time evolution of real space snapshots of the system,
initiated with random homogeneous state. Fig.1 show the time evolution of
local velocity ${v({\bf r},t)}=|{\bf v}|$ (upper panel) and magnitude of local
density ${\rho({\bf r},t)}$ (lower panel). Fig.1($a-c$; $f-h$) shows the plot
for the immortal flock at three different times $t=0$, $900$ and $2000$ and
Fig 1($d-e$; $i-j$) is for the fixed time $t=900$ and different $\lambda$=,
$0.05$ and $0.1$ respectively. As observed from the Fig.1($a-c$), the velocity
field shows that system evolves towards the ordered state with the growing
domains of ordered region with large magnitude of local $v$. Also the domains
of high density region grows with time as shown in Fig.1($f-h$). The growing
domains of ordered velocity and high density is also affected by increasing
$\lambda$ as shown in Fig. 1($d-e$; $i-j$) for fixed time and varying
$\lambda$. The growth is more affected for density Fig. 1($i-j$) in comparison
to velocity field Fig. 1($d-e$).
The characteristics of the growing domains is measured by calculating the two-
point velocity and density correlation functions $C_{v}(r,t)=\langle{\bf
v}({\bf r}_{0},t)\cdot{\bf v}({\bf r}_{0}+{\bf r},t)\rangle$ and
$C_{\rho}(r,t)=\langle\delta\rho({\bf r}_{0},t)\delta\rho({\bf r}_{0}+{\bf
r},t)\rangle$ respectively. Here, $\delta\rho$ represents the fluctuation in
density from the mean $\rho_{0}$ and the $<..>$ means averaging over
directions, reference points ${\bf r}_{0}$ and over $100$ independent
realisations.

Figure 2: (color online) (a,b) represent the $\log-\log$ plot of
characteristic lengths $L_{v,\rho}(t)$ vs. time $t$ for velocity and density
field respectively for different $\lambda$ values. The dashed lines have their
respective slopes as shown on top of them. (c) plot of $z_{\rho}^{as}$ vs.
$\lambda$. The solid $\triangle$ on the $y-$axis is value $1.2$. Figure 3:
(color online) The plot of scaled two-point correlation functions
$C_{v,\rho}(r/L_{v,\rho}(t))$ vs. scaled distance $r/L_{v,\rho}(t)$ for
velocity and density fields. $(a-b)$ is for the IF ($\lambda=0$) and $(c-d)$
is for the MF with $\lambda=0.01$. Different colors are for different times as
shown in the legend. Figure 4: (color online) $(a-d)$, the plot of
$\rho_{||}(x)$ vs. $x$ for $\lambda=0$, $0.0005$, $0.0008$, $0.001$
respectively. Different colors/symbols $\circ$ (black), $\square$ (red),
$\diamond$ (green) and $\triangle$ (blue) in each window represent the
different times ($t_{1}<t_{2}<t_{3}<t_{4}$) respectively in the steady state
at the interval of $t=400$. The arrows in $(a-b)$ show the direction of moving
flock.  Figure 5: (color online) (a) The $\log-\log$ plot of $\frac{\Delta
N}{\sqrt{N}}$ vs. $N$ for different $\lambda$. (b) The plot of
$\beta(N,\lambda)$ vs. $N$. (c) The plot of $\beta(\lambda)$ vs. $\lambda$.
The black data are obtained from numerical simulation and data with line (red)
obtained from Eq. 21 in VI by putting all the constants from the numerical
simulation.

Ordering kinetics:- We define the size of the growing domain of by calculating
the characteristics lengths $L_{v,\rho}(t)$ defined by the distance by which
the the two-point correlation functions (as shown in Fig.1(a-b) (SM1)) decay
to $0.1$ of its value at $r=0$. Fig.2(a-b) respectively shows the plot of the
two characteristic lengths $L_{v,\rho}(t)$ vs. time $t$ for different
$\lambda$’s from $0$ to $1$. We find that the growth of $L_{v}(t)$ remains
almost independent of $\lambda$. For all the cases the asymptotic growth law
is $0.5$ as shown by the dashed line. Hence the kinetics of velocity remains
unaffected due to the birth and death parameter $\lambda$ and remains the same
as for the usual non-conserved growth kinetics of Model A Hohenberg and
Halperin (1977); Das  _et al._ (2018). As shown in Fig.2(b) the growth of
$L_{\rho}(t)$ shows strong dependence on the parameter $\lambda$. For immortal
flock and weak birth and death $\lambda\leq 0.01$, the growth of $L_{\rho}(t)$
clearly shows three regimes. For very early time, the growth is mostly
dominated by background non-conserved continuous velocity field and
$L_{\rho}(t)\simeq t^{0.5}$, as we increase time, density starts to grow much
faster and $L_{\rho}(t)\simeq t$. The fast growth is due to the hydrodynamic
modes present in moving ordered flock Toner and Tu (1995); Toner  _et al._
(2005); Chaté  _et al._ (2008). Then at the late times $L_{\rho}(t)$
approaches to the asymptotic growth law $\simeq t^{5/6}$ predicted for the
continuum theory of polar flock Toner and Tu (1998). On increasing
$\lambda>0.01$, the approach to the asymptotic growth happens early and
asymptotic growth exponent decreases with increasing $\lambda$ and finally for
large $\lambda$’s it approaches to the diffusive limit with $L_{\rho}(t)\sim
t^{1/3}$ for the conserved growth kinetics Lifshitz and Slyozov (1961). In
Fig.2(c) we show the plot of asymptotic growth exponent $z_{\rho}^{as}$ vs.
$\lambda$, where $z_{\rho}^{as}$ is obtained from the late time power law
dependence of $L_{\rho}(t)\simeq t^{1/z_{\rho}^{as}}$ in the main figure. We
found that for very small $\lambda<0.001$ , $z_{\rho}^{as}\simeq 6/5$ and then
it increases with increasing $\lambda$ and saturate to the diffusion limit
$\simeq 3$ for higher $\lambda$. Hence asymptotic growth of the density is
driven by the conventional sound mode propagating in the direction parallel to
the direction of flock; for the IF and flock with small birth and death rate,
whereas it is purely diffusive for the MF.
We further characterise the scaling behaviour of two-point correlation
functions. In Fig.3(a-b) we show the plot of scaled correlations for velocity
and density $C_{v}(r/L_{v}(t))$ and $C_{\rho}(r/L_{\rho}(t)$ respectively for
the IF, $\lambda=0.0$ and in Fig.3$(c-d)$ for the $\lambda=0.01$ and for
different times. We find that for both the cases IF and $\lambda=0.01$, the
velocity field $C_{v}(r/L_{v}(t))$ shows the good scaling collapse, whereas
density $C_{\rho}(r/L_{\rho}(t))$ does not. Non scaling of density field for
immortal flock is also reported in previous studies Das  _et al._ (2018), is
due to the presence of more than one regime of growing domain as shown in
$L_{\rho}(t)$ in Fig.2(b) Mishra  _et al._ (2010).

Steady state:- Now we explore the effect of birth and death on the steady
state properties of the system. One of the characteristics of immortal flock
is the formation of ordered high density bands moving in the low density
disordered background. These bands move as the sound wave travelling along the
ordering direction of the flock Mishra  _et al._ (2010); Chaté  _et al._
(2008); Mishra and Mishra (2022); Bhattacherjee  _et al._ (2015). In SM2, we
show the animations of local density in the steady state for different
$\lambda^{\prime}s$ = $0$, $0.0005$, $0.0008$, $0.001$ for system of size
$L=256$. It can be observed that the density fluctuations as well as the speed
of the moving bands decreases with increasing $\lambda$. To further quantify
the motion of travelling density pattern for different $\lambda$, we define
the one dimensional density profile $\rho_{\parallel}(x)$, by averaging the
local density $\rho({\bf r},t)$ over the direction transverse to the
propagating direction of the band i.e,
$\rho_{\parallel}(x)=\frac{\sum_{j=1}^{L}\rho_{j,\perp}}{L}$; here,
$\rho_{\perp,\parallel}$ is the density along the transverse and longitudinal
direction of the global orientation direction of the flock respectively. $x$
is chosen as the travelling direction (direction of ordering of the flock) of
the bands. In the Fig.4 $(a-d)$, we show the plot of $\rho_{||}(x)$ for four
different $\lambda$’s = $0$, $0.0005$, $0.0008$ and $0.001$ respectively.
Different colors in each plot represents the $\rho_{\parallel}(x)$ at four
different times at the interval of $t=400$. We find that for $\lambda=0$, the
pattern of $\rho_{\parallel}(x)$ shifts towards the direction of propagating
band (shown by the arrow). The shift is proportional to the speed of the
bands. As we introduce $\lambda$, and on increasing it, the shift as well as
the amplitude of fluctuation reduces and for $\lambda>0.0005$, the band fades
away and density approaches to the mean value $\rho_{0}=0.75$ at late times.
We also calculated the sound speed using the linearised hydrodynamic equations
of motion for the two fields (${\bf v}$ and $\rho$). The SM1, contains the
details of the calculation. We find that sound speed very much depends on the
birth and death rate and decreases with increasing $\lambda$ as given in Eq.14
(VI). Also interestingly we find that the sound speed is purely non-dispersive
for IF ($\lambda=0$) and becomes dispersive for finite $\lambda$ Eq.14 (VI)
(although the dependence is quadratic is $\lambda$). In VI we show the plot of
$\rho_{\parallel}(x)$ for three different system sizes $K=175,200,256$ for
$\lambda=0.0005$. Clearly the speed of the band depends on $K$ and increases
on increasing $K$.

Further we also calculate the effect of $\lambda$ on the number fluctuation
$\Delta N$ of particle density and compared it with the density fluctuation
calculated from the linearised approximation. The details of the linear
calculation using coarse-grained hydrodynamic equations is given in VI. In
Fig.5(a) we show the plot of $\frac{\Delta N}{\sqrt{N}}$ vs. $N$ for different
$\lambda$ values. We expect the $\frac{\Delta N}{\sqrt{N}}\simeq N^{\alpha}$,
where the exponent $\alpha>0$ for density fluctuation larger than the
corresponding equilibrium systems Chaikin  _et al._ (1995). We find that for
$\lambda\leq 0.05$, density fluctuation is larger than the equilibrium and for
higher $\lambda>0.05$, it is suppressed. Further in Fig 5(b) we plot
$\beta(N,\lambda)$ vs. $N$, where $\beta(N,\lambda)$ is obtained by
$\beta(N,\lambda)=\frac{d\ln{\frac{\Delta N}{\sqrt{N}}}}{d\ln{N}}$. For small
$N$, $\beta(N,\lambda)$ increases with $N$ and for large $N$ it approaches to
value = $0.4$, for $\lambda\leq 0.05$ and zero for large $\lambda>0.05$. In
Fig. 5(c) we show the plot of $\beta(\lambda)$ for different $\lambda$; where
$\beta(\lambda)$ is obtained from the large $N$ limit of $\beta(N,\lambda)$
from the previous plot. The data (black) shows the calculation from the
numerical results and red data (lines) is the prediction from the linearised
hydrodynamics (VI Eq.21). We find that for small $\lambda$, linearised theory
under predicts the behaviour of $\beta(\lambda)$, whereas for large $\lambda$
it approaches to zero and numerical result matches with theoretical
prediction.

##  IV Summary

In this letter, we have modeled the Malthusian flock with birth and death rate
in two-dimensions using coarse-grained hydrodynamic equations of motion for
local velocity and density. Unlike the previous study of Toner (2012) for high
density, our study is focused on the finite density of particles, such that
density field cannot be ignored. Starting from the random homogeneous state;
the ordering kinetics of the system towards the ordered state is studied using
numerical integration of the coupled non-linear partial differential equations
of motion for the two fields. We find that the ordering kinetics of the
velocity field remains unaffected due to birth and death rate, but density
shows strong dependence. For IF and small birth and death rate the asymptotic
growth law for density is $5/6$ and shows a crossover to diffusive growth law
of $1/3$ for large birth and death rate. The approach to asymptotic growth
becomes faster as we increase the birth and death rate. We also characterised
the steady state properties of the flock using both numerical integration of
full non-linear equations and linearised hydrodynamic about the ordered state
and find that the presence of finite birth and death rate suppresses the speed
of sound wave and large density fluctuations present in the immortal flock
Toner (2012).
Our study shows the effect of birth and death rate on the ordering kinetics
and steady state properties of polar self-propelled particles for the case
when density fluctuations are taken in account. Hence it can be used to
understand the effect of birth and death in various biophysical systems with
moderate densities Voituriez  _et al._ (2005); Kruse  _et al._ (2005); Mather
_et al._ (2010); Karig  _et al._ (2018); Cates  _et al._ (2010). Our finding
can be tested for similar experimentsDell’Arciprete  _et al._ (2018) in the
laboratory.

##  V Acknowledgments

P.J. gratefully acknowledge the DST INSPIRE fellowship for funding this
project. The support and the resources provided by PARAM Shivay Facility under
the National Supercomputing Mission, Government of India at the Indian
Institute of Technology, Varanasi are gratefully acknowledged by all authors.
S.M. thanks DST-SERB India, ECR/2017/000659 and CRG/2021/006945 for financial
support. P.J. and S.M. also thank the Centre for Computing and Information
Services at IIT (BHU), Varanasi.

##  VI appendix

###  VI.1 Ordering kinetics

Figure 6: (color online) The plot of two-point correlation functions
$C_{v,\rho}(r)$ vs. distance $r$ for velocity and density fields. $(a-b)$ is
for the IF ($\lambda=0$) and $(c-d)$ is for the MF with $\lambda=0.01$.
Different colors are for different times as shown in the legend.

###  VI.2 Steady state

Figure 7: (color online) The plot of $\rho_{||}(x)$ vs. $x$ for
$\lambda=0.0005$ for three different system sizes $(a-c)$; $K=175$, $200$ and
$256$ respectively for four different times $t_{1}<t_{2}<t_{3}<t_{4}$ in the
steady state at the time interval of $t=400$. The arrows in each figure
represent the direction of moving flock.

###  VI.3 Linearized approximation

In this section we focus on the linearised hydrodynamics of the system about
the broken symmetry state. The steady state solution of Eq.1 and Eq.2 in the
main text is a homogeneous ordered state with density $\rho=\rho_{0}$ and
order parameter ${\bf v}=v{{}_{0}}\widehat{x}$, where
$v_{0}=\sqrt{\frac{\alpha}{\beta}}$ and
$\alpha=\alpha_{0}(1-\rho_{0}/\rho_{c})$. Here we have chosen $\hat{x}$ as the
direction of broken symmetry. We add small fluctuations to the uniformly
ordered velocity and homogeneous density, and write the velocity field as
${\bf v}$=$v{{}_{0}}\widehat{x}+\delta{\bf v}$. The fluctuation in ${\bf v}$,
$\delta{\bf v}=(v_{||}\widehat{x},v_{\perp}\widehat{y})$, where $\hat{y}$ is
direction perpendicular to the ordering direction. Now we write the equation
of motion for the fluctuation $\delta{v_{||}}$ to the linear order in small
fluctuations $(\delta\rho,{v_{||}},{v_{\perp}})$

$\partial{{}_{t}}v{{}_{\parallel}}=-\sigma{{}_{1}}\partial{{}_{\parallel}}\delta\rho-2\alpha
v{{}_{\parallel}}+\text{irrelevant terms}$ (4)

In the steady state, we can solve the right hand side of Eq. 4 and get the
following solution for $v_{||}$

$v_{\parallel}=-D_{2}\partial_{||}\delta\rho$ (5)

where, $D_{2}=\sigma_{1}/2\alpha$ . Inserting Eq.5 in the linearised equations
of motion for ${v}_{\perp}$ and $\delta\rho$ and neglecting the irrelevent
terms,

$\partial_{t}v_{\perp}+\gamma\partial_{||}v_{\perp}=-\sigma{{}_{1}}\partial_{\perp}\delta\rho+D_{T}\nabla^{2}v_{\perp}+f_{\perp}$
(6)

and density equation in Eq.1(main text),

$\partial_{t}\delta\rho+v_{0}\partial_{||}\delta\rho-
D_{2}\rho_{0}\partial_{||}^{2}\delta\rho-
D_{\rho}\nabla^{2}\delta\rho+\rho_{0}\partial_{\perp}v_{\perp}=-\lambda\rho_{0}\delta\rho$
(7)

where $D_{2},D_{\rho},D_{B}$ and $D_{T}$ are the diffusion constants,
$\gamma=\lambda_{1}v_{0}$. For the given system parameters, they are all
constants. To further study the solution of above linearised equations we
define a vector ${\bf A}({\bf r},t)=(v_{\perp},\delta\rho)$. Eq. 6 and Eq.7,
we write the equations in Fourier space by defining

${\bf A}({\bf r},t)\simeq\int{{\bf A}({\bf q},\omega)\exp(-i\omega t+i{\bf
q}.{\bf r})d{\bf q}d\omega}$

Now we can write the linear equations for $v_{\perp}({\bf q},\omega)$ and
$\delta\rho({\bf q},\omega)$

$[-i(\omega-\gamma
q{{}_{\parallel}})+\Gamma{{}_{L}}]v{{}_{\perp}}+i\sigma{{}_{1}}q{{}_{\perp}}\delta\rho=f{{}_{\perp}}$
(8)

and

$[-i(\omega-v{{}_{0}}q{{}_{\parallel}})+\Gamma{{}_{\rho}}+\lambda\rho_{0}]\delta\rho+i\rho{{}_{0}}q{{}_{\perp}}v{{}_{\perp}}=0$
(9)

where ${\bf q}$ = $(q_{\parallel},q_{\perp})$ =
$q(\cos(\theta),\sin(\theta))$, where $\theta$ is the angle from the direction
of broken symmetry. The Fourier transform of random force $f_{\perp}({\bf
q},\omega)$ has properties $<f_{\perp}({\bf q},\omega)>=0$ and variance
$<f_{\perp}({\bf q},\omega)f_{\perp}({\bf
q}^{\prime},\omega^{\prime})>=\Delta\delta({\bf q}+{\bf
q}^{\prime})\delta(\omega+\omega^{\prime})$; The two damping
$\Gamma_{L,\rho}({\bf q})$ defined as $\Gamma{{}_{L}}({\bf
q})=D{{}_{T}}(q_{\parallel}^{2}+q_{\perp}^{2})$ and $\Gamma_{\rho}({\bf
q})=D{{}_{2}}\rho_{0}q_{\parallel}^{2}+D_{\rho}(q_{\parallel}^{2}+q_{\perp}^{2})$.
The linear equations Eq.8 and Eq.9 for $v_{\perp}$ and $\delta\rho$ can be
solved using following matrix;

$\begin{bmatrix}-i(\omega-\gamma
q_{\parallel})+\Gamma_{L}&i\sigma_{1}q_{\perp}\\\
iq_{\perp}\rho_{0}&-i(\omega-
v_{0}q_{\parallel})+\Gamma_{\rho}+\lambda\rho_{0}\end{bmatrix}\begin{bmatrix}v_{\perp}\\\
\delta\rho\end{bmatrix}=\begin{bmatrix}f_{L}\\\ 0\par\end{bmatrix}$

The solutions for $v_{\perp}$ and $\delta\rho$ is

$\begin{bmatrix}v_{\perp}\\\
\delta\rho\end{bmatrix}=[M]^{-1}\begin{bmatrix}f_{T}\\\ 0\end{bmatrix}$

where the matrix $[M]$ is,

$\begin{bmatrix}M\end{bmatrix}=\begin{bmatrix}-i(\omega-\gamma
q_{\parallel})+\Gamma_{L}&i\sigma_{1}q_{\perp}\\\
iq_{\perp}\rho_{0}&-i(\omega-
v_{0}q_{\parallel})+\Gamma_{\rho}+\lambda\rho_{0}\end{bmatrix}$

and $[M]^{-1}$ is

$\begin{bmatrix}M^{-1}\end{bmatrix}=\frac{1}{\det[M]}\begin{bmatrix}-i(\omega-
v_{0}q_{\parallel})+\Gamma_{\rho}+\lambda\rho_{0}&-i\sigma_{1}q_{\perp}\\\
-iq_{\perp}\rho_{0}&-i(\omega-\gamma q_{\parallel})+\Gamma_{L}\end{bmatrix}$

where

$det[M]=-\omega^{2}+\omega(v_{0}q_{\parallel}+\gamma
q_{\parallel}-i\Gamma_{\rho}-i\Gamma_{L}-i\lambda\rho_{0})-v_{0}\gamma
q_{\parallel}^{2}+iq_{\parallel}(\gamma\Gamma_{\rho}+v_{0}\Gamma_{L})+\Gamma_{\rho}\Gamma_{L}+\rho_{0}\sigma_{1}q_{\perp}^{2}+\Gamma_{L}\lambda\rho_{0}+i\lambda\rho_{0}\gamma
q_{\parallel}$ (10)

The quadratic equation for the $\omega$;

$\omega^{2}-\omega[q_{\parallel}(v_{0}+\gamma)-i(\Gamma_{\rho}+\Gamma_{L}+\lambda\rho_{0})]-iq_{\parallel}(\gamma\Gamma_{\rho}+v_{0}\Gamma_{L})+v_{0}\gamma
q_{\parallel}^{2}-\rho_{0}\sigma_{1}q_{\perp}^{2}-\Gamma_{L}\lambda\rho_{0}-i\lambda\rho_{0}\gamma
q_{\parallel}=0$ (11)

The real part two solutions of $\omega_{\pm}$ are given by;

$\omega_{\pm}=c_{\pm}(\theta_{{\bf
q}})q-\frac{\Delta\Gamma\lambda\rho_{0}q}{\sqrt{(\frac{\gamma-
v_{0}}{2})^{2}\cos^{2}{\theta}+c_{0}^{2}\sin{\theta}}}-\frac{(\lambda\rho_{0})^{2}}{2q{\sqrt{(\frac{\gamma-
v_{0}}{2})^{2}\cos^{2}{\theta}+c_{0}^{2}\sin{\theta}}}}$ (12)

where

$c_{\pm}(\theta_{{\bf q}})=\frac{\gamma+v_{0}}{2}\cos(\theta_{{\bf q}})\pm
c_{2}(\theta_{{\bf q}})$ $c_{2}(\theta_{{\bf q}})=\sqrt{\frac{1}{4}(\gamma-
v_{0})^{2}\cos^{2}(\theta_{{\bf q}})+c_{0}^{2}\sin^{2}(\theta_{{\bf q}})}$
$c_{0}=\sqrt{\sigma_{1}\rho_{0}}$

and $\Delta\Gamma=\widehat{\Gamma_{\rho}}-\widehat{\Gamma_{L}}$,
$\widehat{\Gamma_{L}}=D_{T}$,
$\widehat{\Gamma_{\rho}}=D_{2}\rho_{0}\cos{\theta}+D_{\rho}$.We can calculate
the speed of sound wave $C(q,\lambda,\theta)=\frac{d\omega_{+}}{dt}$ from Eq.
12 as

$C(q,\lambda,\theta)=\frac{\gamma+v_{0}}{2}\cos(\theta_{{\bf
q}})+\sqrt{\frac{1}{4}(\gamma-v_{0})^{2}\cos^{2}(\theta_{{\bf
q}})+c_{0}^{2}\sin^{2}(\theta_{{\bf
q}})}-\frac{\Delta\Gamma\lambda\rho_{0}}{\sqrt{\frac{1}{4}(\gamma-
v_{0})^{2}\cos^{2}(\theta_{{\bf q}})+c_{0}^{2}\sin^{2}(\theta_{{\bf
q}})}}+\frac{(\lambda\rho_{0})^{2}}{q^{2}\sqrt{\frac{1}{4}(\gamma-
v_{0})^{2}\cos^{2}(\theta_{{\bf q}})+c_{0}^{2}\sin^{2}(\theta_{{\bf q}})}}$
(13)

For the case $\theta=0$ , the speed of sound wave from Eq.12 as

$C(q,\lambda)=\frac{d\omega_{+}}{dq}=v_{0}-\frac{2\Delta\Gamma\lambda\rho_{0}}{(\gamma-
v_{0})}+\frac{4(\lambda\rho_{0})^{2}}{q^{2}({\gamma-v_{0}})^{2}}$ (14)

Further we go back to the solution for $v_{\perp}$ and $\delta\rho$

$v_{\perp}({\bf q},\omega)=G_{LL}({\bf q},\omega)f_{\perp}({\bf q},\omega)$
(15) $\delta\rho({\bf q},\omega)=G_{\rho L}({\bf q},\omega)f_{\perp}({\bf
q},\omega)$ (16)

where the propagators $G_{LL}$ and $G_{\rho L}$ are given by;

$G_{LL}({\bf q},\omega)=\frac{(-i(\omega-
v_{0}q_{\parallel})+\Gamma_{\rho}({\bf q})+\lambda\rho_{0})}{(\omega-
c_{+}(\theta_{\bf q})q)(\omega-c_{-}(\theta_{\bf q})q)+i\omega(\Gamma_{L}({\bf
q})+\Gamma_{\rho}({\bf
q})+\lambda\rho_{0})-iq\cos{\theta}(\gamma\Gamma_{L}({\bf
q})+v_{0}\Gamma_{\rho}({\bf q})+2\lambda\rho_{0}(\gamma+v_{0}))}$ (17)
$G_{\rho L}({\bf q},\omega)=\frac{-i\sigma_{1}q_{\perp}}{(\omega-
c_{+}(\theta_{\bf q})q)(\omega-c_{-}(\theta_{\bf q})q)+i\omega(\Gamma_{L}({\bf
q})+\Gamma_{\rho}({\bf
q})+\lambda\rho_{0})-iq\cos{\theta}(\gamma\Gamma_{L}({\bf
q})+v_{0}\Gamma_{\rho}({\bf q})+2\lambda\rho_{0}(\gamma+v_{0}))}$ (18)

Hence, the two-point density-density correlation function $C_{\rho}({\bf
q},\omega)$ with become;

$C_{\rho}(q,\omega)=\frac{\Delta\rho_{0}^{2}q_{\perp}^{2}}{{(\omega-
c_{+}(\theta_{\bf q})q)^{2}(\omega-c_{-}(\theta_{\bf
q})q)^{2}+(\omega(\Gamma_{L}({\bf q})+\Gamma_{\rho}({\bf
q})+\lambda\rho_{0})-q\cos{\theta}(\gamma\Gamma_{L}({\bf
q})+v_{0}\Gamma_{\rho}({\bf q}))+2\lambda\rho_{0}(\gamma+v_{0}))^{2}}}$ (19)

The $C_{\rho}$ has two sharp peaks at $\omega$= $c_{\pm}$q with width
$\mathcal{O}(q^{2})$. The width of these peaks in the limit $q\to 0$, is very
small compared to its displacement from the origin ($\mathcal{O}(\omega)$).
The expression for the structure factor
$S(q)=C_{\rho}(q)=\int_{-\infty}^{+\infty}C_{\rho}({\bf
q},\omega)\,\frac{d\omega}{2\pi}$ is

$S(q)=\frac{\Delta_{0}\rho_{0}^{2}q_{\perp}^{2}+c_{+}^{2}q^{2}\lambda^{2}\rho_{0}^{2}}{[c_{+}q(\Gamma_{L}({\bf
q})+\Gamma_{\rho}({\bf
q})+\lambda\rho_{0})]^{2}}\times[1+\frac{\Gamma_{\rho}({\bf
q})}{\Gamma_{L}({\bf
q})}+\frac{2\lambda\rho_{0}(\gamma+v_{0}+0.5)}{\Gamma_{L}({\bf q})}]$ (20)

Here we have neglected the higher order term of $\mathcal{O}(q^{4})$. We write
the expression for small $\lambda$ such that we can ignore the term of order
$\mathcal{O}(\lambda^{2})$. Writing the different values of constants from the
numerical simulation we get,

$S(q)=\frac{A[q^{2}+B\lambda]}{[Cq^{2}+D\lambda]^{2}}$

where $A=0.056$, $B=2.25$, $C=0.86$, $D=0.75$. We can now calculate the number
fluctuation $\Delta N$ in a volume $V$ as $\sqrt{S(q\to 0)V}$. We get the
expression for $\Delta N$ as

$\Delta N=N\frac{\sqrt{(A(1+BN\lambda))}}{C+DN\lambda}$ (21)

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honest node $wr\text{-}delivers$ some value, any honest node eventually wr-
delivers some value.

###### Proof.

Let node $\mathcal{P}_{i}$ be the first honest node who wr-delivers _h_ in
${avw\mathsf{BRBC}}_{j}$. So $\mathcal{P}_{i}$ must have received $n-f$
_READY(h)_ messages from distinct nodes, among which $f+1$ are honest nodes.
As shown in Figure 8, invoking $abandon()$ only blocks sending _ECHO_
messages. So these $f+1$ honest nodes will eventually transmit their
_READY(h)_ messages to all honest nodes. Thus, any honest node will eventually
receives $f+1$ _READY(h)_ messages, and will tansmit _READY(h)_ if has not
sent it. Therefore, every honest node will eventually receive $n-f$ _READY(h)_
, and $wr\text{-}deliver$ _h_. ∎

######  Lemma 2.

External validity of avw$\mathsf{BRBC}$. For any avw$\mathsf{BRBC}$ instance,
If an honest node outputs a value $v$, then $v$ is valid w.r.t. $Q$, i.e.,
$Q(v)=1$.

###### Proof.

This property is fairly trivial, as if it is violated, there must be $f+1$
honest nodes send _READY(h)_ with $\mathcal{H}(v)=h$, further indicating that
there must be $f+1$ honest nodes send _ECHO(h)_ with $\mathcal{H}(v)=h$, which
causes contradiction because honest nodes wouldn’t send _ECHO(h)_ with
$\mathcal{H}(v)=h$ if $Q(v)\neq 1$. ∎

Security of FIN-$\mathsf{MVBA}$-Q. We now prove that FIN-$\mathsf{MVBA}$-Q
realizes all four desired properties of $\mathsf{MVBA}$, i.e., termination,
agreement, external validity, and 1/2-quality.

######  Lemma 3.

If all honest nodes input some values satisfying $Q$, there exists $n-f$
avw$\mathsf{BRBC}$ instances, such that all honest nodes will eventually
$wr\text{-}deliver$ in them.

###### Proof.

We consider two opposing scenarios: 1$)$ No honest node invoke $abandon()$
before all honest nodes $wr\text{-}deliver$ in the same $n-f$
avw$\mathsf{BRBC}$ instances. 2$)$ Some honest node invoke $abandon()$ before
all honest nodes $wr\text{-}deliver$ in the same $n-f$ avw$\mathsf{BRBC}$
instances. For the former, _Validity_ property ensures all honest nodes
eventually $wr\text{-}deliver$ in $n-f$ avw$\mathsf{BRBC}$ instances leading
by honest nodes. For the latter, if some honest node $\mathcal{P}_{i}$ invoked
$abandon()$, the set $F_{i}$ of $\mathcal{P}_{i}$ must contain $n-f$ _true_
values. So at least $n-f$ avw$\mathsf{BRBC}$ instances have been
$wr\text{-}delivered$ by at least $f+1$ honest nodes. According to the
_Totality_ property of avw$\mathsf{BRBC}$, all honest nodes will eventually
$wr\text{-}deliver$ in these $n-f$ avw$\mathsf{BRBC}$ instances. ∎

######  Lemma 4.

If all honest nodes input some values satisfying $Q$, all honest nodes will
eventually step into iteration phase (line 13-28).

###### Proof.

According to Lemma 3, there exists $n-f$ avw$\mathsf{BRBC}$ instances (denoted
by ${HR}_{n-f}$), such that all honest nodes will eventually
$wr\text{-}deliver$ in them. So for every avw$\mathsf{BRBC}$ instance in
${HR}_{n-f}$, at least $n-f$ nodes will multicast _FIN_ message responding to
it. So all honest nodes will eventually receive $n-f$ _FIN_ messages from
distinct nodes for every avw$\mathsf{BRBC}$ instance in ${HR}_{n-f}$. Thus,
all honest nodes will step into iteration phase. ∎

######  Lemma 5.

Using $H$ to denote the first $f$+$1$ honest nodes invoking $abandon()$. For
any value $v$, its hash value $\mathcal{H}(v)$ won’t be $wr\text{-}delivered$
by any honest node, if responding _VAL(v)_ messages havn’t been received by
any of $H$ before $H$ all invoked $abandon()$.

###### Proof.

Since the reconstruction threshold of $Election()$ we use is $n-f$, at least
$f+1$ honest nodes will have invoked $abandon()$ before adversary can learn
the output of $Election()$. As shown in Figure 8, honest node will reject
sending _ECHO_ message for new received _VAL_ message after invoking
$abandon()$. Thus for any value $v$ whose responding _VAL_ messages havn’t
been received by any of $H$ before they invoked $abandon()$, it can’t gather
enough (i.e. $n-f$) _ECHO( $\mathcal{H}(v)$)_ messages. Without enough _ECHO(
$\mathcal{H}(v)$)_ messages, honest nodes won’t send responding _READY(
$\mathcal{H}(v)$)_ messages. So no honest node can receive sufficient ($n-f$)
_READY( $\mathcal{H}(v)$)_ messages to $wr\text{-}deliver$ $\mathcal{H}(v)$. ∎

######  Lemma 6.

Agreement If two honest nodes output $v$ and $v^{\prime}$, respectively, then
$v=v^{\prime}$.

###### Proof.

According to the _Agreement_ property of $RABA$, all honest nodes will have
same output for every possible running $RABA$ instances. So honest nodes will
end iteration in the same repeat round (denoted by $R_{halt}$). Let $k$ be the
result of $Election()$ of round $R_{halt}$. $RABA$ output 1 in round
$R_{halt}$, so that at least one honest node $raba$-$propose$ or
$raba$-$repropose$ 1\. So that at least one honest node has
$wr\text{-}delivered$ in $avw\mathsf{BRBC}_{k}$. According to Theorem1, all
honest nodes will $wr\text{-}deliver$ the same hash _h_. According to the
collision resistance property of hash function, there can’t be a different
value $v^{\prime}$ that $\mathcal{H}(v)=\mathcal{H}(v^{\prime})=h$. ∎

######  Lemma 7.

External-Validity. If an honest node outputs a value $v$, then $v$ is valid
w.r.t. $Q$, i.e., $Q(v)=1$;

###### Proof.

As shown in Lemma 6, if an honest node outputs a value $v$, at least one
honest node has $wr\text{-}delivered$ $\mathcal{H}(v)$. So at least $f+1$
honest nodes have received message _VALUE(v)_ in responding avw$\mathsf{BRBC}$
instance and sent _ECHO( $\mathcal{H}(v)$)_. Since the condition of sending
_ECHO( $\mathcal{H}(v)$)_ is $Q(v)=1$, the _External-Validity_ property holds.
∎

######  Lemma 8.

1/2-Quality. If an honest node outputs $v$, the probability that $v$ was input
by the adversary is at most 1/2.

###### Proof.

Let node $\mathcal{P}_{i}$ be the first honest node who step into iteration
phase. So at least $n-f$ values in set $F_{i}$ are true. Let $S$ denotes the
set of avw$\mathsf{BRBC}$ instances whose responding values in $F_{i}$ are
true. $H$ denotes the set of avw$\mathsf{BRBC}$ instances in $S$ who is
leading by honest nodes. $A$ denotes the set of avw$\mathsf{BRBC}$ instances
in $S$ who is leading by adversary. $R$ denotes the set of avw$\mathsf{BRBC}$
instances outside $S$. We now discuss the three possible cases of all possible
iterations. Let $k_{r}$ donotes the result of $Election()$ of the $r_{th}$
iteration.

  1. 1.

$k_{r}$ falls in $A$. In this case, at least $f+1$ honest nodes will input 1
for $RABA_{r}$. According to the Totality of avw$\mathsf{BRBC}$, all honest
nodes eventually $wr\text{-}deliver$ in $avw\mathsf{BRBC}_{k_{r}}$. And Lemma
4 shows that all honest nodes will step into iteration phase. So the Biased
termination property is met for $RABA_{r}$. The Biased validity property
ensures all honest nodes output 1 for $RABA_{r}$. Then FIN-MVBA will
eventually output the input value of $avw\mathsf{BRBC}_{k_{r}}$, which is
chosen by the adversary. Note that the probability of this scenario occurring
is at most 1/3.

  2. 2.

$k_{r}$ falls in $H$. Similar to the former case, FIN-MVBA will output the
input value of $avw\mathsf{BRBC}_{k_{r}}$, which is an input chosen by the
honest node. Note that the probability of this scenario occurring is at least
1/3.

  3. 3.

$k_{r}$ falls in $R$. In this case, Lemma 5 shows that no $new$ value can be
delivered after $f+1$ honest nodes invoking $abandon()$. So the attack we
describe in Appendix B won’t work here. Since there is no guarantee on how
many honest nodes $wr\text{-}deliver$ in $avw\mathsf{BRBC}_{k_{r}}$.
$RABA_{r}$ can output 1 or 0 (we will show why $RABA$ eventually outputs in
this scenario in Lemma 9). Outputing 1 won’t help even if
$avw\mathsf{BRBC}_{k_{r}}$ is leading by adversary, as it will reduce the
probability of case(1). So we only care about the scenarion where $RABA_{r}$
outputs 0, whose probability of occurrence is at most 1/3. And honest nodes
will jump to next iteration in this scenario.

Thus, during every possible iteration, $RABA$ outputs 1 with a probability of
up to 2/3. And once $RABA$ outputs 1, the probability that the output of
FIN-$\mathsf{MVBA}$-Q belong to the inputs of adversary is up to 1/2. ∎

######  Lemma 9.

Termination. If all honest nodes input some values satisfying $Q$, then each
honest node would output.

###### Proof.

According to Lemma 4, all honest nodes will eventually step into iteration
phase. So we only care about the Termination property of $RABA_{r}$ in
$r_{th}$ iteration. Let $k_{r}$ donote the result of $Election()$ of the
$r_{th}$ iteration. As shown in Lemma 8, $RABA_{r}$ has at least 2/3
probability to meet Biased termination property of $RABA$, and thus
terminates. We divide the remaining cases into two categories (case(3) in
Lemma 8). 1$)$ All honest nodes $raba$-$propose$ 0 and never
$raba$-$repropose$ 1 for $RABA_{r}$. Thus $RABA_{r}$ terminates due to
Unanimous termination property. 2$)$ Some honest nodes (less than $f+1$) have
$raba$-$propose$ or $raba$-$repropose$ 1 for $RABA_{r}$. Thus at least one
honest node has $wr\text{-}delivered$ in $avw\mathsf{BRBC}_{k_{r}}$. According
to the Totality property of avw$\mathsf{BRBC}$, all honest nodes will
eventually $wr\text{-}delivered$ in $avw\mathsf{BRBC}_{k_{r}}$, and thus
$raba$-$propose$ or $raba$-$repropose$ 1 for $RABA_{r}$. $RABA_{r}$ then
terminates due to the Biased termination property. ∎

##  Appendix D Proof of $\mathsf{FIN}\textrm{-}\mathsf{NG}$

Security of $\mathsf{FIN}\textrm{-}\mathsf{NG}$. We now prove that
$\mathsf{FIN}\textrm{-}\mathsf{NG}$ realizes agreement, total order, and
liveness, not only for these always-honest nodes but also for obliviously
recovered nodes (that was corrupted).

We first prove $\mathsf{FIN}\textrm{-}\mathsf{NG}$ is secure regarding those
always-honest nodes (note that there are at least $n-f$ nodes that are always
honest during the course of execution). Later we call these nodes always-
honest nodes or simply honest nodes.

######  Lemma 10.

If an honest node $\mathcal{P}_{i}$ output $0pt_{e+1}$, it can eventually
$wr\text{-}delivered$ in all w$\mathsf{BRBC}$ instances between $0pt_{e}$ and
$0pt_{e+1}$. Let set ${Digests}_{e+1}$ denote all the digests
$wr\text{-}delivered$ in those w$\mathsf{BRBC}$ instances, and let set
${TXs}_{e+1}$ denote the original data responding to ${Digests}_{e+1}$. Then
$\mathcal{P}_{i}$ can eventually obtain the whole ${TXs}_{e+1}$.

###### Proof.

According to Lemma 7, if $\mathcal{P}_{i}$ output $0pt_{e+1}$ from
${FIN\text{-}\mathsf{MVBA}\text{-}Q}_{e+1}$, then $Q(0pt_{e+1})=1$ for at
least $f+1$ honest nodes. Such that at least $f+1$ honest nodes have
$wr\text{-}delivered$ in all w$\mathsf{BRBC}$ instances between $0pt_{e}$ and
$0pt_{e}+1$. Due to the Totality property of w$\mathsf{BRBC}$, all honest
nodes will eventually $wr\text{-}delivered$ in all w$\mathsf{BRBC}$ instances
between $0pt_{e}$ and $0pt_{e}+1$. Besides, for every $h$
$wr\text{-}delivered$ in those w$\mathsf{BRBC}$ instances, at least $f+1$
honest nodes have received VAL(v) such that $\mathcal{H}(v)=h$. Thus
$\mathcal{P}_{i}$ can use the pulling mechanism described in SectionIV to
obtain missing data. ∎

######  Lemma 11.

$\mathsf{FIN}\textrm{-}\mathsf{NG}$ satisfies agreement, total order, and
liveness properties (regarding these always honest nodes) except with
negligible probability.

###### Proof.

Here we prove the three properties one by one (for those always-honest nodes).

For agreement: Suppose that one honest node $\mathcal{P}_{i}$ output a
transaction $tx$. $\mathcal{P}_{i}$ must have finished two
FIN-$\mathsf{MVBA}$-Q instances with output $0pt_{e}$ and $0pt_{e+1}$, such
that $tx$ belongs to some w$\mathsf{BRBC}$ instance between $0pt_{e}$ and
$0pt_{e+1}$. Due to the Agreement property of FIN-$\mathsf{MVBA}$-Q, if
another honest $\mathcal{P}_{j}$ output in the same two FIN-$\mathsf{MVBA}$-Q
instances, they output the same value. And $\mathcal{P}_{j}$ thus can obtain
some $tx^{\prime}$ responding to the same w$\mathsf{BRBC}$ instance due to
Lemma 10. And the collision resistance property of hash function together with
the Agreement property of w$\mathsf{BRBC}$ ensure that $tx=tx^{\prime}$.
Beside, we also need to prove that if node $\mathcal{P}_{i}$ outputs in some
FIN-$\mathsf{MVBA}$-Q instance, all honest nodes will eventually active this
instance. To prove it, we need the Liveness property of
$\mathsf{FIN}\textrm{-}\mathsf{NG}$ (we will prove it later). In the begining,
all honest nodes will active the first FIN-$\mathsf{MVBA}$-Q instance and
terminate then due to the Termination property of FIN-$\mathsf{MVBA}$-Q. An
the Liveness property of $\mathsf{FIN}\textrm{-}\mathsf{NG}$ ensures all
honest nodes will eventually generate a new input satisfying $Q$, and then
active the next FIN-$\mathsf{MVBA}$-Q instance. So that all honest nodes will
continuously active FIN-$\mathsf{MVBA}$-Q instances one by one. Such that if
$\mathcal{P}_{i}$ output a transaction $tx$, all honest nodes eventually
output $tx$.

For Total-order: In $\mathsf{FIN}\textrm{-}\mathsf{NG}$, all honest nodes
sequential participate in MVBA epoch by epoch, and in each MVBA, all honest
nodes output the same block, so the total-order is trivially hold.

For Liveness: One honest node $\mathcal{P}_{i}$ can start
$w\mathsf{BRBC}_{e+1}$ once it $wr\text{-}delivered$ in $w\mathsf{BRBC}_{e}$.
The Totality property ensures that all honest nodes eventuall
$wr\text{-}deliver$ in $w\mathsf{BRBC}_{e}$. All honest nodes thus can active
$w\mathsf{BRBC}_{e+1}$. So that $\mathcal{P}_{i}$ won’t be stuck. It also
implies at least $n-f$ parallel broadcasts can grow continuously since all
honest nodes won’t be stuck and can continuously active new w$\mathsf{BRBC}$
instance. Such that $\mathcal{P}_{i}$ can eventually generate a new input to
active a new FIN-$\mathsf{MVBA}$-Q instance.

According to the _1/2-Quality_ of FIN-$\mathsf{MVBA}$-Q, the input of honest
nodes can be outputted with a probability not less than 1/2, so even in the
worst case, once a FIN-$\mathsf{MVBA}$-Q instance decides to output the input
of an honest node $\mathcal{P}_{i}$, the broadcasted transactions proposed by
$\mathcal{P}_{i}$’s FIN-$\mathsf{MVBA}$-Q input will be decided as the final
consensus output. It further indicates that the adversary cannot censor the
input payloads that are broadcasted by honest nodes, because the probability
of FIN-$\mathsf{MVBA}$-Q outputting the input of some honest node is at least
1/2, which implies that after $k$ executions of FIN-$\mathsf{MVBA}$-Q, the
probability that a finished broadcast is not yet output would be at most
$(1/2)^{k}$, i.e., becomes negligible after sufficient repetition of
FIN-$\mathsf{MVBA}$-Q.

∎

##  Appendix E Proof of $\mathsf{JUMBO}$

Since the proofs of agreement and total order properties for $\mathsf{JUMBO}$
resembles those of $\mathsf{Dumbo}\textrm{-}\mathsf{NG}$, we only describe the
proof for liveness and fairness here.

W.l.o.g, we assume that nodes disseminate the same number of transactions in
each CBC instance. To enhance the fairness of $\mathsf{JUMBO}$, we adopt a
“speed limit” patch described in Section VI. For a better description, we do
some changes on variables. We use $\delta_{ij}$ to denote how many
transactions output by node $\mathcal{P}_{i}$ in $j$-th broadcast thread since
the height included by last block (from the local view of node
$\mathcal{P}_{i}$). Let $\delta_{i}$ to track the $(f+1)$-th smallest one of
all $\\{\delta_{ij}\\}_{j\in[n]}$. The variable $\delta_{low}$ tracks the
smallest on of all $\\{\delta_{i}\\}_{i\in[n]}$.

######  Lemma 12.

Any MVBA instance won’t be blocked forever. In other words, all honest nodes
will constantly output in MVBA thread.

###### Proof.

We proof this by mathematical induction:

  1. 1.

W.l.o.g, we assume epoch number start from 0. Now we proof that all honest
nodes will output in epoch 0. There are two possible cases:

    1. (a)

All honest nodes stay in epoch 0 (they all output 0 block). We recall the
condition of halt voting. $\mathcal{P}_{i}$ halts voting for the $j$-th
broadcast thread, if $\beta\cdot\delta_{ij}\geq\delta_{i}$ and
$\delta_{ij}>0$. So no honest nodes will halt voting in the first CBC instance
of each honest broadcast thread, as $\delta_{ij}=0$ in their views before
running that instance. Hence honest nodes will at least finish the first CBC
instance of each honest broadcast thread, and thus meet the condition of
generating a new input for a new MVBA instance. As a result, all honest nodes
will eventually active the first MVBA instance. Thus they will all output in
epoch 0 because of the _Termination_ property of MVBA.

    2. (b)

Some honest nodes stay in epoch 0, while the rest stay in higher epochs. We
use $N_{0}$ to denote the set of honest nodes who stay in epoch 0, and
$N_{high}$ to denotes the rest honest nodes. Now we proof that all nodes in
$N_{0}$ can finally generate new inputs for the first MVBA instance.

We assume that $\mathcal{P}_{i}\in N_{high}$ and the $j$-th broadcast thread
is honest. If $\mathcal{P}_{i}$’s outputs contain a higher height than 0 for
the $j$-th broadcast thread. It must have voted in the first CBC instance of
the $j$-th broadcast thread, and thus won’t block other honest nodes there. If
$\mathcal{P}_{i}$’s outputs don’t contain a higher height than 0 for the
$j$-th broadcast thread. Things are going similarly to the former case. As a
result, all honest nodes in $N_{0}$ can at least finish the first CBC instance
of each honest broadcast thread, and thus meet the condition of generating new
a input for a new MVBA instance. Thus they will all output in epoch 0 because
of the _Termination_ property of MVBA.

  2. 2.

W.l.o.g, we assume all honest nodes have output in epoch $e-1$. Now we proof
that all honest nodes will eventually output in epoch $e$. There are two
possible cases:

    1. (a)

All honest nodes stay in epoch $e$ (they output the same number of blocks). We
recall the condition of halt voting. $\mathcal{P}_{i}$ halts voting for the
$j$-th broadcast thread, if $\beta\cdot\delta_{ij}\geq\delta_{i}$ and
$\delta_{ij}>0$. So no honest nodes will halt voting in the first CBC instance
of each honest broadcast thread since last block, as $\delta_{ij}=0$ in their
views before running that instance. Hence honest nodes will at least finish
the first CBC instance of each honest broadcast thread, and thus meet the
condition of generating a new input for a new MVBA instance. As a result, all
honest nodes will eventually active the MVBA instance in epoch $e$. Thus they
will all output in epoch $e$ because of the _Termination_ property of MVBA.

    2. (b)

Some honest nodes stay in epoch $e$, while the rest stay in higher epochs. We
use $N_{low}$ to denote the set of honest nodes who stay in epoch $e$, and
$N_{high}$ to denotes the rest honest nodes. Now we proof that all nodes in
$N_{low}$ can finally generate new inputs for the MVBA instance of epoch $e$.

We use $h_{ej}$ to denote the height of the first CBC instance of the $j$-th
broadcast thread since the block committed in epoch $e-1$. We assume that
$\mathcal{P}_{i}\in N_{high}$ and the $j$-th broadcast thread is honest. If
$\mathcal{P}_{i}$’s outputs contain a higher height than $h_{ej}$ for the
$j$-th broadcast thread. It must have voted in the $h_{ej}$-th CBC instance,
and thus won’t block other honest nodes who havn’t output there. If
$\mathcal{P}_{i}$’s outputs don’t contain a higher height than $h_{ej}$ for
the $j$-th broadcast thread. Things are going similarly to the former case. As
a result, all honest nodes in $N_{low}$ can at least finish the first CBC
instance of each honest broadcast thread, and thus meet the condition of
generating new a input for a new MVBA instance. Thus they will all output in
epoch $e$ because of the _Termination_ property of MVBA.

Thus all honest nodes will constantly output in MVBA thread. ∎

######  Lemma 13.

All honest broadcast threads won’t be blocked forever.

###### Proof.

According the proof of 12, at least the first CBC instance of honest broadcast
threads in each epoch won’t be blocked. We assume that node $\mathcal{P}_{i}$
is honest, and $\mathcal{P}_{i}$ is in epoch $e$. Let $B_{1st}$ be the
transaction block $\mathcal{P}_{i}$ propose in his first CBC instance after
the block committed in epoch $e-1$. Then $B_{1st}$ is guaranteed to be output
by all honest nodes according to the $Validity$ property of CBC. Because all
honest nodes will constantly output in MVBA thread, all honest nodes will add
$B_{1st}$ into their inputs. According to the 1/2-Quality of MVBA, $B_{1st}$
will be output after expected constant number of MVBA instances. Thus
$\mathcal{P}_{i}$ will eventualy step into an epoch where $B_{1st}$ has been
output before. $\mathcal{P}_{i}$ can then start a new CBC instance, which is
guaranteed to be finished by all honesy nodes according to the previous
description. Thus $\mathcal{P}_{i}$ can constantly active new CBC instances.
So the $i$-th broadcast thread won’t be blocked forever, and so does the rest
honest broadcast threads. ∎

W.l.o.g, we assume that a transcation $tx$ is placed in honest node
$\mathcal{P}_{i}$’s input buffer. According to Lemma-13, $\mathcal{P}_{i}$
will eventually propose it in one of his CBC instance. Thus $tx$ will be
output by all honest nodes in that CBC instance. According the proof of 12,
$tx$ will finally be contained in all MVBA input from honest nodes. After
that, $tx$ will be output by consensus in expected constant number of MVBA
instances. Thus the $Liveness$ property for $\mathsf{JUMBO}$ is held.

##  Appendix F Choice of Experimental Environment

We adopt a popular appraoch of using Linux TC to emulate WAN networks among
EC2 instances in the same region [21, 80]. Namley, we apply manually-
controlled delay and bandwidth throttling at each node. In contrast, a few
recent studies [12, 13, 14, 27, 18, 16, 20, 31] adopted a different way to set
up benchmarking by evenly distributing EC2 instances at different AWS regions.
The reasons behind our benchmarking choice are as follows:

  1. 1.

Economic affordability. The price of data transfer between different regions
for Amazon’s EC2 platform is non-negligible, which is around $\$$0.02 per
GigaByte. So if we evaluate protocol at the scale of 256 nodes and 1 Gbps
bandwidth, the cost on data transfer is around $\$$38.4 per minute, as opposed
to $\$$1.5 for computation. If we deploy all nodes at the same region, we can
conduct the same experiments for 25 times longer at the same cost, because
data transfer within the same region is free. Otherwise, if we distribute 256
nodes across a dozen of regions, the cost would become completely unafforable.

  2. 2.

Result reproducibility. The network bandwidth and latency between different
regions vary greatly over time (due to changes of AWS’ network
infrastructure), making it difficult for us to maintain the same network
conditions when repeating the same experiments.

  3. 3.

Better emulation of WANs. If we deploy several hundred of nodes in a dozen of
different regions, then many nodes will inevitably be deployed in the same
region. The network conditions between nodes within the same region are
extremely good, typically with only a few milliseconds of latency and 10 Gbps
bandwidth, which is completely mismatched with the WAN environment. If we
deploy all nodes to the same region, we can use tools like TC to precisely
control the bandwidth and latency of each connection, in order to better
simulate the WAN environment. Additionally, we can more accurately simulate
network fluctuations and other asynchronous adversarial attack behaviors.

  4. 4.

Precise control of bandwidth. The last benefit of our benchmarking choice is
that we can precisely control the bandwidth during each test. This allows us
compare our throughput to line-rate speed to understand the actual bandwidth
utilization of our different protocols.