# Self-Ordering of Buckling, Bending, Bumping Beams

Arman Guerra  Department of Mechanical Engineering, Boston University, Boston,
Massachusetts 02215, USA  Anja Slim  School of Mathematics, Monash University,
Victoria 3800, Australia  School of Earth, Atmosphere and Environment, Monash
University, Victoria 3800, Australia  Douglas P. Holmes  Department of
Mechanical Engineering, Boston University, Boston, Massachusetts 02215, USA
Ousmane Kodio<EMAIL_ADDRESS>Department of Mathematics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA

###### Abstract

A collection of thin structures buckle, bend, and bump into each-other when
confined. This contact can lead to the formation of patterns: hair will self-
organize in curls; DNA strands will layer into cell nuclei; paper, when
crumpled, will fold in on itself, forming a maze of interleaved sheets. This
pattern formation changes how densely the structures can pack, as well as the
mechanical properties of the system. How and when these patterns form, as well
as the force required to pack these structures is not currently understood.
Here we study the emergence of order in a canonical example of packing in
slender-structures, i.e. a system of parallel growing elastic beams. Using
experiments, simulations, and simple theory from statistical mechanics, we
predict the amount of growth (or, equivalently, the amount of compression) of
the beams that will guarantee a global system order, which depends only on the
initial geometry of the system. Furthermore, we find that the compressive
stiffness and stored bending energy of this meta-material is directly
proportional to the number of beams that are geometrically frustrated at any
given point. We expect these results to elucidate the mechanisms leading to
pattern formation in these kinds of systems, and to provide a new mechanical
meta-material, with a tunable resistance to compressive force.

When thin structures pack, there is a competition between elasticity, which
often encourages pattern formation and densification, and geometric
constraints. For example, paper, when crumpled into a ball, forms complex
three-dimensional swirls Cambou and Menon (2011); Deboeuf et al. (2013), and
DNA strands inserted into cell nuclei fold and pack into layers Smith et al.
(2001); Kindt et al. (2001). In some cases, these densification processes are
resisted by friction Alarcón et al. (2016); Alben (2022) and geometrical
incompatibilities in the deformation of the materials Andrejevic et al.
(2021); Domokos et al. (2000); Roman and Pocheau (1999); Pocheau and Roman
(2004). The formation of patterns has been studied thoroughly in cases where
thin structures are adhered to a substrate Biot (1937); Dillard et al. (2018);
Kodio et al. (2017); Oshri et al. (2018), sheets are constrained in a ring
Boué et al. (2006); Boué and Katzav (2007); Adda-Bedia et al. (2010); Alben
(2022), and rods are inserted into a container Donato et al. (2003); Donato
and Gomes (2007); Stoop et al. (2011); Vetter et al. (2014, 2015). However,
the question of to what degree the rods and sheets will order themselves in
these complex and random packing processes is still open.

In structured arrangements of elastic beams, the competition between order and
geometric frustration has been used to great effect in the design of materials
with novel and programmed properties Florijn et al. (2014); Paulose et al.
(2015); Frenzel et al. (2016); Bertoldi et al. (2017). There are many models
in statistical mechanics to rationalize the emergence of order, for example
the Ising model Kang et al. (2014) for ferromagnetism and the Potts model
Feynman (2018) for percolation, or even in the case of thin elastic structures
Bahri et al. (2022); Hanakata et al. (2022); Nelson et al. (2004). However,
these models are insufficient to capture the ordering of beams because of the
contact arising between them and the difficulty of finding the interacting
forces between adjacent elements. For example, consider a simple 1-D version
of the ordering of packed beams, the gills of a mushroom (see Figure 1a). If
the mushroom dries and shrinks, the gills will at first buckle, and then bump
into each other. Reminiscent of the 1-D Ising model for magnetism, the ground
state of this system occurs when all gills point in the same direction (shown
in an analogous experiment in Figure 1b, right). However, here there exists a
hierarchy of disordered meta-stable states in the shallow-post-buckling
regime, where the gills self-organize into “clumps,” and leave “holes” where
they have separated (Figure 1a, middle).

Figure 1:  (a) When an oyster mushroom dries, the gills become compressed
along their length, causing them to buckle (middle) and then align (right).
(b) Experimental observations of a similar phenomenon in a system of slender
parallel plates compressed in an Instron. (c) A numerical experiment of the
same system.

In this Letter, we consider a simplified version of this system: an array of
$N$ parallel elastic beams confined to a vertical space $\ell_{s}$ that is
shorter than their length $\ell$, and equally spaced inside of a box of width
$W$, such that the distance between any two beams is $W/N$ (Figure 1c). We
characterize the control parameter of the system to be the confinement factor
$\Gamma=\ell/\ell_{s}$. If the beams are initially perturbed in random
directions, we observe behaviour reminiscent of a phase transition, where the
initial disordered state (beams buckled in random directions) will gradually
decay to the fully-ordered ground state (all beams aligned in one direction)
as $\Gamma$ increases. We therefore ask the following two questions: First,
when and how much will the beams align; can we predict how order will emerge?
Second, how does the mechanical response of the system depend on the degree of
order and its emergent topology; conversely could we use this emergent
topology in the design of tunable-stiffness materials?

To investigate this alignment experimentally, we built a stiff acrylic two-
piece mount that served to clamp a large number of slender elastic plates at
two of their opposite edges. We inserted sheets of PVS (thickness $h=1.5$ mm,
number $N=26$) into the mounts, and compressed the system uniformly with an
Instron, which allowed us to measure $\Gamma$, as well as the force response
of the arrangement of beams (Figure 1b). We then perform non-thermal quenches
on this ensemble of beams – that is, we iso-statically compress the beams many
times, each time manually biasing each beam by hand to initially buckle to the
right or left based on a random coin-flip. Just as we see in the case of the
mushroom (Figure 1a), any adjacent beams buckling towards each other will
eventually make contact and form clumps (Figure 1b, middle). After enough
confinement, these clumps become unstable and decompose, and the beams
eventually all point in one direction (Figure 1b, right).

We parametrize the direction of the buckling of each beam using what we call
the “tropism” (analogous to the magnetism in the Ising model) where $T_{i}=0$
when beam $i$ is un-buckled, and $T_{i}=+1$ ($-1$) if it is buckled to the
right (left), as shown in Figure 2a. To account for beams that are no longer
in the first buckling mode because of contact with other adjacent beams or
with the walls, we consider that a beam is buckled to the right (left) if the
portion of the beam halfway up the box is farther to the right (left) than its
ends. We can average the behavior of all beams into an overall system tropism

$\bar{T}=\left|\frac{1}{N}\sum_{i=1}^{N}T_{i}\right|.$ (1)

Note that $\bar{T}\approx 0$ when the beams are randomly directed, and
$\bar{T}\approx 1$ when the beams are all aligned (Figure 1b, right). We plot
$\bar{T}$ for increasing $\Gamma$ for a fixed beam spacing, number, thickness,
and initial length in Figure 2b ( ), and find indeed a gradual increase in the
average order of the system. We can compare this with the behavior of two
beams buckling towards each other (Figure 2a ii and iii) and find that for the
many-beam case, the clumps remain stable (and therefore $\bar{T}<1$) at much
higher $\Gamma$ for the same normalized beam spacing $d/\ell_{s}$ (Figure 1c).

Figure 2:  The emergence of order for (a) two beams and (b $\&$ c) many beams.
(a i.) The tropism $T_{i}$ is $+1$ ($-1$) for a beam buckled to the right
(left), which we indicate in blue (orange). The average tropism $\bar{T}$
(Equation 1) quantifies the degree of system order. (a ii.) Experiments
(green) overlaid with simulations (blue and orange) of two beams which buckle
towards each other. There are two meta-stable “clumped” $\bar{T}=0$ states
(vertically and rotationally symmetric) however when $\Gamma>\Gamma_{c}$,
$\bar{T}=1$. (a iii.) $\Gamma_{c}$ increases with the normalized distance
between two beams $d/\ell_{s}$ (experiments – , simulations – , model – black
line (SI)) (b) $\bar{T}$ for experiments and simulations of many beams
($N=26$, $W=140$ mm, $\ell_{s}=54$ mm, $h=1.54$ mm, average $d=5.4$ mm, number
of ensemble measurements: experiment – 25, simulation – 100) (c) $\bar{T}$
increases with $\Gamma$ with a rate highly dependent on $N$, $d$, $W$, $h$,
and $\ell_{s}$.

To investigate how the observed behavior depends on the friction between the
beams, the boundary conditions, the variation in distances between adjacent
beams, and any imperfections in the experimental set-up (including
electrostatic interactions between neighboring plates and gravity), we
supplemented our physical experiments with numerical ones using the Large-
scale Atomic/Molecular Massively Parallel Simulator (LAMMPS Thompson et al.
(2022), with visualizations in Ovito Stukowski (2009)) using a similar
protocol as Guerra and Holmes (2021), which is meant to replicate the physics
of elastic beams. This consisted of simulating a large number of point
particles with diameter $h$ equal to the thickness of the beam with nearest-
neighbor potentials $U=Y(r-r_{0})^{2}+B(1+\cos{\theta})$ where $Y=E_{l}\pi
h/8$ is the stretching modulus, $B=E_{l}\pi h^{3}/64$ is the bending modulus,
$E_{l}$ is the Young’s modulus of the beams, $r$ and $r_{0}$ are the current
and equilibrium nearest-neighbor distances respectively, and $\theta$ is the
angle between any three adjacent beam particles. We note that $r_{0}$ is not
necessarily equal to $h$, in fact in all simulations we use $r_{0}=h/4$ to
allow the beams to slide smoothly against one another.

In Figure 2b, we replicate the geometry of our physical experiments and plot
the ensemble behavior of $\bar{T}$ for these simulations ( ). We include
simulations with varying inter-beam friction, where simulations with
coefficient of sliding friction set to $\mu_{s}=0$ (frictionless, ),
$\mu_{s}=1$ (rubber-like, ), $\mu_{s}=2$ (highly frictional, ). The effect of
boundary conditions is investigated by performing simulations with $\mu_{s}=0$
in which we vary the beam end boundary conditions (pin-pin, ), and the
vertical box edge boundary conditions (periodic, ). Additionally, we performed
simulations where the distances between the beams was uniformly randomly
distributed with a lower bound of $1.2h$ and an upper bound such that the
average distance between the beams is the same as the model experiment ( ). We
find good agreement between experiments and simulations, and furthermore the
tropism $\bar{T}$ is statistically independent of the degree of friction,
boundary conditions, and beam spacing variability. Since the tropism does not
depend on any of these choices, we can select one simulation protocol and vary
the geometric parameters, with the expectation that the results apply broadly.
Hence, without loss of generality, we perform simulations with clamped-
clamped, evenly spaced beams without friction in a periodic box. We vary the
beam density as well as the number of beams, and plot $\bar{T}$ as a function
of $\Gamma$ in Figure 2c, and find that $\bar{T}$ strongly depends on the
geometric parameters of the box.

In Figure 3a we show the evolution of a specific set of LAMMPS simulated beams
as $\Gamma$ increases. Beams that are in contact with a neighbor are colored
red and non-contacting beams are colored by their tropism ($1\rightarrow$
blue, $-1\rightarrow$ orange). Adjacent beams that buckle toward each other
form clumps, while beams that buckle away form holes. We note that these
clumps also exist in the case where the beams are randomly distributed in the
box; they are not just a feature of the evenly spaced beams. We see that as
$\Gamma$ increases, the number of clumps and holes decrease, leading to
increasingly aligned beams. We can uncover the mechanism by which the beams
align by zooming in on the instance shown in Figure 3b. In Figure 3bi we see
that the clump and hole are separated by one beam that is not in contact with
the clump, and that the force on the clump is balanced, that is, there are as
many beams on the right buckled towards the left as there are beams on the
left buckled towards the right. In Figure 3bii, the beam on the right of the
clump has made contact with the clump, but so has a beam on the left of the
clump, so again the clump is balanced. In Figure 3biii however, an additional
beam makes contact with the left of the clump, but is not balanced by the beam
that would have otherwise been on the right of the clump to balance it. The
force on the clump is therefore unbalanced, and the clump rushes into the
hole, leading to clump-hole annihilation, and the alignment of all the beams
(Figure 3biv).

Figure 3:  Using the mechanism behind alignment, we can predict the tropism.
(a) Simulations at increasing $\Gamma$ from top to bottom. Beams which are
part of a clump are colored purple, and all others are colored by their
tropism ($T_{i}=1\rightarrow$ blue, $T_{i}=-1\rightarrow$ orange). (b) A step-
by-step illustration of clump-hole annihilation: (i) before the clump and hole
meet, the lateral force on the clump is balanced by beams on both sides. (ii)
When the final beam between the hole and the clump meets the clump, the force
on the clump is still balanced. (iii) The clump becomes unstable when it is
touched by a beam on the left which does not have a counterpart on the hole
side. (iv) The clump quickly moves into the hole, and all the beams become
aligned. (c) Illustration of the parameters in our mathematical model. (d) If
we compare the sum of the distances defined in (c) with the maximum possible
distance between a clump and a hole, the ensemble value of $\bar{T}$ falls
onto a single curve (red). Error bars are the same as in Figure 2c, left off
for clarity.

We can use this mechanism to predict $\bar{T}$ as a function of $\Gamma$. We
know that $\bar{T}=1$ when the final clump “meets” and annihilates with the
final hole. For simplicity we will consider the case where the edges of the
box are periodic, and as such the maximum possible distance between the final
clump-hole pair is $W/2$. What does it mean for a clump and a hole to meet? A
clump and a hole meet when all of the beams between them are in contact. In
this case, the space between the center of the hole and the center of the
clump comprises three parts as illustrated in Figure 3c. The first is the
horizontal extent of the hole, which we call $d_{h}$. If we approximate the
shape of a beam as a triangle, using the Pythagorean theorem,
$d_{h}\approx\frac{1}{2}\ell_{s}\sqrt{\Gamma^{2}-1}$. The second is the
horizontal deflection of the center beam of the clump, which we call $d_{c}$.
This has the approximate shape of a mode-2 buckled elastica, so we can
approximate this as $d_{c}\approx d_{h}/2$. The third is the sum of the
thicknesses of the beams between the clump and the hole. The number of such
beams $N_{b}$ stays the same as the beams grow, but the horizontal thickness
$\widetilde{h}$ of the beams change, as shown in Figure 3c. If we keep our
triangular approximation for the shape of the beams, we get that
$\widetilde{h}\approx h\Gamma$ and therefore the additional space that each
beam takes up is $\widetilde{h}-h=h(\Gamma-1)$.

Since the largest possible distance between a clump and a hole is $W/2$,
therefore the maximum amount of empty space between the clump and hole is the
distance between the clump and the hole minus the thickness of the beams
between them, or $W/2-Nh/2$. We would expect that when the beams grow and this
space is taken up, that is, $d_{h}+d_{c}+N_{b}h(\Gamma-1)=W/2-Nh/2$, there
will be nothing separating the farthest possible clump and hole, they will
make contact and annihilate, and $\bar{T}\rightarrow 1$. In the case where the
last clump and hole are as far apart as they can be, $N_{b}=N/2$, so the
fraction of the horizontal space between the clump and hole taken up by the
beams, which we will call the “porosity” $\alpha$, is
$\alpha=[d_{h}+d_{c}+N(\widetilde{h}-h)/2]/(W/2-Nh/2)$. In Figure 3d, we plot
$\bar{T}$ against this porosity and find that, as we expect, for all
geometries, $\bar{T}\approx 1$ when $\alpha=1$. More than that however, it
seems that the tropism is approximately equal to $\alpha$, that is,

$\bar{T}\approx\alpha=\frac{\frac{1}{2}(\Gamma-1)hN+\frac{3}{4}\ell_{s}\sqrt{\Gamma^{2}-1}}{W/2-Nh/2}.$
(2)

This comes from the fact that as the beams grow, the number of clumps and
holes which make contact and annihilate is proportional to the space that the
beams take up. We note that, as can be seen from Figure 3d, even when
$\alpha=1$, there are some cases where $\bar{T}$ is slightly less than 1. We
expect that this comes from errors in our approximations of $d_{h}$, $d_{c}$,
and $\widetilde{h}$.

Earlier we showed that $\bar{T}$ does not significantly depend on the boundary
conditions of the system, the friction between beams, or the variation in the
distance between beams. We can now see why: the boundary conditions of the
beams do not significantly affect the validity of our approximations for the
maximum beam deflection, and the box walls and variability in inter-beam
distance affect neither the average distance nor (in the case of an ensemble
average of experiments) the average number of beams between the holes and
clumps. In our simulations we also did not see an effect of friction on the
shape of the beams in the clumps, indicating that the beams in this specific
geometry do not need to slide past each other for order to emerge.

Figure 4:  The compressive stiffness of the system is a function of the degree
of order. (a) Snapshots of a simulation at $\Gamma=1.18$ (i) and $\Gamma=1.38$
(ii), superimposed with the vertical compressive force ($F_{i}$, blue) and
bending energy ($U_{bi}$, red) of each beam normalized by the force and
bending energy of a beam which is not part of a clump ($F_{1}$, $U_{b1}$),
showing that the clumps have a higher resistance to compression, and a larger
average curvature than the non-clumped regions. (b) The total compressive
force and bending energy ($F$, $U_{b}$) of the same system of beams as a
function of $\Gamma$, normalized by the compressive force and bending energy
of the system if all of the beams were buckled in the same direction. The
circles and squares are experiments and simulations with the same initial
conditions, and the diamonds are the predictions from our approximated model
(Equation 3). Vertical lines show places where a clump and hole annihilate.
(c) Normalized force and bending energy against $1+N_{t}/N$ together with our
predictions from Equation 3 (simulation $U_{b}$ – squares, simulation $F$ –
circles, experiment $F$ – diamonds).

Now that we understand when the beams will become ordered, we turn to our
second question: how do the beams respond to the imposed compression when they
are not ordered? This is analogous to why it requires less force to confine
paper to a target volume through folding than through crumpling Deboeuf et al.
(2013): a larger degree of geometrical frustration in thin structures often
leads to more stored energy Sadoc and Mosseri (1999). We can observe this
directly in our system. In Figure 4 we plot the normalized compressive force
$\widetilde{F}_{i}=F_{i}/F_{1}$ of each beam in a simulation, where $F_{i}$ is
the compressive force applied to beam $i$, and $F_{1}$ is the compressive
force applied to a single beam with the same boundary conditions, compressed
to the same $\Gamma$ and no neighboring beams. We simultaneously define and
plot the normalized bending energy $\widetilde{U}_{bi}=U_{bi}/U_{b1}$ in the
same way, where $U_{bi}$ is the bending energy of beam $i$, and $U_{b1}$ is
the corresponding bending energy of the beam from which we derive $F_{1}$. We
superimpose the normalized compressive force graph onto the simulated
configurations and colored beam particles by their bending energy. For beams
that are not in clumps, and therefore have no contacts,
$\widetilde{F}_{i}=\widetilde{U}_{bi}=1$. In contrast, beams that are in
contact with other beams in a clump and are geometrically frustrated Sadoc and
Mosseri (1999) have a higher $\widetilde{F}_{i}$ and $\widetilde{U}_{bi}$, as
they cannot find their lowest-energy state. These results indicate that our
system is behaving like a mechanical meta-material, an object that gets its
properties not only from the material that it is made of, but also its
internal geometry Bertoldi et al. (2017).

Previously, we noted that the beam at the center of a clump has the
approximate shape of a mode-2 elastica. We would expect that $U_{b}\propto n$
where $n$ is the mode number of the beam (since doubling the mode number on
average doubles the curvature in the beam), so for the central beam in the
clump we might expect that $\widetilde{U}_{bi}\approx 2$. This approximation
is confirmed in Figure 4a. Even for very large $\Gamma$ we find that the
central beam in any clump has a normalized bending energy of approximately 2.
To estimate the normalized compressive force of a beam ($F_{i}$) at the center
of a clump, we perform additional simulations of a single beam, where the
slope at the vertical center of the beam was forced to match the slope derived
from our triangular approximation of the beam shape. In these simulations we
find that $\widetilde{F}_{i}\approx 3$ for the approximated shape. In Figure
4a, we find that the maximum value of $\widetilde{F}_{i}$ is indeed
approximately $3$ for most of the simulation, but increases for very large
$\Gamma$, which we expect comes from a deviation of the true beam shape from
our approximation.

From the graphs in Figure 4a we see that the normalized bending energy and
compressive force of the beams in the clump seems to increase linearly from
$\approx 1$ for the beams at the edge of the clump, to the maximum value at
the clump center. If we use our earlier approximations
($\widetilde{U}_{bi}\approx 2$ and $\widetilde{F}_{i}\approx 3$), we might
expect that, on average, a beam which is in a clump has
$\widetilde{U}_{bi}\approx 1.5$ and $\widetilde{F}_{i}\approx 2$. Hence, we
would expect that the total force and bending energy in our arrangement of
beams is

$\begin{split}&F=F_{1}(N+N_{c}),\\\ &U_{b}=U_{b1}(N+N_{c}/2),\end{split}$ (3)

where $N_{c}$ is the number of beams that are in a clump. In Figure 4b, we
plot the total normalized force $F/(NF_{1})$ ( ) and bending energy
$U_{b}/(NU_{b1})$ ( ) of the specific simulation run pictured in Figure 4a,
along with their estimated values from the number of clumped beams as given by
Equation 3 and find good agreement, except at high $\Gamma$, when the
normalized force estimate starts to fail as we expected. In Figure 4c, we plot
$F/(NF_{1})$ (circles) and $U_{b}/(NU_{b1})$ (squares) against $1+N_{c}/N$ and
find that for a wide variety of box geometries, $U_{b}/(NU_{b1})$ collapses to
$(1+N_{c}/N)/2$ for all data, and $F/(NF_{1})$ collapses to $1+N_{c}/N$ for
all data except for that at very high $\Gamma$.

In this work we have studied a system of many parallel, growing beams clamped
at their ends and confined in a box. We have found that, as they grow, clumps
of beams that form when the beams first buckle eventually meet with holes
where the beams have separated, and the beams begin to order themselves. The
degree of order (which we parameterize as the tropism $\bar{T}$) is
proportional to the fraction of the initially empty horizontal space that the
beams take up as they grow. Our results indicate that, for parallel growing
beams constrained in a rectangular box, at high enough $\Gamma$,
$\bar{T}\rightarrow 1$, that is, we had no results indicating that the beams
would “jam,” or lock in place. We expect that with a different geometry,
however, these geometrically frustrated states could become locked in place by
the inter-beam friction, similar to interleaved books Alarcón et al. (2016),
bird nests Weiner et al. (2020), and a beam injected into a tube van Adrichem
and Newman (1993); Gao and Miska (2009); Miller et al. (2015). We also probed
the mechanical properties, and found that the compressive stiffness of the
metamaterial made up of these buckling beams is proportional to the number of
beams in a clump. This provides a system wherein the stiffness can be manually
or automatically tuned by forcing beams into our out of a clump while the
metamaterial is under load. We might expect that, similar to the tropism,
friction could play a major role in the mechanical properties of this kind of
system, as it does in many cases of slender objects in contact Poincloux et
al. (2021), creating an additional route for potentially novel
functionalities.

We have, so far, not considered beams which have any intrinsic thermal motion
Nelson et al. (2004), adhesion Elder et al. (2020); Cranford (2022), or long-
range potentials, all of which would affect the ordering, and mechanical
properties of the system. We also note that, as the beams are compressed, a
transverse force (the compression) turns into a longitudinal transfer, namely
the redirection of the beams. This could provide a method to redirect and
control mechanical waves Wu et al. (2020); Packo et al. (2021).

##  I Acknowledgements

We thank Abigail Plummer, Harold Park, and Dominic Vella for helpful
discussion. We also gratefully acknowledge the financial support from DARPA
(#HR00111810004) and from NSF CMMI–CAREER through Mechanics of Materials and
Structures (#1454153), and the computing resources of the Boston University
Shared Computing Cluster.

##  II SI

###  II.1 Critical Confinement Factor of Two Contacting Beams

In Figure 2 we show simulations and experiments of two beams which come into
contact and then, at a critical confinement factor ($\Gamma_{c}$), one beam
wins, and the other snaps-through such that both beams are pointing in the
same direction. We might expect that when the space between the two beams is
taken up by the extension of a beam ($d_{h}$) and the thickness of the beams
$\widetilde{h}$, the beams will align. In other words, the beams will align
when the spacing between the beams is
$d_{h}+\widetilde{h}=\frac{1}{2}\ell_{s}\sqrt{\Gamma_{c}^{2}-1}+h\Gamma_{c}$.
We plot this in Figure 2a iii. (black line) and find good agreement with
simulated ( ) and experimental ( ) values.

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We also used that, for all $(\mathbf{x},\mathbf{y})\in G_{+1}\cap R\cap
R_{+1}\cap R_{-1}^{\mathsf{c}}\cap\Delta^{\mathsf{c}}$,
$Q(\mathbf{x},\mathbf{N}_{+1}(\mathbf{x}))>0$ and
$Q(\mathbf{y},\mathbf{N}_{-1}(\mathbf{y}))>0$ according to the properties of
the directional neighbourhoods $\\{\mathbf{N}_{+1}(\mathbf{x})\\}$ and
$\\{\mathbf{N}_{-1}(\mathbf{x})\\}$, implying that $G_{+1}\cap R\cap
R_{+1}\cap R_{-1}^{\mathsf{c}}\cap\Delta^{\mathsf{c}}=G_{-1}\cap R\cap
R_{+1}\cap R_{-1}^{\mathsf{c}}\cap\Delta^{\mathsf{c}}$. Finally, we used the
definition of $\mu_{\mathrm{rev.}}^{T}$. In the fifth and sixth equalities, we
used symmetrical arguments as above by recalling that
$(\mathbf{x},\mathbf{y})\in R_{\nu}$ if and only if
$(\mathbf{y},\mathbf{x})\in R_{-\nu}$. In the final equality, we used the
definitions of $\mu_{-1}$ and $T_{-1}$. This concludes the proof. ∎

###### Proof of Theorem 2.

With some abuse of notation, let, for $\mathbf{x}\in\bm{\mathcal{X}}$,
$R_{\mathbf{x}}:=\\{\mathbf{y}\in\bm{\mathcal{X}}:(\mathbf{x},\mathbf{y})\in
R\\}$.

Let us first consider the case where
$\mathbf{x}\in\mathring{\bm{\mathcal{X}}}$. For $A\in\bm{\mathsf{X}}$, we have
that

$\displaystyle
2P_{\mathrm{rev.}}(\mathbf{x},A\setminus\\{\mathbf{x}\\})=2\int_{A\setminus\\{\mathbf{x}\\}}Q_{\mathrm{rev.}}(\mathbf{x},\mathrm{d}\mathbf{y})\,\alpha_{\mathrm{rev.}}(\mathbf{x},\mathbf{y})$
$\displaystyle=\sum_{\nu\in\\{-1,+1\\}}\int_{A\setminus\\{\mathbf{x}\\}}Q_{\nu}(\mathbf{x},\mathrm{d}\mathbf{y})\,\alpha_{\mathrm{rev.}}(\mathbf{x},\mathbf{y})$
$\displaystyle=\sum_{\nu\in\\{-1,+1\\}}\int_{A\cap
R_{\mathbf{x}}\cap\mathbf{N}_{\nu}(\mathbf{x})\cap\mathbf{N}_{-\nu}(\mathbf{x})^{\mathsf{c}}\setminus\\{\mathbf{x}\\}}Q_{\nu}(\mathbf{x},\mathrm{d}\mathbf{y})\,\alpha_{\mathrm{rev.}}(\mathbf{x},\mathbf{y})$
$\displaystyle=\sum_{\nu\in\\{-1,+1\\}}\int_{A\cap
R_{\mathbf{x}}\cap\mathbf{N}_{\nu}(\mathbf{x})\cap\mathbf{N}_{-\nu}(\mathbf{x})^{\mathsf{c}}\setminus\\{\mathbf{x}\\}}Q(\mathbf{x},\mathrm{d}\mathbf{y})\,\frac{\alpha_{\mathrm{rev.}}(\mathbf{x},\mathbf{y})}{Q(\mathbf{x},\mathbf{N}_{\nu}(\mathbf{x}))},$

using the definitions of $P_{\mathrm{rev.}}$ and
$Q_{\mathrm{rev.}}(\mathbf{x},\cdot\,)$, that, for any
$\mathbf{x}\in\mathring{\bm{\mathcal{X}}}$, $Q_{\nu}(\mathbf{x},\cdot\,)$
assigns null measure outside
$\mathbf{N}_{\nu}(\mathbf{x})\cap\mathbf{N}_{-\nu}(\mathbf{x})^{\mathsf{c}}\setminus\\{\mathbf{x}\\}$
(see Section 4.2) with an argument similar as in (12), and that
$\alpha_{\mathrm{rev.}}(\mathbf{x},\mathbf{y})=0$ for all
$(\mathbf{x},\mathbf{y})\notin R$.

For all $(\mathbf{x},\mathbf{y})$ with
$\mathbf{x}\in\mathring{\bm{\mathcal{X}}}$ and $\mathbf{y}\in
R_{\mathbf{x}}\cap\mathbf{N}_{\nu}(\mathbf{x})\cap\mathbf{N}_{-\nu}(\mathbf{x})^{\mathsf{c}}\setminus\\{\mathbf{x}\\}$,

$\displaystyle\frac{\alpha_{\mathrm{rev.}}(\mathbf{x},\mathbf{y})}{Q(\mathbf{x},\mathbf{N}_{\nu}(\mathbf{x}))}=\frac{\varphi(r(\mathbf{x},\mathbf{y})\beta(\mathbf{x},\mathbf{y}))}{Q(\mathbf{x},\mathbf{N}_{\nu}(\mathbf{x}))}$
$\displaystyle\geq\frac{\varphi(r(\mathbf{x},\mathbf{y})(1\wedge\beta(\mathbf{x},\mathbf{y})))}{Q(\mathbf{x},\mathbf{N}_{\nu}(\mathbf{x}))}$
$\displaystyle\geq\frac{\varphi(r(\mathbf{x},\mathbf{y}))(1\wedge\beta(\mathbf{x},\mathbf{y}))}{Q(\mathbf{x},\mathbf{N}_{\nu}(\mathbf{x}))}$
$\displaystyle=\varphi(r(\mathbf{x},\mathbf{y}))\left(\frac{1}{Q(\mathbf{x},\mathbf{N}_{\nu}(\mathbf{x}))}\wedge\frac{1}{Q(\mathbf{y},\mathbf{N}_{-\nu}(\mathbf{y}))}\right)$
$\displaystyle\geq\varphi(r(\mathbf{x},\mathbf{y}))=\alpha(\mathbf{x},\mathbf{y}),$

using the definitions of $\alpha_{\mathrm{rev.}}$ and $\beta$, and Results 1
and 6 of Lemma 1.

Therefore,

$\displaystyle 2P_{\mathrm{rev.}}(\mathbf{x},A\setminus\\{\mathbf{x}\\})$
$\displaystyle\geq\sum_{\nu\in\\{-1,+1\\}}\int_{A\cap
R_{\mathbf{x}}\cap\mathbf{N}_{\nu}(\mathbf{x})\cap\mathbf{N}_{-\nu}(\mathbf{x})^{\mathsf{c}}\setminus\\{\mathbf{x}\\}}Q(\mathbf{x},\mathrm{d}\mathbf{y})\,\alpha(\mathbf{x},\mathbf{y})$
$\displaystyle=\int_{A\cap
R_{\mathbf{x}}\setminus\\{\mathbf{x}\\}}\left(\mathds{1}(\mathbf{y}\in\mathbf{N}_{-1}(\mathbf{x})\cap\mathbf{N}_{+1}(\mathbf{x})^{\mathsf{c}})+\mathds{1}(\mathbf{y}\in\mathbf{N}_{+1}(\mathbf{x})\cap\mathbf{N}_{-1}(\mathbf{x})^{\mathsf{c}})\right)Q(\mathbf{x},\mathrm{d}\mathbf{y})\,\alpha(\mathbf{x},\mathbf{y})$
$\displaystyle=\int_{A\setminus\\{\mathbf{x}\\}}Q(\mathbf{x},\mathrm{d}\mathbf{y})\,\alpha(\mathbf{x},\mathbf{y})=P_{\mathrm{MH}}(\mathbf{x},A\setminus\\{\mathbf{x}\\}),$

using in the first equality that, for any $\mathbf{x}\in\bm{\mathcal{X}}$,

$\mathds{1}(\mathbf{y}\in\mathbf{N}_{-1}(\mathbf{x})\cap\mathbf{N}_{+1}(\mathbf{x})^{\mathsf{c}})+\mathds{1}(\mathbf{y}\in\mathbf{N}_{+1}(\mathbf{x})\cap\mathbf{N}_{-1}(\mathbf{x})^{\mathsf{c}})=1\quad\text{for
\quad$Q(\mathbf{x},\cdot\,)$-almost all $\mathbf{y}$},$

which can be proved in a similar fashion as Lemma 3, and in the second
equality that $\alpha(\mathbf{x},\mathbf{y})=0$ for all
$(\mathbf{x},\mathbf{y})\notin R$.

The case $\mathbf{x}\notin\mathring{\bm{\mathcal{X}}}$ is done similarly after
noticing that, in this case, there exists $\nu\in\\{-1,+1\\}$ such that
$Q(\mathbf{x},\mathbf{N}_{-\nu}(\mathbf{x}))=0$ and
$Q(\mathbf{x},\mathbf{N}_{\nu}(\mathbf{x}))=1$, implying that
$Q(\mathbf{x},A)=Q_{\nu}(\mathbf{x},A)$.

Under the general framework of Section 4, the situation of perfect balance
corresponds to that where
$Q(\mathbf{x},\mathbf{N}_{-1}(\mathbf{x}))=Q(\mathbf{x},\mathbf{N}_{+1}(\mathbf{x}))=1/2$
for all $\mathbf{x}\in\bm{\mathcal{X}}$. In this case,

$\frac{\alpha_{\mathrm{rev.}}(\mathbf{x},\mathbf{y})}{Q(\mathbf{x},\mathbf{N}_{\nu}(\mathbf{x}))}\geq
2\alpha(\mathbf{x},\mathbf{y}),$

from which we can deduce that
$P_{\mathrm{rev.}}(\mathbf{x},A\setminus\\{\mathbf{x}\\})\geq
P_{\mathrm{MH}}(\mathbf{x},A\setminus\\{\mathbf{x}\\})$.

The result

$P_{\mathrm{rev.}}(\mathbf{x},A\setminus\\{\mathbf{x}\\})\geq\frac{1}{2}P_{\mathrm{MH}}(\mathbf{x},A\setminus\\{\mathbf{x}\\}),\quad\text{for
any $\mathbf{x}\in\bm{\mathcal{X}}$ and $A\in\bm{\mathsf{X}}$},$

implies that, for any $f$ such that $\pi(f^{2})<\infty$,

$\mathrm{var}(f,P_{\mathrm{rev.}})\leq
2\mathrm{var}(f,P_{\mathrm{MH}})+\mathbb{V}\mathrm{ar}[f(\mathbf{X})],\quad\mathbf{X}\sim\pi,$

by Lemma 33 in Andrieu et al. (2018).

The proof of

$\mathrm{var}_{\lambda}(f,P_{\mathrm{Lifted}})\leq\mathrm{var}_{\lambda}(f,P_{\mathrm{rev.}}),$

in the case where $\varphi(r)=1\wedge r$ follows from Theorem 7 in Andrieu and
Livingstone (2021). It can be readily verified that the result holds for any
function $\varphi$ considered here using the same proof technique as in
Andrieu and Livingstone (2021). The proof is concluded by taking the limit
$\lambda\rightarrow 1$ on both side of the inequality. ∎

##  Appendix B No Peskun ordering in all generality

In this section, we show using an example that it is not possible to obtain a
Peskun ordering in all generality when considering the general definition of
lifted samplers in Andrieu and Livingstone (2021). The Peskun ordering that we
consider is the generalized version given in Lemma 33 in Andrieu et al.
(2018). Let us consider two ergodic reversible transition kernels $P_{1}$ and
$P_{2}$ operating on the same space $(\bm{\mathcal{X}},\bm{\mathsf{X}})$ and
leaving $\pi$ invariant. We say that there exists a (generalized) Peskun
ordering between $P_{1}$ and $P_{2}$ if there exists some $M>0$ such that for
each $B\in\bm{\mathsf{X}}$,

$P_{2}(\mathbf{x},B\setminus\\{\mathbf{x}\\})\geq
MP_{1}(\mathbf{x},B\setminus\\{\mathbf{x}\\}),$

for all $\mathbf{x}\in\bm{\mathcal{X}}$. Conversely, we say that $P_{1}$ and
$P_{2}$ do not admit a Peskun ordering if for all $M>0$ (as small as we want),
there exists $B_{M}\in\bm{\mathsf{X}}$ such that
$P_{2}(\mathbf{x},B_{M}\setminus\\{\mathbf{x}\\})<MP_{1}(\mathbf{x},B_{M}\setminus\\{\mathbf{x}\\})$
for some $\mathbf{x}\in\bm{\mathcal{X}}$.

We now present the example. Let $\bm{\mathcal{X}}=\operatorname{\mathbb{R}}$
and $\pi=\mathcal{N}(0,1)$. Let
$Q_{+1}(\mathbf{x},\cdot\,)=\mathcal{N}(\mathbf{x},\sigma^{2})$ and
$Q_{-1}(\mathbf{x},\cdot\,)=\mathcal{N}(\mathbf{x},1/\sigma^{2})$ with
$\sigma\in(0,1)$. Let $Q=(1/2)Q_{-1}+(1/2)Q_{+1}$. For any
$B\in\bm{\mathsf{X}}$ such that $0\notin B$, we have that

$\displaystyle P_{\mathrm{MH}}(0,B)$
$\displaystyle=\int_{B}\left(\frac{1}{2}\,Q_{-1}(0,\mathbf{y})+\frac{1}{2}\,Q_{+1}(0,\mathbf{y})\right)\left(1\wedge\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\right)\mathrm{d}\mathbf{y}$
$\displaystyle=\int_{B}\left(\frac{1}{2}\,Q_{-1}(0,\mathbf{y})+\frac{1}{2}\,Q_{+1}(0,\mathbf{y})\right)\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\mathrm{d}\mathbf{y},$

using the symmetry of $Q$.

We can define a lifted version similarly as in Section A.1. For all
$\mathbf{x}\in\bm{\mathcal{X}}$ and $\nu\in\\{-1,+1\\}$, let

$T_{\nu}(\mathbf{x},\mathrm{d}\mathbf{y}):=Q_{\nu}(\mathbf{x},\mathbf{y})\left(1\wedge\frac{\pi(\mathbf{y})\,Q_{-\nu}(\mathbf{y},\mathbf{x})}{\pi(\mathbf{x})\,Q_{\nu}(\mathbf{x},\mathbf{y})}\right)\mathrm{d}\mathbf{y}.$

Note that the definition does not involve splits of neighbourhoods. As in
Section A.1, we can prove that $\mu_{+1}(A\times
B)=\int_{A}\pi(\mathrm{d}\mathbf{x})\,T_{+1}(\mathbf{x},B)$ and
$\mu_{-1}(A\times B)=\int_{B}\pi(\mathrm{d}\mathbf{x})\,T_{-1}(\mathbf{x},A)$
are equal, which implies that

$P_{\mathrm{Lifted}}((\mathbf{x},\nu),\mathrm{d}(\mathbf{y},\nu^{\prime})):=\delta_{\nu}(\nu^{\prime})\,T_{\nu}(\mathbf{x},\mathrm{d}\mathbf{y})+\delta_{-\nu}(\nu^{\prime})\,\delta_{\mathbf{x}}(\mathrm{d}\mathbf{y})\,(1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}}))$

leaves the distribution $\pi\otimes\mathcal{U}\\{-1,+1\\}$ invariant.

The reversible counterpart to the lifted sampler is defined as

$P_{\mathrm{rev.}}(\mathbf{x},\mathrm{d}\mathbf{y}):=\frac{1}{2}\,T_{+1}(\mathbf{x},\mathrm{d}\mathbf{y})+\frac{1}{2}\,T_{-1}(\mathbf{x},\mathrm{d}\mathbf{y})+\delta_{\mathbf{x}}(\mathrm{d}\mathbf{y})\left(1-\frac{1}{2}\,T_{+1}(\mathbf{x},\bm{\mathcal{X}})-\frac{1}{2}\,T_{-1}(\mathbf{x},\bm{\mathcal{X}})\right).$

This definition can be seen as being similar to that in Section A.1 and, like
in that section, it can be proved that $P_{\mathrm{rev.}}$ is
$\pi$-reversible.

For the same $B\in\bm{\mathsf{X}}$ as above with $0\notin B$, we have that

$\displaystyle P_{\mathrm{rev.}}(0,B)$
$\displaystyle=\int_{B}\frac{1}{2}\,Q_{+1}(0,\mathbf{y})\left(1\wedge\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{-1}(\mathbf{y},0)}{Q_{+1}(0,\mathbf{y})}\right)\mathrm{d}\mathbf{y}$
$\displaystyle\qquad+\int_{B}\frac{1}{2}\,Q_{-1}(0,\mathbf{y})\left(1\wedge\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{+1}(\mathbf{y},0)}{Q_{-1}(0,\mathbf{y})}\right)\mathrm{d}\mathbf{y}.$

We have that

$\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{-1}(\mathbf{y},0)}{Q_{+1}(0,\mathbf{y})}=\sigma^{2}\exp\left(\frac{\mathbf{y}^{2}}{2}\left(\frac{1}{\sigma^{2}}-\sigma^{2}-1\right)\right),$

and

$\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{+1}(\mathbf{y},0)}{Q_{-1}(0,\mathbf{y})}=\frac{1}{\sigma^{2}}\exp\left(-\frac{\mathbf{y}^{2}}{2}\left(\frac{1}{\sigma^{2}}+1-\sigma^{2}\right)\right).$

Note that

$\frac{1}{\sigma^{2}}+1-\sigma^{2}>0$

as $\sigma\in(0,1)$. We can choose $\sigma$ small enough so that

$\frac{1}{\sigma^{2}}-\sigma^{2}-1>0.$

We choose $\sigma$ accordingly. Therefore, we can choose
$B=\\{\mathbf{y}\in\operatorname{\mathbb{R}}:|\mathbf{y}|>k\\}$ with $k$ a
large enough constant, so that, for all $\mathbf{y}\in B$, we have that

$\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{-1}(\mathbf{y},0)}{Q_{+1}(0,\mathbf{y})}\geq
1\quad\text{and}\quad\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{+1}(\mathbf{y},0)}{Q_{-1}(0,\mathbf{y})}\leq
1.$

Note that $0\notin B$. With such a set $B$, we have that

$P_{\mathrm{rev.}}(0,B)=\int_{B}\frac{1}{2}\,Q_{+1}(0,\mathbf{y})\,\mathrm{d}\mathbf{y}+\int_{B}\frac{1}{2}\,Q_{+1}(0,\mathbf{y})\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\mathrm{d}\mathbf{y}.$

Let $M>0$. We have that

$\displaystyle P_{\mathrm{rev.}}(0,B)-MP_{\mathrm{MH}}(0,B)$
$\displaystyle\quad=\int_{B}\frac{1}{2}\,Q_{+1}(0,\mathbf{y})\,\mathrm{d}\mathbf{y}+\int_{B}\frac{1}{2}\,Q_{+1}(0,\mathbf{y})\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\mathrm{d}\mathbf{y}$
$\displaystyle\qquad-M\int_{B}\left(\frac{1}{2}\,Q_{-1}(0,\mathbf{y})+\frac{1}{2}\,Q_{+1}(0,\mathbf{y})\right)\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\mathrm{d}\mathbf{y}$
$\displaystyle\quad=\frac{1}{2}\int_{B}Q_{+1}(0,\mathbf{y})\left(1+\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)-M\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{-1}(0,\mathbf{y})}{Q_{+1}(0,\mathbf{y})}-M\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\right)\mathrm{d}\mathbf{y}$
$\displaystyle\quad\leq\frac{1}{2}\int_{B}Q_{+1}(0,\mathbf{y})\left(1+\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)-M\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{-1}(0,\mathbf{y})}{Q_{+1}(0,\mathbf{y})}\right)\mathrm{d}\mathbf{y}$

Note that

$\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{-1}(0,\mathbf{y})}{Q_{+1}(0,\mathbf{y})}=\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{-1}(\mathbf{y},0)}{Q_{+1}(0,\mathbf{y})}=\sigma^{2}\exp\left(\frac{\mathbf{y}^{2}}{2}\left(\frac{1}{\sigma^{2}}-\sigma^{2}-1\right)\right).$

Therefore, regardless the value of $M$, we could have chosen $k$ such that

$\left(1+\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)-M\exp\left(-\frac{\mathbf{y}^{2}}{2}\right)\frac{Q_{-1}(0,\mathbf{y})}{Q_{+1}(0,\mathbf{y})}\right)<0.$

Consequently, $P_{\mathrm{rev.}}(0,B)-MP_{\mathrm{MH}}(0,B)<0$, which
concludes the example.