# Duality, Hidden Symmetry and Dynamic Isomerism in 2D Hinge Structures

Qun-Li Lei<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures
and Department of Physics, Collaborative Innovation Center of Advanced
Microstructures, Nanjing University, Nanjing 210093, China  School of Chemical
and Biomedical Engineering, Nanyang Technological University, 62 Nanyang
Drive, 637459, Singapore  Feng Tang  National Laboratory of Solid State
Microstructures and Department of Physics, Collaborative Innovation Center of
Advanced Microstructures, Nanjing University, Nanjing 210093, China  Ji-Dong
Hu  National Laboratory of Solid State Microstructures and Department of
Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China  Yu-qiang Ma<EMAIL_ADDRESS>National
Laboratory of Solid State Microstructures and Department of Physics,
Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China  Ran Ni<EMAIL_ADDRESS>School of Chemical
and Biomedical Engineering, Nanyang Technological University, 62 Nanyang
Drive, 637459, Singapore

###### Abstract

Recently, a new type of duality was reported in some deformable mechanical
networks which exhibit Kramers-like degeneracy in phononic spectrum at the
self-dual point. In this work, we clarify the origin of this duality and
propose a design principle of 2D self-dual structures with arbitrary
complexity. We find that this duality originates from the _partial central
inversion_ (PCI) symmetry of the hinge, which belongs to a more general end-
fixed scaling transformation. This symmetry gives the structure an extra
degree of freedom without modifying its dynamics. This results in _dynamic
isomers_ , i.e., dissimilar 2D mechanical structures, either periodic or
aperiodic, having identical dynamic modes, based on which we demonstrate a new
type of wave-guide without reflection or loss. Moreover, the PCI symmetry
allows us to design various 2D periodic isostatic networks with hinge duality.
At last, by further studying a 2D non-mechanical magnonic system, we show that
the duality and the associated hidden symmetry should exist in a broad range
of Hamiltonian systems.

## Introduction

Space group symmetries are cornerstones of condensed matter physics
Dresselhaus _et al._ (2007); Zhang _et al._ (2019); Vergniory _et al._ (2019);
Tang _et al._ (2019). Nevertheless, physical systems can also have hidden
symmetries which can not be captured by space group theory Louvet _et al._
(2015); Hou (2013); Hou & Chen (2017, 2018); Zhang (2019); Guarneri _et al._
(2020); Liu & Semperlotti (2021); Guo _et al._ (2022). Some of these hidden
symmetries were previously reported at some specific points in the Brillouin
zone (BZ) Hou (2013); Hou & Chen (2017); Guo _et al._ (2022). Recently, a
hidden symmetry in the full BZ induced by self-duality was discovered in 2D
mechanical isostatic networks Fruchart _et al._ (2020); Lei _et al._ (2021).
These structures have a low coordination number, which creates many local
hinges with unconstrained degrees of freedom Lubensky _et al._ (2015); Liu _et
al._ (2019); Bertoldi _et al._ (2017); Ma _et al._ (2019); McInerney _et al._
(2020). By tuning the open angle of hinges $\vartheta$, one can change these
structures continuously from an open to a folded state Maxwell (1864); Souslov
_et al._ (2009); Sun _et al._ (2012); Mao & Lubensky (2018); Xin _et al._
(2020). Interestingly, these structures with small $\vartheta$ are dual to the
structures with large $\vartheta$, thus there exists a critical
$\vartheta^{*}$, at which the dual counterpart of the structure is itself. At
this self-dual point, hidden symmetry emerges, resulting in Kramers-like
double degeneracy Klein (1952), and many other interesting phenomena, like
mechanical non-abelian spintronics Fruchart _et al._ (2020), the degeneracy of
elastic modulus Fruchart & Vitelli (2020), the critically-tilted Dirac cone
Lei _et al._ (2021), the symmetric boundary effect Gonella (2020), topological
corner states and Maxwell modes Danawe _et al._ (2021, 2022) etc.
Nevertheless, so far, only a few meticulous structures are found to be self-
dual. The origin of the duality and the resulting hidden symmetry remain
mysterious Lei _et al._ (2021); Fruchart _et al._ (2021).

In this work, we illuminate the origin of this duality and propose a design
principle of 2D structures with such duality. We find that this duality, which
we rephrase as ‘hinge duality’, originates from a special _partial central
inversion_ (PCI) symmetry of the hinge, which gives the structure an extra
degree of freedom without changing the Hamiltonian. When multiple hinges are
connected into hinge chains or networks, the combination of local PCI
generates _dynamic isomers_ , i.e., different configurations having exactly
the same dynamic modes. Based on the PCI and resulting hinge duality, we also
propose design rules to generate arbitrarily complicated 2D periodic
structures with self-duality. Furthermore, we show that PCI belongs to a more
general end-fixed scaling transformation, which can further broaden the range
of dynamic isomers. At last, we also demonstrate the existence of hinge
duality in a non-mechanical magnonic system, which suggests this duality is a
generic property of hinge structures, existing in a broad range of Hamiltonian
systems.

---
Figure 1: (a-c): Dual transformation for a single hinge, which is composed by
PCI (a $\rightarrow$ b) and a global $90^{\circ}$ rotation (b $\rightarrow$
c), with respect to the hinge point. The arrows indicate the vibrational
degrees of freedom. (d-g) Dynamic isomers of hinge chain with different hinge
sequences.

## Duality of a single hinge

We first consider a simple hinge composed by two structually identical arms
$a$ and $b$, which can freely rotate around the hinge point in dimension $d=2$
Guest & Fowler (2005); Schulze _et al._ (2014). Each arm is modelled as a
mechanical spring network with $n+1$ nodes. The total degrees of freedom in
the system thus is $(2n+1)d$, with the hinge node shared by two arms. Under
the harmonic approximation, all nodes do small vibrations around their
equilibrium positions, and the time-dependent position of node $i$ in arm $a$,
$b$ can be written as

$\displaystyle{\mathbf{r}}^{a}_{i}(t)={\mathbf{R}}^{a}_{i}+{\mathbf{u}}^{a}_{i}(t);~{}~{}~{}~{}{\mathbf{r}}^{b}_{i}(t)={\mathbf{R}}^{b}_{i}+{\mathbf{u}}^{b}_{i}(t),$
(1)

respectively, where ${\mathbf{R}}^{a}_{i}$ and ${\mathbf{R}}^{b}_{i}$ are the
equilibrated positions, ${\mathbf{u}}^{a}_{i}(t)$ and
${\mathbf{u}}^{b}_{i}(t)$ are the time-dependent vibrational displacements for
nodes $i\in[0,1,2,\cdots,n]$ in arm $a$ and $b$, respectively. The position of
the hinge node is
${\mathbf{r}}_{0}(t)={\mathbf{r}}^{a}_{0}={\mathbf{r}}^{b}_{0}={\mathbf{R}}_{0}+{\mathbf{u}}_{0}(t)$.
Thus ${\mathbf{R}}^{b}_{0}={\mathbf{R}}^{a}_{0}={\mathbf{R}}_{0}$ and
${\mathbf{u}}^{b}_{0}={\mathbf{u}}^{a}_{0}={\mathbf{u}}_{0}$. Since two arms
are structurally identical, they are connected by the rotational operation
${\hat{R}}(\vartheta)$ with respect to the hinge point ${\mathbf{R}}_{0}$,
i.e., ${\mathbf{R}}^{a}_{i}={\hat{R}}(\vartheta)\cdot{\mathbf{R}}^{b}_{i}$
with $\vartheta$ the open angle between two arms. The Hamiltonian of the
system is

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{k}+\sum_{i,j}\frac{\lambda_{ij}}{2}\left\\{\left[({\mathbf{u}}^{a}_{i}-{\mathbf{u}}^{a}_{j})\cdot{\mathbf{e}}^{a}_{ij}\right]^{2}+\left[({\mathbf{u}}^{b}_{i}-{\mathbf{u}}^{b}_{j})\cdot{\mathbf{e}}^{b}_{ij}\right]^{2}\right\\}$

where $H_{k}$ represents the kinetic energy, $\lambda_{ij}$ is the spring
constant for corresponding bond pair in two arms.
${\mathbf{e}}^{k}_{ij}=({\mathbf{R}}^{k}_{i}-{\mathbf{R}}^{k}_{j})/\left|{\mathbf{R}}^{k}_{i}-{\mathbf{R}}^{k}_{j}\right|(k=a,b)$
is the normalized bond vector, with $i,j$ running over all pairwise spring
(bond) connections in each arm. The vibrational displacement vector
${\mathbf{X}}=\\{{\mathbf{u}}_{0},{\mathbf{u}}^{a}_{1},{\mathbf{u}}^{b}_{1},\cdots,{\mathbf{u}}^{a}_{n},{\mathbf{u}}^{b}_{n}\\}$
satisfies the dynamic equation
${\mathcal{M}}\cdot\partial_{t}^{2}{\mathbf{X}}={\mathcal{D}}\cdot{\mathbf{X}}$
with ${\mathcal{D}}$ the dynamic matrix and ${\mathcal{M}}$ the mass matrix
Mao & Lubensky (2011); Po _et al._ (2016); Kane & Lubensky (2014).

Based on general symmetry consideration, when an isolated mechanical system
under the central inversion with respect to arbitrary fixed point, the
Hamiltonian is invariant. However, this is usually not true for a part of the
system. Nevertheless, one can prove that Hamiltonian Eq. (LABEL:Hamiltonian)
is invariant when central inversion is conducted only for one arm with respect
to the hinge point, e.g.,

$\displaystyle{{\mathbf{R}}^{b}_{i}}^{\prime}=2{\mathbf{R}_{0}}-{\mathbf{R}}^{b}_{i},~{}~{}~{}~{}i\in[0,1,2,\cdots,n].$
(3)

We call this transformation _partial central inversion_ (PCI), as depicted in
Fig. 1 a$\rightarrow$b. This transformation leaves the vibrational degrees of
freedom ${\mathbf{u}}^{a}_{i}$ intact. In addiction, rotation symmetry
guarantees the invariance of the system in a $90^{\circ}$ rotation around the
hinge point (Fig. 1b$\rightarrow$c). Therefore, by defining these two
consecutive transformations as ${\hat{V}_{0}}$, we have the commutation
relationship $[{\hat{V}_{0}},\hat{H}]=0$ or $[{\hat{V}_{0}},\mathcal{D}]=0$.
Equivalently, ${\hat{V}_{0}}$ can also be interpreted as the combination of
operator $\hat{K}$ that changes the open angle of the hinge from $\vartheta$
to $\vartheta^{*}=\pi-\vartheta$, i.e.,
$\hat{K}\mathcal{D}(\vartheta)\hat{K}^{-1}={\mathcal{D}}(\vartheta^{*})$, and
operator $\hat{U}_{0}$ that switches the corresponding nodes in two arms
(${\mathbf{r}}^{a}_{i}\rightleftharpoons{\mathbf{r}}^{b}_{i}$) and rotates all
vibrational degrees of freedom by $90^{\circ}$ at the same time, namely,
${\hat{V}_{0}}={\hat{K}}{\hat{U}}_{0}$. In the case of $n=2$, which is shown
in Fig. 1a, $\hat{U}_{0}$ can be written as Sup (????)

$\displaystyle{\hat{U}_{0}}=\begin{pmatrix}{{\hat{r}}_{\blacksquare}}&0&0&0&0\\\
0&0&{{\hat{r}}_{\blacksquare}}&0&0\\\ 0&{{\hat{r}}_{\blacksquare}}&0&0&0\\\
0&0&0&0&{{\hat{r}}_{\blacksquare}}\\\ 0&0&0&{{\hat{r}}_{\blacksquare}}&0\\\
\end{pmatrix}$ (4)

where ${\hat{r}}_{\blacksquare}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ is
the $90^{\circ}$ rotation operator. As can be seen, $\hat{U}_{0}$ performs the
node switching $(a,1)\rightleftharpoons(b,1)$ and
$(a,2)\rightleftharpoons(b,2)$, while leaves the position of hinge node
unchanged. Since $\hat{K}$ commutes with $\hat{U}_{0}$, from
$[\hat{V}_{0},{\mathcal{D}}]=0$ one has

$\displaystyle\hat{U}_{0}{\mathcal{D}}(\vartheta)\hat{U}_{0}^{-1}$
$\displaystyle=$ $\displaystyle{\mathcal{D}}(\vartheta^{*}).$ (5)

Eq.(5) expresses the dual relationship between two hinge configurations at
open angles $\vartheta$ and $\vartheta^{*}$, which we call “hinge duality”.
Especially, $\vartheta^{*}=\vartheta$ corresponds to the self-dual point at
which the hinge structure remains intact under $\hat{K}$, but the dynamic
modes are transformed by $\hat{U}_{0}$. One can generally prove that the
vibrational modes before and after this transformation are orthogonal to each
other, i.e., ${\mathbf{X}}\cdot(\hat{U}_{0}\cdot{\mathbf{X}})=0.$ This
guarantees that at the self-dual point, an arbitrary vibration energy
(frequency) level of the system is at least double degenerated. One can also
prove that changing the mass of the corresponding node pair, or the spring
constant $\lambda_{ij}$, does not affect the hinge duality. Moreover, this
hinge duality is independent of the number of nodes in the arm. This means
that even in the continuous limit of the arm ($n\rightarrow\infty$), the hinge
duality is still preserved.

## Dynamic isomerism

From the above analysis, one can see that the PCI gives the hinge an extra
degree of freedom which preserves the dynamic modes of the system. When
multiple hinges are inter-connected to form a hinge chain, these extra degrees
of freedom are addictive, i.e., a hinge chain with $N$ hinges has $2^{N}$
dissimilar configurations whose dynamic modes are exactly the same. We call
these configurations _dynamic isomers_ , which can be labelled by binary
sequences like ”10110011…” of length $N$. Here, 1 and 0 indicate two dual
states with open angle $\vartheta$ and $\vartheta^{*}$ for a single hinge. In
Fig. 1d-g, we show several dynamically isomeric chains with different
sequences, where sequence ”000000…” or ”111111…” represents two simplest
periodic hinge chains (Fig. 1d, f). We also show an intermediate configuration
between these two states in Fig. 1e, and a more disordered chain configuration
in Fig. 1g. It’s surprising that such a disordered chain has the same
vibrational eigenmodes as that of the periodic chains. It should be noticed
that mode propagation direction in Fig. 1f is rotated by $90^{\circ}$ compared
with Fig. 1d. In fact, one can also construct other kinds of periodic hinge
chain, e.g., that with sequence “101010…”, in which the vibration mode
propagates in a different direction. These intriguing properties can be
utilized to build a new type of flexible wave-guides without reflection or
loss, as demonstrated by Movie S1, S2 in Sup (????). It also allowed us to
design more complicated 2D dynamic isomeric networks (Fig. S1 in Sup (????)).
These findings have potential applications in building phononic circuits Yu
_et al._ (2018).

---
Figure 2: (a-b) Directly applying the dual transformation ${\hat{V}}_{0}$ to
the unit cell of the periodic hinge chain reverses the wave propagation
direction. (c) Phononic spectrums of the 2D periodic hinge chain at three
different open angles, where $\vartheta=90^{\circ}$ is the self-dual point at
which the hidden symmetry induces the double degeneracy in the whole BZ.

## Duality in periodic hinge chains

One can see that the two minimal periodic hinge chain with sequence ‘000000…’
and ‘111111…’ in Fig. 1d,f are connected by a sequence of PCI transformations
and a global $90^{\circ}$ rotation of the whole chain. These two steps
actually define the dual transformation for the periodic hinge chain, which is
equivalent to directly applying the dual transformation $\hat{V}_{0}$ to the
unit cell of the periodic structure. To see this more clearly, we use $l$ to
mark different cells and $(l,i)$ represents the $i$ node in cell $l$. As shown
in Fig. 2a, nodes $(l,1)$ and $(l,2)$ are on arm $a$, while nodes $(l,3)$ and
$(l+1,1)$ are on arm $b$ in cell $l$. When operator $\hat{V}_{0}$ acts on the
unit cell, one has the node switching: $(l,2)\leftrightarrows(l,3)$ and
$(l,1)\leftrightarrows(l+1,1)$, Importantly, $(l,1)\leftrightarrows(l+1,1)$
reverses the propagation direction of vibrational modes and shifts the node
$(l,1)$ one period distance $\mathbf{a}_{1}$. Thus, this transformation for
periodic hinge chain should be written as $\hat{V}_{1}=\hat{K}\hat{U}_{1}$
with

$\displaystyle{\hat{U}}_{1}=\begin{pmatrix}{r_{\blacksquare}}&0&0&0\\\
0&0&{r_{\blacksquare}}&0\\\ 0&{r_{\blacksquare}}&0&0\\\
0&0&0&{{\hat{T}}_{a_{1}}r_{\blacksquare}}\\\ \end{pmatrix}{\mathcal{I}}$ (6)

Here, the switching between $(l,1)$ and $(l+1,1)$ is expressed as the
combination of shifting operator
${\hat{T}}_{a_{1}}=e^{-i{\mathbf{q}}\cdot{\mathbf{a}_{1}}}$ and complex
conjugation $\mathcal{I}$ which reverses the sign of wave vector $\mathbf{q}$.
Therefore, $\hat{U}_{1}$ is an anti-unitary matrix satisfying
$\hat{U}_{1}^{2}=-1$. Since $\hat{V}_{1}$ commutes with $\mathcal{D}$, we have
the dual transformation relationship for the periodic chain

$\displaystyle\hat{U}_{1}{\mathcal{D}}(\vartheta,{\mathbf{q}})\hat{U}_{1}^{-1}$
$\displaystyle=$ $\displaystyle{\mathcal{D}}(\vartheta^{*},{\mathbf{q}}).$ (7)

Similar to that in single hinge, the above dual relationship is also
independent of the number of nodes in the arms. In Fig. 2b, we show the
phononic spectrum for the periodic hinge chains in Fig. 2a. We find the
identical spectrum for systems at $\vartheta=60^{\circ}$ and
$\vartheta^{*}=120^{\circ}$, as well as the double degeneracy in the whole BZ
zone at $\vartheta=90^{\circ}$, a hallmark of self-dual point Fruchart _et
al._ (2020).

---
Figure 3: By constructing different hinges in the unit cell, one can obtain
various 2D periodic structures with hinge duality. (a) A minimal 2D hinge
network, (b) generalized twisted Kagome, (c) dynamic isomer of twisted kagome
with two hinge arms having different size, (d) p2g twisted kagome, (e) snub
square lattice, (f) pmg lattice Lei _et al._ (2021). The corresponding bond
pairs are drawn in the same color in each figure, which have the same spring
constant.

## Duality in 2D periodic hinge networks

From the above analysis, one can see that the existence of a hinge and
corresponding dual transformation in the unit cell are the requisites for
periodic chains to achieve hinge duality. This inspires us to propose design
rules of 2D periodic networks with hinge duality: i) constructing a single
hinge in which each arm of the hinge has more than three nodes; ii) making the
corresponding nodes in two arms as a node pair; iii) choosing two node pairs
and constructing two vectors that connect the nodes in each pair. These two
vectors define the two lattice vectors $\mathbf{a_{1}}$ and $\mathbf{a_{2}}$
of 2D periodic networks. Following the above procedure, one can generate
arbitrarily complex 2D periodic structures with hinge duality by designing the
hinge arm. In Fig. 3a, we show the minimal 2D periodic networks in which the
hinge arm is composed by three nodes and two bonds. When the hinge arm is a
three-node triangle, one obtains a generalized twisted Kagome structure Kane &
Lubensky (2014) (Fig. 3b). All these structures have the same dual
transformation $\hat{V}_{2}=K\hat{U}_{2}$ with

$\displaystyle\hat{U}_{2}=\begin{pmatrix}r_{\blacksquare}&0&0\\\
0&{{\hat{T}}_{a_{2}}r_{\blacksquare}}&0\\\
0&0&{{\hat{T}}_{a_{1}}r_{\blacksquare}}\\\ \end{pmatrix}{\mathcal{I}},$ (8)

where the nodes switchings $(l_{1},l_{2},1)$ $\rightarrow$ $(l_{1}+1,l_{2},1)$
and $(l_{1},l_{2},2)$ $\rightarrow$ $(l_{1},l_{2}+1,2)$ in unit cell
$(l_{1},l_{2})$ are related with the shifting operator ${\hat{T}}_{a_{1}}$ and
${\hat{T}}_{a_{2}}$ along $\mathbf{a_{1}}$ and $\mathbf{a_{2}}$, respectively.
If the hinge arm is a hourglass made up by two equilateral triangles, one has
a lattice with p2g space group symmetry (Fig. 3d), which can be obtained by
twisting the standard Kagome lattice Sun _et al._ (2012). When the hinge arm
is a perfect square, a snub square lattice with p4g symmetry is obtained (Fig.
3e). The phononic spectrums for these structures (Fig. S2-S6) and a general
proof of the dual relationship in 2D periodic hinge structures is also
provided in Sup (????).

The above design principles can also have more complicated variation. In Fig.
2f, we show the network with pmg symmetry reported in Lei _et al._ (2021). The
unit cell of this structure is a hinge made up by two rhombus arms with a
‘dangling’ bond on each arm. Nevertheless, the two ‘dangling’ bonds are
arranged in an opposite direction, making two arms unable to map to each other
by rotation. In fact, the dual transformation in this structure involves
additional PCI transformation for the dangling bond (see the dashed blue bond
in Fig. 3f). This example suggests that there exist more complicated design
rules for hinge-dual structures based on multiple PCIs. Moreover, we find that
PIC in Eq. (3) belongs to a more general scaling transformation, with scaling
ratio $\xi=-1$, i.e.,

$\displaystyle{{\mathbf{R}}^{k}_{i}}^{\prime}={\mathbf{R}_{0}}-\xi({\mathbf{R}_{0}}-{\mathbf{R}}^{k}_{i}),~{}~{}~{}~{}~{}i\in[0,1,2,\cdots,n]$
(9)

and $k=a,b$. In this transformation, the hinge point is fixed. One can prove
that Eq. (9) with arbitrary $\xi$ also preserves the Hamiltonian as long as
$\lambda_{ij}$ remains unchanged during the transformation, which can be
realized easily in mechanical systems. Under this condition, we can have 2D
periodic dynamic isomers composed by hinge arms that have different sizes
(Fig. 3c). In these dynamic isomers, the propagation directions of vibrational
waves are also modified Sup (????).

It should be mentioned that structures in Fig. 3 are all deformable networks,
some of which are Maxwell (isostatic) structures. In fact, the existence of
free hinge in the unit cell guarantees the deformability of these structures,
thus there must be an corresponding Guest-Hutchinson mode Mao & Lubensky
(2018); Guest & Hutchinson (2003). Nevertheless, as discussed in Sup (????),
structures with hinge duality are not necessary to be deformable. Hinge
structures with either node pinning or bond orientation constraint can also
have hinge duality as shown in Fig. S7-8 Sup (????). This broadens the
functionality and potential application scenarios of these structures.

---
Figure 4: Magnonic spectrums of p31m twisted Kagome lattice at three different
open angles. Note that spectrums at $\vartheta=60^{\circ}$ and $\ 120^{\circ}$
are numerically indistinguishable.

At last, we also consider a non-mechanical magnon system, whose Hamiltonian
can be written as

$\displaystyle H=-2J\sum_{i,j}{\mathbf{s}}_{i}\cdot{\mathbf{s}}_{j}$ (10)

where $(i,j)$ indicates the nearest spin pairs. Magnons are the collective
magnetic excitations (spin waves) associated with the precession of the spin
moments Gurevich & Melkov (2020). In the weak perturbation regime,
$s_{x},s_{y}\ll s_{z}\simeq s$, magnons obey the first-order dynamic equation
rather than the second-order one for phonons, i.e.,

$\displaystyle{\partial}_{t}{\mathbf{S}}_{\perp}=\frac{2Js}{{\hbar}}{\mathcal{D}}_{m}\cdot\widetilde{\mathbf{S}}_{\perp}$
(11)

with ${\mathbf{S}}_{\perp}=(s_{1}^{x},s_{1}^{y},s_{2}^{x},s_{2}^{y},\cdots)$
the spin moments in $xy$ plane and
$\widetilde{\mathbf{S}}_{\perp}=(s_{1}^{y},-s_{1}^{x},s_{2}^{y},-s_{2}^{x},\cdots)$
and ${\mathcal{D}}_{m}$ the dynamic matrix (see Sup (????) for details). In
Fig.4, we show the magnonic spectrums for the p31m twisted Kagome network
under different open angles. Similar to the phononic systems, we observe the
identical magnonic spectrum at two dual open angles $\vartheta=60^{\circ}$ and
$\vartheta^{*}=120^{\circ}$, as well as the Kramers-like degeneracy at the
self-dual point $\vartheta=90^{\circ}$. These results suggest that the hinge
duality should be a generic property of hinge structures, insensitive to
specific Hamiltonian or dynamics of the system.

## Discussion and Conclusion

In conclusion, we unveil the origin of hinge duality and the resulting hidden
symmetry in 2D Hamiltonian hinge systems. We find the hinge structure has a
unique PCI symmetry related with the end-fixed scaling, based on which the
hinge can have a Hamiltonian-invariant dual transformation between two
different configurations. This intriguing property leads to 2D periodic or
aperiodic dynamic isomers of hinge chains/networks, which can be utilized to
build flexible wave-guides or networks with applications in phononic circuits
Yu _et al._ (2018). It is also interesting to explore the dynamic isomerism at
the molecular scale Quinlivan & Parkin (2017); Écija _et al._ (2013).
Furthermore, we propose simple rules to design self-dual 2D periodic
structures with arbitrary complexity, which provides a guideline to fabricate
self-dual metamaterials with protected new topological phases or other
unconventional mechanical/phononic properties Guo _et al._ (2022); Fruchart
_et al._ (2020); Fruchart & Vitelli (2020); Lei _et al._ (2021); Danawe _et
al._ (2021, 2022); Gonella (2020). At last, we show that the hinge duality
also exists in non-mechanical magnonic system. Thus we expect it a generic
property of a broad range of Hamiltonian or dynamic systems. In fact, during
the preparing of this work, we noticed Ref. Fruchart _et al._ (2021) which
discussed this topic from a different point of view. We expect further
breakthrough in discovering other kinds of duality-induced hidden symmetry,
especially in aperiodic 2D Zhou _et al._ (2019) and periodic 3D isostatic
systems Rocklin & Mao (2014).

###### Acknowledgements.

### Acknowledgments:

This work is supported by the National Natural Science Foundation of China
(Nos. 11474155, 11774147 and 12104215), by Singapore Ministry of Education
through the Academic Research Fund MOE2019-T2-2-010 and RG104/17 (S), by
Nanyang Technological University Start-Up Grant (NTU-SUG: M4081781.120), by
the Advanced Manufacturing and Engineering Young Individual Research Grant
(A1784C0018) and by the Science and Engineering Research Council of Agency for
Science, Technology and Research Singapore.

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#  Adiabatic Approximation of Coarse Grained Second Order Response

Fenna Müller and Max von Renesse111Fakultät für Mathematik und Informatik,
Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany. E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose estimators for dynamic second order response theory in coarse
grained variables for driven out-of-equilibrium subsystems. The error is
controlled through the the notion of subsystem spectral gap for the
convergence of coarse grained observables.

##  1 Introduction and main result

This note addresses the problem of forecasting the dynamic behaviour of driven
out-of-equilibrium subsystems, using only equilibrium subsystem measurements.
As a prototype consider a tracer interacting with an else unknown and
unobserved particle system which is driven out of equilibrium by an additional
force acting only on the tracer, e.g. [15, 10]. How can one make a sensible
prediction on the dynamics of the forced tracer from probing it in the
unforced regime when the whole system is still unknown but can be assumed in
equilibrium?

We treat this as a nonparametric statistical problem for a
$\mathds{R}^{d}$-valued solution $x^{\varepsilon}_{\cdot}$ to an overdamped
Langevin SDE

$\displaystyle\mathrm{d}x_{t}^{\varepsilon}=-\nabla
U(x_{t}^{\varepsilon})\mathrm{d}t-\varepsilon
f(x_{t}^{\varepsilon})\mathrm{d}t+\sqrt{2}\mathrm{d}W_{t}$ $\displaystyle
x_{0}^{\varepsilon}\sim\mu=\frac{1}{Z}e^{-U}dx,$

where only $f:\mathds{R}^{d}\mapsto\mathds{R}^{d}$ is known and only partial
observations $\pi(x)\in\mathds{R}^{m}$, $m\leqslant d$, of the states $x$ are
possible under a fixed measurable (’coarse graining’) map

$\pi:\mathds{R}^{d}\mapsto\mathds{R}^{m},\quad m\leqslant d,$

describing the passage to the observable subsystem. Assuming that
$f(x)=f(\pi(x))$ is also coarse grained, given
$\kappa:\mathds{R}^{m}\mapsto\mathds{R}$ we want to estimate

$\mathbb{E}_{\mu}(\kappa(\pi(x^{\varepsilon}_{t}))),$

in terms of quantities of the form

$\mathbb{E}_{\mu}(\Phi(\pi(x^{0}_{s})_{s\in[0,t]})),$

where $\Phi\in(\mathds{R}^{m})^{[0,t]}$ are suitably chosen path functionals.
In practice such expectations of $\Phi(\pi(x_{s})_{s\in[0,t]})$ at equilibrium
can be obtained from corresponding time-averaged subsystem observations, using
the ergodicity of $x$.

This inference problem from unperturbed $\pi(x^{0})$ to perturbed
$\pi(x^{\varepsilon})$ is non-trivial due to the interaction with the hidden
degrees of freedom. Standard response theory (c.f. e.g. [1, 14]) does not give
an answer since it relies on full system measurements at equilibrium which
might not be available or too costly in practice. The aim of this note is to
suggest a simple and practical work around for such a situation.

Using a suitable semigroup expansion and a variant of adiabatic approximation,
in case when $f=\nabla V$ with $V(x)=v(\pi(x))$ we propose the approximation

$\displaystyle\mathbb{E}_{\mu}(k(\pi(x_{T}^{\varepsilon}))\approx\left<k\right>+\varepsilon\left[\langle
Vk\rangle-\langle VP_{T}k\rangle\right]$
$\displaystyle+\frac{\varepsilon^{2}}{2}\left[\left<V^{2}k\right>-\left<VP_{T}Vk\right>+\left<V\right>\left(\left<P_{T/2}k,V\right>-\left<kV\right>\right)+\left<k\right>\left(\left<P_{T/2}V,V\right>-\left<V^{2}\right>\right)\right],$

which can be estimated empirically up to arbitrary precision from equilibrium
$\pi(x_{\cdot})$-observations. Hence only some structural information about
the full systems and the precise form of the perturbation needs to be known,
and no effective model for the dynamics of the tracer e.g. by stochastic delay
differential equations or via avaraging as in e.g. [7, 13, 4] is used. The
estimator above extends the standard first order expansion from [1, 14, 17]
and is complimentary to a previous approach via linearization of higher order
correlations in [2, 16]. Its accuracy depends critically on the mixing of the
(generally non-Markovian) subsystem $\pi(x_{\cdot})$, and which we quantify
through the notion of subsystem spectral gap, see definition 3.1 and
proposition 3.5 below. In particular, the subsystem and not the environment
needs to be fast for reasonable accuracy, contrary to the standard assumptions
in e.g. averaging.

The method is illustrated by numerical simulations towards the end.

##  2 Dyson formula and semigroup expansion

We will treat the given statistical inference problem in the framework of
perturbations of overdamped Langevin dynamics and their associated Markov
semigroups, where for simplicity we assume full regularity of the coefficients
throughout. To this aim, for $\varepsilon\geqslant 0$ we consider the unique
strong solutions to the SDE in $\mathds{R}^{d}$

$\displaystyle\begin{split}\mathrm{d}x_{t}^{\varepsilon f}&=-\nabla
U(x_{t}^{\varepsilon f})\mathrm{d}t-\varepsilon f_{t}(x_{t}^{\varepsilon
f})\mathrm{d}t+\sqrt{2}\mathrm{d}W_{t}\\\ x_{0}^{\varepsilon
f}&=\xi\end{split}$ (1)

where $U:\mathds{R}^{d}\mapsto\mathds{R}$ and
$f:\mathds{R}^{d}\times\mathds{R}_{\geqslant 0}\mapsto\mathds{R}^{d}$ satisfy
appropriate growth and regularity assumptions, $W$ is a
$\mathrm{d}$-dimensional Brownian motion, $\xi$ is independent of $W$. We
shall assume also that the solutions have infinite life time and induce the
family of time-inhomogeneous Feller semigroups $(P_{s,t}^{\varepsilon
f})_{0\leqslant s\leqslant t}$ on the Banach space $C_{0}(\mathds{R}^{d})$ via

$P_{s,t}^{\varepsilon f}g(x)=\mathbb{E}\left[g(x^{\varepsilon
f}_{t})|x^{\varepsilon}_{s}=x\right],$

and write $P_{s,t}^{0}=P_{t-s}$. The corresponding family of generators
$(L^{\varepsilon f}_{t})_{t\geqslant 0}$ read

$L^{\varepsilon f}_{t}g(x)=L^{0}-\varepsilon f_{t}(x)\cdot\nabla g(x),$

where

$\quad L^{0}g(x)=-\nabla U(x)\cdot\nabla g(x)+\Delta g(x).$

In this situation one verifies the following implicit relation between
$(P_{s,t}^{\varepsilon f})_{0\leqslant s\leqslant t}$ and
$(P_{s,t}^{0})_{0\leqslant s\leqslant t}$.

######  Proposition 2.1 (’Dyson Formula’).

For regular $U$ and $f$ it holds that

$\displaystyle P^{\varepsilon
f}_{s,t}g=P^{0}_{t-s}g-\varepsilon\int_{s}^{t}P^{0}_{s_{1}-s}(f_{s_{1}}\cdot\nabla
P_{s_{1},t}^{\varepsilon}g)\mathrm{d}s_{1}$ (2)

Nested iteration of $\eqref{eqn:Dyson}$ produces explicit expressions for the
power series expansion of $P^{\varepsilon f}_{s,t}$ in $\varepsilon$. Here we
record the first and second order terms, given that $\xi$ is distributed
according to the equilibrium invariant measure $\mu$. We denote by
$\left<\cdot\right>$ the expectation w.r.t. $\mu$.

######  Proposition 2.2.

Let $(x^{\varepsilon f}_{t\in[0,T]})_{\varepsilon,f}$, $(P_{s,t}^{\varepsilon
f})_{0\leqslant s\leqslant t}$ and $L^{\varepsilon f}_{t}$ as above with
$f_{t}=-h_{t}\nabla V$ and $\xi\sim\mu$ with
$\mu(\mathrm{d}x)=\frac{e^{-U(x)}}{Z}\mathrm{d}x$. Then,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}_{|{\varepsilon=0}}\mathbb{E}_{\mu}\bigl{[}k(x_{T}^{\varepsilon
f})\bigr{]}=-\int_{0}^{T}h_{t}\left<VL^{0}P_{T-t}^{0}k\right>\mathrm{d}t$

and

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}\varepsilon^{2}}_{|{\varepsilon=0}}\mathbb{E}_{\mu}\bigl{[}k(x_{T}^{\varepsilon
f})\bigr{]}$
$\displaystyle=-2\int_{0}^{T}\int_{0}^{t_{1}}h_{t_{1}}h_{t_{2}}\left<(L^{0}V)P_{t_{1}-t_{2}}(\nabla
V\cdot\nabla P_{T-t_{1}}k)\right>\mathrm{d}t_{2}\mathrm{d}t_{1}$

###### Proof.

As for the first order term, differentiation of (2) w.r.t. $\varepsilon$
yields

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}_{|{\varepsilon=0}}\mathbb{E}_{\mu}\left[{k(x_{T}^{\varepsilon
f})}\right]$ $\displaystyle=\int_{0}^{T}h_{t}\langle P^{0}_{t}(\nabla
V\cdot\nabla P_{T-t}^{0}k)\rangle\mathrm{d}t$
$\displaystyle=\int_{0}^{T}h_{t}\langle\nabla V\cdot\nabla
P_{T-t}^{0}k\rangle\mathrm{d}t$ $\displaystyle=-\int_{0}^{T}h_{t}\langle
V,L^{0}P_{T-t}^{0}k\rangle\mathrm{d}t,$

where used the $P^{0}_{t}$-invariance of $\mu$ and integration by parts in the
second resp. third line. As for the second order term, we expand the time
integral on the r.h.s. of (2) by another application of Dyson’s formula and
differentiate twice w.r.t. $\varepsilon$. This yields

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}\varepsilon^{2}}_{|\varepsilon=0}$
$\displaystyle\mathbb{E}_{\mu}\left[{k(x_{T}^{\varepsilon
f})}\right]=2\int_{0}^{T}\int_{0}^{t_{1}}h_{t_{1}}h_{t_{2}}\left<\nabla
V\cdot\nabla P_{t_{1}-t_{2}}(\nabla V\cdot\nabla
P_{T-t_{1}}k)\right>\mathrm{d}t_{2}\mathrm{d}t_{1}$

and hence the claim via another integration by parts.∎

In case of a conservative and time-homogeneous perturbation $f(t,x)=-\nabla
V(x)$ the expressions above simplify further.

######  Lemma 2.3.

In a setting as in proposition 2.2 but for $h_{t}=1$ for all $t\geqslant 0$,
the response simplifies to

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}_{|{\varepsilon=0}}\mathbb{E}_{\mu}\left[k(x_{T}^{\varepsilon
f})\right]=\langle Vk\rangle-\langle VP_{T}^{0}k\rangle$

and

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}\varepsilon^{2}}_{|\varepsilon=0}\mathbb{E}_{\mu}\left[k(x_{T}^{\varepsilon})\right]=\left<V^{2}k\right>-\left<VP_{T}^{0}(Vk)\right>+\int_{0}^{T}\left<VP_{t}^{0}(L^{0}VP_{T-t}^{0}k)\right>\mathrm{d}t\,.$
(3)

###### Proof.

Starting from proposition 2.2 in case of $h=1$

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}_{|{\varepsilon=0}}\mathbb{E}_{\mu}\left[k(x_{T}^{\varepsilon
f})\right]$ $\displaystyle=-\int_{0}^{T}\langle
VL^{0}P_{T-t}^{0}k\rangle\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\frac{\mathrm{d}}{\mathrm{d}t}\langle
VP_{T-t}^{0}k\rangle\mathrm{d}t$ $\displaystyle=\langle Vk\rangle-\langle
VP_{T}k\rangle.$

Likewise, for the second order term

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}\varepsilon^{2}}_{|\varepsilon=0}$
$\displaystyle\mathbb{E}_{\mu}\left[k(x_{T}^{\varepsilon})\right]=-2\int_{0}^{T}\int_{0}^{t_{1}}\left<(L^{0}V)P^{0}_{t_{1}-t_{2}}(\nabla
V\cdot\nabla P^{0}_{T-t_{1}}k)\right>\mathrm{d}t_{2}\mathrm{d}t_{1}$
$\displaystyle=$ $\displaystyle 2\int_{0}^{T}\left<V\nabla V\cdot\nabla
P^{0}_{T-t_{1}}k\right>\mathrm{d}t_{1}-2\int_{0}^{T}\left<VP^{0}_{t_{1}}(\nabla
V\cdot\nabla P^{0}_{T-t_{1}}k)\right>\mathrm{d}t_{1}$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\left<V^{2}L^{0}P^{0}_{T-t}k\right>\mathrm{d}t-\int_{0}^{T}\frac{\mathrm{d}}{\mathrm{d}t_{1}}\left<VP^{0}_{t_{1}}(VP^{0}_{T-t_{1}}k)\right>\mathrm{d}t$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{0}^{T}\left<VP^{0}_{t}(L^{0}VP^{0}_{T-t}k)\right>\mathrm{d}t$
$\displaystyle=$
$\displaystyle\left<V^{2}k\right>-\left<VP^{0}_{T}(Vk)\right>+\int_{0}^{T}\left<VP^{0}_{t}(L^{0}VP^{0}_{T-t}k)\right>\mathrm{d}t_{1}\leavevmode\nobreak\
.$

Here, we have used that $2V\nabla V=\nabla V^{2}$ and
$\frac{\mathrm{d}}{\mathrm{d}t}\left<VP_{t}^{0}(VP^{0}_{T-t}k)\right>$
$=\left<VP_{t}^{0}(L^{0}VP_{T-t}k)\right>+2\left<VP^{0}_{t}(\nabla
V\cdot\nabla(P^{0}_{T-t}k))\right>$. ∎

##  3 Subsystem spectral gap and coarse graining

The above mentioned inference problem corresponds to the situation when both
the observable $k$ and the potential $V$ are coarse grained, i.e. when

$k(x)=\kappa(\pi(x))\mbox{ and }V(x)=v(\pi(x))$

for some measurable map $\pi:\mathds{R}^{d}\mapsto\mathds{R}^{m}$, $m\leqslant
d$, and where $\kappa,v:\mathds{R}^{m}\mapsto\mathds{R}$ are known functions.
A natural idea is to use power series expansion in $\varepsilon$ and use lemma
2.3. However, the time integral in the the second order coefficient (3)
involves $L^{0}V$ which in general is not coarse grained, i.e. not a subsystem
observable and hence cannot be estimated from time-averaged equilibrium
$\pi(x^{0}_{\cdot})$-measurements. To overcome this problem, we shall propose
an approximation by certain subsystem observables below. To estimate the
induced error, we introduce a new quantity which controls the rate of
convergence of subsystem observables.

Here and in the sequel we shall assume for simplicity that the generator $L$
of the equilibrium process has discrete spectrum.

######  Definition 3.1.

Let $L$ be the generator of an $\mathds{R}^{d}$-valued $\mu$-reversible Markov
process $x_{\cdot}$ and let
$\pi:\mathds{R}^{d}\mapsto\mathds{R}^{m},m\leqslant d$ measurable. Let

$S=L^{2}_{\mu}(\mathds{R}^{d},\sigma(\pi))\subset
L^{2}_{\mu}(\mathds{R}^{d}),$

where $\sigma(\pi)$ denotes the sigma algebra generated on $\mathds{R}^{d}$ by
$\pi$, be the closed subspace of square integrable subsystem observables and
denote by $\mathrm{Sp}(-L)$ the spectrum of $-L$, then the quantity

$\lambda_{\pi}=\inf\left\\{\lambda\in\mathrm{Sp}(-L)\backslash\\{0\\}|\neg(Eig(\lambda)\perp
L^{2}(\mathds{R}^{d},\sigma(\pi),\mu))\right\\}$

is called subssystem spectral gap of $\pi(x^{0}_{\cdot})$.

The coarse grained process $(\pi(x^{0}_{t}))_{t\geqslant 0}$ is not assumed to
be Markovian in the definition above. The following statement is a simple
consequence and yields a criterion when the passage to the subsystem improves
the spectral gap.

######  Corollary 3.2.

In the situation as above let $\lambda_{*}$ be the spectral gap of the full
system $x_{\cdot}^{0}$, then

$\lambda\geqslant\lambda_{*}$

with strict inequality if and only if

$\mathbb{E}_{\mu}(g|\sigma(\pi))=0\quad\forall g\in\mathop{\rm
Eig}(\lambda_{*}).$

######  Example 3.3.

Consider the time-continuous Markov chain $(x_{t})_{t\geqslant 0}$ on the
state space $E=\\{1,2,3,4\\}$ with generator

$\displaystyle Q=\begin{pmatrix}-\frac{9}{10}&\frac{8}{10}&0&\frac{1}{10}\\\
\frac{8}{10}&-\frac{9}{10}&\frac{1}{10}&0\\\ 0&\frac{1}{10}&-1&\frac{9}{10}\\\
\frac{1}{10}&0&\frac{9}{10}&-1\end{pmatrix}$

$Q$ is symmetric w.r.t. the invariant distribution invariant distribution
$\mu=\frac{1}{4}(1,1,1,1)$ with spectral gap $\lambda_{*}=\frac{1}{5}$.
Dividing the system into two subsystems $E_{1}=\\{1,4\\},E_{2}=\\{2,3\\}$
yields a new (non Markovian) stochastic process $\pi(x_{t})=1_{\\{x_{t}\in
E_{1}\\}}-1_{\\{x_{t}\in E_{2}\\}}$ on the state space
$E^{\prime}=\\{1,-1\\}$, which has a subsystem spectral gap of
$\lambda_{\pi}=\frac{18-\sqrt{2}}{10}$. In fact, the other eigenvectors other
than $\mu$ for $Q$ are
$v_{2}^{T}=(-1,-1,1,1),v_{3}^{T}=(1+\sqrt{2},-1-\sqrt{2},-1,1),v_{4}=(1-\sqrt{2},-1+\sqrt{2},-1,1)$
with eigenvalues
$-\lambda_{2}=-\frac{1}{5},-\lambda_{3}=-\frac{18-\sqrt{2}}{10},-\lambda_{4}=-\frac{18+\sqrt{2}}{10}$
and $v_{2}\perp
L^{2}_{\mu}(E,\sigma(\pi))=\\{(u,v,v,u)^{T}\in\mathds{R}^{4}\,|\
u,v\in\mathds{R}\\}$. $\Box$

The quantity $\lambda_{\pi}$ controls the exponential decay to equilibrium for
subsystem observables in $L^{2}$-sense.

######  Proposition 3.4.

For $f\in L^{2}_{\mu}(\mathds{R}^{d},\sigma(\pi))$ with $\left<f\right>=0$, it
holds that

$\displaystyle\left<(P_{t}f)^{2}\right>\leqslant
e^{-2\lambda_{\pi}t}\left<f^{2}\right>$ (4)

Proof. Denote by $\lambda_{0}=0<\lambda_{1}=\lambda_{*}<\lambda_{2}<\cdots$
the set of eigenvalues of $-L$ and $(\varphi_{n})_{n\in\mathds{N}}\subset
L^{2}(\mu)$ a set of orthonormal eigenfunctions. Then, for any $f\in
L^{2}_{\mu}(\mathds{R}^{d},\sigma(\pi))$ with $\langle f\rangle=0$, by the
spectral theorem

$\displaystyle P_{t}f$
$\displaystyle=\sum_{n=1}^{\infty}\left<\varphi_{n},f\right>e^{-\lambda_{n}t}\varphi_{n}$
$\displaystyle=\sum_{\stackrel{{\scriptstyle n\geqslant
1}}{{\varphi_{n}\not\perp
L^{2}_{\mu}(R^{d},\sigma(\pi))}}}\left<\varphi_{n},f\right>e^{-\lambda_{n}t}\varphi_{n},$

hence

$\displaystyle\left<(P_{t}f)^{2}\right>$
$\displaystyle=\sum_{\stackrel{{\scriptstyle n\geqslant
1}}{{\varphi_{n}\not\perp
L^{2}_{\mu}(R^{d},\sigma(\pi))}}}\left<\varphi_{n},f\right>^{2}e^{-2\lambda_{n}t}$
$\displaystyle\leqslant e^{-2\lambda_{\pi}t}\sum_{\stackrel{{\scriptstyle
n\geqslant 1}}{{\varphi_{n}\not\perp
L^{2}_{\mu}(R^{d},\sigma(\pi))}}}\left<\varphi_{n},f\right>^{2}$
$\displaystyle=e^{-2\lambda_{\pi}t}\left<f^{2}\right>.$ $\Box$

We can now introduce our estimator for the dynamic second order response in
terms of subsystem observables as follows.

######  Proposition 3.5.

Let $(x_{t}^{\varepsilon})_{t\geqslant 0}$ be the solution to (1) with
$f_{t}(x)=-\nabla V$ with $V(x)=v(\pi(x))$ and let $(P_{t})$ denote the
semigroup associated to $x^{0}_{\cdot}$. Then, for all $T>0$ and
$k(x)=\kappa(\pi(x))$

$\displaystyle\begin{split}\frac{\mathrm{d}^{2}}{\mathrm{d}\varepsilon^{2}}_{\bigl{|}\varepsilon=0}\mathbb{E}_{\mu}\left[k(x_{T}^{\varepsilon})\right]=&\left<V^{2}k\right>-\left<VP_{T}Vk\right>+\left<V\right>\left(\left<P_{T/2}k,V\right>-\left<kV\right>\right)\\\
&+\left<k\right>\left(\left<P_{T/2}V,V\right>-\left<V^{2}\right>\right)+f_{k}(T)+f_{V}(T)\end{split}$

where

$\displaystyle|f_{V}(T)$ $\displaystyle+f_{k}(T)|$
$\displaystyle\leqslant\left(\lVert k\rVert_{L^{\infty}}\lVert
V\rVert_{L^{2}}+\lVert V\rVert_{L^{\infty}}\lVert k\rVert_{L^{2}}\right)\lVert
LV\rVert_{L^{2}}\frac{1}{\lambda_{\pi}}\left(e^{-\lambda_{\pi}T/2}-e^{-\lambda_{\pi}T}\right)$

and $\lambda_{\pi}$ denotes the subsystem spectral gap associated to
$\pi(x^{0}_{\cdot})$.

###### Proof.

We need to approximate the remaining integral

$\displaystyle\int_{0}^{T}\left<VP_{t}\left((LV)P_{T-t}k\right)\right>\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\left<VP_{T-t}\left((LV)P_{t}k\right)\right>\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\left<(P_{T-t}V)(P_{t}k)LV\right>\mathrm{d}t$
$\displaystyle=\int_{0}^{T/2}\left<V\right>\left<(P_{t}k)LV\right>\mathrm{d}t+\int_{0}^{T/2}\left<\left(P_{T-t}V-\left<V\right>\right)(P_{t}k)LV\right>\mathrm{d}t$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\int_{T/2}^{T}\left<k\right>\left<(P_{T-t}V)LV\right>\mathrm{d}t+\int_{T/2}^{T}\left<(P_{t}k-\left<k\right>)(P_{T-t}V)LV\right>\mathrm{d}t$
$\displaystyle\begin{split}&=\left<V\right>\left(\left<P_{T/2}k,V\right>-\left<kV\right>\right)+f_{V}(T)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\left<k\right>\left(\left<P_{T/2}V,V\right>-\left<V^{2}\right>\right)+f_{k}(T)\end{split}$

with

$\displaystyle f_{V}(T)$
$\displaystyle=\int_{0}^{T/2}\left<(P_{T-t}V-\left<V\right>)(P_{t}k)LV\right>\mathrm{d}t$
and $\displaystyle|f_{V}(T)|$ $\displaystyle\leqslant\int_{0}^{T/2}\lVert
P_{T-t}V-\left<V\right>\rVert_{L^{2}}\lVert P_{t}k\rVert_{L^{\infty}}\lVert
LV\rVert_{L^{2}}\mathrm{d}t$ $\displaystyle\leqslant\lVert
LV\rVert_{L^{2}}\lVert
k\rVert_{L^{\infty}}\int_{0}^{T/2}e^{-\lambda_{\pi}(T-t)}\lVert
V\rVert_{L^{2}}\mathrm{d}t$ $\displaystyle=\lVert LV\rVert_{L^{2}}\lVert
k\rVert_{L^{\infty}}\frac{1}{\lambda_{\pi}}\left(e^{-\lambda_{\pi}T/2}-e^{-\lambda_{\pi}T}\right)\lVert
V\rVert_{L^{2}}$

where we have used Cauchy-Schwarz and the contractivity of the semigroup
$(P_{t})_{t}$. Similarly,

$\displaystyle f_{k}(T)$
$\displaystyle=\int_{T/2}^{T}\left<(P_{t}k-\left<k\right>)(P_{T-t}V)LV\right>\mathrm{d}t,$
hence $\displaystyle|f_{k}(T)|$ $\displaystyle\leqslant\lVert
LV\rVert_{L^{2}}\lVert V\rVert_{L^{\infty}}\lVert
k\rVert_{L^{2}}\frac{1}{\lambda_{\pi}}\left(e^{-\lambda_{\pi}T/2}-e^{-\lambda_{\pi}T}\right).$

Summing all terms yields the claim. ∎

######  Remark 3.6.

In the event that the potential is unbounded, we can still make an estimate,
namely

$\displaystyle|f_{k}(T)|$
$\displaystyle\leqslant\int_{T/2}^{T}\lVert(P_{T-t}V)(LV)\rVert_{L^{2}}\lVert
P_{t}k-\left<k\right>_{\mu_{0}}\rVert_{L^{2}}$
$\displaystyle\leqslant\int_{T/2}^{T}\lVert(P_{T-t}V)\rVert_{L^{2p}}\lVert
LV\rVert_{L^{2q}}\lVert P_{t}k-\left<k\right>_{\mu_{0}}\rVert_{L^{2}}$
$\displaystyle\leqslant\lVert V\rVert_{L^{2p}}\lVert LV\rVert_{L^{2q}}\lVert
k\rVert_{L^{2}}\frac{1}{\lambda_{\pi}}\left(e^{-\lambda_{\pi}T/2}-e^{-\lambda_{\pi}T}\right)$

where we again applied Hölder inequality again and used the fact that the
$L^{\infty}$ contractivity of the semigroup implies contractivity $L^{p}$ for
any $1\leqslant p\leqslant\infty$, c.f. [5].

##  4 Numerical experiments

As a consequence of lemma 2.3 and proposition 3.5 we arrive at the following
estimator for $\mathbb{E}_{\mu}(k(\pi(x_{t}^{\varepsilon}))$

$\displaystyle
e_{\mu}(k(\pi(x_{T}^{\varepsilon}))=\left<k\right>+\varepsilon\left[\langle
Vk\rangle-\langle VP_{T}k\rangle\right]$
$\displaystyle+\frac{\varepsilon^{2}}{2}\left[\left<V^{2}k\right>-\left<VP_{T}Vk\right>+\left<V\right>\left(\left<P_{T/2}k,V\right>-\left<kV\right>\right)+\left<k\right>\left(\left<P_{T/2}V,V\right>-\left<V^{2}\right>\right)\right],$

which is composed entirely of equilibrium subsystem observables. Hence
$e_{\mu}(k(\pi(x_{t}^{\varepsilon}))$ can be estimated from ergodic avarages
of corresponding subsystem measurements at equilibrium. The statistical error
in this ergodic approximation is also controlled by the subsystem spectral
gap, which however may be difficult to estimate in practice. We present two
examples of the performance of this approach, using a straightforward Euler-
Maruyama scheme for the numerical simulation of the full system trajectories
$x^{0}$ and $x^{\varepsilon}$.

###  4.1 A system with two time scales

Let $x_{t}^{\varepsilon}\in\mathds{R}^{2}$ be the solution to the Ito SDE

$\displaystyle\begin{split}\mathrm{d}x_{t}^{\varepsilon}&=-\left(ax_{t}^{\varepsilon}+V^{\prime}(\pi^{1}x_{t}^{\varepsilon})\right)\mathrm{d}t+\mathrm{d}W_{t}\\\
x_{0}^{\varepsilon}&=\xi\quad\xi\sim e^{-U(x)}\mathrm{d}x\end{split}$ (5)

given a 2 dimensional BM $(W_{t})_{t}$ and

$\displaystyle a=\begin{pmatrix}2&-r\\\ -r&2r\end{pmatrix}$ (6)

Hence, the (strong) solution to the equilibrium equation is an Ornstein
Uhlenbeck process

$\displaystyle
x_{t}^{0}=e^{-at}\xi+\sqrt{2}\int_{0}^{t}e^{-a(t-s)}\mathrm{d}W_{s}\leavevmode\nobreak\
.$ (7) We consider the projection $\pi(x)=e_{1}\cdot x$. $\displaystyle
P_{t}^{0}\pi(x)=e_{1}^{T}\cdot e^{-at}x\quad\forall\leavevmode\nobreak\
x\in\mathds{R}^{2}\leavevmode\nobreak\ .$ (8)

It is useful to know that the eigenvalues of $a$ are
$\lambda_{1/2}=\mp\sqrt{2r^{2}-2r+1}+r+1$. Note that $\lambda_{1}(r=0)=0$ so
that the spectral gap vanishes when $r=0$.

$\displaystyle a^{-1}=\frac{1}{4r-r^{2}}\begin{pmatrix}2r&r\\\
r&2\end{pmatrix}$

The error of observing the position of the projected observable $k(x)=\pi(x)$
can be estimated by proposition 3.5 by

$\displaystyle|f_{V}(T)+f_{k}(T)|\leqslant$ $\displaystyle\left(\lVert
k\rVert_{L^{2p}}\lVert V\rVert_{L^{2}}\lVert LV\rVert_{L^{2q}}+\lVert
V\rVert_{L^{\infty}}\lVert k\rVert_{L^{2}}\right)\lVert
LV\rVert_{L^{2}}\frac{1}{\lambda}\left(e^{-\lambda T/2}-e^{-\lambda T}\right)$

where we estimate the constant by

$\displaystyle\left(\lVert k\rVert_{L^{2p}}\lVert V\rVert_{L^{2}}\lVert
LV\rVert_{L^{2q}}+\lVert V\rVert_{L^{\infty}}\lVert
k\rVert_{L^{2}}\right)\lVert LV\rVert_{L^{2}}$ $\displaystyle\leqslant$
$\displaystyle\left(3\left(\frac{2r}{4-r}\right)^{2}\right)^{\frac{1}{4}}\lVert\left((2x_{1}-rx_{2})\sin(x_{1}-\frac{\pi}{4})-\cos(x_{1}-\frac{\pi}{4})\right)\rVert_{4}$
$\displaystyle+\frac{2}{\sqrt{4-r}}\lVert\left((2x_{1}-rx_{2})\sin(x_{1}-\frac{\pi}{4})-\cos(x_{1}-\frac{\pi}{4})\right)\rVert_{2}$
$\displaystyle\leqslant$
$\displaystyle\left(3\left(\frac{2r}{4-r}\right)^{2}\right)^{\frac{1}{4}}\left<\left((2x_{1}-rx_{2})\right)^{4}\right>^{1/4}+\frac{2}{\sqrt{4-r}}\left<(2x_{1}-rx_{2})^{2}\right>^{\frac{1}{2}}$
$\displaystyle\leqslant$
$\displaystyle\left(3\left(\frac{2r}{4-r}\right)^{2}\right)^{\frac{1}{4}}\left(\frac{3}{(4r-r^{2})^{2}}\left(64r^{2}+32r^{3}+20r^{4}-8r^{5}\right)\right)^{1/4}+\frac{2}{\sqrt{4-r}}\left(\frac{2r+4}{4-r}\right)^{\frac{1}{2}}$
$\displaystyle\approx$ $\displaystyle 2.564$

for the choice of $r=0.1$ with $p=q=2$. The spectral gap in this case is
$\lambda=\lambda_{1}\approx 0.194$. Note that nevertheless, the approximation
of the second order response yields an improved estimate of the nonequilibrium
trajectory as seen from figure 1.

(a) The error bound from equation 1 is indicated by the blue opaque area.

(b) A closeup shows that the approximation of proposition 3.5 yields much
better results than predicted by the error bound.

Figure 1: The trajectory of a system given by equation (5).

###  4.2 Evolution in a multiwell potential

For SDEs we construct a potential landscape mimicking the behaviour of the
Markov chain in example 3.3:

$\displaystyle U$
$\displaystyle=U_{\Gamma_{1}}+U_{\Gamma_{2}}+\sum_{i=1}^{4}U_{i}+U_{C}$
$\displaystyle U_{\Gamma_{1}}\left(x,y\right)$
$\displaystyle=-\frac{\sqrt{2\pi}}{\sigma}e^{-\frac{1}{2\sigma}(y-1)^{2}}\quad
U_{\Gamma_{2}}\left(x,y\right)=-\frac{\sqrt{2\pi}}{\sigma_{m}}e^{-\frac{1}{2\sigma_{m}}(y+1)^{2}}$
$\displaystyle U_{1}\left(x,y\right)$
$\displaystyle=-\frac{1}{2\sigma_{1}}e^{-(x-1)^{2}-(y-1)^{2}}\quad
U_{2}\left(x,y\right)=-\frac{1}{2\sigma_{1}}e^{-(x+1)^{2}-(y-1)^{2}}$
$\displaystyle U_{3}\left(x,y\right)$
$\displaystyle=-\frac{1}{2\sigma_{2}}e^{-(x+1)^{2}-(y+1)^{2}}\quad
U_{4}\left(x,y\right)=-\frac{1}{2\sigma_{2}}e^{-(x-1)^{2}-(y+1)^{2}}$
$\displaystyle U_{C}\left(x,y\right)$
$\displaystyle=(x^{2}+y^{2})/10\leavevmode\nobreak\ .$

As a coarse-grained variable, consider the projection onto the $x$ axis
$\pi(x,y)=x$. Figure 2 shows the mean value of $k(x,y)=\pi(x,y)=x$ under the
perturbing potential, which increases the probability to remain in one of the
right hand side minima:

$\displaystyle V(x,y)=v(x)=\exp\left(-(x-1)^{2}/4\right)$

(a) The potential $U$ of the unperturbed dynamics with
$\sigma=\frac{1}{2},\leavevmode\nobreak\ \sigma_{m}=\frac{1}{3}$ and
$\sigma_{1}=\sigma_{2}=\frac{1}{10}$.

(b) The linear and second order response of the $x$ coordinate in a system
evolving in the potential landscape on the left hand side to the potential
$v(x)=\exp\left(-(x-1)^{2}/4\right)$ (blue).

Figure 2: Evolution in a multiwell potential

By Kramer’s rule the relaxation along the $x$-axis is much faster than the one
along the $y$-axis. Therefore, projecting onto the $x$ axis should yield a
better approximation of the second order response than projecting onto the $y$
axis. Furthermore, figure 2 (b) is consistent with the error bound of 3.5: the
approximation up to second order is very good for small times, and large times
but cannot fully capture the dynamics on intermediate time scales.

Acknowledgments. We thank Klaus Kroy for very valuable remarks which helped to
improve this paper and Victor Marx for his input at an early stage of this
work.

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