# Explicit generators of the Steinberg Module of the mapping class group

Ingrid Irmer  SUSTech International Center for Mathematics
Southern University of Science and Technology
Shenzhen, China  Department of Mathematics
Southern University of Science and Technology
Shenzhen, China<EMAIL_ADDRESS>
###### Abstract.

A conjecture of Broaddus is proven, giving a simple characterisation of a
representative of the unique orbit of the action of the mapping class group on
the homology of Harvey’s complex of curves for any genus surface. As an
application, the kernel of the action of the mapping class group of a genus
$g$ surface on the Steinberg module is shown to be trivial.

Mathematics Classification. Primary 57K20. Secondary 57R70.
Key words and phrases: Mapping class groups, Moduli space, Morse Theory.

###### Contents

  1. 1 Introduction
  2. 2 Assumptions and Background
  3. 3 From critical points to spheres
  4. 4 The Set of Minima
  5. 5 Proof of the Conjecture

##  1\. Introduction

It was shown in [6], Theorem 3.5, that the complex of curves,
$\mathcal{C}_{g}$, of a closed, orientable topological surface
$\mathcal{S}_{g}$ of genus $g\geq 2$ has the homotopy type of an infinite
wedge of spheres of dimension $2g-2$. In [4], an algorithm for constructing a
homotopically nontrivial sphere was given, and an explicit example was
computed in genus 2. This paper proves a conjecture showing that a
considerably simpler construction of spheres: “Broaddus spheres” - of which
there is one in each genus - could be used.

The reduced homology group
$\tilde{H}_{2g-2}\bigl{(}\mathcal{C}_{g};\mathbb{Z}\bigr{)}$ is also referred
to as the Steinberg module $\mathrm{St}(\mathcal{S}_{g})$. Since there is a
simplicial action of the mapping class group $\Gamma_{g}$ of $\mathcal{S}_{g}$
on $\mathcal{C}_{g}$, $\mathrm{St}(\mathcal{S}_{g})$ has the structure of a
$\Gamma_{g}$-module. It was also shown in [4] that
$\mathrm{St}(\mathcal{S}_{g})$ is cyclic modulo the action of the mapping
class group.

######  Theorem 1.1.

$\mathrm{St}(\mathcal{S}_{g})$ is generated by the mapping class group orbit
of the Broaddus sphere of genus $g$.

A Poincaré duality group is a group whose homology and cohomology satisfy a
duality property analogous to the Poincaré duality between the homology and
cohomology of a compact manifold. This definition was generalised in [3] to
the notion of a duality group. Informally, a duality group is a group whose
homology and cohomology satisfy a property analogous to Poincaré duality, with
a nontrivial “dualising module” taking the place of the orientation module. It
was shown in Theorem 6.6 of [7] that the mapping class group of a closed
orientable surface is a virtual duality group, and that the dualising module
is $\mathrm{St}(\mathcal{S}_{g})$.

For genus $g$ at least 3, the center $Z_{g}$ of $\Gamma_{g}$ is trivial, as
shown in Theorem 3.10 of [5]. In Section 3.4 of [5] it was shown that the
center of $Z_{2}$ is generated by involutions that act trivially on
$\mathcal{C}_{g}$. The techniques of this paper can be used to relate the
stabiliser subgroup of the action of $\Gamma_{g}$ on a generator of
$\mathrm{St}(\mathcal{S}_{g})$.

######  Corollary 1.2.

The kernel of the action of $\Gamma_{g}/Z_{g}$ on
$\mathrm{St}(\mathcal{S}_{g})$ is trivial.

The systole function $f_{sys}:\mathcal{T}_{g}\rightarrow\mathbb{R}_{+}$ is the
piecewise smooth map whose value at any point of $\mathcal{T}_{g}$ is the
length of the systoles. The systole function is known to be a topological
Morse function, [1]. These are defined in [9], and can be used to construct
cell decompositions of topological spaces analogous to those of (smooth) Morse
functions.

The central idea is that the Broaddus sphere can be described using the set of
“admissible boundary points” of a set of minima, as defined in [12].
Informally, the sets of minima in this paper behave like “unstable manifolds”
of critical points of $f_{sys}$. A set of minima, or a critical point, is
labelled by a set of curves. The mapping class group maps sets of minima to
sets of minima, and critical points to critical points. As this action is
determined by the action of the mapping class group on curves, the action of
the mapping class group on the Broaddus spheres is identical to the action on
the sets of minima that they bound in $\mathcal{T}_{g}$. The Broaddus spheres
are shown to represent nontrivial homology classes in
$\mathrm{St}(\mathcal{S}_{g})$ by studying the way the fixed point sets of the
action of the $\Gamma_{g}$ on the sets of minima restrict homotopies.

Outline of the paper. Section 2 first introduces some background and
notations. For the benefit of readers who are not be familiar with Schmutz’s
sets of minima, a survey of their properties is given. The construction from
[4] of a generator of $\mathrm{St}(\mathcal{S}_{g})$, as well as a related
example of a critical point due to Schmutz are given in 3. The relationship
between these ingredients is made clear in Section 4. The “unstable manifold”
of the critical point in the form of a certain set of minima, is identified
with the ball in Teichmüller space bounded by the sphere corresponding to
Broaddus’s conjectured generator of $\mathrm{St}(\mathcal{S}_{g})$. The
vertices of this triangulated sphere are shown to correspond to particular
boundary points of the set of minima. In Section 5 symmetries are used to show
that the homology class of the sphere is nontrivial.

##  2\. Assumptions and Background

The orientable, closed, compact, connected topological surface of genus $g$
will be denoted by $\mathcal{S}_{g}$; when this surface is given a marked
hyperbolic structure corresponding to a point in Teichmüller space, it will be
denoted by $S_{g}$.

The Teichmüller space of $\mathcal{S}_{g}$ will be denoted by
$\mathcal{T}_{g}$, the mapping class group by $\Gamma_{g}$, moduli space by
$\mathcal{M}_{g}$, and Harvey’s curve complex by $\mathcal{C}_{g}$. For
convenience, the simplicial complex $\mathcal{C}_{g}$ will be confused with
its geometric realisation.

A curve is a homotopy class of embeddings of $S^{1}$ into $\mathcal{S}_{g}$.
When there is no possibility of misunderstanding, the word curve will also
refer to a particular representative of the homotopy class, such as a
geodesic. A set of curves is said to fill $\mathcal{S}_{g}$ if the complement
of a set of representatives in minimal positition cuts the surface into
polygons.

A curve $c$ determines an analytic map
$L(c):\mathcal{T}_{g}\rightarrow\mathbb{R}_{+}$ whose value at any point is
given by the length of the geodesic representative of $c$. A length function
$L(A,C)$ is a linear combination of such functions with positive coefficients
where $C$ is a finite set of $|C|$ curves, and
$A=(a_{1},\ldots,a_{|C|})\in\mathbb{R}^{|C|}_{+}$ the coefficients. Length
functions satisfy many convexity properties, for example they are strictly
convex along Weil-Petersson geodesics, [14]. The Weil-Petersson metric will be
assumed whenever a metric is needed on $\mathcal{T}_{g}$.

The systoles of $S_{g}$ is the set of shortest curves; this is always finite.
The set of points in $\mathcal{T}_{g}$ at which the set of systoles is exactly
$C$ will be denoted by $\mathrm{Sys}(C)$. The Thurston spine is the CW complex
contained in $\mathcal{T}_{g}$ consisting of the set of points at which the
systoles fill $\mathcal{S}_{g}$.

Teichmüller space has a decomposition into the thick part, where $f_{sys}$ is
less than or equal to the Margulis constant, and the thin part, which is the
complement of the thick part. For convenience, a decomposition into a
$\delta$-thick part and $\delta$-thin part will be used, where $\delta>0$
might be less than the Margulis constant. The $\delta$-thick part of
$\mathcal{T}_{g}$ is the subset of $\mathcal{T}_{g}$ on which $f_{sys}$ is
greater than or equal to $\delta$, and will be denoted by
$\mathcal{T}_{g}^{\delta}$.

It was shown in [7] that $\mathcal{C}_{g}$ is $\Gamma_{g}$-equivariantly
homotopy equivalent to the boundary of the thick part of Teichmüller space.
The idea behind this is very simple. In the thin part of $\mathcal{T}_{g}$,
the systoles are pairwise disjoint; at a given point they determine a
multicurve $m$. The intersection of $\mathrm{Sys}(m)$ with the boundary of the
thick part of $\mathcal{T}_{g}$ corresponds to the cell labelled by $m$ in the
dual of $\mathcal{C}_{g}$. The same argument as in [7] gives an embedding of
$\mathcal{C}_{g}$ in the boundary of $\mathcal{T}_{g}^{\delta}$.

Sets of minima. Let $C$ be a finite set of filling curves. From Lemma 1 of
[12], it is known that every length function given by a strictly positive
linear combination of the lengths of curves in $C$ has a unique minimum at
some point of $\mathcal{T}_{g}$. The set $\mathrm{Min}(C)$ defined in [12]
consists of all points in $\mathcal{T}_{g}$ at which a length function of the
form $L(A,C)$ for some $A\in\mathbb{R}_{+}^{|C|}$ has its minimum. A reference
for sets of minima is Section 2 of [12].

There is a surjective map
$\phi_{C}:\mathbb{R}^{|C|}_{+}\twoheadrightarrow\mathrm{Min}(C)$, where here
$\phi(A)$ is the point in $\mathrm{Min}(C)$ at which $L(A,C)$ has its minimum.
Note that $\phi_{C}$ is not injective. This paper will be concerned with
specific examples of sets $\mathrm{Min}(C)$ that are known to be
differentiable cells with empty boundary.

Suppose $C=\\{c_{1},\ldots,c_{k}\\}$ is a set of filling curves. The lengths
of curves in $C$ are said to parameterise $\mathrm{Min}(C)$. For any $x_{1}$,
$x_{2}$ in $\mathrm{Min}(C)$, the map
$F:\mathrm{Min}(C)\rightarrow\mathbb{R}_{+}^{k}$ given by
$x\mapsto(L(c_{1})(x),\ldots,L(c_{k})(x))$ has the property that
$F(x_{1})=F(x_{2})$ only if $x_{1}=x_{2}$. When the rank of the Jacobian of
$F(C)$ is constant on $\mathrm{Min}(C)$, as will always be the case in this
paper, Section 2 of [12] showed how to use the lengths of the curves in $C$ to
obtain a set of coordinates on $\mathrm{Min}(C)$.

######  Lemma 2.1 (Lemma 4 of [2], or Proposition 1 from [13]).

Let $C^{\prime}$ be any collection of curves on a surface that do not fill.
Then at any point of $\mathcal{T}_{g}$, there are tangent vectors
corresponding to directions in which the lengths of all the geodesics
representing curves in $C^{\prime}$ are increasing.

It follows from Lemma 14 of [12] that points on $\partial\mathrm{Min}(C)$ in
the interior of $\mathcal{T}_{g}$ are minima of strictly positive linear
combinations of filling subsets of $C$. For a length function of the form
$L(A,C^{\prime})$, where $C^{\prime}$ is a nonfilling subset of $C$, Lemma 2.1
implies that any infima are realised as limits of sequences of points in
$\mathrm{Min}(C)$. These are Schmutz’s “admissible boundary points”, [12],
Section 2.

##  3\. From critical points to spheres

This section begins by surveying an important family of examples of critical
points of the topological Morse function $f_{sys}$. These examples will later
be related to the construction of $2g-2$-dimensional spheres in
$\mathcal{C}_{g}$, whose construction is also given in this section.

######  Example 3.1 (A critical point of $f_{sys}$ of index $2g-1$, from
Theorem 36 of [12]).

Take a regular, right angled hyperbolic polygon $T$ with $2g+2$ sides. Four
copies of $T$ can be glued together along four pairs of edges with a common
vertex. This is done in such a way that each copy of $T$ shares one edge with
two others. A right angled hyperbolic polygon with $8g-4$ sides is obtained.
The construction with $g=2$ is shown in Figure 1. This gives a fundamental
domain of a surface. The sides of the fundamental domain are glued together in
such a way that each edge of a copy of $T$ lies along a geodesic loop on the
surface obtained by traversing exactly two edges, each on different copies of
$T$. It transpires that the gluing maps are completely determined by this
condition. When given a marking, this surface corresponds to a point in
$\mathcal{T}_{g}$ that will be denoted by $p$.

It was shown in [12] that $p$ is a critical point of index $2g-1$ and that the
set $C$ of systoles at $p$ is the set of curves that can be embedded in the
graph on $S_{g}$ consisting of the edges of the four copies of $T$. This gives
$2g+2$ systoles, each of which intersects exactly two other systoles. Let
$c_{1}$ and $c_{2}$ be a pair of systoles that intersect. It was shown that
$\mathrm{Min}(C{\smallsetminus}\\{c_{1},c_{2}\\})=\mathrm{Min}(C)$. For every
$x\in\mathrm{Min}(C)$, the gradients $\\{\nabla L(c)\ |\ c\in C\\}$ lie in a
$2g-1$ dimensional subspace of $T_{x}\mathcal{T}_{g}$. As
$C{\smallsetminus}\\{c_{1},c_{2}\\}$ has no proper filling subsets and the
rank of the Jacobian of the map $\mathcal{T}_{g}\rightarrow\mathbb{R}^{2g+2}$
given by $x\mapsto\bigl{(}L(c_{1}),\ldots,L(c_{2g+2})\bigr{)}$ is constant and
equal to $2g-1$ everywhere on $\mathrm{Min}(C)$, it follows from Corollary 13
and Lemma 14 of [12] that $\mathrm{Min}(C)$ is a continously differentiable
cell with empty boundary.

The cell $\mathrm{Min}(C)$ behaves like the unstable manifold of the critical
point $p$. The definition of $\mathrm{Min}(C)$ implies that any vector in the
tangent space to $\mathrm{Min}(C)$ at $p$ gives a direction in which the
length of at least one curve in $C$ is decreasing. On a neighbourhood of $p$,
the systoles are contained in $C$, so $p$ is a local maximum for $f_{sys}$ in
$\mathrm{Min}(C)$.

The point $p$ is an element of a $\Gamma_{g}$-orbit of critical points. The
systole function is invariant under the action of $\Gamma_{g}$. A
$\gamma\in\Gamma_{g}$ maps a critical point with set of systoles $C$ to a
critical point with systoles given by the image of $C$ under $\gamma$.
Similarly, $\Gamma_{g}$ acts on sets of minima, mapping $\mathrm{Min}(C)$ to
the set of minima of the image of $C$ under $\gamma$.

Figure 1. The left side of this figure (not drawn to scale!) shows a
fundamental domain of the genus 2 surface in Example 3.1. The systoles lie
along the boundary and the edges shown. The numbers on the edges are intended
to indicate the gluing maps. The right side of the figure shows the systoles
on the surface.  Figure 2. This figure is reproduced from [4] and shows the
1-skeleton of the triangulated sphere in $\mathcal{C}_{2}$ constructed by the
algorithm below. Theorem 4.1 shows this is homotopic in $\mathcal{C}_{g}$ to
the intersection of $\mathrm{Min}(C)$ from Example 3.1 with the embedding of
$\mathcal{C}_{g}$ in $\partial\mathcal{T}_{g}^{\delta}$.

Construction of the Sphere. The $2g-2$ dimensional sphere in $\mathcal{C}_{g}$
from Conjecture 4.9 of [4] is constructed as follows: Let $C$ be the set of
$2g+2$ curves from Example 3.1 and let $K_{2g+2}$ be a $(2g+2)$-gon. Choose a
bijection between the vertices of $K_{2g+2}$ and the elements of $C$ with the
property that adjacent vertices of $K_{2g+2}$ are labelled by intersecting
curves. The symbol $\Phi_{g}$ will be used to denote the symplicial complex
whose vertices correspond to diagonals of $K_{2g+2}$ and whose simplices
correspond to sets of disjoint diagonals of $K_{2g+2}$.

An associahedron $\mathcal{K}_{n}$ is a polytope of dimension $n$ whose
vertices correspond to all the distinct ways of parenthesising the word
$x_{0}x_{1}\ldots x_{n+1}$. As shown in [8], Theorem 1, the simplicial complex
$\Phi_{g}$ is the boundary of the dual of the $2g-1$-dimensional associahedron
$\mathcal{K}_{2g-1}$, and is therefore a triangulation of the
$2g-2$-dimensional sphere.

The barycentric subdivision of $\mathcal{C}_{g}$ will be denoted by
$\mathcal{C}_{g}^{\circ}$. Recall that each vertex $v$ of $\Phi_{g}$ is
labelled by pair of nonintersecting curves $c(v)$ in $C$. A simplicial map

$q^{\circ}_{g}:\Phi_{g}\rightarrow\mathcal{C}_{g}^{\circ}$ (1)

is contructed, where $q^{\circ}_{g}$ maps a vertex $v$ to the vertex in
$\mathcal{C}_{g}^{\circ}$ labelled by the multicurve
$m\bigl{(}C{\smallsetminus}c(v)\bigr{)}$ consisting of the set of nontrivial
homotopy classes of curves on the boundary of the surface obtained by cutting
$S_{g}$ along the geodesic representatives of the curves
$C{\smallsetminus}c(v)$. Informally, $m(C^{\prime})$ is contained in the
boundary of the subsurface filled by $C^{\prime}$.

The map $q^{\circ}_{g}$ can be seen to map simplices to simplices. The
condition that the simplices of $\Phi_{g}$ correspond to sets of disjoint
diagonals of $K_{2g+2}$ implies that the image of a simplex under $\Phi_{g}$
is a simplex in $\mathcal{C}_{g}^{\circ}$.

Defining the map $\mathbf{q_{g}:\Phi_{g}\rightarrow\mathcal{C}_{g}}$. For the
purposes of this paper, it will be simplest to work with the image in
$C_{g}^{\circ}$ of $\Phi_{g}$ under $q^{\circ}_{g}$. This is a matter of
taste; if one wants to obtain a sphere in $\mathcal{C}_{g}$, for any vertex
$v$ with the property that $m\bigl{(}C{\smallsetminus}c(v)\bigr{)}$ has more
than one connected component, choose a single curve
$m_{1}\bigl{(}C{\smallsetminus}c(v)\bigr{)}$ in
$m\bigl{(}C{\smallsetminus}c(v)\bigr{)}$. It was shown in [4] that
$q^{\circ}_{g}$ is homotopic to a map from $S^{2g-2}$ to a subcomplex with the
vertex in the image of $q^{\circ}_{g}$ labelled by
$m\bigl{(}C{\smallsetminus}c(v)\bigr{)}$ replaced by the vertex labelled by
$m_{1}\bigl{(}C{\smallsetminus}c(v)\bigr{)}$. This homotopy pushes simplices
in the interior of a simplex of $\mathcal{C}_{g}^{\circ}$ onto the boundary of
the simplex, as illustrated in Figure 3. The map $q^{\circ}_{g}$, composed
with one such choice of homotopy for every vertex not in $\mathcal{C}_{g}$,
gives the map $q_{g}$.

For each $g\geq 2$ there is a map $q_{g}$, the homotopy class of which will be
referred to as a Broaddus sphere.

Figure 3. Lower dimensional analogue of homotoping a subcomplex of
$\mathcal{C}_{g}^{\circ}$ onto a subcomplex of $\mathcal{C}_{g}$. Edges and
vertices of $\mathcal{C}_{g}^{\circ}$ not in $\mathcal{C}_{g}$ are shown in
grey.

##  4\. The Set of Minima

The purpose of this section is to show that the Broaddus sphere in genus $g$
can be embedded in $\partial\mathcal{T}_{g}^{\delta}$ in such a way that it
makes sense to think of it as the boundary of $\mathrm{Min}(C)$, where $C$ is
the set of curves from Example 3.1.

A thickening of the embedded curve complex.  The dimension $3g-4$ of
$\mathcal{C}_{g}$ is smaller than the dimension of $\mathcal{T}_{g}$, namely
$6g-6$. In order to use the embedding of $\mathcal{C}_{g}$ into
$\partial\mathcal{T}_{g}^{\delta}$ from [7] it will be convenient to define a
regular neighbourhood of the image of the embedding. An abuse of notation will
be used to simplify the exposition; the image of the embedding of
$\mathcal{C}_{g}$ or $\mathcal{C}_{g}^{\circ}$ in
$\partial\mathcal{T}_{g}^{\delta}$ will also be referred to as
$\mathcal{C}_{g}$ or $\mathcal{C}_{g}^{\circ}$.

Ivanov’s embedding of $\mathcal{C}_{g}$ in $\partial\mathcal{T}_{g}^{\delta}$
has the property that a vertex labelled by a curve $c$ is mapped into
$\mathrm{Sys}(\\{c\\})$, an edge labelled by $\\{c_{1},c_{2}\\}$ is mapped
into the union of $\mathrm{Sys}(\\{c_{1}\\})$, $\mathrm{Sys}(\\{c_{2}\\})$ and
$\mathrm{Sys}(\\{c_{1}\cup c_{2}\\})$, etc.

For each multicurve $m$ labelling a top dimensional simplex of
$\mathcal{C}_{g}$, the intersection of $\mathrm{Sys}(m)$ with
$\partial\mathcal{T}_{g}^{\delta}$ is an embedded submanifold without
boundary. For a multicurve $m^{\prime}$ with fewer connected components, the
boundary of $\mathrm{Sys}(m^{\prime})\cap\partial\mathcal{T}_{g}^{\delta}$
lies along
$\\{\mathrm{Sys}(m^{\prime\prime})\cap\partial\mathcal{T}_{g}^{\delta}\ |\
m^{\prime}\subsetneq m^{\prime\prime}\\}$. As these extend to embedded
submanifolds with boundary in a neighbourhood of
$\partial\mathcal{T}_{g}^{\delta}$, the same arguments as in [7] show that it
is possible to embed a regular neighbourhood
$\pi:\mathcal{N}_{g}\rightarrow\mathcal{C}_{g}^{\circ}$ into the thin part of
$\mathcal{T}_{g}$ in such a way that the duality between the sets of systoles
and the cells of the regular neighbourhood is preserved.

Construction of the map $\mathbf{q_{C,g}}$. Recall the definition of
$q^{\circ}_{g}:\Phi_{g}\rightarrow\mathcal{C}_{g}^{\circ}$ from the previous
section, where $\Phi_{g}$ is a triangulation of a $2g-2$ dimensional sphere.
Another map $q_{C,g}$ from $\Phi_{g}$ to $\mathcal{C}_{g}^{\circ}$ will now be
constructed. This map factors through a map homotopic to the inclusion map
into $\partial\mathcal{T}_{g}^{\delta}$ of a $2g-2$-dimensional sphere in the
intersection $\partial\mathcal{T}_{g}^{\delta}\cap\mathrm{Min}(C)$.

The next construction is motivated by the discussion of admissible boundary
points from Section 2. Let $c_{1},c_{2}$ be a pair of disjoint curves in $C$,
and $C^{\prime}:=C{\smallsetminus}\\{c_{1},c_{2}\\}$ a nonfilling set of
curves. The curves $\\{c_{1},c_{2}\\}$ correspond to a diagonal of $K_{2g+2}$.
Then the length function $L(C^{\prime})$ obtained by taking the sum of the
lengths of curves in $C^{\prime}$ has no minimum in $\mathcal{T}_{g}$ by Lemma
2.1. It can be thought of as having its infimum at the limit of a sequence of
points whose parameters $(a_{1},\ldots,a_{2g+2})\in\mathbb{R}^{2g+2}_{+}$
approach $\frac{1}{2g}(0,0,1,1,\ldots,1)$.

For convenience, from now on the normalisation factors on the parameters in
$\mathbb{R}_{+}^{2g+2}$ will be dropped. For the arguments given here, it is
only of interest which of the parameters are close to zero.

For small $\delta$, the length function $L(C^{\prime})$ nearly reaches its
infimum on $\mathrm{Min}(C)\cap\partial\mathcal{T}_{g}^{\delta}$. Denote by
$v(C^{\prime})$ a point at which $L(C^{\prime})$ is minimised on
$\mathrm{Min}(C)\cap\partial\mathcal{T}_{g}^{\delta}$. Similarly, for any
other vertex of $\Phi_{g}$, define a corresponding point in
$\mathrm{Min}(C)\cap\partial\mathcal{T}_{g}^{\delta}$. The only restriction
imposed here is that any choices are made in such a way that the embedding is
invariant under the action of the subgroup of $\Gamma_{g}$ that maps
$\mathrm{Min}(C)$ to itself.

Imposing the restriction that any choices in the construction of the embedding
are made in a $\Gamma_{g}$-equivariant way is not a problem, for reasons that
will now be explained. The subgroup of $\Gamma_{g}$ that maps
$\mathrm{Min}(C)$ to itself permutes the curves in $C$ in a way that preserves
or reverses the cyclic ordering. This subgroup could only map a minimum of
$L(C^{\prime})$ on $\mathrm{Min}(C)\cap\partial\mathcal{T}_{g}^{\delta}$ to a
different minimum if it maps $C^{\prime}$ to itself. This is only possible if
$c_{1}$ and $c_{2}$ represent a pair of opposite vertices of $K_{2g+2}$. In
this case, $m(C^{\prime})$ consists of a single separating curve that is
invariant under the action of the subgroup. It follows from Riera’s theorem
that the minima of $L(C^{\prime})$ on
$\mathrm{Min}(C)\cap\partial\mathcal{T}_{g}^{\delta}$ occur where
$m(C^{\prime})$ is a systole. By symmetry, such points are therefore invariant
under the action of the subgroup of $\Gamma_{g}$ that maps $\mathrm{Min}(C)$
to itself. It follows that any choices can be made in a
$\Gamma_{g}$-equivariant way.

Every edge of $\Phi_{g}$ determines a pair of endpoints $v(C^{\prime})$ and
$v(C^{\prime\prime})$ with $m(C^{\prime})$ and $m(C^{\prime\prime})$ disjoint
multicurves. The midpoint of this edge is represented by a tuple in
$\mathbb{R}^{2g+2}_{+}$ with parameters close to zero in 3 or 4 positions, and
all other parameters close to 1. The parameters that are small consist of the
union of the parameters that are small at the vertices.

To be more precise: $C^{\prime}=C{\smallsetminus}\\{c_{i}\cup c_{k}\\}$ and
$C^{\prime\prime}=C{\smallsetminus}\\{c_{l},c_{m}\\}$ where $c_{l}$ or $c_{m}$
(but not both) might be in $\\{c_{i},c_{j}\\}$. Then the midpoint of the edge
connecting $v(C^{\prime})$ and $v(C^{\prime\prime})$ has parameters
$a_{i},a_{j},a_{l}$ and $a_{m}$ close to zero, and all others close to 1. The
midpoint of this edge can be thought of as a minimum of the length function
$L(C{\smallsetminus}\\{c_{i},c_{j},c_{l},c_{m}\\})$ on
$\partial\mathcal{T}_{g}^{\delta}$. Between the midpoint and each of the
endpoints, the edge is defined by taking a linear interpolation of the
parameters. Maps of higher dimensional simplices of $\Phi_{g}$ into
$\mathrm{Min}(C)$ are defined analogously, with midpoints contained in
$\partial\mathcal{T}_{g}^{\delta}\cap\mathrm{Min}(C)$.

The image of the simplices of $\Phi_{g}$ are not guaranteed to be contained in
$\partial\mathcal{T}_{g}^{\delta}$, but for sufficiently small $\delta$,
$\mathcal{N}_{g}$ can be chosen in such a way that they are contained in
$\mathcal{N}_{g}$. Since $\Phi_{g}$ is a $2g-2$-sphere, and $\mathrm{Min}(C)$
a $2g-1$-ball, inside $\mathcal{N}_{g}$ the embedding of $\Phi_{g}$ can be
homotoped to have image in the intersection
$\partial\mathcal{T}_{g}^{\delta}\cap\mathrm{Min}(C)$. Composing with
$\pi:\mathcal{N}_{g}\rightarrow\mathcal{C}_{g}^{\circ}$ gives a homotopy in
$\mathcal{C}_{g}^{\circ}$. In this way an embedding of a $2g-2$-sphere in
$\mathcal{C}_{g}^{\circ}$ denoted by $q_{C,g}$ is obtained.

######  Lemma 4.1.

For $g\geq 2$, $q^{\circ}_{g}$ is $\Gamma_{g}$-equivariantly homotopic to
$q_{C,g}$.

###### Proof.

Suppose $C_{i}$ for $i=1,\ldots,n$ are nonfilling subsets of $C$, each of
which is obtained by deleting a pair of disjoint curves from $C$, and such
that $\\{m(C_{1}),\ldots,m(C_{n})\\}$ are pairwise disjoint.

Claim 1 - Suppose $\sigma$ is a top dimensional simplex in the image of the
embedding of $\Phi_{g}$ by $q_{C,g}$ defined above, with vertices given by
$\\{v(C_{1}),\ldots,v(C_{n})\\}$. Then for sufficiently small $\delta$,
everywhere on $\sigma$ the length of any multicurve in the set
$\\{m(C_{1}),\ldots,m(C_{n})\\}$ is less than the diameter of the collar of a
geodesic of length $\delta$.

Note that in the $\delta$-thin part of $\mathcal{T}_{g}^{\delta}$, the
statement that a given curve has length less than the diameter of a collar of
a geodesic of length $\delta$ amounts to saying that the curve is disjoint
from the systoles or is a systole.

Proof of Claim 1: Using Keen’s collar lemma, the bound on the Margulis
constant given in [15] and some hyperbolic trigonometry, for example Theorem
3.5.4 of [10], it is not hard to check that the length of the systoles at $p$
in $\mathrm{Min}(C)$ is less than the diameter of the collar of a geodesic of
length $\delta$. Similarly, for sufficiently small $\delta$, the length of
$m(C^{\prime})$ at $p$ is less than the diameter of the collar of a geodesic
of length $\delta$. Moreover, Lemma 2.1 implies that the length function
$L(C^{\prime})$, for nonfilling $C^{\prime}\subsetneq C$, has some values on
$\partial\mathcal{T}_{g}^{\delta}$ smaller than at $p$.

Denote by $\Sigma(C_{i})$ the union of cells each containing the vertex
$v(C_{i})$. At the vertex $v(C_{i})$ the systoles are disjoint from every
curve in $C_{i}$ and hence from $m(C_{i})$. Otherwise, for sufficiently small
$\delta$, one of the systoles would intersect a curve in $C_{i}$, making
$L(C_{i})$ larger than its value at $p$. The systoles are therefore contained
in the multicurve $m(C_{i})$ at $v(C_{i})$. Any curves in $m(C_{i})$ that are
not systoles at the vertex must also have length less than the diameter of the
collar of a geodesic of length $\delta$. This is because such curves are
homotopic to curves lying along arcs of the curves $C_{i}$ and hence cannot be
too long without forcing $L(C_{i})$ to be larger than its value at $p$.

At the vertex $v(C_{j})$ of $\Sigma(C_{i})$, each curve in $m(C_{i})$ is
either homotopic to a curve in $m(C_{j})$ or to a piecewise smooth curve lying
along some of the geodesic representatives of the curves in $C_{j}$. Either
way, for sufficiently small $\delta$, the length of $m(C_{j})$ is less than
the diameter of the collar of a geodesic of length $\delta$. The same is true
for any vertex or midpoint of a simplex in $\Sigma(C_{i})$. This can then be
extended to $\Sigma(C_{i})$, because the length functions minimised over the
cells are linear combinations of the length functions minimised at the
vertices and midpoints of cells. Since $\sigma$ is in the intersection

$\cap_{j=1}^{n}\Sigma(C_{j})$

the claim follows.

Claim 2 - Suppose $\sigma$ is a top dimensional simplex with vertices
$\\{v(C_{1}),\ldots,v(C_{n})\\}$ as in Claim 1. Then for sufficiently small
$\delta$, if $c$ is a systole somewhere on $\sigma$, $c$ is contained in one
of the multicurves $\\{m(C_{1}),\ldots,m(C_{n})\\}$.

Claim 2 is a corollary of Claim 1. Since $\sigma$ is top dimensional, it
follows from the construction that the multicurves
$\\{m(C_{1}),\ldots,m(C_{n})\\}$ determine a pants decomposition of
$\mathcal{S}_{g}$. If $c$ were a systole on $\sigma$ intersecting one of the
multicurves $\\{m(C_{1}),\ldots,m(C_{n})\\}$, this would contradict Claim 1.

A $\Gamma_{g}$-equivariant homotopy between $q_{C,g}$ and $q^{\circ}_{g}$ will
now be constructed. Let $\sigma$ be the image under $q_{C,g}$ of a top
dimensional cell of the barycentric subdivision of $\Phi_{g}$. The vertices of
$\sigma$ are already on $\mathcal{C}_{g}^{\circ}$. A vertex $v(C^{\prime})$
will be mapped to the vertex $v(C^{\prime})^{\mathcal{C}}$ of
$\mathcal{C}_{g}^{\circ}$ labelled by $m(C^{\prime})$. Recall that by Claim 2,
the systoles at $v(C^{\prime})$ are a submulticurve of $m(C^{\prime})$. So
$v(C^{\prime})$ is already contained in a cell in $\mathcal{C}_{g}$ containing
$v(C^{\prime})^{\mathcal{C}}$. The homotopy shifts $v(C^{\prime})$ to
$v(C^{\prime})^{\mathcal{C}}$ along a straight line contained within this
simplex.

Now suppose $e(C^{\prime},C^{\prime\prime})$ is an edge of the embedding of
the barycentric subdivision of $\Phi_{g}$. Here $v(C^{\prime})$ is a vertex
and $v(C^{\prime\prime})$ the midpoint. Then by Claim 2, the edge
$e(C^{\prime},C^{\prime\prime})$ is contained in a thickened simplex
containing the edge $e(C^{\prime},C^{\prime\prime})^{\mathcal{C}}$ in the
image of $\mathcal{C}_{g}^{\circ}$ joining $v(C^{\prime})^{\mathcal{C}}$ and
$v(C^{\prime\prime})^{\mathcal{C}}$. The homotopy maps
$e(C^{\prime},C^{\prime\prime})$ to
$e(C^{\prime},C^{\prime\prime})^{\mathcal{C}}$ by shifting points along
straight lines within this thickened simplex, extending the homotopy taking
the endpoints of $e(C^{\prime},C^{\prime\prime})$ to the endpoints of
$e(C^{\prime},C^{\prime\prime})^{\mathcal{C}}$.

The homotopy can be defined inductively, where the homotopy on the
$n+1$-skeleton extends the homotopy of the $n$-skeleton. ∎

##  5\. Proof of the Conjecture

This section gives a proof of the main theorem and its corollary.

######  Theorem 5.1.

$\mathrm{St}(S_{g})$ is generated by the $\Gamma_{g}$-orbit of the image in
$\mathcal{C}_{g}$ of the map $q_{g}$.

###### Proof.

The theorem will be proven by first proving the lemma below. As it was shown
in Theorem 4.2 of [4] that $\mathrm{St}(\mathcal{S}_{g})$ is cyclic modulo the
action of $\Gamma_{g}$, it then sufficies to show primitivity of the homology
class with representative given by the image of the map $q_{g}$.

######  Lemma 5.2 (Conjecture 4.9 of [4]).

The image of the map $q_{g}$ represents a nontrivial element of
$\tilde{H}_{2g-2}(\mathcal{C}_{g};\mathbb{Z})$.

For convenience, the map $q^{\circ}_{g}$ with image in
$\mathcal{C}_{g}^{\circ}$ will be used in place of $q_{g}$. This can be done
because it was shown that the two maps are homotopic.

The cases $g=1$ and $g=2$ of the lemma were proven in [4], where the first
nontrivial case $g=2$ was worked through explicitly. The sphere obtained in
the case $g=2$ is shown in Figure 2.

For simplicity, balls and spheres in
$\mathcal{N}_{g}\cap\partial\mathcal{T}_{g}^{\delta}$ will be confused with
corresponding objects in $\mathcal{C}_{g}$. This is done by identifying the
image of the embedding of $\mathcal{C}_{g}$ with $\mathcal{C}_{g}$, and
projecting $\mathcal{N}_{g}$ onto this image via the systole-preserving map
$\pi$.

Let $C$ be the set of curves from Example 3.1. It follows from Lemma 4.1 and
the definition of $q_{C,g}$ that the image of $q^{\circ}_{g}$ represents a
boundary in $\mathcal{C}_{g}^{\circ}$ iff the ball $\mathcal{B}_{g}$ in
$\mathrm{Min}(C)$ bounded by the image of the embedding of $\Phi_{g}$ can be
homotoped relative to its boundary into $\partial\mathcal{T}_{g}^{\delta}$.

Suppose there exists a homotopy of $\mathcal{B}_{g}$ relative to its boundary
into $\partial\mathcal{T}_{g}^{\delta}$. Denote by
$\mathcal{B}_{g}^{\mathcal{C}}$ a ball in $\mathcal{C}_{g}^{\circ}$ homotopic
to $\mathcal{B}_{g}$. It is possible to assume without loss of generality that
$\mathcal{B}_{g}^{\mathcal{C}}$ is invariant under the action of the finite
subgroup of $\Gamma_{g}$ that stabilises
$\partial\mathcal{B}_{g}^{\mathcal{C}}$. If $\mathcal{B}_{g}^{\mathcal{C}}$
could not be chosen this way, Theorem 4.2 of [4] would imply that
$\mathrm{St}(S_{g})$ is finite modulo the action of the mapping class group.

The critical point $p$ from Example 3.1 has an automorphism group that acts
transitively on the curves in $C$. This automorphism group therefore
corresponds to a subgroup of $\Gamma_{g}$ that stabilises $\mathrm{Min}(C)$.
Since the action of the mapping class group preserves injectivity radius, this
subgroup also stabilises $\mathcal{B}_{g}$ and its boundary. This group will
be called $G_{C}$ from now on. $G_{C}$ contains a cyclic element of order
$2g+2$ that permutes the systoles, preserving the cyclic ordering. In
addition, it contains elements that correspond to a reflection of the
$(2g+2)$-gon $K_{2g+2}$ from Section 3.

A unique path $\gamma$ that is the fixed point set of this subgroup of $G_{C}$
will now be described. This is an example of a type of path constructed in
[11].

At the critical point $p$ in Example 3.1, the systoles intersect at right
angles, and cut the surface into right angled $(2g+2)$-gons. The set $C$ can
be decomposed into two multicurves, each with $g+1$ elements; these are the
“A-curves” and the “B-curves”. The edges of each $(2g+2)$-gon alternately lie
along A-curves and B-curves. Each systole lies along two edges of this
tesselation, and these two edges are on the boundary of different
$(2g+2)$-gons.

It is possible to deform the hyperbolic structure such that the tesselation by
right angled $(2g+2)$-gons becomes a tesselation by $(2g+2)$-gons with all
edge lengths equal and angles that alternate between $\pi/2+t$ and $\pi/2-t$
for $t\in[0,\pi/2)$. This is done in such a way that all angles at the
vertices of the tesselation still sum to $2\pi$ and opposite pairs of edges at
vertices continue to meet at angle $\pi$. The hyperbolic structure on the
$(2g+2)$-gons extends to a hyperbolic structure on the tesselated surface. A
map $\gamma:(-\pi/2,\pi/2)\rightarrow\mathcal{T}_{g}$ is obtained. Here
$t\in(-\pi/2,\pi/2)$ is mapped to the point in $\mathcal{T}_{g}$ represented
by the hyperbolic surface tesselated by the complements of the geodesics in
$C$, where the tesselation consists of $(2g+2)$-gons with angles alternating
between $\pi/2+t$ and $\pi/2-t$.

The path $\gamma$ is the fixed point set of the subgroup $G_{C}$ of
$\Gamma_{g}$. Since $G_{C}$ acts by isometry, it is a necessary condition that
the lengths of the curves in $C$ are all equal at every point of the fixed
point set, similarly for the angles of intersection, where these are defined
appropriately using an orientation convention as in the definition of
$\gamma$. The action of $G_{C}$ at $\gamma(t)$ performs the same permutation
of the $(2g+2)$-gons of the tesselation of $\gamma(t)$ and their edges as at
the critical point $\gamma(0)$. The fixed point sets of the reflections
contain $\gamma$, and extend radially outwards from $\gamma$, intersecting
$\partial\mathcal{B}_{g}$. The Brouwer fixed point theorem implies that the
fixed point set of the cyclic subgroup of $G_{C}$ must intersect
$\mathcal{B}_{g}^{C}$, similarly for the fixed point set of the reflections.
It follows that $\gamma$ intersects $\mathcal{B}_{g}^{C}$.

The quotient $\mathcal{T}_{g}/G_{C}$ is therefore an orbifold with trivial
fundamental group and cone singularities along $\gamma$ and radiating outwards
from $\gamma$, as explained above. The homotopy from $\mathcal{B}_{g}$ to
$\mathcal{B}_{g}^{\mathcal{C}}$ can therefore be assumed to be
$G_{C}$-equivariant; its projection to $\mathcal{T}_{g}/G_{C}$ is a homotopy
between two singular balls with the same boundary. This homotopy shifts points
on each fixed point set along the fixed point set.

Again due to the fact that $G_{C}$ acts by isometry, $\gamma(t)$ is a geodesic
with respect to any $\Gamma_{g}$-equivariant metric on $\mathcal{T}_{g}$. In
particular, it is a geodesic with respect to the Teichmüller metric. If
$\gamma$ is the axis of a pseudo-Anosov (calculations suggest this is not the
case), the image of $\gamma$ in the moduli space is a closed loop in the
$\delta$-thick part of $\mathcal{T}_{g}$ for some $\delta$. Since a
$\Gamma_{g}$-equivariant homotopy of the ball $\mathcal{B}_{g}$ can only shift
$\mathcal{B}_{g}$ in such a way that the image of the critical point stays on
$\gamma$, this contradicts the existence of a homotopy.

When $\gamma$ is not the axis of a pseudo-Anosov, it intersects
$\partial\mathcal{T}_{g}^{\delta}$ in a discrete set of points. This follows
from the observation that $\gamma$ is also a Weil-Petersson geodesic, and
curve lengths are strictly convex along Weil-Petersson geodesics, [14].

If $\mathcal{B}_{g}$ is homotopic into $\partial\mathcal{T}_{g}^{\delta}$, by
symmetry, there must be at least two $G_{C}$-equivariant homotopies; one
moving a point of $\mathcal{B}_{g}$ in one direction along $\gamma$, and the
other in the opposite direction along $\gamma$. These homotopies take
$\mathcal{B}_{g}$ to distinct balls $\mathcal{B}_{g}^{L}$ and
$\mathcal{B}_{g}^{R}$ in $\mathcal{C}_{g}$, as demonstrated by the fact that
the set of systoles at the two corresponding points of intersection of
$\gamma$ with $\partial\mathcal{T}_{g}^{\delta}$ are different. Both these
balls in $\mathcal{C}_{g}$ have dimension $2g-1$, as they have
$2g-2$-dimensional boundary.

As $\mathcal{C}_{g}$ has the homotopy type of a wedge of spheres of dimension
$2g-2$, the $(2g-1)$-sphere in $\mathcal{C}_{g}$ obtained by gluing
$\mathcal{B}_{g}^{L}$ and $\mathcal{B}_{g}^{R}$ together along their
boundaries must bound a ball in $\mathcal{C}_{g}$. As this is also the case in
the projection to $\mathcal{T}_{g}/G_{C}$, the $\Gamma_{g}$ orbits of
$\mathcal{B}_{g}^{L}$ and $\mathcal{B}_{g}^{R}$ must be $G_{C}$-equivariantly
homotopic in $\mathcal{C}_{g}$. However, this contradicts the fact that
$\gamma$ only intersects $\partial\mathcal{T}_{g}^{\delta}$ in a discrete set
of points. It follows that there can be no homotopy, proving Lemma 5.2.

It remains to show primitivity of the image of $q_{g}$. By Lemma 4.1, the
image of $q_{g}$ maps to an embedded sphere in the moduli space
$\mathcal{M}_{g}$, representing the primitive homology class of the
$\delta$-thin part of $\mathcal{M}_{g}$ that is killed by attaching the
$(2g-1)$-handle corresponding to the $\Gamma_{g}$-orbit of critical points or
sets of minima from Example 3.1. If the image of $q_{g}$ does not represent a
primitive homology class, it is a multiple of a primitive homology class
$[q_{g}^{prim}]$ with stabiliser subgroup $G_{C}^{prim}$ strictly containing
$G_{C}$.

It follows from the Nielsen Realisation Theorem that every finite subgroup of
$\Gamma_{g}$ has a fixed point set. This observation, together with the
invariance of injectivity radius and the Weil-Petersson metric under the
action of $\Gamma_{g}$, can be used to construct a ball
$\mathcal{B}_{g}^{prim}$ in $\mathcal{T}_{g}^{\delta}$, invariant under
$G_{C}^{prim}$ and with boundary projecting onto a representative of
$[q_{g}^{prim}]$ in the boundary of the $\delta$-thick part of
$\mathcal{M}_{g}$.

The fixed point set of $G_{C}^{prim}$ must be on $\gamma$. Since $G_{C}$ is
the largest subgroup that fixes $\gamma$, the fixed point set of
$G_{C}^{prim}$ must be a point $p_{fix}$ on $\gamma$. This is a point at which
the images of $\gamma$ under the mapping classes in
$G_{C}^{prim}{\smallsetminus}G_{C}$ intersect. There is a $G_{C}$-equivariant
homotopy of $\mathrm{Min}(C)$ along $\gamma$, taking the critical point $p$ to
$p_{fix}$, whose restriction to $\partial\mathcal{T}_{g}^{\delta}$ is a
$G_{C}$-equivariant homotopy. The image of $\mathrm{Min}(C)$ under this
homotopy, call it $\mathrm{Min}(C)^{fix}$, is invariant under the action of
$G_{C}$. The balls in the orbit of $\mathrm{Min}(C)^{fix}$ under the action of
$G_{C}^{prim}$ intersect along the largest dimensional ball invariant under
$G_{C}^{prim}$. However, this ball has dimension strictly less than
$\mathrm{Min}(C)$, so its boundary cannot project to a primitive homology
class of the $\delta$-thin part of $\mathcal{M}_{g}$. This concludes the proof
of primitivity, and hence of the theorem.

∎

######  Corollary 5.3.

Denote by $Z_{g}$ the center of the mapping class group $\Gamma_{g}$. Then the
stabiliser subgroup of the action of $\Gamma_{g}/Z_{g}$ on the homology class
$[q_{g}]$ in $\mathrm{St}(S_{g})$ with representative the image of $q_{g}$ is
isomorphic to the dihedral group $D_{2g+2}$. Moreover, the kernel of the
action of $\Gamma_{g}/Z_{g}$ on $\mathrm{St}(S_{g})$ is trivial.

###### Proof.

Recall the construction of the simplicial $2g-2$ dimensional sphere $\Phi_{g}$
via the $(2g+2)$-gon $K_{2g+2}$. It follows that the automorphism group of the
associahedron $\mathcal{K}_{2g-1}$ is isomorphic to the automorphism group of
$K_{2g+2}$, namely the dihedral group $D_{2g+2}$.

Figure 4. In genus 2, the mapping class in the proof of Corollary 5.3 is a
composition of reflections through the red curve shown in part (a), followed
by the red multicurve in part (b). When the genus is odd, the first reflection
is through a multicurve with two connected components, as shown in part (c).

The dihedral group $D_{2g+2}$ is generated by reflections. Let $r$ be one such
reflection. Since $2g+2$ is even, $r$ leaves invariant a pair of vertices of
$K_{2g+2}$; let $c_{1}$ and $c_{2}$ be the curves in $C$ labelling these
edges. There is an element $\alpha(r)$ of the mapping class group that
realises $r$ in the sense that the induced action of $\alpha(r)$ on the curves
of $\mathcal{S}_{g}$ determines the same permutation of the elements of $C$ as
the action of $r$ on the vertices of $K_{2g+2}$. The mapping class $\alpha(r)$
is a composition of two reflections of $\mathcal{S}_{g}$, as illustrated in
Figure 4. The first reflection is through a multicurve that intersects $c_{1}$
and $c_{2}$ twice each, and cuts each of the $(2g+2)$-gons in
$S_{g}{\smallsetminus}C$ into two $(g+2)$-gons. This multicurve is a curve
when the genus is even, and has 2 connected components when the genus is odd,
as illustrated in Figure 4 (a) and (c). The second reflection is through the
set of curves obtained as the boundary curves of a subsurface obtained by
gluing two adjacent $(2g+2)$-gons in $S_{g}{\smallsetminus}C$ along their
common edges as shown in Figure 4 (b). This concludes the proof of the first
part of the Corollary.

The kernel of the action of $\Gamma_{g}/Z_{g}$ on $\mathrm{St}(S_{g})$ is
contained in the intersection of the stabiliser subgroups of the generators,
each of which can be taken to be in the $\Gamma_{g}$ orbit of a fixed
$[q_{g}]$ by Theorem 5.1. These stabiliser subgroups consist of all the
conjugates of the stabiliser subgroup of $[q_{g}]$. This intersection can
readily be seen to be trivial; consider for example the conjugate by a large
power of a pseudo-Anosov. The intersection with the stabiliser subgroup of
$[q_{g}]$ is zero, because the images of the reflections fix different curves.

∎

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# Relaxation-speed crossover in anharmonic potentials

Jan Meibohm  Department of Physics and Materials Science, University of
Luxembourg, L-1511 Luxembourg, Luxembourg  Danilo Forastiere  Department of
Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg,
Luxembourg  Tunrayo Adeleke-Larodo  Department of Physics and Materials
Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg  Karel
Proesmans  Department of Physics and Materials Science, University of
Luxembourg, L-1511 Luxembourg, Luxembourg  Hasselt University, B-3590
Diepenbeek, Belgium

###### Abstract

In a recent Letter [A. Lapolla and A. Godec, Phys. Rev. Lett. 125, 110602
(2020)], thermal relaxation was observed to occur faster from cold to hot
(heating) than from hot to cold (cooling). Here we show that overdamped
diffusion in anharmonic potentials generically exhibits both faster heating
and faster cooling, depending on the initial temperatures and on the
potential’s degree of anharmonicity. We draw a relaxation-speed phase diagram
that localises the different behaviours in parameter space. In addition to
faster-heating and faster-cooling regions, we identify a crossover region in
the phase diagram, where heating is initially slower but asymptotically faster
than cooling. The structure of the phase diagram is robust against the
inclusion of a confining, harmonic term in the potential as well as moderate
changes of the measure used to define initially equidistant temperatures.

Many thermal relaxation processes in nature and industry occur out of
equilibrium, and thus outside of the realm of the quasistatic approximation.
As a consequence, nonequilibrium thermal relaxation gives rise to anomalous
effects, such as ergodicity breaking Bray (2002) or the Mpemba effect Mpemba
and Osborne (1969). The latter describes the surprising observation that some
systems cool down faster, when relaxing from a higher initial temperature. A
better understanding of such anomalous relaxation effects in out-of-
equilibrium systems is important, because it may allow us to use these
phenomena to our advantage, for instance, for increasing the rate of heating
and cooling.

Although a complete understanding of anomalous relaxation in macroscopic
systems appears elusive at present, much progress has been made recently in
reproducing anomalous relaxation phenomena on mesoscopic scales. This has led
to several important results such as new theoretical Lu and Raz (2017); Klich
_et al._ (2019); Walker and Vucelja (2021); Chétrite  _et al._ (2021) and
experimental Kumar and Bechhoefer (2020); Kumar  _et al._ (2021) insights into
the Mpemba effect, strategies to increase the rate at which systems can be
cooled Gal and Raz (2020); Prados (2021); Carollo  _et al._ (2021), and an
information-theoretic bound on the speed of relaxation to equilibrium
Shiraishi and Saito (2019).

Within a setup closely related to, yet slightly different from, the Mpemba
effect, a recent study Lapolla and Godec (2020) reported an asymmetry in the
rate at which systems heat up and cool down. According to this study, and
subsequent works by other authors, heating occurs faster than cooling for
diffusive systems with harmonic potentials Lapolla and Godec (2020) and for
discrete-state two-level systems Manikandan (2021); Van Vu and Hasegawa
(2021). On the other hand, it was shown that this relaxation asymmetry is non-
generic for diffusion in potentials with multiple minima Lapolla and Godec
(2020) or in discrete-state systems with more than two states Manikandan
(2021); Van Vu and Hasegawa (2021). However, it appears to be widely believed
that the described effect is a general property of overdamped, diffusive
systems with stable single-well potentials Lapolla and Godec (2020);
Manikandan (2021); Van Vu and Hasegawa (2021).

In this Letter, we study the relaxation asymmetry for overdamped diffusion in
anharmonic potentials. Opposing common belief, we show that these systems
exhibit both behaviours, faster heating and faster cooling, even for stable
single-well potentials. Based on these results, we draw a phase diagram
locating the different regions of “faster heating” and “faster cooling” in
parameter space. These two regions are separated by a crossover region where
cooling occurs faster at first, but heating overtakes at a finite time. Our
results suggest that the relative speed of thermal relaxation to equilibrium
can be substantially increased by varying the anharmonicity of the potential.
This should be testable in experiments and has potential applications in the
optimisation of cooling strategies for small-scale systems Gal and Raz (2020).

To specify the problem, consider two equilibrium systems, otherwise identical,
but at different temperatures $T_{c}<T_{h}$. We call the system at temperature
$T_{c}$ cold and that at temperature $T_{h}$ hot. At time $t=0$, both systems
experience an instantaneous temperature quench to the same final temperature
$T_{f}$, where $T_{c}<T_{f}<T_{h}$. The relaxation of the two systems toward
equilibrium is monitored by their nonequilibrium free-energy difference
Lapolla and Godec (2020),

$\displaystyle\mathcal{F}_{i}(t)=$ $\displaystyle\left\langle\Delta
E_{i}(t)\right\rangle-T_{f}\left\langle\Delta S_{i}(t)\right\rangle\,,$
$\displaystyle=$ $\displaystyle
k_{\text{B}}T_{f}\int_{-\infty}^{\infty}\\!\\!\text{d}x\,p_{i}(x,t)\ln\left[\frac{p_{i}(x,t)}{p_{f}(x)}\right]\,,$
(1)

with respect to the equilibrium distribution $p_{f}$ at final temperature
$T_{f}$. Here, $\left\langle\Delta E_{i}(t)\right\rangle$ and
$\left\langle\Delta S_{i}(t)\right\rangle$ are the average differences in the
energy and entropy of the (cold or hot) system at time $t$ and its equilibrium
state at temperature $T_{f}$; $k_{\text{B}}$ denotes the Boltzmann constant.
The index $i$ in Eq. (Relaxation-speed crossover in anharmonic potentials)
takes the values $c$ and $h$, and $p_{c}$ and $p_{h}$ denote the probability
densities of the initially cold and hot system, respectively.

In order to quantitatively compare the distances $\mathcal{F}_{i}(t)$ from
equilibrium, the temperatures $T_{c}$ and $T_{h}$ at $t=0$ are chosen so that
$\mathcal{F}_{c}(0)=\mathcal{F}_{h}(0)$ Lapolla and Godec (2020). We call such
a temperature quench “$\mathcal{F}$-equidistant,” i.e., at equal distance with
respect to the temperature measure (Relaxation-speed crossover in anharmonic
potentials). A comparison between this setup and the Markovian Mpemba effect
Lu and Raz (2017); Klich  _et al._ (2019) is made in Sec. I of the
Supplemental Material (SM) SM .

The specific measure (Relaxation-speed crossover in anharmonic potentials) is
used for two reasons. First, $\mathcal{F}_{i}$ is a thermodynamic quantity for
systems at equilibrium and hence for $t<0$ and in the limit $t\to\infty$.
Second, it remains well defined out of equilibrium and thus for all finite
times $t$.

In the long-time limit, both the cold and the hot system relax to equilibrium
so that $\mathcal{F}_{c}(t)$ and $\mathcal{F}_{h}(t)$ tend to zero
asymptotically. The relative distance from equilibrium of the two systems is
conveniently measured by the logarithmic ratio

$\displaystyle\mathcal{R}(t)\equiv\ln\left[\frac{\mathcal{F}_{h}(t)}{\mathcal{F}_{c}(t)}\right]\,.$
(2)

For overdamped diffusion in a harmonic potential, one can prove that
$\mathcal{R}(t)>0$, i.e., $\mathcal{F}_{h}(t)>\mathcal{F}_{c}(t)$ during the
relaxation Lapolla and Godec (2020), i.e., heating occurs faster than cooling;
$\mathcal{R}(t)<0$ corresponds to the opposite case, that of faster cooling.
Note also that $\mathcal{R}(0)=0$ by definition of $\mathcal{F}$ equidistance,
$\mathcal{F}_{c}(0)=\mathcal{F}_{h}(0)$. Hence, the momentary, relative
distance from equilibrium is determined by the sign of $\mathcal{R}(t)$.

We study the evolution of $\mathcal{R}(t)$ for overdamped diffusion in an
anharmonic potential $V(x)$. For simplicity, we analyse the case of one
spatial dimension and assume $V(x)$ to be of the form $V(x)=\lambda
x^{2}+k|x|^{\alpha}$, where we consider parameter values $\lambda$, $k$ and
$\alpha$ for which $V$ is confining, $V(x)\to\infty$ as $x\to\pm\infty$. We
move to a dimensionless formulation by defining a timescale $\tau$ and a
length scale $\ell$ as

$\tau=\frac{1}{\mu
k_{\text{B}}T_{f}}\left(\frac{k}{k_{\text{B}}T_{f}}\right)^{-2/\alpha}\,,\quad\ell=\left(\frac{k}{k_{\text{B}}T_{f}}\right)^{-1/\alpha}\,.$
(3)

Here, $\mu$ is the mobility. In the dimensionless coordinates, times are
measured in units of $\tau$, lengths in units of $\ell$, and energies in units
of $k_{\text{B}}T_{f}$. In particular, the transformation
$t\to\tilde{t}=t/\tau,x\to\tilde{x}=x/\ell$,
$V(x)\to\tilde{V}=V/(k_{B}T_{f})$, to dimensionless coordinates yields, after
dropping the tildes, the potential

$V(x)=\sigma x^{2}+|x|^{\alpha}\,,$ (4)

with the dimensionless parameter $\sigma=\lambda
k^{-2/\alpha}(k_{\text{B}}T_{f})^{2/\alpha-1}$. The parameter $\sigma$
quantifies the importance of the harmonic term $x^{2}$ compared to the
anharmonic term $|x|^{\alpha}$. We focus here on either monomial potentials
with $\sigma=0$, or on the case where $\sigma$ is small. Small $\sigma$ occurs
whenever (1) $\lambda$ is small, i.e, the harmonic coupling is weak, or (2)
$k$ is large, corresponding to strong anharmonic coupling. In addition, one
has the cases (3) $0<\alpha<2$ and small $T_{f}$, where the behaviour is
dominated by the (anharmonic) shape of the potential close to the origin, and
(4) $\alpha>2$ and large $T_{f}$, i.e., the dynamics takes place in the
anharmonic wings of the potential $V(x)$.

The Fokker-Planck equation Risken (1996) that determines the evolution of the
probability density during the relaxation reads, in the new coordinates,
$\partial_{t}p_{i}(x,t)=\mathcal{L}p_{i}(x,t)$ with

$\displaystyle\mathcal{L}=\partial_{x}\left[V^{\prime}(x)+\partial_{x}\right]\,,$
(5)

and initial conditions,

$\displaystyle
p_{i}(x,0)=\frac{\exp[-V(x)/{\mathcal{T}_{i}}]}{Z_{\mathcal{T}_{i}}}\,.$ (6)

Here, we introduced the dimensionless temperature ratios $\mathcal{T}_{i}$
that are either $\mathcal{T}_{h}\equiv T_{h}/T_{f}$ or $\mathcal{T}_{c}\equiv
T_{c}/T_{f}$, depending on whether the initial temperature before the quench
is $T_{h}$ or $T_{c}$. Note that for the final-temperature ratio
$\mathcal{T}_{f}\equiv T_{f}/T_{f}=1$. The constants $Z_{\mathcal{T}_{i}}$ in
Eq. (6) are obtained from normalising the probability density.

In the limit $t\to\infty$, the densities $p_{i}(x,t)$ relax to the equilibrium
distribution, $p_{f}(x)=\exp[-V(x)]/Z_{1}$. Hence, after the
$\mathcal{F}$-equidistant temperature quench at $t=0$, the evolution of the
relative distance from equilibrium, measured by $\mathcal{R}(t)$ [Eq.
(Relaxation-speed crossover in anharmonic potentials)], is a function of the
parameters $\sigma$, and $\alpha$ of the potential $V(x)$ [Eq. (4)] and of the
temperature ratios $\mathcal{T}_{i}$ that enter in the initial conditions (6).

Prior to the temperature quench, the hot and cold systems are prepared at
$\mathcal{F}$ equidistance so that their free-energy differences match. This
condition implicitly relates the hot and cold temperature ratios, so that we
can write $\mathcal{T}_{c}(\mathcal{T}_{h})$, with

$\mathcal{F}_{h}(0)=\mathcal{F}_{c}(0)\big{|}_{\mathcal{T}_{c}(\mathcal{T}_{h})}\equiv\mathcal{F}_{0}\,.$
(7)

Because $\mathcal{F}$ has a single minimum at equilibrium where
$\mathcal{T}=\mathcal{T}_{f}=1$ and $\mathcal{F}=0$, there is always exactly
one solution to Eq. (7) for which
$\mathcal{T}_{c}(\mathcal{T}_{h})<1<\mathcal{T}_{h}$. Figure 1(a) shows
schematically how the free-energy difference relates the different
temperatures.

At $t=0$, the formula for the dimensionless free energy difference
$\mathcal{F}_{0}$ at equidistance [Eq. (Relaxation-speed crossover in
anharmonic potentials) in units of $k_{\text{B}}T_{f}$] can be conveniently
written as

$\displaystyle\mathcal{F}_{0}=$
$\displaystyle\left[1+(1-\mathcal{T})\partial_{\mathcal{T}^{-1}}\right]\ln\left(\frac{Z_{\mathcal{T}}}{Z_{1}}\right)\,,$
(8)

where $\mathcal{T}=\mathcal{T}_{h}$ when $\mathcal{T}>1$ and
$\mathcal{T}=\mathcal{T}_{c}$ when $\mathcal{T}<1$. Hence, in order to obtain
the required $\mathcal{F}$-equidistant temperatures, we need to solve and
invert Eq. (8). This can be done analytically for $\sigma=0$, where we find

$\displaystyle\mathcal{F}_{0}=\frac{1}{\alpha}\left[\mathcal{T}-(1+\ln\mathcal{T})\right]\,,$
(9)

and by taking the inverse

$\mathcal{T}_{h}=-W_{-1}\left(-\mathcal{T}_{c}e^{-\mathcal{T}_{c}}\right)\,,\quad\mathcal{T}_{c}=-W_{0}\left(-\mathcal{T}_{h}e^{-\mathcal{T}_{h}}\right)\,.$
(10)

Figure 1: (a) Free-energy difference $\mathcal{F}_{0}$ at time $t=0$ for hot
(red line) and cold (blue line) temperatures, $\mathcal{T}_{h}$ and
$\mathcal{T}_{c}$, respectively. The $\mathcal{F}$-equidistance relation (7)
is represented by the grey lines. (b) $\mathcal{T}_{h}(\mathcal{T}_{c})$ (red
line) and $\mathcal{T}_{c}(\mathcal{T}_{h})$ (blue line) for $\sigma=0$, Eqs.
(10). The grey lines and coloured labels indicate a temperature pair
$(\mathcal{T}^{\prime}_{c},\mathcal{T}^{\prime}_{h})$, related by
$\mathcal{F}$ equidistance.

Here, $W_{n}(x)$, $n=-1,0$ denotes Lambert (or product-log) function DLMF .
Figure 1(b) shows $\mathcal{T}_{h}(\mathcal{T}_{c})$ (red line) and
$\mathcal{T}_{c}(\mathcal{T}_{h})$ (blue line) from Eqs. (10). For $\sigma\neq
0$ the implicit condition (7) must to be inverted numerically but the curves
remain almost unchanged (not shown).

After preparing the hot and cold systems at $\mathcal{F}$-equidistant
temperatures, both systems are put in contact with the same heat bath with
$\mathcal{T}_{f}=1$. At finite time $t>0$, the probability densities
$p_{i}(x,t)$ that enter $\mathcal{F}_{i}(t)$ and thus $\mathcal{R}(t)$ are
obtained from the Fokker-Planck equation by

$\displaystyle p_{i}(x,t)=e^{\mathcal{L}t}p_{i}(x,0)\,.$ (11)

In other words, in order to compute $\mathcal{R}(t)$ we must evaluate the
operator exponential in Eq. (11). This can be done in the short- and long-time
limits, leading to precise asymptotic results for $\mathcal{R}(t)$. As we show
below, the asymptotics of $\mathcal{R}(t)$ provide an excellent
characterisation of the dynamics, also at finite $t$.

For short times $t\ll 1$, the logarithmic ratio (2) reads

$\mathcal{R}(t)\sim\mathcal{\dot{R}}(0)t=\frac{\mathcal{\dot{F}}_{h}(0)-\mathcal{\dot{F}}_{c}(0)}{\mathcal{F}_{0}}t\,,$
(12)

where the dot denotes a time derivative and $\mathcal{F}_{0}$ is the initial
free-energy difference given in Eq. (8). Through
$\mathcal{\dot{F}}_{i}(0)=\int_{-\infty}^{\infty}\\!\\!\text{d}x\partial_{t}\,p_{i}(x,0)\ln[p_{i}(x,0)/p_{f}(x)]$,
the short-time evaluation of $\mathcal{R}(t)$ depends on the time derivative
$\partial_{t}p(x,0)$, evaluated at $t=0$. By expanding the exponential in Eq.
(11) for $t\ll 1$, we obtain
$\partial_{t}p_{i}(x,0)=(1-\mathcal{T}_{i})\partial_{x}^{2}p_{i}(x,0)$,
leading us, after integration by parts, to the following integral expression
for $\mathcal{\dot{F}}_{i}(0)$:

$\mathcal{\dot{F}}_{i}(0)=-\frac{(1-\mathcal{T}_{i})^{2}}{\mathcal{T}_{i}}\int_{-\infty}^{\infty}\text{d}x\,p_{i}(x,0)V^{\prime\prime}(x)\,.$
(13)

Evaluating Eqs. (13) and (8) for $i=h,c$, we obtain $\mathcal{R}(t)$ in the
short-time limit; see Eq. (12). For $\sigma=0$ we solve Eq. (13) explicitly,
which gives

$\mathcal{\dot{F}}_{i}(0)=(1-\alpha)\frac{(\mathcal{T}_{i}-1)^{2}}{\mathcal{T}_{i}^{2/\alpha}}\frac{\Gamma(1-1/\alpha)}{\Gamma(1+1/\alpha)}\,,$
(14)

where $\Gamma(x)$ denotes the gamma function DLMF . According to Eq. (12),
whether the hot or the cold system relaxes faster at short times is determined
by the sign of $\mathcal{\dot{R}}(0)$. As a function of $\alpha$ and
$\mathcal{T}_{h}$, where $\mathcal{T}_{c}$ follows from $\mathcal{F}$
equidistance, it is therefore instructive to draw a “phase diagram,” marking
the different regions in parameter space of initially faster heating
[$\mathcal{\dot{R}}(0)>0$] and initially faster cooling
[$\mathcal{\dot{R}}(0)<0$].

Figure 2(a) shows the short-time phase diagram for $\sigma=0$, spanned by
$\mathcal{T}_{h}$ and $\alpha$. It separates into an upper and a lower part
with different short-time behaviours. In the lower part,
$\mathcal{\dot{R}}(0)>0$ so that $\mathcal{R}(t)$ is initially positive for
all pairs $\mathcal{T}_{i}$; heating is faster than cooling. In the upper
part, cooling is initially faster than heating. The two parts are separated by
a critical line (red, dash-dotted line) where $\mathcal{\dot{R}}(0)=0$ so that
$\mathcal{R}(t)$ vanishes to first order in time,
$\mathcal{R}(t)\sim\mathcal{O}(t^{2})$. For $\sigma=0$, the critical line is
obtained by equating $\mathcal{\dot{F}}_{c}(0)=\mathcal{\dot{F}}_{h}(0)$, and
solving for $\alpha$. The smallest critical $\alpha$ value is found to be
$\alpha=3$, approached for infinitesimal temperature quenches,
$\mathcal{T}_{i}\to 1$. We note that this value, and the location of the
critical line in general, depends on the choice of temperature measure
$\mathcal{F}$. However, the existence of the critical line is robust against
moderate changes of $\mathcal{F}$; see Sec. II of the SM SM .

Similarly, small variations of $\sigma$ away from zero leave the the topology
of the short-time phase diagram unchanged. The generic effect of $\sigma>0$ on
the critical line is shown by the green, dashed lines and black arrows in Fig.
2(a), for values of $\sigma$ up to unity. We observe that slightly increasing
$\sigma$ moves the critical line to higher values but does not change the
phase diagram qualitatively.

When $\sigma$ is decreased to negative values, a more complex behaviour
emerges, shown by the blue, dotted lines and white arrows in Fig. 2(a): For
initial temperatures close to equilibrium $\mathcal{T}_{i}\approx 1$, the
critical line decreases slightly, to $\alpha$ values below $3$. For quenches
far from equilibrium, on the other hand, the critical line shifts to higher
$\alpha$.

Figure 2: Relaxation-speed phase diagram for short and long times. (a) Short-
time phase diagram, calculated from Eq. (13). Critical line for $\sigma=0$
(dash-dotted line), $\sigma=0.5$ and $1$ (dashed lines), and $\sigma=-0.1$,
$-0.15$, and $-0.2$ (dotted lines). The black and white arrows indicate how
the critical line changes as $\sigma$ is increased and decreased,
respectively, from zero. (b) Long-time phase diagram, calculated using the
eigenvalue decomposition (17). Critical line for $\sigma=0$ (solid line),
$\sigma=0.2$ and $0.4$ (dashed lines), and $\sigma=-0.1$, $-0.15$, and $-0.2$
(dotted lines). As in Fig. 2(a), the black and white arrows indicate how the
critical line changes.

We now turn to the analysis of the long-time limit $t\gg 1$ which requires
different methods. When the spectrum of $\mathcal{L}$ is discrete, the
relaxation of the probability densities $p_{i}$ to $p_{f}$ is exponential in
the long-time limit. As a result, the densities $p_{i}$ are determined by the
leading right eigenfunctions of $\mathcal{L}$ and their corresponding
eigenvalues Lu and Raz (2017), obtained from the non-Hermitian eigenvalue
problem

$\mathcal{L}r_{\mu}=\lambda_{\mu}r_{\mu}\,,\qquad\mathcal{L}^{\dagger}l_{\mu}=\lambda_{\mu}l_{\mu}\,,$
(15)

where $l_{\mu}$ and $r_{\mu}$ are the left and right eigenfunctions,
respectively, and $\lambda_{\mu}$ with
$\lambda_{0}=0>\lambda_{1}>\lambda_{2}>\ldots$ are the associated eigenvalues.
Note that the right eigenfunction $r_{0}$ with eigenvalue $\lambda_{0}=0$ is
given by the steady-state distribution $r_{0}=p_{f}$ and $l_{0}=1$. The
eigenfunctions form a complete biorthogonal basis with orthonormality
relations

$\displaystyle\langle
l_{\mu}|r_{\nu}\rangle=\int_{-\infty}^{\infty}\\!\\!\text{d}x\,l_{\mu}(x)r_{\nu}(x)=\delta_{\mu\nu}\,.$
(16)

Expanding $p_{i}$ in Eq. (11) in the right eigenbasis of $\mathcal{L}$ we
obtain in the long-time limit $t\gg 1$,

$p_{i}(x,t)\sim p_{f}(x)+c_{i,\mu}e^{\lambda_{\mu}t}r_{\mu}(x)\,,$ (17)

where $\mu$ is the lowest number for which $c_{i,\mu}\equiv\langle
l_{\mu}|p_{i}(0)\rangle\neq 0$. Because our problem is symmetric with respect
to the parity operation $x\to-x$, $c_{i,1}$ vanishes, so that $\mu=2$; see
Sec. III in the SM SM  for the case of a harmonic potential. All higher-order
terms in Eq. (17) that play a role at finite times are exponentially
suppressed in the long-time limit considered here. Using Eqs. (2) and (17) we
find that $\mathcal{R}(t)$ approaches a constant $\mathcal{R}_{\infty}$ for
$t\gg 1$ that depends only on the coefficients $c_{i,2}$:

$\displaystyle\mathcal{R}(t)\sim
2\ln\left(\left|\frac{c_{h,2}}{c_{c,2}}\right|\right)\equiv\mathcal{R}_{\infty}\,.$
(18)

Hence, the relative magnitude of the free-energy differences is determined by
the initial overlap between the left eigenvector $l_{2}$ of $\mathcal{L}$ and
the initial distributions $p_{i}(x,0)$ before the temperature quench Lu and
Raz (2017).

We determine $c_{i,2}$ by solving the eigenvalue problem (15) numerically,
discretising it on an evenly spaced, finite lattice with small lattice
spacing. Equations (15) then become matrix eigenvalue problems involving
large, non-symmetric matrices, whose left and right eigenvectors are
approximations of the left and right eigenfunctions $r_{\mu}$ and $l_{\mu}$.

Figure 2(b) shows the long-time phase diagram for $\sigma=0$ obtained from
numerically computing $c_{i,2}$ and evaluating $\mathcal{R}_{\infty}$ in Eq.
(18). The general structure of the long-time phase diagram is qualitatively
similar to that of the short-time phase diagram in Fig. 2(a), featuring
regions of faster heating ($\mathcal{R}_{\infty}>0$) and faster cooling
($\mathcal{R}_{\infty}<0$). For long times, however, the critical line [solid
line in Fig. 2(b)] is located at slightly higher values. Consequently, the
minimum of the critical line, attained for close-to-equilibrium quenches,
takes the slightly larger value $\alpha\approx 3.31$. As in the short-time
limit, the long-time critical line is only weakly perturbed by moderate
changes of the temperature measure $\mathcal{F}$; see Sec. II of the SM SM
for details.

Upon increasing $\sigma$, we again observe no qualitative change of the phase
diagram; the critical line is merely pushed to larger $\alpha$ values [green,
dashed lines in Fig. 2(b)]. Negative $\sigma$, on the other hand, leads to a
qualitative change: For $\sigma<0$, the region of asymptotically faster
cooling becomes finite and is completely enclosed by the critical line [blue,
dotted lines in Fig. 2(b)]. The sensitive dependence of the relaxation
dynamics on negative values of $\sigma$, observed both in the short- and long-
time limits, must be due to the emergence of bistability of the potential
$V(x)$, Eq. (4). The existence of two potential minima gives rise to multiple
relaxation timescales associated with the relaxation within the same minimum
and across the two minima.

From the general structure of the phase diagrams we conclude that
asymptotically steep potentials (large $\alpha$) lead to faster cooling,
compared to $\mathcal{F}$-equidistant heating, when the initial temperature
differences are not too large. For small $\alpha$, the opposite is true.
Intuitively, this may be explained by noting that for an initially hot system,
more probability is located in the tails of the distribution. The steeper the
potential, the faster this tail probability is advectively transported toward
the potential minimum, leading to faster cooling. For small $\alpha$, this
advection effect is weaker, so that it is outperformed by the diffusive
broadening of the bulk of the distribution of the cold system, thus resulting
in faster heating.

Our analysis reveals the existence of distinct critical lines in the short-
and long-time limits. This results in an overlap between the faster-heating
and faster-cooling regions at short and long times, giving rise to a crossover
region in the phase diagram. In the crossover region, the hot system initially
relaxes faster [$\mathcal{R}(t)<0$], but is eventually overtaken by the
initially colder system [$\mathcal{R}(t)>0$]. Hence, there must be at least
one finite time $t_{c}>0$ where $\mathcal{R}(t_{c})=0$, i.e., the system
crosses over from faster cooling to faster heating.

Figure 3: (a) Superimposed short- and long-time phase diagrams for $\sigma=0$,
featuring the critical lines in the short-time (dash-dotted line) and long-
time (solid line) limits. The crossover region is shown by the cross-hatched
region. The coloured dots correspond to the parameter values for the plots in
Fig. 3(b). (b) $\mathcal{R}(t)$ from different numerical methods for
$\mathcal{T}_{h}=3$ and $\alpha=3$, $3.3$ and $3.5$, in blue, green and red,
respectively. The dash-dotted lines show results obtained from Eq. (11), by
numerically calculating the spectrum of $\mathcal{L}$. The solid lines are
computed from numerical simulations of the Langevin equation. The black,
dashed lines correspond to the short- and long-time asymptotics.

Figure 3(a) shows the superimposed short- and long-time phase diagrams for
$\sigma=0$ featuring the crossover region (cross-hatched area). The dash-
dotted and solid lines show the critical lines from Figs. 2(a) and (b),
respectively.

In order to study the behaviour of $\mathcal{R}(t)$ in the crossover region,
and to validate our previous results, we perform a numerical analysis of the
finite-time evolution of $\mathcal{R}(t)$. We focus on a few points in the
phase diagram, shown as the differently coloured dots in Fig. 3(a), where we
expect qualitatively different behaviours: For the parameter sets represented
by the blue and red dots, we expect heating and cooling, respectively, to be
faster, both for short and for long times. By contrast, for the parameters of
the green dot we expect at least one finite-time crossover from faster cooling
to faster heating.

For the finite-time analysis we use two different numerical methods. First, we
obtain an approximation of $p_{i}(x,t)$ by using the discretised analogue of
Eq. (11) obtained with the discretisation scheme discussed earlier.

The second method approximates the probability density $p_{i}(x,t)$ by means
of a Langevin approach Van Kampen (1992): We simulate a large number of
trajectories $x_{i}(t)$, $i=h,c$, following the dynamics
$\dot{x}_{i}(t)=-V^{\prime}(x)+\xi(t)$, where $\xi(t)$ is a Gaussian white-
noise signal with correlation function
$\langle\xi(t)\xi(t^{\prime})\rangle=2\delta(t-t^{\prime})$. The initial
values $x_{i}(0)$ are sampled from the equilibrium distributions $p_{i}(x,0)$
prior to the temperature quench. The Langevin equation is solved numerically
using an Euler-Maruyama scheme Kloeden and Platen (1992) with a small time
step. The probability densities $p_{i}(x,t)$ are then computed by generating
histograms over all locations $x_{i}(t)$ at discrete times $t$.

These methods, whose parameters are summarised in Sec. IV of the SM SM , yield
two independent numerical approximations $p_{i}(x,t)$ from which we then
calculate $\mathcal{R}(t)$. Figure 3(b) shows the so-obtained
$\mathcal{R}(t)$, where the colours of the curves correspond to the colours of
the dots in Fig. 3(a). The dash-dotted lines show $\mathcal{R}(t)$ calculated
from the discretised operator $\mathcal{L}$. The lighter, solid lines show the
corresponding results from the Langevin approach. Also shown are the short-
and long-time asymptotes (dashed lines). We observe that the asymptotes
represent a good characterisation of the dynamics of $\mathcal{R}(t)$ for all
times. In particular, there are no finite-time crossings
$\mathcal{R}(t_{c})=0$ for the parameter values outside of the crossover
region in Fig. 3(a), i.e., for the blue and red curves. Inside the crossover
region [see green curve in Fig. 3(b)] we observe only a single crossing.

Furthermore, there is good agreement between the results from the different
numerical methods and the asymptotic results. Note that the deviations between
the equally coloured curves become larger for longer times. The reason is that
for long times, the individual free-energy differences $\mathcal{F}_{i}(t)$ in
Eq. (2) become exponentially small, so that the relative errors increase as
$t$ becomes large. Due to this numerical difficulty, we were unable to
evaluate $\mathcal{R}(t)$ until convergence, as can be seen by the discrepancy
between our numerical results and the long-time asymptotics [horizontal,
dashed lines in Fig. 3(b)].

Finally, we note that far from equilibrium, for $\mathcal{T}_{c}\approx
0.0229$ and $\mathcal{T}_{h}\approx 5.50$, the short- and long-time critical
lines cross [see Fig. 3(a)] which implies the existence of an inverted
crossover region very far from equilibrium where heating is initially faster
but asymptotically slower than cooling.

In conclusion, $\mathcal{F}$-equidistant thermal relaxation of overdamped
diffusions in anharmonic potentials $V(x)$ allows for both faster heating and
faster cooling, even when $V(x)$ has a single minimum. As a consequence, the
short- and long-time phase diagrams [Figs. 2(a) and (b)], spanned by the
$(\mathcal{T}_{h},\alpha)$-parameter space, are nontrivial, exhibiting regions
of faster heating and faster cooling. Both for short and for long times, we
found that cooling is faster than heating for sufficiently large $\alpha$, and
heating is faster than cooling for small $\alpha$. This can be explained in
terms of a competition between the advective relaxation of the tail
probability of the hot system, and the diffusive broadening of the bulk-
probability in the cold system. Despite the similarities between the short-
and long-time phase diagrams, we found that their critical lines are
different, and that the faster-heating and faster-cooling regions overlap.
Superimposing the two, we localised a crossover region [Fig. 3(a)] where
cooling is initially faster but the rate of heating eventually overtakes.
Outside of the crossover region, we found no crossings, suggesting that the
short- and long-time asymptotics faithfully characterise the relative
relaxation speeds. The critical lines separating the parameter regions with
different behaviours are only weakly perturbed by moderate changes of the
temperature measure $\mathcal{F}$ or by an additional harmonic term in the
potential $V(x)$, as long as the latter remains single-well, i.e., $\sigma>0$.

It would be interesting to test the relaxation-speed crossover in experiments
and thus to reproduce our phase diagram under experimental conditions. This
requires tracking the changes in energy and entropy of the system throughout
the experiment which is possible in state-of-the-art setups Kumar and
Bechhoefer (2020); Ciliberto (2017). On the theoretical side, it would be
desirable to understand the precise dynamical origin of the different
relaxation behaviours 111In Sec. V of SM , we trace back the long-time
relaxation asymmetry in harmonic potentials to the convexity of the inverse of
the temperature measure $\mathcal{F}$.. This might lead to optimisation
methods for the potential to achieve faster heating or cooling, perhaps in the
spirit of first-passage time optimisation Palyulin and Metzler (2012); Chupeau
_et al._ (2020).

###### Acknowledgements.

We thank John Bechhoefer and Massimiliano Esposito for discussions. Funding
from the European Research Council within the project “NanoThermo”
(ERC-2015-CoG Agreement No. 681456) and the Foundational Questions Institute
within the project “Information as a fuel in colloids and superconducting
quantum circuits” (Grant No. FQXi-IAF19-05) is gratefully acknowledged.

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