The culture of providing supplemental material: A thriving future for the
field of audiovisual idiom design also depends on the communities’ culture
when providing demos for their designs as supplemental material. Out of 57
articles in our surveyed literature, only 29 provided a demo that is still
available today. Five provided a demo that is no longer available, and 23
provided no demo at all. Online repositories such as Zenodo, hosted by CERN,
the platform OSF, maintained by the Center for Open Science, or long-term
archives such as Phaidra, hosted by the University of Vienna, allow the upload
of videos and even the assignment of DOIs. While making sound available in a
paper used to be an explicit challenge to the sonification field, it is not
the case anymore. An additional visual representation of the sound can also be
included in the paper, using a spectrogram of the sound itself or an iconic
representation of it.

Evaluating audiovisual idioms: It is often challenging to conduct systematic,
well-informed, and, above all, properly performed and reported evaluations of
audiovisual idioms, especially when the goal is to compare different studies.
As mentioned earlier, Rönnberg and Forsell suggested the standardization of
questionnaires tailored to the field in order to increase the comparability
between different studies [RF22]. Establishing this comparability is
particularly difficult because the two senses are used simultaneously in an
audiovisual display, and the evaluation methods must adapt to this
characteristic of the design. Specifically, the existing methods focus on
several aspects. One is user performance, which studies quantitative metrics
such as task completion times and precision of responses. Another is user
experience, which focuses more on qualitative metrics such as user engagement
and memorability. From the data we collected in our STAR, we can draw high-
level conclusions regarding the type of evaluations that have been used within
the corpus of literature.

Among the papers that included a user evaluation (21 of the 57 papers), most
reported a subjectively stated benefit of sonification in perceiving and
interpreting the visualization. Some subjective ratings and feedback also
suggested that the use of sonification was engaging (six papers considered
this [HK22, RS22, EEBR21, KAV21, Rön21, DCM∗18]), could provide emotional
content (mainly two papers [RS22, Rön21]), was experienced as interesting
(five papers [HK22, RS22, EEBR21, Rön21, DCM∗18]), as well as pleasant (four
papers [RS22, EEBR21, Rön21, DCM∗18]), and in perceived improved quality of
visual information (six papers [PFH∗22, EEBR21, KAV21, FN18, RJ16, BF12]).
Other papers also reported subjective and objective measurements of improved
interpretation and higher understanding of data when sonification and
visualization were used in combination (16 papers [PFH∗22, CWM21, EEBR21,
DCM∗18, FN18, GDAS∗18, YH18, BBV16, GRK∗16, NSC16, RJ16, PBM15, PBV14, ASH∗12,
BF12, Bea11]). This was shown in terms of higher scores or increased accuracy
when sonification was used. However, this increase in accuracy was reported to
be related to increased task completion times (see, for example, in [RJ16]).
Similar effects have also been shown in other studies not included in this
systematic review [MBKSM16, Rön19a, Rön19b]. The case might be that the use of
the additional information provided by sonification that makes the increased
accuracy possible takes a longer time to process and assess. While
traditionally increased task completion times are considered a disadvantage,
in some contexts, this can be seen as positive. Slowing down the user,
resulting in them spending more time with their data, studying it from more
and from different perspectives (including the auditory perspective), might
reveal structures or patterns in a dataset that would have stayed silent and
unseen if analyzed with a conventional unimodal display [HAS11].

These results indicate that there is, in general, a benefit of combining
visualization and sonification for data exploration and analysis, regardless
of the specific visualization and/or sonification design. Nevertheless, we
consider it a major task for the future research of our community to
systematically study what kinds of benefits the integration of sonification
and visualization can provide.

The potential of unconventional task distribution: The data from our STAR
shows that reading levels [Ber83] are not equally distributed to the two
senses: The vast majority – 50 out of 57 papers used the reading level “whole”
for their visualization (see Table 2), while only about half of the papers use
the same reading level for their sonification parts. This observation might be
directly related to the way audiovisual idioms distribute the tasks of
overview and detail to the two senses. The currently more popular distribution
seems to be to use visualization to display an overview of the data, while
sonification is employed to provide details. While this distribution seems
like an intuitive choice, future research should study the potential of
switching the roles between sonification and visualization. It is our daily
lives and the way we experience the world around us that suggests sonifying an
overview and visualizing details. Our auditory sense constantly screens our
360° environment while our visual sense actively focuses on details in our
environment. The field of auditory process monitoring makes use of exactly
this situation when alerting us of a status, regardless of our current visual
focus. Nevertheless, we identified only three cases where the reading level
for the sonification was “whole” and the one for the visualization was “group”
[DLVDCG22, FBC12, YH18].

A second phenomenon we observe in the data is that many audiovisual display
idioms employ the same search level [Mun15] (see Figure 8), and most of them
employ the search level of “explore” (see Table 2). Again, it seems like a
fruitful future endeavor to study the opportunities of designs that explicitly
break such a pattern.

The potential of integrating sonification and scientific visualization:
Scientific visualization typically displays inherently spatial phenomena and
is often used to represent data that varies over time. The inherently spatio-
temporal quality of such displays constitutes their potential to be integrated
with sonification, as sonification is a display technique tailored toward
temporal data structures [GGB05, ERI∗23]. We relate this argument back to the
real-world example from our introduction, where the combination of our visual
and auditory senses offers a better understanding of a phenomenon than each
sense alone: Rain falling outside of a closed window. Opening the window to
not only see but also hear the sound of the rain gives us a better estimation
of the amount and intensity of the rainfall. Abstractly, in such a situation,
we perceive a spatio-temporal phenomenon. It is most plausible that adding
sound to a time series of scientific visualization can increase its holistic
interpretability in an ecologically valid manner [Neu04]. This phenomenon is
shown in two publications in the database, in the work by MacDonald et al.
[MNW∗18], and the work by Temor et al. [TMN∗21]. Similarly, these ideas also
apply to animations in information visualization, where temporal aspects of
the data might be visualized and sonified.

Design frameworks: When it comes to the possibilities of designing audiovisual
display idioms, most researchers, as well as most domain experts, will not be
able to develop their own designs. Interdisciplinary knowledge bridging
visualization, sonification, interactive design, and human perception is
necessary to design effective, engaging, and re-usable audiovisual display
idioms. Therefore, we cannot expect a domain expert to be able to quickly
draft a prototype the same way they may be able to do using established
visualization-only frameworks such as matplotlib [Hun07].

In our database, we identified several design frameworks tackling this
challenge [PC19, PCB23, LF21, DLVDCG22, KLTW17, CWM21]. Their primary focus is
the design of the sonification part of the idiom, which is why we cannot speak
of truly balanced contributions to both senses. A promising endeavor to enable
truly balanced designs, with both the visualization and the sonification being
equally well-informed, might be to not design them using one but two separate
frameworks, each specialized for its purpose. In this regard, the recent work
by Reinsch and Hermann [HR21, RH22, RH23] is promising, as it provides a
sonification design framework embedded in the Python environment and is
conceptually inspired by visualization design tools such as matplotlib. With
both the sonification design and the visualization design happening in the
same environment, in this case, for example, a Jupyter Notebook, both can be
developed with the required quality to meet their individual standards.
Regarding the introduction of users to a new design environment, be it for
visualization, sonification, or their combination, effective “onboarding” is a
pressing issue. In the visualization community, the topic has been studied in
recent years [SPA23, SWG∗23], and the same will be necessary for sonification
and audiovisual display idioms. Again, when it comes to sonification as a
technique to display data, the prior experience of domain experts or the
general public is low. This lack of experience can, in the short term, only be
met with carefully designed onboarding processes.

Adjacent topics as inspiration to the field: Our brief look into the adjacent
fields of monitoring, accessibility, and arts shows a special potential for
inspiration for our communities’ future work. The field of monitoring offers
established evaluation methods that could potentially be adopted to support
the evaluation of audiovisual idioms tailored toward exploratory data
analysis. During our unstructured investigation into the field, we also
identified the potential of a systematic STAR as a future contribution to the
field of monitoring.

Also, the adjacent field of accessibility can inspire future research, where
understanding the analysis patterns of visually impaired individuals can
inform design decisions for idioms tailored towards sighted people [WM10].
While the field of accessibility can be informative to the field of
audiovisual idioms, the same holds true the other way around. Especially the
designs from our STAR that use redundant mapping to both the visual and the
auditory senses can inspire future designs for visually impaired users.
Furthermore, such designs have the potential to foster successful
collaborative data analysis involving both sighted and blind users.

The identified artistic contributions show great potential to be inspirational
to the field of audiovisual idiom design as well. In the future, effort should
be made by the research community to develop a specific framework for the
evaluation and analysis of audiovisual idioms at the crossing of art and
public engagement. Such efforts need to explore how specific design strategies
(e.g., interactivity and embodiment, as used in many of the cases presented in
subsection 5.3) can be combined to support both affect and sense-making.
Again, a systematic STAR dedicated to the artistic perspective would be a
highly timely contribution to the community.

An audiovisual analytics community: With this state-of-the-art report, we hope
to contribute to the future establishment of systematic research on
audiovisual display idiom design. The current development concerning the
studied co-author network and the temporal development of publication numbers
in the field is promising. We hope to reach researchers from both the
visualization and sonification communities, inspiring them for future
interdisciplinary collaborations, and to open their ears and take a look.

## Supplemental Material

As supplemental material to this STAR, we provide at
https://phaidra.fhstp.ac.at/o:5541:

  * •

corpus.bib and corpus.rdf: publication metadata of the surveyed literature in
BibTeX and Zotero RDF format.

  * •

corpus-tagging.csv: a table holding the surveyed literature and all of the
used tags. Interested readers may use the table to identify articles that fall
into specific combinations of classes.

  * •

authors.csv: a table holding the names of the authors in our STAR corpus and
their primary discipline. Interested readers may use it to identify potential
future collaborators.

  * •

corrmat.pdf: a high-resolution version of Figure 8.

  * •

authornet.ows: the Orange Data Mining workflow file, used to generate the co-
author network in Figure 9 (incl. the used color scheme file
authornet_colorscheme.colors).

In addition, the corpus of relevant papers with tags, publication metadata,
and full text of open access work is available as a Zotero library at
https://www.zotero.org/groups/integrationsonificationvisualization/items.

## Acknowledgments

We would like to thank PerMagnus Lindborg, School of Creative Media, City
University of Hong Kong, for valuable feedback. This research was funded in
part by the Austrian Science Fund (FWF): 10.55776/P33531 and the Gesellschaft
für Forschungsförderung Niederösterreich (GFF) SC20-006, as well as by the
Knut and Alice Wallenberg Foundation (grant KAW 2019.0024).

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# Thermodynamic Bounds on the Asymmetry of Cross-Correlations with Dynamical
Activity and Entropy Production

Jie Gu<EMAIL_ADDRESS>Chengdu Academy of Education Sciences, Chengdu
610036, China

###### Abstract

Entropy production and dynamical activity are two complementary aspects in
nonequilibrium physics. The asymmetry of cross-correlation, serving as a
distinctive feature of nonequilibrium, also finds widespread utility. In this
Letter, we establish two thermodynamic bounds on the normalized asymmetry of
cross-correlation in terms of dynamical activity and entropy production rate.
These bounds demonstrate broad applicability, and offer experimental
testability.

††preprint: APS/123-QED

Introduction.—Entropy production plays a pivotal role in nonequilibrium
thermodynamics and statistical mechanics, serving as a fundamental quantity of
interest [1]. The quest to establish connections between entropy production
and measurable parameters has been a central focus in this field. Any
manifestation of nonequilibrium should be connected with entropy production or
dissipation. For instantce, maintaining instantaneous equilibrium necessitates
infinitely slow driving, implying that entropy production accompanies any
finite-time process. This has led to the notion of thermodynamic geometry [2,
3, 4, 5] and speed limits in terms of entropy production [6, 7, 8, 9]. In
equilibrium, currents vanish, so nonvanishing currents are also a signature of
nonequilibrium, leading to the existence of bounds on entropy production in
terms of nonvanishing currents normalized by its variance, known as
thermodynamic uncertainty relations [10, 11, 12]. Additionally, equilibrium
conditions give rise to the principle of microscopic reversibility, implying
the symmetry of cross-correlations. Consequently, corresponding asymmetry also
serves as a distinguishing feature of nonequilibrium steady states [13], and
is presumably associated with entropy production [14].

Recent interest has resurfaced regarding this asymmetry of cross-correlations.
Ohga et al. have reported a fundamental inequality that explores the
relationship between the normalized asymmetry of cross-correlation and the
thermodynamic forces driving the system out of equilibrium [15]. To illustrate
their theory, the authors proved that the number of coherent biochemical
oscillations is equivalent to the normalized asymmetry of cross-correlation
between certain observables, confirming the conjecture that the coherence of
biochemical oscillations is bounded by the driving force [16]. Building upon
another conjecture stating that the average entropy production per oscillation
is bounded from below by the number of coherent oscillations if at least one
oscillation is visible [17], their result also reveals a connection between
asymmetry of cross-correlation and entropy production. Following their idea,
Shiraishi found that the normalized asymmetry is bounded from above by the
entropy production per characteristic maximum oscillation time [18].
Extensions to finite-time domain for classical [19] and quantum systems have
been made [20].

On the other hand, the dynamical activity [21, 22, 23, 24, 25] is a crucial
but less explored component in nonequilibrium physics. Its significance only
emerges beyond linear order around equilibrium, and has recently been
highlighted in studies on out-of-equilibrium fluctuation-response relations
[22, 26], classical speed limits [6, 7, 8], thermodynamic (kinetic)
uncertainty relations [27, 28, 29, 30], inference of entropy production with
lacking data [31, 32], power-efficiency trade-off in heat engines [33], upper
bound on entropy production [34], and the thermodynamic correlation inequality
[35]. In essence, the dynamical activity quantifies the frequency of
transitions, exhibiting a time-symmetric characteristic. In contrast, the
entropy production is time-antisymmetric, changing its sign upon time
reversal, thereby inverting the fluxes. Hence, these two quantities naturally
emerge as complementary facets. Despite its importance, the connection between
the dynamical activity and entropy production remains elusive. Complementary
to previous studies [15, 18, 19, 20], this Letter establishes two
thermodynamic bounds on the normalized asymmetry of cross-correlations in
terms of dynamical activity and entropy production rate [Eqs. (8) and (9)]. We
prove the first bound for unicyclic systems and the second bound for general
cases, and present the condition for saturation. We also provide numerical
evidence to support the validity of the first bound for arbitrary network
topology. All the quantities involved are measurable, making the inequalities
experimentally testable.

Setup.— Consider a stochastic Markov jump process with finite $\mathcal{N}$
states. The dynamics of the probability distribution
${\bm{p}}=[p_{1},p_{2},\ldots,p_{N}]^{\mathsf{T}}$ is described by the master
equation [36]

$\frac{\text{d}{p}_{m}}{\text{d}t}=\sum_{n}W_{mn}p_{n},$ (1)

where $p_{m}$ is the probability of state $m$, $W_{mn}(m\neq n)$ is the time-
independent transition rate from state $n$ to $m$, and the rate of leaving
state $m$ is $W_{mm}=-\sum_{n(\neq m)}W_{nm}$. Thermodynamic consistency is
assumed, i.e., whenever $W_{mn}\neq 0$, $W_{nm}$ is nonzero too. Physically,
this assumption means that the transition between two states is either
forbidden or bidirectional. We also assume that the system is in a
nonequilibrium steady state $\bm{p}^{\text{ss}}$ satisfying
$\bm{W}\bm{p}^{\text{ss}}=0$. After defining the one-way flux
$\mathcal{T}_{mn}=W_{mn}p^{\text{ss}}_{n}$, the probability current
$\mathcal{J}_{mn}$ and local activity (also called traffic) $\mathcal{A}_{mn}$
can be defined in terms of $\mathcal{T}_{mn}(m\neq n)$ respectively as

$\displaystyle\mathcal{J}_{mn}\equiv\mathcal{T}_{mn}-\mathcal{T}_{nm},$ (2)
$\displaystyle\mathcal{A}_{mn}\equiv\mathcal{T}_{mn}+\mathcal{T}_{nm}.$

For later use, we also define a quantity $\gamma$ that characterizes the scale
separation of local activities, given by

$\gamma\equiv\frac{\max\mathcal{T}_{mn}}{\min\mathcal{T}_{mn}},\quad\text{for}\,\,\mathcal{T}_{mn}>0.$
(3)

According to stochastic thermodynamics [37, 38, 39, 30], the dynamical
activity $\mathcal{A}$ and the entropy production rate $\sigma$ can be
expressed as

$\displaystyle\mathcal{A}=\sum_{m\neq
n}\mathcal{T}_{mn}=\sum_{m>n}\mathcal{A}_{mn},$ (4)
$\displaystyle\sigma=\sum_{m\neq
n}\mathcal{T}_{mn}\ln\frac{\mathcal{T}_{mn}}{\mathcal{T}_{nm}}.$

The two-time correlation between observables $a$ and $b$ at time lag $\tau$ is

$C_{ba}^{\tau}\equiv\langle b(t+\tau)a(t)\rangle,$ (5)

where $\langle\cdot\rangle$ denotes average over trials, and the central
quantity is the normalized asymmetry of cross-correlation $\chi_{ab}$ defined
as

$\chi_{ab}\equiv\frac{\alpha_{ab}}{\sqrt{D_{a}D_{b}}}\equiv\frac{(\partial_{\tau}C_{ba}^{\tau}-\partial_{\tau}C_{ab}^{\tau})/2}{\sqrt{\partial_{\tau}C_{aa}^{\tau}\partial_{\tau}C_{ba}^{\tau}}}.$
(6)

The numerator $\alpha_{ab}$, i.e., the slope of cross-correlation asymmetry
(also called stationary fluctuation oscillation in Ref. [18]), vanishes in
equilibrium, so a nonzero asymmetry implies nonequilibrium. The slopes of
auto-correlations, $D_{aa}$ and $D_{bb}$, are a measure of diffusion [15].
Explicitly, the slopes of cross-correlation asymmetry and average auto-
correlation can be expressed as

$\displaystyle\alpha_{ab}=\frac{1}{2}\sum_{m,n}(a_{n}b_{m}-a_{m}b_{n})\mathcal{T}_{mn}=\sum_{m<n}\Omega_{mn}\mathcal{J}_{mn},$
(7)
$\displaystyle\frac{{D_{a}+D_{b}}}{2}=\frac{1}{4}\sum_{m,n}[(a_{m}-a_{n})^{2}+(b_{m}-b_{n})^{2}]\mathcal{T}_{mn}$
$\displaystyle\qquad\quad=\frac{1}{4}\sum_{m<n}L_{mn}^{2}\mathcal{A}_{mn},$

with $\Omega_{mn}=(a_{n}b_{m}-a_{m}b_{n})/2$ and
$L_{mn}=\sqrt{(a_{m}-a_{n})^{2}+(b_{m}-b_{n})^{2}}$. Both $\Omega_{mn}$ and
$L_{mn}$ have geometric meaning: $\Omega_{mn}$ is the oriented area of the
triangle with $m,n$ and the origin, and $L_{mn}$ the edge between $m$ and $n$.
With all these relevant quantities, we present our main result below.

Figure 1:  (a)-(c): Three topologies of networks with four states. (d):
Scatter plot of $|\chi_{ab}|$ vs. the first bound. All data points lie below
the diagonal, which validates the bound.

Main result.— Our main results are two thermodynamic bounds that connect the
normalized asymmetry of cross-correlations $\chi_{ab}$, dynamical activity
$\mathcal{A}$ and entropy production rate $\sigma$ for a Markov jump process
with $\mathcal{N}$ states, given by

$|\chi_{ab}|\leq\frac{\gamma}{\tan(\pi/\mathcal{N})}\sqrt{\frac{\sigma}{2\mathcal{A}}}$
(8)

and

$|\chi_{ab}|\leq\sqrt{\frac{\mathcal{N}^{*}\gamma}{2\pi\tan(\pi/\mathcal{N})}}\sqrt{\frac{\sigma}{\mathcal{A}}},$
(9)

where $\mathcal{N}^{*}$ is the number of nonzero local activities. The
derivation of Eq. (8) for unicyclic networks and Eq. (9) for general cases is
deferred to the end of the Letter. We present numerical results to support Eq.
(8) in Fig. 1(d), with a total of $10^{6}$ data points and three distinct
topologies as in Figs. 1(a)-(c). Each data point is generated as follows: we
draw nonzero transition rates $W_{mn}$’s from the uniform distribution,
calculate the diagonals of the transition rate matrix, and randomly sample the
observables $a_{m}$ and $b_{m}(m=1,\ldots,\mathcal{N})$ from the interval
$[-1,1]$. Subsequently, we calculate the bound and $|\chi_{ab}|$ using Eqs.
(3)–(7). The scatter plot clearly validates the first bound for four-state
networks, and networks with a different number of states show qualitatively
similar results. The three topologies are explicitly considered because it is
inherently impossible to obtain non-fully-connected networks [Figs. 1(b)-(c)]
through random sampling, but they are qualitatively different from each other.
This is because the bound is discontinuous at $W_{mn}=0$, arising from the
presence of $\gamma$. For example, assume that $\mathcal{T}_{mn}$’s are finite
for any $mn\neq 13$ and $31$, and all $p^{\text{ss}}_{m}$’s are finite as
well. If $W_{13}=W_{31}=0$, then $\gamma=\max\mathcal{T}_{mn}/\min_{mn\neq
13,31}\mathcal{T}_{mn}$. However, if $W_{13}\ll 1$ and $W_{31}$ is finite,
then $\gamma=\max\mathcal{T}_{mn}/\mathcal{T}_{13}$, which can be arbitrarily
large.

It can be seen from the derivation that bound (8) saturates in unicyclic
networks when the transition rates are uniform, the observables form a regular
polygon, and the system is close to equilibrium. This saturation condition is
similar to that of the thermodynamic uncertainty relation [10]. In contrast,
bound (9) is saturated in unicyclic networks when the transition rates are
uniform, the observables form a regular polygon, and the number of states
$\mathcal{N}$ approaches infinity. Under these conditions, the two bounds are
asymptotically equivalent. Otherwise, either bound can be tighter depending on
the parameters.

Several remarks are in order regarding the implications of our findings.
Inequality (8) reveals a thermodynamic bound on the normalized cross-
correlation asymmetry, hinging upon the dynamical activity and entropy
production rate—specifically, the square root of their quotient. This
structure is natural, as will be discussed later. The cross-correlation
asymmetry has emerged as a versatile tool widely employed to investigate an
array of phenomena spanning directed interactions, non-equilibrium
oscillations, nonreciprocal motion and so on, as summarized in Ref. [15]. Our
result is applicable as long as the dynamics can be modeled by a Markov jump
process, irrespective of the underlying network topology. This includes
chemical reactions, biochemical systems, and quantum transport, among others.
All the quantities constituting the bound are experimentally measurable,
rendering our findings amenable to empirical validation. Correlations can be
quantified through techniques such as fluorescence cross-correlation
spectroscopy [40, 41]. As for the right-hand side of the inequalities, the
quantities $\gamma$ and $\mathcal{A}$ can be obtained by counting jumps in a
sufficiently long trajectory, and $\sigma$ can be measured through the
energetics of the environment. Therefore, this far-from-equilibrium relation
is in principle experimentally testable.

Connection and comparison with previous works.— We begin by comparing our
findings with the seminal work of Ohga et al. [15]. Both studies impose an
upper bound on the normalized asymmetry of cross-correlation, while a crucial
distinction emerges: the bound established in Ref. [15] relies on the
employment of the maximum cycle affinity as a thermodynamic quantity, while
our bound focuses on the entropy production rate. The bounds presented in
Refs. [18] and [20] incorporate the entropy production rate as a key factor,
too. However, a notable distinction between our result and theirs lies in the
observable dependency of their bounds, while our derived bound is independent
of any specific observables. This distinction, akin to the approach employed
in Ref. [15], arises from the consideration of the normalized asymmetry, in
which the normalization factor in the denominator already encapsulates the
information of observables. In contrast, the bounds proposed in Refs. [18] and
[20] pertain directly to the asymmetry itself, rendering them inherently
observable-dependent.

Regarding the dynamical activity and entropy production rate, both of them are
greater than the pseudo entropy production rate [30]. The reciprocal of the
relative fluctuation can be proved to be less than the pseudo entropy
production [42, 30], so the thermodynamic uncertainty relation and the kinetic
uncertainty relation follow immediately. This shows the “duality” between the
activity and entropy production rate, but not the relation between them. The
product of $\mathcal{A}$ and $\sigma$ appear in several studies. For example,
in the classical speed limit [6], $\mathcal{W}^{2}\leq
2\bar{\mathcal{A}}\sigma\tau$, where $\mathcal{W}$ is the Wasserstein distance
between the initial and final probability distributions, $\bar{\mathcal{A}}$
the time-averaged activity, and $\tau$ the evolution time duration. By
contrast, the quotient of $\sigma$ over $\mathcal{A}$ appears in this study,
which arises naturally from two perspectives. From a dimensional analysis
standpoint, $\chi_{ab}$ is dimensionless, while $\sigma$ has the dimension of
rates. The dynamical activity $\mathcal{A}$ quantifies the inherent time-scale
of the system with the dimension of rates too, so it seems intuitive to employ
$\mathcal{A}$ as the denominator. Furthermore, the numerator of the left-hand
side, representing the asymmetry of cross-correlation, exhibits time-
antisymmetry, while the denominator, corresponding to the geometric average of
auto-correlation, displays time-symmetry. As discussed earlier, the entropy
production (dynamical activity) also exhibits (anti)symmetry with respect to
time. By selecting the dynamical activity as the denominator, the structure of
the normalized asymmetry is preserved.

Figure 2: (a)–(d): Ratio between $|\chi_{ab}|$ and the bound as a function of
the cycle affinity $\mathcal{F}$ (or $k^{+}$). The left (right) column
corresponds to Example 1 (2), and the upper (lower) row corresponds to the
first (second) bound. (e) Schematic of the model in Example 2, where three
states are grouped into a cell.

Example 1.— As the first example, we consider the standard model of
biochemical signal transduction as in Ref. [15]. The system comprises an
upstream receptor and a downstream protein. The upstream receptor undergoes
stochastic switching between “OFF” and “ON” states, corresponding to the
observable $a=0,1$. Similarly, the downstream protein stochastically switches
between inactive and active states, corresponding to the observable $b=0,1$.
The dynamics of this system is modeled by a four-state unicyclic Markov
network [c.f. Fig. 1(c)], whose transition rate matrix is given by

$\begin{pmatrix}-k_{b}^{+,\text{OFF}}-k_{a}^{+}&k_{b}^{-}&0&k_{a}^{-}\\\
k_{b}^{+,\text{OFF}}&-k_{b}^{-}-k_{a}^{+}&k_{a}^{-}&0\\\
0&k_{a}^{+}&-k_{a}^{-}-k_{b}^{-}&k_{b}^{+,\text{ON}}\\\
k_{a}^{+}&0&k_{b}^{-}&-k_{a}^{-}-k_{b}^{+,\text{ON}}\end{pmatrix}.$ (10)

According to stochastic thermodynamics [38], the cycle affinity is given by
$\mathcal{F}=\ln(k_{b}^{+,\text{OFF}}/k_{b}^{+,\text{ON}})$.

We validate the bounds with Figs. 2(a) and (b). Each curve in the figure is
obtained as follows: $k_{a}^{+},k_{a}^{-},k_{b}^{+,\text{OFF}}$ and
$k_{b}^{-}$ are sampled randomly from the uniform distribution, and the ratio
of $|\chi_{ab}|$ to the bound is plotted versus the affinity $\mathcal{F}$ by
varying $k_{b}^{+,\text{ON}}$. This procedure is then repeated for $10^{3}$
times. From Figs. 2(a) and (b), it can be seen that the two bounds are
validated, and the first bound seems to be relatively tighter. Cusps can be
observed in nearly every curve. Their presence is not a result of the
discretization process used for plotting. These cusps actually emerge due to
the influence of the term $\gamma$ in the bounds: Approaching a cusp point, a
distinct combination of $\max\mathcal{A}_{mn}$ and $\min\mathcal{A}_{mn}$
takes over and alters the overall trend. This is similar to how free energy
changes at a first-order phase transition. By following the trend in one
curve, it seems that without this mechanism, some ratios tend to surpass
unity.

Example 2.— As an illustrative example of multicyclic networks, we examine a
network simplified from a model of molecular motors [43]. The corresponding
schematic is depicted in Fig. 2(e). In this network, three internal states are
grouped into a “cell,” and the transitions occurring between adjacent cells
signify either a forward or a backward step taken by the motor. We consider a
ring structure consisting of four such cells. For simplicity, we assume that
the transition rates occurring both within the cell and between neighboring
cells are uniform, as shown in the figure. Following a similar approach as in
Example 1, we generate each curve shown in Figs. 2(c) and (d) by sampling from
the uniform distribution and subsequently fixing the values of $w^{+}$,
$w^{-}$, and $k^{-}$. The quantity $|\chi_{ab}|$ is calculated as
$|\lambda^{\prime\prime}/\lambda^{\prime}|$, where $\lambda^{\prime}$
$(\lambda^{\prime\prime})$ is the real (imaginary) part of the eigenvalue
(with the largest nonzero real part) of the transition matrix $\bm{W}$ [15].
The plotted quantity corresponds to the ratio of $|\chi_{ab}|$ to its bound,
with $k^{+}$ being systematically varied. This entire procedure is repeated a
total of $10^{3}$ times. The validity of the two bounds is demonstrated in
Figs. 2(c) and (d), where behaviors similar to those observed in Example 1 are
evident. As expected, the bound cannot be saturated for multicyclic networks.

Derivation.— For unicyclic networks $(1\to 2\to\ldots\to\mathcal{N}\to 1)$,
The steady-state currents are uniform, i.e.,
$\mathcal{J}_{i,i+1}\equiv\mathcal{J}$. Following the line of reasoning in
Ref. [15], it can be assumed that $a$ and $b$ are scaled to satisfy
$D_{a}D_{b}=(D_{a}+D_{b})^{2}/4$. With this assumption, and employing Eqs.
(2)-(7), we arrive at

$\displaystyle\frac{\mathcal{A}\alpha_{ab}^{2}}{(D_{a}+D_{b})^{2}/4}$
$\displaystyle=\frac{16\sum_{i}\mathcal{A}_{i,i+1}(\sum_{i}\Omega_{i,i+1})^{2}\mathcal{J}^{2}}{(\sum_{i}\mathcal{A}_{i,i+1}L_{i,i+1}^{2})^{2}}$
(11)
$\displaystyle\leq{16\gamma^{2}}\left(\sum_{i}\frac{\mathcal{J}^{2}}{\mathcal{A}_{i,i+1}}\right)\frac{(\sum_{i}\Omega_{i,i+1})^{2}}{(\sum_{i}L_{i,i+1}^{2})^{2}}$
$\displaystyle\leq\frac{\gamma^{2}\sigma}{2\tan^{2}(\pi/\mathcal{N})},$

which is equivalent to Eq. (8). In the derivation of the first inequality, we
used the inequality

$\frac{\sum x_{i}}{(\sum x_{i}y_{i})^{2}}\leq\left(\frac{\max x_{i}}{\min
x_{i}}\right)^{2}\frac{\sum x_{i}^{-1}}{(\sum y_{i})^{2}},$ (12)

where $x_{i},y_{i}>0$. The equality holds if the local activities are uniform,
i.e., $\mathcal{A}_{12}=\mathcal{A}_{23}=\ldots=\mathcal{A}_{\mathcal{N}1}$. A
sufficient condition for this is uniform transition rates, i.e.,
$W_{12}=W_{23}=\ldots=W_{\mathcal{N}1}$ and
$W_{21}=W_{32}=\ldots=W_{1\mathcal{N}}$. The term
$2\sum_{i}{\mathcal{J}^{2}}/{\mathcal{A}_{i,i+1}}$ can be identified as the
pseudo entropy production rate [6, 42, 30], which is always equal to or less
than the actual entropy production rate. They coincide when both vanish. With
the pseudo entropy production rate, to obtain the last inequality, we first
used the Cauchy-Schwarz inequality
$\sum_{i}L_{i,i+1}^{2}\geq(\sum_{i}L_{i,i+1})^{2}/{\mathcal{N}}$, followed by
the isoperimetric inequality

$\left(4\mathcal{N}\tan\frac{\pi}{\mathcal{N}}\right)\bigg{|}\sum_{i}\Omega_{i,i+1}\bigg{|}\leq\bigg{(}\sum_{i}L_{i,i+1}\bigg{)}^{2}.$
(13)

The equalities in Cauchy-Schwarz and isoperimetric inequality hold
simultaneously if and only if the points $(a_{i},b_{i})$ form a regular
polygon.

For bound (9), we begin with Eq. (S31) in Ref. [15]:

$\left|\chi_{ba}\right|\leq\frac{4\sum_{c\in\mathcal{C}^{*}}\mathcal{J}_{c}\left(\sum_{e\in
c}L_{e}\right)^{2}\left[4n_{c}^{\prime}\tan\left(\pi/n_{c}^{\prime}\right)\right]^{-1}}{\sum_{c\in\mathcal{C}^{*}}\mathcal{J}_{c}\left(\sum_{e\in
c}L_{e}\right)^{2}\left[n_{c}\tanh\left(\mathcal{F}_{c}/2n_{c}\right)\right]^{-1}}.$
(14)

Since $\left[4n_{c}^{\prime}\tan\left(\pi/n_{c}^{\prime}\right)\right]^{-1}$
is a monotonically increasing function of $n_{c}^{\prime}$
$(n_{c}^{\prime}>2)$ and
$\left[n_{c}\tanh\left(\mathcal{F}_{c}/2n_{c}\right)\right]^{-1}$ a decreasing
function of $n_{c}$ $(n_{c}>0)$, we have

$\displaystyle\left|\chi_{ba}\right|$
$\displaystyle\leq\frac{4\sum_{c\in\mathcal{C}^{*}}\mathcal{J}_{c}\left(\sum_{e\in
c}L_{e}\right)^{2}\left[\mathcal{N}\tanh\left(\mathcal{F}_{c}/2\mathcal{N}\right)\right]}{\sum_{c\in\mathcal{C}^{*}}\mathcal{J}_{c}\left(\sum_{e\in
c}L_{e}\right)^{2}\left[4\mathcal{N}\tan\left(\pi/\mathcal{N}\right)\right]}$
(15)
$\displaystyle\leq\frac{\sum_{c\in\mathcal{C}^{*}}\sigma_{c}\left(\sum_{e\in
c}L_{e}\right)^{2}}{2\pi\sum_{c\in\mathcal{C}^{*}}\mathcal{J}_{c}\left(\sum_{e\in
c}L_{e}\right)^{2}},$

where we have used $n_{c}^{\prime}\leq n_{c}\leq\mathcal{N}$,
$\mathcal{N}\tanh(\mathcal{F}_{c}/2\mathcal{N})/[4\mathcal{N}\tan(\pi/\mathcal{N})]\leq\mathcal{F}_{c}/8\pi$
and $\sigma_{c}=\mathcal{F}_{c}\mathcal{J}_{c}$. For the definitions of
$n_{c}^{\prime}$, $n_{c}$ and $\mathcal{C}^{*}$, please refer to Ref. [15].
The equality holds for unicyclic systems when the observables form a regular
polygon in the large $\mathcal{N}$ limit.

On the other hand, the normalized asymmetry is also bounded by

$\displaystyle\left|\chi_{ba}\right|$
$\displaystyle\leq\frac{4\sum_{c\in\mathcal{C}^{*}}\mathcal{J}_{c}\left|\sum_{e\in
c}\Omega_{e}\right|}{\sum_{c\in\mathcal{C}^{*}}\sum_{e\in
c}\mathcal{A}_{e}L_{e}^{2}}$ (16)
$\displaystyle\leq\frac{4\sum_{c\in\mathcal{C}^{*}}\mathcal{J}_{c}\left(\sum_{e\in
c}L_{e}\right)^{2}\left[4n_{c}^{\prime}\tan\left(\pi/n_{c}^{\prime}\right)\right]^{-1}}{\min\mathcal{A}_{mn}\sum_{c\in\mathcal{C}^{*}}\sum_{e\in
c}L_{e}^{2}}$
$\displaystyle\leq\frac{\sum_{c\in\mathcal{C}^{*}}\mathcal{J}_{c}\left(\sum_{e\in
c}L_{e}\right)^{2}}{\tan(\pi/\mathcal{N})\min\mathcal{A}_{mn}\sum_{c\in\mathcal{C}^{*}}\left(\sum_{e\in
c}L_{e}\right)^{2}},$

where we have used the monotonicity of
$\left[4n_{c}^{\prime}\tan\left(\pi/n_{c}^{\prime}\right)\right]^{-1}$, the
isoperimetric inequality and Cauchy-Schwarz inequality again. The equality
holds for unicyclic systems when the observables form a regular polygon.

Multiplying these two bounds on $|\chi_{ab}|$ and employing
$\mathcal{A}\leq\mathcal{N}^{*}\max\mathcal{A}_{mn}$ results in

$\displaystyle\chi_{ba}^{2}$
$\displaystyle\leq\frac{\sum_{c\in\mathcal{C}^{*}}\sigma_{c}\left(\sum_{e\in
c}L_{e}\right)^{2}}{2\pi\tan(\pi/\mathcal{N})\min\mathcal{A}_{mn}\sum_{c\in
C^{*}}\left(\sum_{e\in c}L_{e}\right)^{2}}$ (17)
$\displaystyle\leq\frac{\mathcal{N}^{*}\gamma}{2\pi\tan(\pi/\mathcal{N})}\frac{\sigma}{\mathcal{A}},$

which is equivalent to Eq. (9). Here, $\mathcal{N}^{*}$ is the number of
nonzero $\mathcal{A}_{mn}$’s. By definition
$\mathcal{A}_{mn}=\mathcal{A}_{nm}$, so each pair is counted as one. For
unicyclic networks, $\mathcal{N}^{*}=\mathcal{N}$.

Discussion.— In summary, we report two thermodynamic bounds [Eqs. (8) and (9)]
on the normalized asymmetry of cross-correlation in terms of the dynamical
activity and entropy production. Identifying a simpler expression that
directly relates these three quantities seems to be a challenge. This bound
exhibits broad applicability, regardless of the underlying network topology,
and offers experimental testability. We also anticipate that this bound will
provide valuable insights in addressing the conjecture proposed by Oberreiter
et al. [17]. Rigorously establishing the validity of Eq. (8) for arbitrary
network topologies does not appear to be straightforward, but to this end, the
utilization of uniform cycle decomposition may serve as a crucial technique
[44, 15]. A future research direction entails investigating whether this bound
holds true or how it should be modified when applied to partially observed
Markov networks [45, 46, 31]. Additionally, exploring the influence of quantum
effects, such as quantum coherence, on this relation represents an intriguing
avenue for further investigation.

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