the supports of $F_{z,t,n}$ and $F_{z,t,m}$ are disjoint. Thus, define
$F_{z,t}=F_{z,t,1}\circ F_{z,t,2}\circ\cdots$ that is a diffeomorphism such
that $F_{z,t}\circ e_{0,n}$ is a standard embedding and moreover it is a
contactomorphism on a small open neighborhood, i.e.
$F_{z,t}^{*}\xi^{z,t}=\xi^{{\operatorname{std}}}$ for the domain
$\bigcup_{n\in{\mathbb{Z}}^{+}}{\mathcal{O}p}({\mathbb{S}}^{2}(n))$.

Recall that Theorem 3.4.1 is a relative homotopy equivalence. Thus,
$e_{z,0,n}=e$ and $F_{z,0}={\operatorname{Id}}$. Therefore there is a
$1$-parametric family $\xi^{z,t}$ of contact structures, i.e. we consider $t$
as the time and $z\in K$ a fixed parameter. We may try to apply Gray’s
Theorem. In order for it to work, we need to make sure that Gray’s flow is
complete. But this is the case because the contact structures coincide in the
family of spheres ${\mathbb{S}}^{2}(n)$ and this forces the Gray flow to
remain confined in the complementary annuli. Thus, there exists a family
$G^{z,t}$ of diffeomorphisms satisfying that $G^{z,t}_{*}\xi^{z,0}=\xi^{z,t}$.

Composing we obtain that $F^{z,t}\circ G^{z,t}$ is the required
contactomorphism. ∎

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††thanks: Present address: Kandi Raj College, Kandi, Murshidabad, West Bengal
742161, India††thanks: Present address: Experimental Physics I, Universität
Bayreuth,Universitätsstraße 30, 95447 Bayreuth,Germany.

# Irreversibility of mesoscopic processes with hydrodynamic interactions

Biswajit Das<EMAIL_ADDRESS>Department of Physical Sciences, Indian
Institute of Science Education and Research Kolkata, Mohanpur Campus,
Mohanpur, West Bengal 741246, India  Sreekanth K Manikandan
<EMAIL_ADDRESS>Department of Chemistry, Stanford University, Stanford,
CA, USA 94305  Shuvojit Paul  Department of Physical Sciences, Indian
Institute of Science Education and Research Kolkata, Mohanpur Campus,
Mohanpur, West Bengal 741246, India  Avijit Kundu  Department of Physical
Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur
Campus, Mohanpur, West Bengal 741246, India  Supriya Krishnamurthy
<EMAIL_ADDRESS>Department of Physics, Stockholm University, Stockholm,
Sweden  Ayan Banerjee<EMAIL_ADDRESS>Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, Mohanpur Campus,
Mohanpur, West Bengal 741246, India

###### Abstract

Microscopic colloidal particles are often used as probes to study the non-
equilibrium activity of living matter or other complex systems. In many of
these contexts hydrodynamic interactions between the probe particle and the
system of interest play an important role. However little is known about what
effect such interactions could have on the overall non-equilibrium
characteristics of the system of interest. In this paper we study two simple
models experimentally and theoretically, which demonstrate that hydrodynamic
interactions could either diminish or enhance the total entropy production of
the combined system. Importantly, we show that, our method of calculating
entropy production helps identify heat flows consistently, even in the
presence of hydrodynamic interactions. The results indicate that interactions
can be finely tuned to optimize not only dynamic properties but also
irreversibility and energy dissipation, thereby opening new avenues for
tailored control and design of driven mesoscale systems.

††preprint: APS/123-QED

Introduction.- Non-equilibrium systems operating in a steady state are
characterized by a continuous dissipation of heat, which in turn leads to an
increase in entropy production in the environment [1, 2]. This entropy
generation rate also serves as a viable measure of the irreversibility of the
underlying processes [3]. Computing this quantity for systems with many
interacting degrees of freedom and understanding how the rate of entropy
production depends on the strength of the interactions are critical open
problems. Addressing these issues can offer crucial insights into the
emergence of irreversibility and spatiotemporal order in complex processes,
especially in biophysical contexts [4].

The effect of interactions is also an important consideration when the non-
equilibrium characterization of a complex dynamical system involves
interactions with a different system serving as a probe [5, 6, 7]. For
example, fluctuations of probe particles within cytoskeletal networks offer
insights into nonequilibrium active mechanics and responses to architectural
changes [8, 9]. Carbon nanotubes similarly serve as probes for inferring
spatiotemporal nonequilibrium activity within cells [10]. In addition,
colloidal passive particles immersed in bacterial baths show enhanced local
dynamics, revealing nonequilibrium responses [11]. Further, microrheological
measurements with attached probe particles help to infer the active nature of
red blood cell flickering [12]. Recent works have shown that such probe-based
approaches can be combined with model-dependent estimates to infer dissipation
from red blood cell flickering data [13]. A probe, even a passive one, is
however an invasive tool and can substantially change the characteristics of
the system it is measuring via hydrodynamic or other interactions.

Naturally, there has been a significant theoretical interest in understanding
how interactions, defined in a broad sense, contribute to the energetics and
irreversibility of dynamical processes in complex interacting systems. A
recent study addressed this issue by decomposing the statistical estimate of
irreversibility into terms in successive orders of interactions [14, 15].
These works show that - given the fact that no two degrees of freedom of the
system can change their state simultaneously (multipartite dynamics), the
knowledge of higher-order interactions between the degrees of freedom of the
system can only improve the estimation of the total entropy production rate.
Moreover, they provide an algorithm which can decompose the estimate of the
entropy production rate into a sum of non-negative terms corresponding to
different orders of interactions. Such an approach, however, does not indicate
how, for a fixed kind and order of interaction between various degrees of
freedom of the system (or a system and the probe), the irreversibility of the
process and energy dissipation depends on the strength of the interaction
parameters.

Here, we demonstrate that for a specific class of processes described using
overdamped Langevin equations, even interactions of the same kind could have
varying implications for estimated dissipation depending on the nature of non-
equilibrium driving. We explore systems with co-evolving degrees of freedom in
a single heat bath or attached to multiple reservoirs, where non-multipartite
dynamics can arise due to non-diagonal friction, mobility, and diffusion
tensors [16] resulting from hydrodynamic interactions [17, 18, 19, 20]. These
interactions are widespread in nature and pivotal in the self-organization of
biological materials, such as protein folding [21]. Additionally, they are
known to facilitate the synchronization of microscopic oscillators involved in
collective ciliary or flagellar motion [22, 23]. We demonstrate, through
explicit examples of both experimentally and analytically tractable systems,
that such interactions can both suppress and enhance the overall
irreversibility (energy dissipation) as compared to the non-interacting limit,
and can lead to situations with non-monotonic dependencies. We also show that
our estimates can be used to identify heat flow at the level of single
trajectories for these systems.

Results.- We first consider a system with two hydrodynamically coupled
particles trapped in two separate parabolic potentials while the mean position
of one of the traps is modulated by the active Ornstein-Uhlenbeck (OU) noise.
We call it ‘OU-noise driven model’ for further discussions. In this system,
detailed balance is always violated, leading to a non-equilibrium steady state
(NESS) over time, even when particles are well separated with minimal
hydrodynamic interaction. [24, 25, 26, 27]. In addition, the strength of the
hydrodynamic interaction can be tuned by changing the mean separation between
the two particles.

To realise the system experimentally, two trapping potentials with a
separation $d$ are created by tight-focusing two separate Gaussian beams
(wavelength, $\lambda=1064\ nm$) - emanating from solid-state lasers of
opposite linear polarisation states - through an objective lens of high
numerical aperture (NA $=1.3$, $100X$, oil-immersion) positioned in a
conventional inverted microscope (Olympus XI). One of the trapping beams is
passed through an acousto-optic modulator (AOM) which is externally modulated
by active OU noise of different amplitudes and timescales. Polystyrene
microparticles (diameter, $2a=3\ \mu m$) - sparsely dispersed in double-
distilled water - are trapped inside a sample holder placed on the stage of
the microscope. To detect the position fluctuations of the trapped
microparticles, two detection beams of different wavelengths ( $650\ nm$ and
$780\ nm$), co-propagating with the trapping beams, are loosely focused onto
the trapped particles. The back-scattered light from both particles is
separately projected on separate ‘balanced-detection’ systems constructed
using high-gain photo-diodes [28]. In this way, the trajectories of both
particles are independently recorded at $1\ nm-10\ kHz$ spatio-temporal
resolution.

Figure 1: (a) Schematic of the hydrodynamically coupled particles in dual
traps of different stiffness constants. The mean position of the trap with
stiffness constant ‘$k_{1}$’ is modulated with an Ornstein-Uhlenbeck noise
(‘$\lambda(t)$’) while the trap with stiffness constant ‘$k_{2}$’ remains
unperturbed. $k_{1}=17.2\pm 0.2\ pN/\mu m$ and $k_{2}=12\pm 0.4\ pN/\mu m$ are
fixed throughout the experiments. (b) The probability density functions of
normalised positional fluctuations of the particles trapped in two traps
($d=4.2\ \mu m$) are plotted. The experimentally recorded fluctuations of both
particles are normalised with the equilibrium standard deviations
($\sigma_{i}=\sqrt{k_{B}T/k_{i}}$) of corresponding traps. (c) The non-zero
probability current corresponding to the particle in the driven trap is shown.
(d) The probability current also appears to be non-zero corresponding to the
particle in the fixed trap.

The dynamics of the system can be written as a multidimensional linear
Langevin equation:
$\mathbf{\dot{x}}(t)=-\mathbf{F}.\mathbf{x}(t)+\bm{\xi}(t)$, with
$\langle\bm{\xi}(t):\bm{\xi}(t^{\prime})\rangle=2\delta(t-t^{\prime})\mathbf{D}$.
Here, $\mathbf{x}(t)\equiv[x_{1}(t),x_{2}(t),\lambda(t)]^{T}$ consists of the
fluctuating positions ($x_{1}(t),x_{2}(t)$) of the two particles measured with
respect to the center of each optical traps having stiffness constants $k_{1}$
and $k_{2}$, and $\lambda(t)$ is the OU noise that is exponentially correlated
with the relaxation timescale $\tau_{e}$ and amplitude $D_{e}$ with
$\langle\lambda(t)\lambda(t^{\prime})\rangle=\frac{D_{e}}{\tau_{e}}\exp(-\frac{t-t^{\prime}}{\tau_{e}})$.
The vector $\bm{\xi}(t)\equiv[\eta_{1}(t),\eta_{2}(t),\eta_{3}(t)]^{T}$
contains the random Brownian forces of the system. The drift ($\mathbf{F}$)
and diffusion ($\mathbf{D}$) tensors can be expressed in terms of the
hydrodynamic coupling constant $\epsilon\equiv\frac{3a}{2d}$ (considering
first order of the Oseen tensor [18]) and the Stokes friction coefficient of
the medium $\gamma\equiv 6\pi\eta a$ with viscosity $\eta$, such that

$\mathbf{F}=\begin{pmatrix}k_{1}/\gamma&\epsilon k_{2}/\gamma&-k_{1}/\gamma\\\
\epsilon k_{1}/\gamma&k_{2}/\gamma&-\epsilon k_{1}/\gamma\\\
0&0&\frac{1}{\tau_{e}}\end{pmatrix},\ \mathbf{D}=\begin{pmatrix}D_{0}&\epsilon
D_{0}&0\\\ \epsilon D_{0}&D_{0}&0\\\
0&0&\frac{D_{e}}{\tau_{e}^{2}}\end{pmatrix},$ (1)

as $D_{0}=\frac{k_{B}T}{\gamma}$ ($k_{B}$ is the Boltzmann’s constant). The
rationale behind the dynamical equations is explicitly discussed in SI .1.

Figure 2: Entropy production rate ($\Pi$) of the ‘OU-noise driven model’: (a)
Inferred $\Pi$ from the experimental data monotonically increases with the
strength of the external noise (parametrised as $\theta$) at two different
separations between the particles. The entropy production rate of the system
decreases with the increasing hydrodynamic interaction ($\epsilon$) as shown
for two different noise strengths (b) $\theta=0.42$ and (c) $\theta=0.49$.
Corresponding theoretical estimations are shown with the solid lines.

A schematic of the system of our interest is shown in Fig.1(a). We primarily
focus on the entropy production rate of the system after it reaches a
nonequilibrium steady state defined by the joint probability distribution
($P_{ss}(\mathbf{x})$) and non-zero probability current
($\mathbf{j}_{ss}(\mathbf{x})$) corresponding to the whole phase space of the
system. The marginal steady-state distributions of both the particles in the
‘driven’ ($x_{1}$) and ‘fixed’ ($x_{2}$) traps are Gaussian. The variance of
the particle in the ‘driven’ trap is expectedly larger than the same in a
fixed trap as shown in Fig.1(b) for a typical experimental trajectory. The
recorded positional fluctuations of both particles are expressed as
dimensionless quantities $z_{1,2}\equiv x_{1,2}/\sigma_{1,2}$ as
$\sigma_{1,2}=\sqrt{k_{B}T/k_{1,2}}$. The non-zero probability current (as
shown in Fig.1(c)) is strongly prevalent in $(x_{1},\lambda)$ space as the
particle in the first trap is driven by the OU noise. However, the probability
current also prevails in the $(x_{2},\lambda)$ space (Fig.1(d)), even though
the external noise does not directly drive the particle in the second trap.
This indicates the appearance of induced nonequilibrium dynamics through
hydrodynamic interactions.

To check the effects of interactions, we perform the experiments at two
different separations ($d=4.2\ \mu m$ and $d=7.5\ \mu m$) between the traps
($k_{1}=17.2\pm 0.2\ pN/\mu m$ and $k_{2}=12\pm 0.4\ pN/\mu m$) by varying the
strength of the external noise (parameterized by
$\theta=\frac{D_{e}/\tau_{e}^{2}}{D_{0}}$). Then we estimate the entropy
production rate from the experimental and numerical data using the short-time
inference technique [29, 30, 31, 26, 32] based on the thermodynamic
uncertainty relation [33]. This technique is model-independent and
particularly advantageous to estimate the entropy production rate from the
experimental trajectories as it does not require any calibration factor to
transform the measurements in positional units [26]. The technique is also
briefly discussed in SI .2.

We find that the entropy production rate is monotonically increasing with
$\theta$ at both separations (Fig.2(a)). However, the entropy generation rate
for the separation $d=4.2\ \mu m$ at any $\theta$ is found to be slightly
lower than the same for $d=7.5\ \mu m$ as shown in Fig.2(b) and Fig.2(c). This
observation suggests that if the separation between two particles is reduced,
the total entropy production rate of the system will also be decreased, even
though the hydrodynamic coupling strength is enhanced in this process.

The analytically calculated total entropy production rate (in units of
$k_{B}/s$) for the system [34],

$\begin{split}\Pi=&\int
d\mathbf{x}\frac{\mathbf{D}^{-1}\mathbf{j}_{ss}^{2}(\mathbf{x})}{P_{ss}(\mathbf{x})}\\\
=&\frac{D_{e}k_{1}^{2}(\gamma+(1-\epsilon^{2})k_{2}\tau_{e})}{D_{0}\gamma\tau_{e}(\gamma^{2}+\gamma(k_{1}+k_{2})\tau_{e}+(1-\epsilon^{2})k_{1}k_{2}\tau_{e}^{2})},\end{split}$
(2)

also corroborates our observation as shown in the plots of Fig.2. If the
hydrodynamic interaction becomes negligible ($\epsilon\rightarrow 0$), the
entropy production rate will be,

$[\Pi]_{\epsilon\rightarrow
0}=\frac{D_{e}k_{1}^{2}}{D_{0}\gamma\tau_{e}(\gamma+k_{1}\tau_{e})},$ (3)

which is greater than $\Pi$ \- indicating the reduction of entropy production
rate in the presence of hydrodynamic interactions. This occurs since the
motion of the ‘driven’ particle gets constricted due to the other particle
placed close to it - which results in a lower entropy production rate. Similar
observations were made in Ref.[26] where the energy dissipation in a driven
particle was found to be reduced due to hydrodynamic flows close to a
microbubble.

Note that we could think of our two-particle system as a composite system
consisting of the driven particle ($x_{1}$) - which has an intrinsic energy
dissipation rate $[\Pi]_{\epsilon\rightarrow 0}$ \- being the system of
interest, and the other particle ($x_{2}$) being a probe brought close to it.
This demonstrates that interactions with a probe, which by itself does not
dissipate energy, could significantly reduce the energy dissipation to the
environment from nonequilibrium systems.

Figure 3: ‘Two-temperature model’: (a) Schematic of the hydrodynamically
coupled particles in dual traps of different stiffness constants. The particle
trapped in the potential with stiffness constant ‘$k_{1}$’ is in temperature
$T+\Delta T$ while the particle in the trap with stiffness constant ‘$k_{2}$’
remains in temperature $T$. (b) Theoretical estimation of the entropy
production rate of the system is plotted with $\Delta T$ and hydrodynamic
coupling strength $\epsilon$. [$k_{1}=17.2pN/\mu m$, $k_{2}=12pN/\mu m$] (c)
Estimation of entropy production rate as a function of $\epsilon$ \- using the
short-time inference scheme from numerical trajectories of the system for
fixed $\Delta T$. The solid lines indicate corresponding theoretical
predictions. The non-monotonic feature of $\Pi_{tw}$ with $\epsilon$ is shown
as the insets.

Additionally, it is important to note that the energy balance of the
interacting system should hold such that input power estimated as,
$\langle\dot{W}\rangle=k_{1}\langle\lambda(t)\circ\dot{x}_{1}(t)\rangle$ is
exactly equal to the total dissipated heat.This can be easily computed from
the increase in entropy of the medium as $-\langle\dot{Q}\rangle=-\langle
Q_{1}+Q_{2}\rangle=k_{B}T\Pi_{m}\equiv
k_{B}T\langle\mathbf{D}^{-1}\mathbf{(Fx)\circ\dot{x}}\rangle$) (see SI .3 for
the details of the calculation). Note that it would not have been possible to
establish energy balance in this system through the approach of applying
trajectory energetics to langevin equations of individual variables, as has
been previously done in Refs. [35, 36, 37, 38]. Since the thermal noises
involved in hydrodynamically coupled systems are cross-correlated (this leads
to the non-diagonal form of the diffusion matrix), defining heat flow at the
subsystem level ($Q_{1}$ and $Q_{2}$) for this system still remains unclear
[16].

Next, we theoretically discuss the role of hydrodynamic interactions in a
system of two particles where the particles are at different temperatures. We
call this model the ‘two-temperature model’ for further discussion. The
dynamics of this system can be similarly expressed as a coupled Langevin
equation with drift ($\mathbf{F}_{tw}$) and diffusion ($\mathbf{D}_{tw}$)
tensors of following forms [35] -

$\mathbf{F}_{tw}=\begin{pmatrix}k_{1}/\gamma&\epsilon k_{2}/\gamma\\\ \epsilon
k_{1}/\gamma&k_{2}/\gamma\\\ \end{pmatrix},\
\mathbf{D}_{tw}=\begin{pmatrix}\frac{k_{B}(T+\Delta
T)}{\gamma}&\epsilon\frac{k_{B}(T+\Delta T)}{\gamma}\\\
\epsilon\frac{k_{B}(T+\Delta T)}{\gamma}&\frac{k_{B}(T+\epsilon^{2}\Delta
T)}{\gamma}\\\ \end{pmatrix},$ (4)

where the particle in the trap of stiffness $k_{1}$ feels the temperature
$T+\Delta T$ and the other particle in the trap of stiffness $k_{2}$ feels a
lower temperature $T$. The underlying rationale for the dynamical equations of
this system is elegantly explained in Refs. [35, 39]. This model is also
experimentally viable and studied in detail in several Refs. [35, 36]. In
these studies, the two-temperature configuration was created by forcing one of
the particles by random white noise while the other particle was kept in close
proximity. The effect of the random white noise is assimilated as an
‘effective temperature’ which is attributed to the different temperatures of
the two particles. The total entropy production rate of this system can be
analytically computed as (with the same formalism used for the ‘OU-noise
driven model’),

$\Pi_{tw}=\frac{k_{2}^{2}\Delta
T^{2}\epsilon^{2}(1-\epsilon^{2})}{(k_{1}+k_{2})\gamma T(T+\Delta T)},$ (5)

and is found to be non-monotonic in $\epsilon$ with a maxima at
$\epsilon=1/\sqrt{2}\equiv 0.707$ as shown in Fig.3(b) and Fig.3(c). This
indicates that the entropy production rate of the system is enhanced with the
increased hydrodynamic coupling up to a certain separation before it goes down
even though the interaction is enhanced.

Interestingly, by combining this formula with the energy balance condition, we
can straightforwardly obtain an expression for the rate of heat dissipated by
the hot particle $=-\langle\dot{Q}_{1}\rangle$ (which, if energy balance
holds, will be the same as the heat absorbed by the cold particle
$\langle\dot{Q}_{2}\rangle$ with a minus sign) as,

$\displaystyle\begin{split}\langle\dot{Q}_{1}\rangle=-\langle\dot{Q}_{2}\rangle&=k_{B}\frac{\Pi_{tw}}{\frac{1}{T}-\frac{1}{T+\Delta
T}},\\\
&=\frac{k_{2}^{2}\epsilon^{2}(1-\epsilon^{2})}{(k_{1}+k_{2})\gamma}k_{B}\Delta
T\\\ &\equiv-\langle\dot{Q}\rangle.\end{split}$ (6)

Note that this approach was not taken in previous studies, and it was argued
that there is a violation of the First Law in such systems with some amount of
heat being lost in interactions, except for the case where $k_{1}=k_{2}$ [36,
37]. We, however, compute the entropy production rate and assume that the
First Law should hold, which enables us to obtain an expression for heat
consistently. The derived expression is the same as the expression for heat
flow from hot particle to cold particle previously obtained in Refs. [36, 37].
In addition, we argue that the estimated heat flow needs to be entirely
associated with the heat dissipated by the hot particle, as well as the heat
absorbed by the cold particle such that the energy balance condition is
automatically satisfied (See SI .3 for further details).

This system will reach an equilibrium steady state if the two particles are
well-separated with negligible hydrodynamic interaction. The hydrodynamic
interaction combined with the difference in temperatures breaks the detailed
balance, resulting in the transfer of heat from the relatively ‘hot’ particle
to the ‘cold’ one as the particles are placed closer to each other. If the
separation between particles is so small that the motion of the particles is
affected, the entropy production rate decreases, resulting in a non-monotonic
dependence of entropy production on the interaction strength.

As a corollary, note that - considering either one of the interacting
particles as a probe (having interactions with the other) - our particular
system highlights the importance of ensuring that the probe’s temperature
matches that of the system. This guarantees that the interaction with the
probe alone does not make the system appear out of equilibrium, not does it
add additional contributions to dissipation.

Conclusions.- In conclusion, we unravel the effects of hydrodynamic
interactions influencing the irreversibility of two mesoscopic systems with
different origins of nonequilibrium characteristics. We specifically find that
the entropy production rate shows completely different features for the two
systems as a function of the strength of the hydrodynamic interactions.
Moreover, we resolve the issues related to the energy balance for
hydrodynamically coupled systems with the energetics estimated directly from
the entropy production rate.

Our observations suggest that even first-order interactions between subsystems
can bring forth surprising behaviours in the irreversibility of a system with
only a few degrees of freedom. In this context, the details of the dynamics
are crucial in discerning the role of certain interactions in nonequilibrium
systems with multiple reservoirs or co-evolving degrees of freedom in a single
heat bath. Notably, it will be interesting to test the effects of higher-order
interactions on the irreversibility of out-of-equilibrium processes in similar
steady states as well as in different time-dependent configurations [40].
Also, the impact of such interactions in the context of information transfer
in biochemical signalling networks [41, 42] and optimising protocols for non-
equilibrium control problems [43, 44, 45] will be quite interesting to
investigate.

Acknowledgements.-The work is supported by IISER Kolkata, an autonomous
teaching and research institute supported by the Ministry of Education,
Government of India, and the Science and Engineering Research Board,
Department of Science and Technology, Government of India, through the
research grant CRG/2022/002417. BD is thankful to the Ministry Of Education of
Government of India for financial support through the Prime Minister’s
Research Fellowship (PMRF) grant. SKM acknowledges the Knut and Alice
Wallenberg Foundation for financial support through Grant No. KAW 2021.0328.
S.K acknowledges the support of the Swedish Research Council through the grant
2021-05070.

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## Supplementary Materials

###  .1 OU-noise driven model: Dynamical descriptions

The dynamics of the two identical microscopic particles (radius $a$) trapped
in parabolic potentials separated by a distance $d$ with different stiffness
constants $k_{1}$ and $k_{2}$ can be expressed by a coupled $Langevin$
equation [17, 36, 19],

$\begin{split}\dot{x}_{1}(t)=&\mathcal{H}_{11}[-k_{1}x_{1}(t)+\xi_{1}(t)]+\mathcal{H}_{12}[-k_{2}x_{2}(t)+\xi_{2}(t)]\\\
\dot{x}_{2}(t)=&\mathcal{H}_{21}[-k_{1}x_{1}(t)+\xi_{1}(t)]+\mathcal{H}_{22}[-k_{2}x_{2}(t)+\xi_{2}(t)],\end{split}$
(7)

where $\mathcal{H}_{ij}$ $(i,j=1,2)$ are the constant elements of a
hydrodynamic coupling tensor of the form
$\mathbf{\mathcal{H}}=\begin{pmatrix}1/\gamma&\epsilon/\gamma\\\
\epsilon/\gamma&1/\gamma\\\ \end{pmatrix}$ \- considering that the
displacements of the particles ($x_{1}(t)$, $x_{2}(t)$) are small compared to
the mean separation between the traps ($d$). Here
$\epsilon\equiv\frac{3a}{2d}$ denotes the coupling coefficient taken as the
lower order component of the Oseen tensor [17, 18] for longitudinal motions
and $\gamma\equiv 6\pi\eta a$ is the viscous drag coefficient of the medium.
The terms $\xi_{1}(t)$ and $\xi_{2}(t)$ are delta-correlated random Brownian
forces such that $\langle\xi_{i}(t)\rangle=0,\
\langle\xi_{i}(t)\xi_{j}(t)\rangle=2k_{B}T(\mathcal{H})^{-1}_{ij}\delta(t-t^{\prime})$.
$k_{B}$ is the Boltzmann’s constant.

Without any external perturbation, the coupled particles will be in thermal
equilibrium with the bath of temperature $T$. Following Ref. [35], Eq.(7) can
be rewritten as,

$\begin{split}\dot{x}_{1}(t)&=b_{1}(x_{1}(t),x_{2}(t))+\eta_{1}(t)\\\
\dot{x}_{2}(t)&=b_{2}(x_{1}(t),x_{2}(t))+\eta_{2}(t),\end{split}$ (8)

with

$\begin{split}b_{i}(x_{i},x_{j})&=-\frac{1}{\gamma}k_{i}x_{i}-\frac{\epsilon}{\gamma}k_{j}x_{j},\\\
\eta_{1}&=\frac{1}{\gamma}(\xi_{1}+\epsilon\xi_{2}),\\\
\eta_{2}&=\frac{1}{\gamma}(\xi_{2}+\epsilon\xi_{1}).\end{split}$ (9)

The Fokker-Planck equation corresponding to the time evolution of the joint
probability distribution ($\rho_{eq}(x_{1},x_{2},t)$) of this (thermally)
equilibrium configuration can be written as,

$\partial_{t}\rho_{eq}(x_{1},x_{2},t)=-\partial_{x_{1}}(b_{1}\rho_{eq})-\partial_{x_{2}}(b_{2}\rho_{eq})+\alpha_{11}\partial_{x_{1}}^{2}\rho_{eq}+\alpha_{22}\partial_{x_{2}}^{2}\rho_{eq}+2\alpha_{12}\partial_{x_{1}x_{2}}^{2}\rho_{eq},$
(10)

with $\alpha_{11}=\alpha_{22}\equiv\frac{k_{B}T}{\gamma}$ and
$\alpha_{12}=\alpha_{21}\equiv\epsilon\frac{k_{B}T}{\gamma}$ are the elements
of the equilibrium diffusion matrix ($\mathbf{D}_{eq}$) of the system such
that
$\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=2(\mathbf{D}_{eq})_{ij}\delta(t-t^{\prime})$.

To drive the system out of equilibrium, the particle trapped in the potential
with stiffness constant $k_{1}$ is modulated with an Ornstein-Uhlenbeck (OU)
noise ($\lambda(t)$). The external modulation is exponentially correlated with
the relaxation timescale $\tau_{e}$ and amplitude $D_{e}$ as
$\langle\lambda(t)\lambda(t^{\prime})\rangle=\frac{D_{e}}{\tau_{e}}\exp(-\frac{t-t^{\prime}}{\tau_{e}})$
and it is derivable from following dynamical equation,

$\dot{\lambda}(t)=-\frac{\lambda(t)}{\tau_{e}}+\frac{\sqrt{2D_{e}}}{\tau_{e}}\eta_{3}(t),~{}~{}~{}~{}~{}\langle\eta_{3}(t)\rangle=0,\langle\eta_{3}(t)\eta_{3}(t^{\prime})\rangle=\delta(t-t^{\prime}).$
(11)

Now, the dynamics of the perturbed system with the external OU modulation can
be expressed with a system of Langevin equations with $\lambda(t)$ as another
degree of freedom in addition to $x_{1}(t)$ and $x_{2}(t)$ such that,

$\begin{split}\dot{x}_{1}(t)=&\mathcal{H}_{11}[-k_{1}(x_{1}(t)-\lambda(t))+\xi_{1}(t)]+\mathcal{H}_{12}[-k_{2}x_{2}(t)+\xi_{2}(t)]\\\
\dot{x}_{2}(t)=&\mathcal{H}_{21}[-k_{1}(x_{1}(t)-\lambda(t))+\xi_{1}(t)]+\mathcal{H}_{22}[-k_{2}x_{2}(t)+\xi_{2}(t)]\\\
\dot{\lambda}(t)=&-\frac{\lambda(t)}{\tau_{e}}+\frac{\sqrt{2D_{e}}}{\tau_{e}}\eta_{3}(t).\end{split}$
(12)

Here,
$\langle\xi_{1}(t)\eta_{3}(t^{\prime})\rangle=\langle\xi_{2}(t)\eta_{3}(t^{\prime})\rangle=0\implies\langle\eta_{1}(t)\eta_{3}(t^{\prime})\rangle=\langle\eta_{2}(t)\eta_{3}(t^{\prime})\rangle=0$.
Moreover, Eq.(12) can be rewritten in the following matrix form,

$\begin{pmatrix}\dot{x}_{1}(t)\\\ \dot{x}_{2}(t)\\\
\dot{\lambda}(t)\end{pmatrix}=-\begin{pmatrix}k_{1}/\gamma&\epsilon
k_{2}/\gamma&-k_{1}/\gamma\\\ \epsilon k_{1}/\gamma&k_{2}/\gamma&-\epsilon
k_{1}/\gamma\\\ 0&0&\frac{1}{\tau_{e}}\end{pmatrix}\begin{pmatrix}x_{1}(t)\\\
x_{2}(t)\\\ \lambda(t)\end{pmatrix}+\begin{pmatrix}\eta_{1}(t)\\\
\eta_{2}(t)\\\ \eta_{3}(t)\end{pmatrix}$ (13)
$\implies\mathbf{\dot{x}(t)}=-\mathbf{F}.\mathbf{x(t)}+\bm{\xi(t)},$ (14)

with
$\langle\bm{\xi}(t):\bm{\xi}(t^{\prime})\rangle=2\delta(t-t^{\prime})\mathbf{D}$.
Therefore, the drift ($\mathbf{F}$) and diffusion ($\mathbf{D}$) matrices for
the ‘OU-noise driven model’ are of following forms,

$\mathbf{F}=\begin{pmatrix}k_{1}/\gamma&\epsilon k_{2}/\gamma&-k_{1}/\gamma\\\
\epsilon k_{1}/\gamma&k_{2}/\gamma&-\epsilon k_{1}/\gamma\\\
0&0&\frac{1}{\tau_{e}}\end{pmatrix},\ \mathbf{D}=\begin{pmatrix}D_{0}&\epsilon
D_{0}&0\\\ \epsilon D_{0}&D_{0}&0\\\
0&0&\frac{D_{e}}{\tau_{e}^{2}}\end{pmatrix},$ (15)

with $D_{0}=\frac{k_{B}T}{\gamma}$.

Steady state distribution.- The probability of finding the particle in a
configuration $\mathbf{x}$ at a certain time $t$ can be determined in terms of
probability distribution function $P(\mathbf{x},t)$ which follows the Fokker-
plank equation of the form

$\begin{split}\partial_{t}P(\mathbf{x},t)&=-\nabla\cdot(-\mathbf{F}\mathbf{x}P({\mathbf{x},t})-\mathbf{D}\nabla
P(\mathbf{x},t))\\\ &\equiv-\nabla\cdot\mathbf{j}(\mathbf{x},t)\end{split}$
(16)

where, $\mathbf{j}(\mathbf{x},t)$ denotes the probability current in the phase
space.

Starting from an arbitrary initial condition for $\mathbf{x}$, the system will
reach a non-equilibrium steady state in the long time ($\lim
t\rightarrow\infty$) with a characteristic probability distribution and
current given by,

$\displaystyle\begin{split}P_{ss}(\mathbf{x})=(2\pi\sqrt{\det\mathbf{C}})^{-1}e^{-\frac{1}{2}\mathbf{x}^{T}\mathbf{C}^{-1}\mathbf{x}}\\\
\mathbf{j}_{ss}(\mathbf{x})=(-\mathbf{F}\mathbf{x}+\mathbf{D}\mathbf{C}^{-1}\mathbf{x})P_{ss}(\mathbf{x}),\end{split}$
(17)

in terms of the long-time limit of the covariance matrix $\mathbf{C}(t)$.
Covariance matrix.- Note that the diffusion tensor is not proportional to an
identity matrix as $D_{ii}\neq D_{jj},D_{ij}\neq 0$, which indicates the
system to be non-multipartite [16]. Following the technique introduced in
[46], the steady-state covariance matrix $\mathbf{C}$ is given by

$\mathbf{C}=\mathbf{F}^{-1}(\mathbf{D}+\mathbf{Q}),$ (18)

where $\mathbf{Q}$ is an antisymmetric $3\times 3$ matrix that can be uniquely
determined by

$\mathbf{FQ}+\mathbf{QF}^{T}=\mathbf{FD}-\mathbf{DF}^{T}.$ (19)

If $\mathbf{Q}$ is nonzero in the NESS, it implies the violation of the
detailed balance in the system. Now the elements of the steady state ($\lim
t\rightarrow\infty$) covariance matrix for this system will be,

$\displaystyle\begin{split}&C_{11}\equiv\langle
x_{1}x_{1}\rangle=\frac{D_{0}\gamma}{k_{1}}+\frac{D_{e}k_{1}(\gamma(k_{1}+k_{2}-\epsilon^{2}k_{2})+(1-\epsilon^{2})k_{2}(k_{1}+k_{2})\tau_{e})}{(k_{1}+k_{2})(\gamma^{2}+\gamma(k_{1}+k_{2})\tau_{e}+(1-\epsilon^{2})k_{1}k_{2}\tau_{e}^{2})},\\\
&C_{12}\equiv\langle x_{1}x_{2}\rangle=\frac{\epsilon D_{e}\gamma
k_{1}^{2}}{(k_{1}+k_{2})(\gamma^{2}+\gamma(k_{1}+k_{2})\tau_{e}+(1-\epsilon^{2})k_{1}k_{2}\tau_{e}^{2})},\\\
&C_{13}\equiv\langle
x_{1}\lambda\rangle=\frac{D_{e}k_{1}(\gamma+(1-\epsilon^{2})k_{2}\tau_{e})}{(\gamma^{2}+\gamma(k_{1}+k_{2})\tau_{e}+(1-\epsilon^{2})k_{1}k_{2}\tau_{e}^{2})},\\\
&C_{21}\equiv\langle x_{2}x_{1}\rangle=C_{12},\\\ &C_{22}\equiv\langle
x_{2}x_{2}\rangle=\frac{D_{0}\gamma}{k_{2}}+\frac{\epsilon^{2}D_{e}\gamma
k_{1}^{2}}{(k_{1}+k_{2})(\gamma^{2}+\gamma(k_{1}+k_{2})\tau_{e}+(1-\epsilon^{2})k_{1}k_{2}\tau_{e}^{2})},\\\
&C_{23}\equiv\langle x_{2}\lambda\rangle=\frac{\epsilon D_{e}\gamma
k_{1}}{\gamma^{2}+\gamma(k_{1}+k_{2})\tau_{e}+(1-\epsilon^{2})k_{1}k_{2}\tau_{e}^{2}},\\\
&C_{31}\equiv\langle\lambda x_{1}\rangle=C_{13},\\\
&C_{32}\equiv\langle\lambda x_{2}\rangle=C_{23},\\\
&C_{33}\equiv\langle\lambda\lambda\rangle=\frac{D_{e}}{\tau_{e}}.\end{split}$
(20)

If the particles are well separated such that hydrodynamic coupling is
negligible ($\epsilon\rightarrow 0$), the covariance matrix will be,

$\mathbf{C}=\begin{pmatrix}\frac{D_{0}\gamma}{k_{1}}+\frac{D_{e}k_{1}}{\gamma+k_{1}\tau_{e}}&0&\frac{D_{e}k_{1}}{\gamma+k_{1}\tau_{e}}\\\
0&\frac{D_{0}\gamma}{k_{2}}&0\\\
\frac{D_{e}k_{1}}{\gamma+k_{1}\tau_{e}}&0&\frac{D_{e}}{\tau_{e}}\end{pmatrix}$
(21)

Moreover, we can find the covariance matrix corresponding to the out-of-
equilibrium situation with the hydrodynamic coupling where the external
modulation is Gaussian white noise - by taking $\tau_{e}\rightarrow 0$ (with
$D_{e}\neq 0$) in Eq.(20),

$\begin{split}&[C_{11}]_{tw}=\frac{D_{0}\gamma}{k_{1}}+\frac{D_{e}k_{1}^{2}}{\gamma
k_{1}}-\epsilon^{2}\frac{k_{2}}{k_{1}}\frac{D_{e}k_{1}^{2}}{\gamma(k_{1}+k_{2})},\\\
&[C_{12}]_{tw}=\epsilon\frac{D_{e}k_{1}^{2}}{\gamma(k_{1}+k_{2})},\\\
&[C_{22}]_{tw}=\frac{D_{0}\gamma}{k_{2}}+\epsilon^{2}\frac{D_{e}k_{1}^{2}}{\gamma(k_{1}+k_{2})}.\end{split}$
(22)

If we consider $D_{e}k_{1}^{2}=k_{B}\Delta T\gamma$ to match the noise
correlation of the external Gaussian modulation of Ref. [35], Eqs.(22) will be
transformed to,

$\begin{split}&[C_{11}]_{tw}=\frac{k_{B}T}{k_{1}}+\frac{k_{B}\Delta
T}{k_{1}}-\epsilon^{2}\frac{k_{2}}{k_{1}}\frac{k_{B}\Delta
T}{(k_{1}+k_{2})},\\\ &[C_{12}]_{tw}=\epsilon\frac{k_{B}\Delta
T}{(k_{1}+k_{2})},\\\
&[C_{22}]_{tw}=\frac{k_{B}T}{k_{2}}+\epsilon^{2}\frac{k_{B}\Delta
T}{(k_{1}+k_{2})},\end{split}$ (23)

which match exactly with the Eqs.(14) of Ref.[35]. Moreover, Eq. (23) are the
elements of the steady state covariance matrix of the ‘two-temperature model’
we discussed in the main text.

It is also clear that the steady-state cross-variances will be non-zero only
in the out-of-equilibrium configuration. In an equilibrium situation
($D_{e}=0$ or $\Delta T=0$), the two particles will be statistically
independent with the variance $\frac{k_{B}T}{k_{1}}$ and
$\frac{k_{B}T}{k_{2}}$ respectively.

###  .2 Entropy production rate and Short-time inference technique

The entropy production rate for a nonequilibrium system in a steady state with
the probability density $\rho_{ss}(\mathbf{x})$ and current
$\mathbf{j}_{ss}(\mathbf{x})$ is analytically estimated as [34]

$\Pi=\int
d\mathbf{x}\frac{\mathbf{D}^{-1}\mathbf{j}_{ss}^{2}(\mathbf{x})}{P_{ss}(\mathbf{x})},$
(24)

which we use to compute the entropy production rate for both models and
discuss their dependency on the interaction strength. However, estimating the
entropy production rate from the experimental or numerical trajectories is
nontrivial using the above formula. We have used an indirect technique based
on the thermodynamic uncertainty relation [33]. The short-time inference
technique was first introduced for overdamped $Langevin$ dynamics in Ref. [29]
and rigorously proved in Refs. [30, 31]. It was subsequently used to estimate
the entropy production rate from experimental trajectories in Ref. [26].
Moreover, this technique is effectively applicable to systems with nonlinear
potential landscapes [32] and to inferring the activity of biological cells
[47]. Using this technique, we can estimate the entropy production rate of a
system in a non-equilibrium steady state as

$\Pi=\lim_{dt\rightarrow 0}\max_{J}\left[\frac{2k_{B}\langle J\rangle^{2}}{dt\
\text{Var}(J)}\right],$ (25)

where $J$ is a weighted scalar current - that can be computed from the time
discretized experimental or numerical trajectory data ($\mathbf{x}^{i}$)
sampled at an interval of $dt$ \- as

$J=\mathbf{f}\left(\frac{\mathbf{x}^{i+1}+\mathbf{x}^{i}}{2}\right)\cdot\left(\mathbf{x}^{i+1}-\mathbf{x}^{i}\right).$
(26)

Motivated by the linearity of the systems, we have approximated
$\mathbf{f(x)}$ as the linear combination of linear basis functions
$\psi_{m}(\mathbf{x})$ of different dimensions (equal to the number of degrees
of freedom considered in a model) as
$\mathbf{f(x)}=\sum_{m=1}^{M}\mathbf{a}_{m}\psi_{m}(\mathbf{x})$. For any such
adequate representation of $\mathbf{f(x)}$, an analytical solution to the
maximisation problem of Eq.(25) is known [30] and given by,

$\Pi=\frac{2\langle\psi_{k}\rangle(\bm{\Xi}^{-1})_{k,l}\langle\psi_{l}\rangle}{dt},$
(27)

where,
$(\bm{\Xi})_{k,l}=\langle\psi_{k}\psi_{l}\rangle-\langle\psi_{k}\rangle\langle\psi_{l}\rangle$.
To find more details about the method, we refer to Sec. II B of Ref. [30] and
to Sec. III of Ref. [32].

###  .3 Validation of thermodynamic laws

For the ‘OU-noise driven model’, the input power can be estimated as,

$\begin{split}\langle\dot{W}\rangle=&k_{1}\langle\lambda(t)\circ\dot{x}_{1}(t)\rangle\\\
=&\frac{D_{e}k_{1}^{2}(\gamma+(1-\epsilon^{2})k_{2}\tau_{e})}{\tau_{e}(\gamma^{2}+\gamma(k_{1}+k_{2})\tau_{e}+(1-\epsilon^{2})k_{1}k_{2}\tau_{e}^{2})},\end{split}$
(28)

which should be exactly equal to the total rate of heat flow between the
particles to validate the first law of thermodynamics. since there is no
change in the internal energy of the particles in a steady state. However, we
find that heat fluxes due to the respective particles, defined as
$Q_{i}(t_{m})=\int_{0}^{t_{m}}dt\
(\gamma\dot{x}_{i}-\eta_{i})\circ\dot{x}_{i}$, do not add up to the total
input power of the system. Since the thermal noises involved in
hydrodynamically coupled systems are cross-correlated (this leads to the non-
diagonal form of the diffusion matrix), defining heat flow at the subsystem
level for a system with even few degrees of freedom is not at all
straightforward [16] and that may lead to the wrong estimation of the total
rate of heat flow defined as a direct sum of such heat fluxes corresponding to
the individual subsystems. To avoid the anomaly regarding the first law of
thermodynamics, it will be useful to look at the increase in entropy of the
medium as it directly relates to the heat dissipation of a steady-state system
[34, 16] such that,

$\begin{split}\frac{\langle\dot{Q}\rangle}{k_{B}T}=&\Pi_{m}\equiv\langle\mathbf{D}^{-1}\mathbf{(Fx)\circ\dot{x}}\rangle\\\
\implies\Pi_{m}=&\int d\mathbf{x}\
\mathbf{D}^{-1}\mathbf{(Fx)}\mathbf{j}_{ss}(\mathbf{x}).\end{split}$ (29)

The medium entropy production rate for the ‘OU-noise driven model’ is
estimated to be,

$\begin{split}\Pi_{m}=&-\frac{D_{e}k_{1}^{2}(\gamma+(1-\epsilon^{2})k_{2}\tau_{e})}{D_{0}\gamma\tau_{e}(\gamma^{2}+\gamma(k_{1}+k_{2})\tau_{e}+(1-\epsilon^{2})k_{1}k_{2}\tau_{e}^{2})},\\\
\implies\langle\dot{Q}\rangle=&D_{0}\gamma\Pi_{m}\\\
=&-\frac{D_{e}k_{1}^{2}(\gamma+(1-\epsilon^{2})k_{2}\tau_{e})}{\tau_{e}(\gamma^{2}+\gamma(k_{1}+k_{2})\tau_{e}+(1-\epsilon^{2})k_{1}k_{2}\tau_{e}^{2})},\end{split}$
(30)

as $D_{0}\gamma=k_{B}T$. It is now evident that the estimated heat dissipation
validates the first law of thermodynamics as,
$-\langle\dot{Q}\rangle=\langle\dot{W}\rangle$ [Eq. (28) with appropriate sign
convention].

For the ‘two-temperature model’, heat fluxes due to the respective particles
was estimated as $Q_{i}(t_{m})=\int_{0}^{t_{m}}dt\
(\gamma\dot{x}_{i}-\eta_{i})\circ\dot{x}_{i}$ in Refs. [35, 36, 37]. These
studies showed that the heat released from the ‘hot’ particle was not fully
exhausted by the ‘cold’ one if the particles were trapped in potentials with
different stiffnesses - leading to the ‘apparent’ violation of the energy
balance relation. It was also indicated that energy would not be conserved for
this system due to the dissipative nature of the coupling and energy
conservation could be restored if the two particles are trapped in potentials
with equal stiffnesses as the system will then be indistinguishable from those
with conservative coupling [37].

Here we compute the heat fluxes of the particles from the knowledge of the
total entropy production rate of the system. If the rate of heat dissipated by
the ‘hot’ particle is $\langle\dot{Q}_{1}\rangle$ and the rate of heat
absorbed by the ‘cold’ particle is $\langle\dot{Q}_{2}\rangle$, the total
entropy production rate for the system ($\Pi_{tw}$) in NESS will be,

$\Pi_{tw}=-\frac{\langle\dot{Q}_{1}\rangle}{k_{B}(T+\Delta
T)}-\frac{\langle\dot{Q}_{2}\rangle}{k_{B}T}.$ (31)

Now, according to the energy balance condition,
$-\langle\dot{Q}_{1}\rangle=\langle\dot{Q}_{2}\rangle=\langle\dot{Q}\rangle$.
So Eq. (31) reduces to,

$\begin{split}-\langle\dot{Q}\rangle\left(\frac{1}{T}-\frac{1}{T+\Delta
T}\right)=k_{B}\Pi_{tw}\\\
\implies\langle\dot{Q}\rangle=-k_{B}\Pi_{tw}\frac{T(T+\Delta T)}{\Delta T}\\\
=-\frac{k_{2}^{2}\epsilon^{2}(1-\epsilon^{2})}{(k_{1}+k_{2})\gamma}k_{B}\Delta
T.\end{split}$ (32)

The derived expression is the same as the expression for heat flow from hot
particle to cold particle previously obtained in Refs. [36, 37]. However, in
addition, we argue that it needs to be entirely associated with the heat
dissipated by the hot particle, and the heat absorbed by the cold particle
such that the energy balance condition is automatically satisfied.