# Enhanced directionality of active processes in a viscoelastic bath

Biswajit Das1, Shuvojit Paul1, Sreekanth K. Manikandan2, and Ayan Banerjee1
1Department of Physical Sciences, Indian Institute of Science Education and
Research Kolkata, Mohanpur Campus, Mohanpur, West Bengal 741246, India.
2Department of Chemistry, Stanford University, Stanford, CA, USA 94305.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, , , and .

###### Abstract

Active fluctuations are known to play a significant role in the intracellular
transport of passive objects. However, the effect of viscoelasticity of the
environment in shaping such processes is relatively less understood. Here,
with a minimal experiment using a driven colloid in a viscoelastic bath, we
show that viscoelasticity significantly increases the mean injected power to
the passive object ($\sim 50\%$ compared to a viscous medium), for the same
strength of the external driving. Additionally, we observe a notable reduction
in negative work fluctuations across a wide range of driving amplitudes. These
findings collectively suggest an enhanced directionality in driven processes
within a viscoelastic bath, which we attribute to the emergence of
interactions between the colloid and the viscoelastic medium.

##  1 Introduction

Microscopic biophysical processes are vital for the functioning of living
organisms. Many of these processes, such as intracellular transport of passive
objects [1, 2], are non-equilibrium in nature, mostly driven by ATP-consuming
molecular motors such as kinesins [3, 4]. Developing accessible models for
these processes is of utmost importance as it provides novel perspectives into
the functioning of the fundamental components of life within complex
environments.

In a zeroth order approximation, such processes can be modelled assuming a
separation of time-scales between the environmental and system degrees of
freedom, leading to, effectively, a Markovian description for the dynamics of
the system. The thermodynamics of such processes has been extensively studied
in recent decades, particularly in the context of stochastic thermodynamics
[5, 6]. This field has focused on various aspects, including fluctuation
relations [7, 8, 9], as well as advancements in stochastic inference
techniques [10, 11, 12, 13, 14, 15, 16, 17, 18]. These theories have now been
well tested in experiments, and have provided us with accurate estimates of
thermodynamic quantities, such as equilibrium free energy differences and
dissipation for specific biophysical systems [19, 20] and efficiencies of
molecular motors [21, 22].

A vast majority of these experiments are carried out in in-vitro conditions
where the experimental medium is usually a homogeneous viscous solution and
the Markovian assumption is valid. This is also the case for most artificially
synthesized molecular motors, Brownian Carnot engines and other nanoscale
devices [22, 23]. Contrary to this, in reality, microscopic biophysical
processes differ from simplified models by two key features. Firstly, living
cells possess inherent activity and dynamic behavior, contributing to complex
fluctuations that primarily govern the underlying processes, except in some
situations where passive diffusion also may play big roles. For example,
diffusive processes with directed motility of various intracellular elements
are crucial in positioning and distributing organelles inside a cell [24, 25].
However, overall, the hidden nature of many such sub-cellular processes leads
to non-Markovian dynamics for the observable microscopic processes. Secondly,
the intracellular medium is highly viscoelastic. This viscoelasticity arises
from the intricate composition and structure of the cellular environment,
characterized by a combination of viscous and elastic properties [26, 27].
This also leads to memory effects due to the large relaxation times, which
again breaks the effective Markovian description for the dynamics of the
system [28, 29, 30, 31, 32, 33].

Quite naturally, both of these issues have individually generated significant
recent interest. Studies on various systems, such as bacterial suspensions or
self-propelled particles, have revealed that active fluctuations can induce
non-trivial transport phenomena inside cells [34, 35], enhance mixing [36] and
diffusion [37, 38], and even drive the emergence of collective behaviour [39].
Studies have also shown that active fluctuations can drive the self-assembly
of colloidal systems [40], lead to fluctuation induced phase transitions [41],
and enhance the efficiency of microscopic heat engines [42]. Yet another
interesting observation was made in Ref. [43], which showed that the molecular
motor kinesin accelerates in the presence of an active noise in the
environment, implying active fluctuations in cells are utilized to promote
various physiological processes. This finding was further explored in detail
recently in Ref. [44].

Similarly, recent experiments have shown several intriguing phenomena related
to the dynamics of an embedded particle as well as swimmers in viscoelastic
fluids [45, 46]. These observations include the increase in Kramer’s rate of
transitions [47, 48, 49], circular motion of active particles instead of only
persistent random walks [50, 51], increase in the rotational diffusion
coefficient of an active Brownian particle by orders of magnitude compared to
that in a viscous fluid [51], and very interestingly, the possibility of
attaining the Carnot efficiency in microscopic sterling engine [52]. Several
recent studies also show that viscoelasticity is essential for facilitating
various biological processes [53, 35, 54].

Compared to the vast literature on the individual fields, the study of the
combined effects of active processes in viscoelastic baths is relatively
underexplored. Some notable findings in this emerging area include the
swelling of a polymer in an active and viscoelastic bath, where the extent of
swelling depends on the viscoelastic properties of the environment [55], the
study of active diffusion of model chromosomal loci driven by athermal noise,
revealing the influence of the polymeric effect on the anomalous diffusion
exponent [56], the reduction of quasi-static efficiency in Brownian engines
due to the viscoelasticity of the active suspension [57], and the implications
of complex multi-scale effects leading to time-dependent temperature [58].

Despite these recent progresses, two interconnected open questions remain:
Does the viscoelasticity of the environment affect the performance of micro-
engines driven by active processes, and does it influence the arrow of time
and fluctuations associated with such processes? In this work, we address
these issues through a minimal experiment in a viscoelastic environment. We
consider a colloidal particle in a harmonic trap, whose mean position is
modulated according to an Ornstein-Uhlenbeck (OU) process. The usage of this
type of driving is very common in the context of active matter systems, where
the OU process is typically used to mimic the interaction of the passive
colloid with the surrounding active particles (such as molecular motors) [59,
60, 57, 61]. The stochastic thermodynamics of this system in a viscous medium
has been extensively studied both experimentally [62, 14, 63] and
theoretically [64, 65, 66, 67, 68, 60].

In this paper, we study this model in a homogeneous viscoelastic medium, for
which there are no known results to the best of our knowledge. We first
analyze the particle dynamics experimentally and theoretically using a
generalised Langevin model. The model is solved analytically and the solutions
match well with our experimental results. We then proceed to characterizing
the non-equilibrium thermodynamics of the system. We calculate the average
power supplied to the particle by stochastic driving, which in this case is
the same as the thermodynamic arrow of time or total entropy production of the
process, up to a scaling factor of the temperature of the environment.
Interestingly, coupling the system with a viscoelastic medium, as opposed to a
viscous medium with the same zero-shear viscosity, leads to approximately
$50\%$ enhancement in the mean value of this quantity. This suggests that,
rather counter intuitively, viscoelasticity amplifies the process’s
directionality under identical external driving forces. Furthermore, based on
a recent theory for interacting particle systems, we propose that this
increase in entropy production in a viscoelastic bath is a generic property,
independent of the system’s specifics. Next, we look into the fluctuations of
the injected power both experimentally and theoretically. In particular, we
obtain previously unknown, exact analytical solutions for the work
distribution at short-times, which are non-Gaussian, yet completely determined
in terms of the mean and variance of the single-step work fluctuations. The
solution obeys a modified fluctuation relation
$P(W)/P(-W)=\exp(-2\beta_{ws}\;W)$, where $2\beta_{ws}\ll 1$. We also obtain
an approximate solution for arbitrary time work distribution, which agree
excellently with the experimental data. Interestingly, we observe a
significant reduction in negative work fluctuations, both at short and
arbitrary times, in the viscoelastic environment compared to their viscous
counterparts. This finding is consistent with the improved arrow of time we
observed, highlighting the influence of viscoelasticity on the directionality
of the process.

##  2 Results

Our experimental system primarily consists of an optically trapped, spherical
polystyrene particle of diameter $2a_{0}=3\ \mu m$ (Sigma Aldrich) dispersed
in a $0.05\%\ (w/v)$ aqueous solution of poly-ethylene oxide (PEO) [Fig.
1(a)]. PEO is a polymer that creates a stable configuration in water, which
provides elasticity along with its inherent viscosity and renders the solution
viscoelastic. The stiffness of the trap is measured to be $12\pm 0.5~{}pN/\mu
m$. To mimic the active noise in the background of different strengths, we
spatially modulate the trapping laser using an acousto-optic modulator (AOM)
connected to a signal generator that generates the Orstein-Uhlenbeck (OU)
noise of various amplitudes. The trajectory we measure consists of the
position of the laser spot $x_{0}(t)$, as well as the one-dimensional position
of the particle $x(t)$ at a spatio-temporal resolution of $1nm-10kHz$. The
readings are obtained using a balanced detector for $\sim 100s$ (see E for
details).

Figure 1: (a) Schematic of a spherical particle trapped in a viscoelastic bath
by optical potential whose mean position $({x}_{0})$ is externally modulated
by Ornstein-Uhlenbeck noise. (b) Typical trajectories of the particle in
$({x},{x}_{0})$ plane for different strengths of the external noise (Gray
dots) with corresponding non-zero probability flux (Red arrows).

It is known that $x(t)$ is well described using a generalized stochastic
differential equation [69]:

$m\ddot{x}(t)=-k[x(t)-x_{0}(t)]-\int_{-\infty}^{t}\Gamma(t-t^{\prime})\dot{x}(t^{\prime})dt^{\prime}+\eta^{\prime}(t).$
(1)

Here the function $\Gamma(t)$ is referred to as the memory kernel. Further,
the noise term $\eta^{\prime}(t)$ satisfied
$\langle\eta^{\prime}(t)\rangle=0$, and has correlations
$\langle\eta^{\prime}(t)\eta^{\prime}(t^{\prime})\rangle=k_{B}T\Gamma(t-t^{\prime})$;
where $k_{B}$ is the Boltzmann constant and $T$ is the temperature of the bath
[69]. We further modulate $x_{0}(t)$ according to the Ornstein-Uhlenbeck
process:

$\dot{x}_{0}(t)=-\frac{x_{0}(t)}{\tau_{e}}+\frac{\sqrt{2A}}{\tau_{e}}\phi_{ext}(t).$
(2)

Here $\phi_{ext}(t)$ is a zero-mean delta-correlated Gaussian noise with
standard deviation equal to 1, i.e., $\langle\phi_{ext}(t)\rangle=0$,
$\langle\phi_{ext}(t)\phi_{ext}(t^{\prime})\rangle=\delta(t-t^{\prime})$. $A$
is the strength of that noise (which we experimentally vary using the AOM),
and $\tau_{e}$ is the correlation time of that process. The correlation of
$x_{0}$ is given by $\langle
x_{0}(t)x_{0}(t^{\prime})\rangle=\frac{A}{\tau_{e}}\exp\left(-\frac{|t-t^{\prime}|}{\tau_{e}}\right)$.

For the memory kernel $\Gamma$, we consider the Jeffrey’s fluid model of
viscoelasticity [30, 50, 51, 47, 52]. According to this model [69, 70, 71,
72]:

$\Gamma(t-t^{\prime})=2\gamma_{0}\delta(t-t^{\prime})+\frac{\gamma}{\tau}\exp\left(-\frac{|t-t^{\prime}|}{\tau}\right).$
(3)

Here $\tau$ is the relaxation time of the fluid, $\gamma_{0}$ and $\gamma$ are
the solvent and the polymer contributions of the frictional coeffecient,
respectively [70]. Notably, the memory kernel will be viscous-like (delta-
correlated) in the $\tau\rightarrow 0$ limit as discussed in Refs. [47, 52].
Hereafter, we refer to this limit as the viscous limit.

There are two distinct advantages of using this model for the viscoelastic
bath. Firstly, we can identify the parameters that appear in this model using
independent measurements. One such mapping is due to a technique developed in
Ref. [32]. In this approach, one measures the phase response of a particle
embedded in the fluid to an external perturbation and computes the relevant
parameters by fitting it to theoretical predictions obtained using the
Jeffrey’s fluid model. By this method, the phase response of the particle over
a wide frequency range can be obtained from a single trajectory using a
$3~{}Hz$ square wave as the perturbation (For the details, see D). We perform
this analysis in our setup and the results are shown in Fig. 8 of D. We obtain
the estimates $\tau=0.018\pm 0.002s$ and the ratio between the polymer
contribution to the viscosity and solvent contribution to the viscosity,
$\mu_{r}(\equiv\gamma/\gamma_{0})=1.805\pm 0.10$. Secondly, as we show in
Appendix AI, using the values of parameters inferred in the previous step, and
by introducing an additional auxiliary variable $X(t)$, we can write down a
Markovian evolution equation in a three dimensional space, whose projection to
one dimension results in the Langevin equation in Eq. (1). The same approach
has also been used previously in other works to convert a non-Markovian
process into Markovian so that the dynamics may be determined conveniently
[69, 72]. Moreover, it is to be noted that this method can also be extended
easily to other more general viscoelastic Maxwellian fluids with more than one
relaxation time [73].

Introducing the auxiliary variable $X(t)$, the total dynamical system can then
be described using a three-dimensional, linear, stochastic differential
equation,

$\bm{\dot{Y}}(t)=-\bm{\lambda}\bm{Y}(t)+\bm{D}\bm{\Phi}(t).$ (4)

Where $\bm{Y}(t)=[x(t),\;X(t),\;x_{0}(t)]^{T}$, and,

$\displaystyle\bm{\lambda}$
$\displaystyle=\begin{bmatrix}\frac{k}{\gamma_{0}}+\frac{\gamma}{\gamma_{0}\tau}&-\frac{\gamma}{\gamma_{0}\tau}&-\frac{k}{\gamma_{0}}\\\
-\frac{1}{\tau}&\frac{1}{\tau}&0\\\ 0&0&\frac{1}{\tau_{e}}\end{bmatrix},\hskip
28.45274pt\bm{D}=\begin{bmatrix}\sqrt{2D_{0}}&0&0\\\ 0&\sqrt{2D_{1}}&0\\\
0&0&\sqrt{2D_{ext}}\end{bmatrix}.$ (5)

Here $\bm{\Phi}(t)=[\phi_{0}(t),\;\phi_{1}(t),\;\phi_{ext}]^{T}$ is a $3\times
1$ dimensional Gaussian-distributed white noise with
$\langle\phi_{i}(t)\rangle=0$ and
$\langle\phi_{i}(t)\phi_{j}(s)\rangle=\delta_{ij}\delta(t-s)$. The
corresponding diffusion constants are given by
$D_{0}=\frac{k_{B}T}{\gamma_{0}}$, $D_{1}=\frac{k_{B}T}{\gamma}$ and
$D_{ext}=\frac{A}{\tau_{e}^{2}}$. For the convenience of our discussion, we
further define a dimensionless parameter
$\theta=\frac{D_{ext}}{D_{0}}=\frac{A/\tau_{e}^{2}}{k_{B}T/\gamma_{0}}$. As we
keep all other parameters except $A$ fixed in our experiment, the variation of
$\theta$ directly maps to the variation of $A$ (i.e., the strength of the OU
noise).

Eq. (4) is an example of a linear multivariate OU process, and the
corresponding Fokker-Planck equation can be solved using standard techniques
[74, 69] to compute the steady state distribution $P_{ss}(\bm{Y})$ (see the
Appendix AI). Due to the presence of the Ornstein-Uhlenbeck driving, this
steady state will be not be in equilibrium, being characterized by a nonzero
probability flux $\bm{J}_{ss}(\bm{Y})$, as shown in Fig.1(b).

Figure 2:  (a) Mean squared displacement (MSD) of the probe particle in the
viscoelastic fluid. (b) Comparison of the MSD of the particle between viscous
and viscoelastic fluid for $\theta=0.56$. The solid lines denote the
theoretical predictions of all the cases. (c) Probability distribution of
position fluctuations of the trapped particle in the viscoelastic fluid
plotted for different strengths of the external noise. (d) The probability
distribution of position fluctuations of the trapped particle in the
viscoelastic and in viscous fluid plotted for a fixed strength of the external
noise. $\theta=0.56$. (e) Theoretical estimation of the variance of the
displacement of the trapped particle at different noise strengths plotted
against the relaxation timescale of the fluid ($\tau$). The dash-dotted line
denotes the timescale of the fluid used in the experiment. (Inset) The
nonmonotonic behaviour of the variance with $\tau$.

Using this analytical solution, we can further obtain the variance of the
particle position as,

$\text{Var}(x(t))=\frac{k_{B}T}{k}+\frac{Ak}{(\gamma+\gamma_{0}+k\tau)}-\frac{Ak^{2}(\tau_{e}^{2}-\tau^{2})}{(\gamma+\gamma_{0}+k\tau)(\gamma\tau_{e}+\gamma_{0}\tau+\gamma_{0}\tau_{e}+k\tau_{e}^{2}+k\tau\tau_{e})}.$
(6)

Note that we can obtain the variance of $P(x)$ for an active viscous bath
(Eq.(25) of Appendix AI) by considering the limit $\tau\rightarrow 0$ in Eq.
(6) which is exactly similar to the related expression given in Ref.[59].

Our experimental measurements agree very well with the model. In Fig. 2(a), we
show the experimentally measured mean-square displacement functions (MSD)
along with the corresponding theoretical lines for two different amplitudes of
the OU noise. We also show the theoretical MSD line in grey colour for zero
external applied noise. Each of these MSD flattens at large times due to the
spatial confinement of the particle by optical tweezers, and the corresponding
value depends linearly on the effective temperature (defined from the variance
of the probability distribution function as
$T_{\text{eff}}=k\text{Var}\left(x(t)\right)/k_{B}$).

In the $\theta\rightarrow 0$ limit, it reaches the equilibrium value
$2k_{B}T/k$. In Fig. 2(b) we show experimental measures of MSDs in the
viscoelastic and viscous fluids for a fixed value of $\theta$ along with
corresponding theory curves. For experiments in the viscous fluid, we have
chosen a solution of ultra-pure glycerol in water with volume fraction $\sim
0.254$ to reach the viscosity $\sim 2452~{}\mu$P.s at the room temperature
$20^{\circ}$C, which is very close to the zero-shear viscosity (when
$\tau\rightarrow 0$) of the viscoelastic fluid, i.e.,
$2496~{}\mu\text{P.s}=(\gamma_{0}+\gamma)/6\pi a_{0}$ [75, 76]. Note that due
to the change in the refractive index when glycerol is mixed in water, the
trap stiffness gets changed. To bring the stiffness back to our fixed value,
we changed the trapping laser power appropriately.

It is clear in Fig. 2(b) that the effective temperature in a viscoelastic
fluid is slightly higher as compared to that in a viscous fluid. Further, due
to the change in the fluid time scale, a prominent difference in the short-
time region is also visible. Fig. 2(c) represents the measured steady-state
probability distributions of $x(t)$ for the same two noise amplitudes as in
Fig. 2(a) in the viscoelastic fluid. Certainly, due to higher effective
temperature for $\theta=0.56$, the broadening of the corresponding
distribution is higher compared to that of $\theta=0.12$. Interestingly, the
higher effective temperature is also visible in Fig. 2(d) which shows the
experimental and theoretical steady-state probability distribution of $x(t)$
in viscous and viscoelastic fluids, for a fixed value of $\theta$.

In Fig. 2(e), we plot $\text{Var}(x(t))$ as a function of $\tau$. We find that
for large values of $\tau$, $\text{Var}(x(t))$ saturates to a constant value,
which is higher than the corresponding value in the viscous ($\tau\rightarrow
0$) limit. Interestingly, however, $\text{Var}(x(t))$ is non-monotonic in
$\tau$ and features a minimum at an intermediate $\tau$ value, that depends on
the two other inherent time-scales of the system only and not on $\theta$ (see
Eq.(24) of the Appendix AI for the expression of $\tau$ corresponding to the
minima). We show this in the inset of Fig. 2(e). Note that this is purely an
effect of viscoelasticity of the medium since the effect of the OU driving
alone does not result in this non-monotonic behaviour (see Ref. [59]).

Now we move on to study the non-equilibrium thermodynamics of this system. To
this end, we look at the statistics of the injected mechanical work done by
the external forcing on the colloidal particle, defined as [77, 6, 65, 66,
63]:

$W_{\tau_{m}}=\frac{k}{k_{B}T}\int_{t}^{t+\tau_{m}}\dot{x}(t^{\prime})x_{0}(t^{\prime})dt^{\prime}.$
(7)

Using this expression, the average power delivered to the particle can be
straightforwardly computed from the experimental data both in viscous and
viscoelastic fluids. The results are shown in Fig. 3(a), as a function of
noise-strength $\theta$. We find that $\langle W_{\tau_{m}}\rangle$ is
positive as expected from the Second Law, and increases linearly as a function
of $\theta$. The estimates also agree well with the analytically computed
estimate (solid lines in Fig. 3(a), see Appendix AII for details) from the
model, given by,

$\langle
W_{\tau_{m}}\rangle=\frac{k\tau_{m}}{k_{B}T}\frac{Ak(\tau+\tau_{e})}{\tau_{e}(\gamma_{0}\tau+\gamma\tau_{e}+\gamma_{0}\tau_{e}+k\tau_{e}^{2}+k\tau\tau_{e})}.$
(8)

Interestingly, the obtained average power in a viscoelastic bath is
significantly higher compared to the same for a ‘memory-less’ Newtonian bath
with our chosen parameters. This implies that for the same amplitude of
external driving, a higher power is delivered to a particle in a viscoelastic
bath compared to its viscous counterpart (See the Eq.(30) and Eq.(31) of
Appendix AII for corresponding expressions in terms of $\theta$ both for
viscous and viscoelastic fluids for a comparison).

Figure 3: (a) Rate of mean work done by the external modulation on the trapped
Brownian particle in viscoelastic bath and viscous bath ( $\tau\rightarrow 0$)
plotted against the strength of the external noise. The solid lines denote the
theoretical predictions. (b) Variation of work with the relaxation timescale
of the medium plotted for different strengths of the external noise along with
experimental points. The dash-dotted line denotes the timescale of the fluid
used in the experiment.

Even though our finding is for a particular example of a non-equilibrium
system, it is possible to argue that this is a generically expected property
for any non-equilibrium system in a viscoelastic environment. The argument is
based on the observation that the average injected power for this system is
the same as the average total entropy production, irrespective of the
rheological properties of the bath (See C). This is because the injected power
is the only time extensive source of dissipation in the system. Furthermore,
for systems in contact with a heat reservoir, this quantity can be considered
as representative of the global arrow of time, defined as,

$\text{Arrow of
time}=\frac{1}{T}\sum_{\bm{Y}(\cdot)}P[\bm{Y}(\cdot)]\log\frac{P[\bm{Y}(\cdot)]}{P[\tilde{\bm{Y}}(\cdot)]},$
(9)

where $\bm{Y}(\cdot)=\\{\bm{Y}(t)\\}_{t=0}^{T}$ is the trajectory of the
system measured for a time duration $T$, and $\tilde{\bm{Y}}(t)$ is its time
reversed counterpart defined as $\tilde{\bm{Y}}(t)=\bm{Y}(T-t)$. It was
recently shown that the presence of interactions increases the evidence of the
global arrow of time [78]. In a viscoelastic bath, we know already that the
polymers in the viscoelastic solution contribute to additional interactions
and that can significantly affect the statistics of the position of the
particle. Thus we argue that the increase in entropy production is due to the
interaction of the colloidal particle with polymers in the solution. We note
that a similar argument can be put forward on the basis of the theoretical
model, where the random variable $X(t)$ is the additional interacting degree
of freedom which is absent in a viscous bath.

A natural question that then arises is whether the arrow of time of the
process can be continuously increased by working with a medium of increased
viscoelastic response. Although this is experimentally challenging to address,
we can seek an answer using the expressions derived using the theoretical
model. In Fig. 3(b), we show that in the limit of increasing relaxation time
$\tau$ of the viscoelastic bath, $\langle W_{\tau_{m}}\rangle$ increases and
saturates to a constant value that depends only on the properties of the trap
parameters. However, in contrast to the non-monotonic behavior observed for
$\text{Var}(x(t))$, we find that $\langle W_{\tau_{m}}\rangle$ exhibits a
monotonically increasing function of $\tau$. This observation reinforces the
suggestion that hidden interactions play a significant role in the
amplification of the thermodynamic arrow of time.

Finally, we look at the fluctuations of the injected power by computing the
probability distribution $P(W_{\tau_{m}})$. In certain cases, it is known that
this probability distribution obeys a fluctuation theorem of the form:

$\ln\frac{P(W_{\tau_{m}}=w)}{P(W_{\tau_{m}}=-w)}\rightarrow\frac{w}{k_{B}T},\
\ \ \tau_{m}>>\tau_{l},$ (10)

where $\tau_{l}$ denotes the largest time scale of the system and $k_{B}$ is
the Boltzmann constant. The work fluctuation theorem has already been tested
and validated in diverse experiments and numerical simulations. Similar setups
to ours include an optically trapped particle dragged with a constant velocity
[79], Brownian oscillator with deterministic periodic driving in the non-
markovian bath[80], a colloidal particle trapped in a bi-stable potential
subjected to periodic modulation [81], Brownian particle trapped in an optical
trap with discontinuously varying trap stiffness [82, 83] and also in other
cases such as electronic circuits [84], mechanical oscillators [85], etc.
However, violations are also found in many systems, e.g., a colloidal particle
under the action of a Gaussian distributed white and coloured noise [86, 87,
62, 68], wave turbulence [88], an RC circuit driven with a stochastic voltage
[89], a rotating active Brownian particle in a viscoelastic medium [90], etc.
These violations have been explored very well, and it is known that the
primary reason for the violation is the presence of time intensive
contributions to the total entropy production, which can significantly
contribute to fluctuations at short times and in continuous state space
systems [91, 92].

An exact expression for the work probability distribution that appears in the
fluctuation relations is known only in a handful of cases. For example, in the
case of a colloidal particle in a harmonic trap with the mean position
externally controlled, for any arbitrary deterministic driving, the work
distribution can be shown to be a Gaussian [93, 94]. The case where the
stiffness is time dependently controlled is one of the simplest models showing
non-Gaussian work fluctuations [95]. Another system for which an exact
calculation of the work distribution has been carried out is the colloidal
particle in a logarithmic-harmonic potential [96], which also has non-Gaussian
work fluctuations. A similar occurrence of non-Gaussian work fluctuations in a
complex fluid in the context of an optically trapped particle with
deterministically varying trap stiffness has also been discussed by Carberry
et.al. in Ref.[83]. In [97], exact work statistics have also been obtained for
a colloidal particle in the uniformly dragged two dimensional confining
potential, and in the presence of additional non-conservative forces such as
torques. In this case, however, the work distribution is found to be a
Gaussian, although it deviates from the conventional work fluctuation theorem.
A general framework for computing the large-time form of the work
distribution, for linear Langevin processes, is described in Refs. [80, 64,
65]. The method is based on the calculation of the corresponding moment-
generating function using a large deviation technique and gives closed form
expressions exact to sub leading order in $1/t$. The methods developed in
Refs. [66, 67] provides exact analytical expressions for the moment generating
function for any value of $t$ for the same class of systems. However,
inverting them to obtain a closed form expression for $P(W,t)$ is challenging.

Figure 4: (a)-(d) Probability density function of single-step work done
(calculated from experimental data) by the external modulation on the trapped
particle in the viscoelastic bath plotted for different strengths of the
external noise. Corresponding probability ratios are plotted in (e)-(h).
(i)-(l) The numerically calculated probability density functions of single-
step work done are shown for different strengths of the modulation. (m)-(n)
Numerically estimated probability ratios for the corresponding cases.
Theoretical estimations in all the cases are represented by solid red lines.
Figure 5:  Variation of $\beta_{ws}$ with $\tau$. (Inset) $\beta_{ws}$ shows
extremely slow variation with $\theta$ for both viscous and viscoelastic
baths.

Here, using a novel technique, we are able to compute a closed form expression
of the probability distribution $P(W_{\tau_{m}})$ in the single-step (short-
time) limit. This limit has received a lot of attention recently, primarily in
the context of saturation of the thermodynamic uncertainty relation [11, 13,
12, 14, 15, 98, 99]. In particular, it is known that in the strict limit of
$\tau_{m}\rightarrow 0$ fluctuations of any time-integrated current in an
overdamped diffusive system is Gaussian. The single-step limit we consider
here is the closest to the short-time limit (that is experimentally
accessible), where the distribution is expected to be non-Gaussian. The
technique we use is based on the fact that the single-step work $W$ appears as
the product of two zero-mean Gaussian distributed quantities $\dot{x}(t)$ and
$x_{0}$(t) which are cross-correlated (see Eq. (7)). The probability
distribution of $W$, therefore, can be obtained directly from the joint
probability distribution function of $\dot{x}(t)$ and $x_{0}$(t) as a skewed
zeroth-order modified Bessel function [100]

$P(W)=\mathcal{A}_{ws}\exp{(\beta_{ws}W)}\text{K}_{0}(\alpha_{ws}|W|),$ (11)

where

$\displaystyle\begin{split}\beta_{ws}&=\frac{\langle\text{W}\rangle}{\text{Var}(\text{W})-2\langle\text{W}\rangle^{2}},\\\
\alpha_{ws}&=\left[\beta_{ws}^{2}+\frac{\beta_{ws}}{\langle\text{W}\rangle}\right]^{\frac{1}{2}},\\\
\mathcal{A}_{ws}&=\frac{\sqrt{\alpha_{ws}^{2}-\beta_{ws}^{2}}}{\pi}.\end{split}$
(12)

The subscript ‘$ws$’ stands for single-step work. See Appendix AIII for a
detailed derivation of the expression above.

Clearly, the expression above is non-Gaussian. However, interestingly, it is
entirely determined in terms of the mean and variance of single-step work.
Further note that, except for an exponential factor of
$e^{\beta_{ws}\text{W}}$, the distribution $P(W)$ does not depend on the sign
of $W$. Hence, it is straightforward to write down a modified fluctuation
relation, of the form:

$R(W)=\ln{\frac{P(W)}{P(-W)}}=2\beta_{ws}W.$ (13)

In Fig. 4 (a) – (d) and (i) – (l), we show the excellent agreement between
this analytically computed expression with the experimental as well as
numerical data. In Fig. 4 (e) – (h), and (m) – (p) we again verify this
fluctuation theorem using both experimental and numerical data.

By definition, $\beta_{ws}$ corresponds to the relative likelihood of having a
positive work fluctuation over the negative one of the same magnitude. It is
hence interesting to check how this factor is affected by the viscoelasticity
of the medium. In Fig. 5, we plot $\beta_{ws}$ as a function of the
viscoelastic fluid time-scale $\tau$ and find that it monotonically increases
with it. This trend is qualitatively similar to that observed for the average
entropy production rate (see Fig. 3b), and shows that for a fixed strength of
non-equilibrium driving, a positive work fluctuation is relatively more
favoured in a viscoelastic bath with a higher value of $\tau$. We have further
found that the effect of viscoelasticity on $\beta_{ws}$ is much more
significant as compared to the effect of $\theta$ on the same. In the inset of
Fig. 5, we analytically compare $\beta_{ws}$ obtained for a range of $\theta$
values for the viscoelastic case and viscous case. We find that $\beta_{ws}$
for the viscoelastic case is always higher than $\beta_{ws}$ for the viscous
case, but in both cases, it varies negligibly with $\theta$.

Figure 6: (a) Probability density function of work done (calculated from
experimental data) by the external modulation on the trapped particle in the
viscoelastic bath plotted for different strengths of the external noise.
Theoretical estimations in all the cases are represented by solid lines. (b)
Probability of positive fluctuations of work are enhanced for the viscoelastic
bath as seen from the comparison of the same for the viscous bath. Theoretical
estimations (Eq.(53) of Appendix IV) in all the cases are represented by solid
lines. (c) $R(W_{\tau_{m}})$ (estimated from the experimental data) plotted as
a function of $W_{\tau_{m}}$ for different strengths of the external
modulation when the particle is in the viscoelastic environment. (d)
$R(W_{\tau_{m}})$ for the viscoelastic bath is found to be deviating from the
usual work fluctuation theorem (black dashed line) with smaller external noise
than the same for the viscous bath.

Next, we consider finite time work distributions, which represent the
distributions of cumulative work $W_{\tau_{m}}$ obtained using Eq. (7). To
approximate this distribution, we assume an uncorrelated time series of short-
time work and utilize the probability distribution function from Eq. (11).
However, in reality, the short-time work time series is not perfectly
uncorrelated, rendering the expression invalid. Nevertheless, we find that by
modifying our approximations based on the mean and variance of $W_{\tau_{m}}$
directly calculated from the Langevin dynamics of the process, we can achieve
good agreement with experimentally and numerically measured distributions,
particularly for typical work fluctuations (see Eq.(53) of Appendix AIV). This
is illustrated in Fig. 6(a) and (b). We also observe that positive work
fluctuations are significantly more favoured in the viscoelastic case as
compared to the corresponding viscous limit. In Fig. 6 (c) and (d), we further
compute the ratio of probabilities [$R(W_{\tau_{m}})$] for positive and
negative work fluctuations. While the ratio is not linear for all work values,
it shows linear behaviour with a unit slope for small work values, deviating
for larger ones. The deviations are greater in a viscoelastic bath compared to
a viscous environment, which again substantiates stronger deviations from
equilibrium.

##  3 Conclusions

In summary, we have shown that the viscoelasticity of the environment has a
significant impact on the thermodynamics and fluctuations of a driven
colloidal particle. Specifically, we have found a substantial increase (around
$50\%$) in the mean injected power to the system when compared to the same
system in a viscous bath under the same external driving strength, even though
the steady-state variance of the position fluctuations gets nominally (around
$10\%$) enhanced. Additionally, we have observed lower negative work
fluctuations across a wide range of driving amplitudes. Since the injected
power corresponds to the thermodynamic arrow of time (entropy production) in
this process, these findings suggest an enhanced directionality in a driven
process when it takes place in a viscoelastic bath rather than a viscous bath
with the same zero-shear viscosity.

In light of recent work [78], our findings suggest that the influence of
viscoelasticity on directionality and thermodynamics may extend beyond the
specific experimental setup studied in this work, potentially affecting a wide
range of driven processes in biological contexts. For example, it will be
interesting to study whether the viscoelasticity of the medium affects the
directionality of molecular motors studied in vitro. Our results also suggest
that viscoelastic environments may provide a more controlled and predictable
environment for driven processes, potentially leading to improved performance
and reliability in practical applications, such as the assembly of nanoscale
materials [101]. We plan to address some of these issues in a future work.

## Acknowledgments

The work was 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. SP is
supported financially by the postdoctoral research fellowship of the IISER
Kolkata. SKM acknowledges the Knut and Alice Wallenberg Foundation for
financial support through Grant No. KAW 2021.0328.

## Conflict of Interest

The authors have no conflicts to disclose.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A  Analytical way to calculate PDF of stochastic work by external
force

###  I Dynamics of the system

We model the one-dimensional dynamics of an externally forced spherical
Brownian particle of mass $m$ confined in a harmonic potential of stiffness
$k$ in a homogeneous and isotropic viscoelastic medium with the following
stochastic differential equation [69]:

$m\ddot{x}(t)=-k[x(t)-x_{0}(t)]-\int_{-\infty}^{t}\Gamma(t-t^{\prime})\dot{x}(t^{\prime})dt^{\prime}+\eta^{\prime}(t)$
(14)

where $x_{0}(t)$ is the random position of the potential minimum at time $t$
originated from the equation

$\dot{x}_{0}(t)=-\frac{x_{0}(t)}{\tau_{e}}+\frac{\sqrt{2A}}{\tau_{e}}\phi_{ext}(t).$
(15)

$\phi_{ext}(t)$ is a zero-mean delta-correlated Gaussian noise with unit
standard deviation, and $A$ is the strength of that noise. Note that
$x_{0}(t)$ is an Ornstein-Uhlenbeck (OU) process with correlation time
$\tau_{e}$, with the correlation given by $\langle
x_{0}(t)x_{0}(t^{\prime})\rangle=\frac{A}{\tau_{e}}\exp\left(-\frac{|t-t^{\prime}|}{\tau_{e}}\right)$.
$\eta^{\prime}(t)$ is the Gaussian-distributed stochastic noise representing
the thermal agitation of the fluid molecules. To satisfy the fluctuation-
dissipation theorem (FDT) in equilibrium, the correlation of the noise is
related to the memory kernel $\Gamma(t)$ as
$\langle\eta^{\prime}(t)\eta^{\prime}(t^{\prime})\rangle=k_{B}T\Gamma(t-t^{\prime})$,
where $k_{B}$ is the Boltzmann constant and $T$ is the temperature of the
bath. As a viscoelastic model, we consider Jeffrey’s fluid model where the
memory Kernel is of the kind:

$\Gamma(t-t^{\prime})=2\gamma_{0}\delta(t-t^{\prime})+\frac{\gamma}{\tau}\exp\left(-\frac{t-t^{\prime}}{\tau}\right),$
(16)

where $\tau$ is the relaxation time of the fluid; $\gamma_{0}$ and $\gamma$
are the solvent and the polymer contributions respectively. The thermal noise
$\eta^{\prime}(t)$ can be represented as the sum of two zero-mean independent
Gaussian processes as
$\eta^{\prime}(t)=\eta^{\prime}_{0}(t)+\eta^{\prime}_{1}(t)$, where the
correlations of these two processes are given by
$\langle\eta^{\prime}_{0}(t)\eta^{\prime}_{0}(t^{\prime})\rangle=2k_{B}T\gamma_{0}\delta(t-t^{\prime})$
and
$\langle\eta^{\prime}_{1}(t)\eta^{\prime}_{1}(t^{\prime})\rangle=k_{B}T\frac{\gamma}{\tau}\exp\left(-\frac{|t-t^{\prime}|}{\tau}\right)$
respectively with $\gamma_{0}=6\pi a_{0}\mu_{s}$ and $\gamma=6\pi
a_{0}\mu_{p}$. $\mu_{p}$ and $\mu_{s}$ are the viscosities of the polymer and
the solvent, respectively. At the overdamped limit (where the inertial effect
is negligible), after carrying out an integration by parts, Eq. (14) can be
written as

$\dot{x}(t)=-\left(\frac{k}{\gamma_{0}}+\frac{\gamma}{\gamma_{0}\tau}\right)x(t)+\frac{\gamma}{\gamma_{0}\tau^{2}}\int_{-\infty}^{t}\exp\left(-\frac{t-t^{\prime}}{\tau}\right)x(t^{\prime})dt^{\prime}+\eta_{0}(t)+\eta_{1}(t)+\frac{k}{\gamma_{0}}x_{0}(t).$
(17)

We define $\eta_{0}(t)=\eta^{\prime}_{0}(t)/\gamma_{0}$ and
$\eta_{1}(t)=\eta^{\prime}_{1}(t)/\gamma_{0}$. Now, in order to model the
system as Markovian we introduce an auxiliary variable

$X(t)=\frac{1}{\tau}\int_{-\infty}^{t}\exp\left(-\frac{t-t^{\prime}}{\tau}\right)\left[x(t^{\prime})+\tau\sqrt{\frac{2k_{B}T}{\gamma}}\phi_{1}(t^{\prime})\right]dt^{\prime}$
(18)

and combining Eqns. (15), (17) and (18) we can write in the matrix form,

$\begin{bmatrix}\dot{x}(t)\\\ \dot{X}(t)\\\
\dot{x}_{0}(t)\end{bmatrix}=-\begin{bmatrix}\frac{k}{\gamma_{0}}+\frac{\gamma}{\gamma_{0}\tau}&-\frac{\gamma}{\gamma_{0}\tau}&-\frac{k}{\gamma_{0}}\\\
-\frac{1}{\tau}&\frac{1}{\tau}&0\\\
0&0&\frac{1}{\tau_{e}}\end{bmatrix}\begin{bmatrix}x(t)\\\ X(t)\\\
x_{0}(t)\end{bmatrix}+\begin{bmatrix}\sqrt{2D_{0}}&0&0\\\ 0&\sqrt{2D_{1}}&0\\\
0&0&\sqrt{2D_{ext}}\end{bmatrix}\begin{bmatrix}\phi_{0}(t)\\\ \phi_{1}(t)\\\
\phi_{ext}(t)\end{bmatrix}$ (19)

where $\phi_{0}$, $\phi_{1}$ and $\phi_{ext}$ are Gaussian-distributed white
noises with mean zero and unit standard deviation.
$D_{0}=\frac{k_{B}T}{\gamma_{0}}$, $D_{1}=\frac{k_{B}T}{\gamma}$ and
$D_{ext}=\frac{A}{\tau_{e}^{2}}$. Considering

$\displaystyle\bm{Y}(t)=\begin{bmatrix}x(t)\\\ X(t)\\\
x_{0}(t)\end{bmatrix},\hskip
2.84526pt\bm{\lambda}=\begin{bmatrix}\frac{k}{\gamma_{0}}+\frac{\gamma}{\gamma_{0}\tau}&-\frac{\gamma}{\gamma_{0}\tau}&-\frac{k}{\gamma_{0}}\\\
-\frac{1}{\tau}&\frac{1}{\tau}&0\\\ 0&0&\frac{1}{\tau_{e}}\end{bmatrix},$
$\displaystyle\hskip 2.84526pt\bm{D}=\begin{bmatrix}\sqrt{2D_{0}}&0&0\\\
0&\sqrt{2D_{1}}&0\\\ 0&0&\sqrt{2D_{ext}}\end{bmatrix},\hskip
2.84526pt\bm{\Phi}(t)=\begin{bmatrix}\phi_{0}(t)\\\ \phi_{1}(t)\\\
\phi_{ext}(t)\end{bmatrix}$

we can rewrite the Eq. (19) as the following:

$\bm{\dot{Y}}(t)=-\bm{\lambda}\bm{Y}(t)+\bm{D}\bm{\Phi}(t).$ (20)

Clearly, Eq. (20) is a multivariate Ornstein-Uhlenbeck process with
correlation matrix [69]

$\bm{C}=\exp(-\bm{\lambda}t)\bm{\sigma}.$ (21)

$\bm{\sigma}$ is the covariance matrix determined by the stationary condition
of the process in terms of Lyapunov equation
$\bm{\lambda}\bm{\sigma}+(\bm{\lambda}\bm{\sigma})^{T}=\bm{D}\bm{D}^{T}$ [74].
The corresponding steady-state probability distribution ($P_{ss}(\mathbf{Y})$)
and the corresponding current ($\mathbf{J}_{ss}(\mathbf{Y})$) can be described
as:

$\displaystyle\begin{split}P_{ss}(\mathbf{Y})=(2\pi\sqrt{\det\bm{\sigma}})^{-1}e^{-\frac{1}{2}\mathbf{Y}^{T}\bm{\sigma}^{-1}\mathbf{Y}}\\\
\mathbf{J}_{ss}(\mathbf{Y})=(-\bm{\lambda}\mathbf{Y}+\mathbf{D}_{f}\bm{\sigma}^{-1}\mathbf{Y})P_{ss}(\mathbf{Y}),\end{split}$
(22)

where $\mathbf{D}_{f}=\frac{1}{2}\mathbf{D}\mathbf{D}^{T}$. The variance of
$x(t)$ is given by $\sigma_{11}$ and after solving the stationary condition,
we get the following expression:

$\text{Var}(x)=\sigma_{11}=\frac{k_{B}T}{k}+\frac{Ak}{(\gamma+\gamma_{0}+k\tau)}-\frac{Ak^{2}(\tau_{e}^{2}-\tau^{2})}{(\gamma+\gamma_{0}+k\tau)(\gamma\tau_{e}+\gamma_{0}\tau+\gamma_{0}\tau_{e}+k\tau_{e}^{2}+k\tau\tau_{e})}.$
(23)

Interestingly, $\text{Var}(x)$ shows non-monotonic dependence (Fig.2(e) of
main text) on the relaxation timescale of the fluid ($\tau$) with the minima
at $\tau=\tau_{min}$ such that,

$\tau_{min}=\tau_{e}\
\frac{[(\gamma_{0}+\gamma)(2\gamma_{0}+\gamma+2k\tau_{e})]^{1/2}-(\gamma_{0}+\gamma)}{\gamma_{0}+2k\tau_{e}}$
(24)

It is clear that $\tau_{min}$ has a non-trivial dependence on the other
timescales of the system and is independent of the strength of external
perturbation ($A$).

The expression for the variance of $P(x)$ for a Newtonian fluid can be easily
calculated by considering the limit $\tau\rightarrow{0}$ in $\sigma_{11}$,
which becomes

$\sigma_{N}=\frac{k_{B}T}{k}+\frac{Ak}{\gamma+\gamma_{0}+k\tau_{e}}.$ (25)

Note that this expression is similar to that obtained in an active viscous
bath given in Ref. [59]. The position auto-correlation function
($\text{ACF}_{x}$) of the particle is $C_{11}(t)$, i.e., $\langle
x(t)x(t^{\prime})\rangle=C_{11}(t-t^{\prime})$, that can be easily calculated
from Eq. (21). The mean-square displacement (MSD) of the particle for our
system is related to the $\text{ACF}_{x}$ as
$\langle(x(t)-x(0))^{2}\rangle=2[C_{11}(0)-C_{11}(t)]$ [70].

###  II Work fluctuations

The stochastic mechanical work by the external force over a time $\tau_{m}$ is
defined as the following [77, 6, 65, 66, 63]:

$W_{\tau_{m}}=\frac{k}{k_{B}T}\int_{t}^{t+\tau_{m}}\dot{x}(t^{\prime})x_{0}(t^{\prime})dt^{\prime},$
(26)

and the mean of work is given by,

$\langle
W_{\tau_{m}}\rangle=\frac{k}{k_{B}T}\int_{t}^{t+\tau_{m}}\langle\dot{x}(t^{\prime})x_{0}(t^{\prime})\rangle
dt^{\prime}.$ (27)

Considering $c=\begin{bmatrix}x_{0}(t)&0&0\end{bmatrix}$ and multiplying with
Eq. (20) from the left we get

$\langle\dot{x}(t)x_{0}(t)\rangle=-\left(\lambda_{11}\langle
x(t)x_{0}(t)\rangle+\lambda_{12}\langle
X(t)x_{0}(t)\rangle+\lambda_{13}\langle
x_{0}(t)^{2}\rangle\right)+2D_{0}\langle x_{0}(t)\phi_{0}(t)\rangle.$

As, $\langle x_{0}(t)\phi_{0}(t)\rangle=0$ and $\langle
x(t)x_{0}(t)\rangle=\sigma_{13}$, $\langle X(t)x_{0}(t)\rangle=\sigma_{23}$,
$\langle x_{0}(t)^{2}\rangle=\sigma_{33}$; we finally have,

$\displaystyle\langle\dot{x}(t)x_{0}(t)\rangle$ $\displaystyle=$
$\displaystyle-\left(\lambda_{11}\sigma_{13}+\lambda_{12}\sigma_{23}+\lambda_{13}\sigma_{33}\right)$
(28) $\displaystyle=$
$\displaystyle\frac{Ak(\tau+\tau_{e})}{\tau_{e}(\gamma_{0}\tau+\gamma\tau_{e}+\gamma_{0}\tau_{e}+k\tau_{e}^{2}+k\tau\tau_{e})}$

Therefore,

$\langle
W_{\tau_{m}}\rangle=\frac{k\tau_{m}}{k_{B}T}\frac{Ak(\tau+\tau_{e})}{\tau_{e}(\gamma_{0}\tau+\gamma\tau_{e}+\gamma_{0}\tau_{e}+k\tau_{e}^{2}+k\tau\tau_{e})}$
(29)

The mean rate of work done ($\langle W_{\tau_{m}}\rangle/\tau_{m}$) can
further be expressed in terms of
$\theta(=\frac{A/\tau_{e}^{2}}{k_{B}T/\gamma_{0}})$ as,

$\frac{\langle
W_{\tau_{m}}\rangle}{\tau_{m}}=\frac{k^{2}\tau_{e}}{\gamma_{0}}\frac{\theta(1+\tau/\tau_{e})}{[\gamma+\gamma_{0}(1+\tau/\tau_{e})+k\tau_{e}(1+\tau/\tau_{e})]}.$
(30)

The corresponding expression for the viscous environment can be obtained by
taking $\tau\rightarrow 0$ limit as

$\left[\frac{\langle W_{\tau_{m}}\rangle}{\tau_{m}}\right]_{\tau\rightarrow
0}=\frac{k^{2}\tau_{e}}{\gamma_{0}}\frac{\theta}{(\gamma+\gamma_{0}+k\tau_{e})}\equiv\frac{Ak^{2}}{k_{B}T}\frac{1}{\tau_{e}(\gamma+\gamma_{0}+k\tau_{e})},$
(31)

where $(\gamma+\gamma_{0})$ is the effective viscosity of the viscous fluid.

Now the variance of the work is the following:

$\displaystyle\begin{split}\text{Var}(W_{\tau_{m}})=\langle(W_{\tau_{m}}-\langle
W_{\tau_{m}}\rangle)^{2}\rangle=\left(\frac{k}{k_{B}T}\right)^{2}\Bigg{[}\tau_{m}\Delta
t\text{ACF}_{x_{0}\dot{x}}(0)&+2\tau_{m}\int_{\Delta
t}^{\tau_{m}}\text{ACF}_{x_{0}\dot{x}}(t^{\prime})dt^{\prime}\\\
&-2\int_{\Delta
t}^{\tau_{m}}t^{\prime}\text{ACF}_{x_{0}\dot{x}}(t^{\prime})dt^{\prime}\Bigg{]},\end{split}$
(32)

where, $\Delta t$ is the time step between two observations and
$\text{ACF}_{x_{0}\dot{x}}(t)$ is the auto-correlation function of the product
of $x_{0}(t)$ and $\dot{x}(t)$, i.e., $\text{ACF}_{x_{0}\dot{x}}(t)=\langle
x_{0}(t)\dot{x}(t)-x_{0}(0)\dot{x}(0)\rangle$. $\text{ACF}_{x_{0}\dot{x}}(t)$
can be related to the auto and cross correlation functions of $x_{0}(t)$ and
$x(t)$, i.e., $\text{ACF}_{x}$, $\text{ACF}_{x_{0}}$, $\text{XCF}_{x,x_{0}}$
and $\text{XCF}_{x_{0},x}$, which is described in the subsection below.

### Relating $\text{ACF}_{x_{0}\dot{x}}$ to $\text{ACF}_{x}$,
$\text{ACF}_{x_{0}}$, $\text{XCF}_{x,x_{0}}$ and $\text{XCF}_{x_{0},x}$:

To find the auto-correlation function of $x_{0}(t)\dot{x}(t)$, note that the
steady-state probability density function of $\bm{Y}$ (see Eq. (20)) is a
multivariate Gaussian distribution. If $\bm{\chi}$ is a multivariate Gaussian
distribution with components $\chi_{i}$, $\chi_{j}$, $\chi_{k}$ and $\chi_{l}$
then

$\langle\chi_{i}\chi_{j}\chi_{k}\chi_{l}\rangle=\text{Cov}(\chi_{i},\chi_{j})\text{Cov}(\chi_{k},\chi_{l})+\text{Cov}(\chi_{i},\chi_{k})\text{Cov}(\chi_{j},\chi_{l})+\text{Cov}(\chi_{i},\chi_{l})\text{Cov}(\chi_{j},\chi_{k}).$

Therefore, we can write

$\displaystyle\begin{split}\langle
x_{0n}x_{n}x_{0(n+p)}x_{(n+p)}\rangle=\text{Cov}(x_{0n},x_{n})\text{Cov}(x_{0(n+p)},x_{(n+p)})&+\text{Cov}(x_{0n},x_{0(n+p)})\text{Cov}(x_{n},x_{(n+p)})\\\
&+\text{Cov}(x_{0n},x_{(n+p)})\text{Cov}(x_{n},x_{0(n+p)}),\end{split}$ (33)

where $x_{0n}$ and $x_{n}$ are the n-th points of $x_{0}$ and $x$ time series
respectively, and ”Cov” represents covariance. Now, note that the auto-
correlation function of $x_{0}(t)x(t)$ is represented in the discrete form for
lag-time $p\Delta t$ as $\left[\langle
x_{0n}x_{n}x_{0(n+p)}x_{(n+p)}\rangle-\langle x_{0n}x_{n}\rangle^{2}\right]$.
In the continuous form, from Eq. (33), we have the autocorrelation function of
$x_{0}(t)x(t)$,

$\text{ACF}_{x_{0}x}(t)=\text{ACF}_{x_{0}}(t)\text{ACF}_{x}(t)+\text{XCF}_{x_{0},x}(t)\text{XCF}_{x,x_{0}}(t),$
(34)

where, ”XCF” represents cross-correlation function. Similarly, we can write in
the discrete form,

$\displaystyle\text{ACF}_{x_{0}\dot{x}}(p\Delta t)$ $\displaystyle=\langle
x_{0n}\dot{x}_{n}x_{0(n+p)}\dot{x}_{(n+p)}\rangle-\langle
x_{0n}\dot{x}_{n}\rangle^{2}$ (35)
$\displaystyle=\Bigg{\langle}x_{0n}\left(\frac{x_{n}-x_{(n-1)}}{\Delta
t}\right)x_{0(n+p)}\left(\frac{x_{(n+p)}-x_{(n-1+p)}}{\Delta
t}\right)\Bigg{\rangle}-\Bigg{\langle}x_{0n}\left(\frac{x_{n}-x_{(n-1)}}{\Delta
t}\right)\Bigg{\rangle}^{2}$

when $\Delta t\rightarrow 0$. After rearranging carefully and using Eq. (34),
we get the expression of the auto-correlation function of $x_{0}(t)\dot{x}(t)$
as the following:

$\text{ACF}_{x_{0}\dot{x}}(t)=\text{ACF}_{x_{0}}(t)\text{ACF}_{\dot{x}}(t)-\text{XCF}_{x_{0},\dot{x}}(t)\text{XCF}_{\dot{x},x_{0}}(t).$
(36)

Where,

$\text{ACF}_{\dot{x}}(t)=-\frac{d^{2}}{dt^{2}}\\{\text{ACF}_{x}(t)\\},$ (37)
$\text{XCF}_{x_{0},\dot{x}}(t)=\frac{d\\{\text{XCF}_{x_{0},x}(t)\\}}{dt},$
(38)

and

$\text{XCF}_{\dot{x},x_{0}}(t)=\frac{d\\{\text{XCF}_{x,x_{0}}(t)\\}}{dt}.$
(39)

Further, from Eq. (21), we have, $\text{ACF}_{x}(t)=C_{11}(t)$,
$\text{XCF}_{x_{0},\dot{x}}(t)=C_{31}(t)$ and
$\text{XCF}_{\dot{x},x_{0}}(t)=C_{13}(t)$. Therefore the corresponding
expressions can be derived from the correlation matrix $\bm{C}$.

###  III Short-time work probability density function

We can write Eq. (26) in the discrete form as

$W_{\tau_{m}}=\frac{k}{k_{B}T}\sum_{i=1}^{n(=\tau_{m}/\Delta
t)}x_{0i}\dot{x}_{i}\Delta t=\sum_{i=1}^{n(=\tau_{m}/\Delta t)}W_{i}.$ (40)

where $W_{i}$ is the i-th point of the short-time work $W$, which is defined
as

$W_{i}=\frac{k}{k_{B}T}x_{0i}\dot{x}_{i}\Delta t.$ (41)

Now, note that $x_{0}$ and $\dot{x}\Delta t$ are two Gaussian distributed
cross-correlated quantities which are given by
$\mathcal{N}(0,\sigma_{33}^{2})$ and $\mathcal{N}(0,\Delta
t^{2}\text{ACF}_{\dot{x}}(0))$ respectively with the normalized cross-
correlation
$\rho=\frac{\text{XCF}_{x_{0},\dot{x}}(0)}{\sigma_{33}\sqrt{\text{ACF}_{\dot{x}}(0)}}$.
Therefore, the probability density function of $z=x_{0}\dot{x}\Delta t$, can
be calculated from the joint probability density function $f$ of these two
quantities from the relation [100]

$\text{P}(z)=\int_{-\infty}^{\infty}\frac{1}{\left|y\right|}f(y,z/y)dy.$ (42)

Here,

$\displaystyle f(x_{0},\dot{x}\Delta t)$
$\displaystyle=\frac{1}{2\pi\sigma_{33}\sqrt{\Delta
t^{2}\text{ACF}_{\dot{x}}(0)}\sqrt{1-\rho^{2}}}\times$ (43)
$\displaystyle\exp\left[-\frac{1}{2(1-\rho^{2})}\left(\frac{x_{0}^{2}}{\sigma_{33}^{2}}-\frac{2\rho
x_{0}\dot{x}\Delta t}{\sigma_{33}\sqrt{\Delta
t^{2}\text{ACF}_{\dot{x}}(0)}}+\frac{(\dot{x}\Delta t)^{2}}{\Delta
t^{2}\text{ACF}_{\dot{x}}(0)}\right)\right].$

Using Eq. (43) for Eq. (42), we get,

$\displaystyle P(z)=\frac{\exp\left(\frac{\rho z}{\sigma_{33}\sqrt{\Delta
t^{2}\text{ACF}_{\dot{x}}(0)}(1-\rho^{2})}\right)}{2\pi\sigma_{33}\sqrt{\Delta
t^{2}\text{ACF}_{\dot{x}}(0)}\sqrt{1-\rho^{2}}}\int_{-\infty}^{\infty}\frac{1}{|y|}\exp\left[-\frac{1}{2(1-\rho^{2})}\left(\frac{y^{2}}{\sigma_{33}^{2}}+\frac{z^{2}}{\Delta
t^{2}\text{ACF}_{\dot{x}}(0)y^{2}}\right)\right]dy.$ (44)

The integral can be solved using the formula given in [102], and we calculate,

$\displaystyle P(z)=\mathcal{A}\exp(\beta z)\text{K}_{0}(\alpha|z|),$ (45)

where, $\mathcal{A}=\frac{1}{\pi\sigma_{33}\sqrt{\Delta
t^{2}\text{ACF}_{\dot{x}}(0)}\sqrt{1-\rho^{2}}}$,
$\beta=\frac{\rho}{\sigma_{33}\sqrt{\Delta
t^{2}\text{ACF}_{\dot{x}}(0)}(1-\rho^{2})}$,
$\alpha=\frac{1}{\sigma_{33}\sqrt{\Delta
t^{2}\text{ACF}_{\dot{x}}(0)}(1-\rho^{2})}$ and $\text{K}_{0}$ is the zeroth
order modified Bessel function. The moment-generating function corresponding
to $P(z)$ is

$\displaystyle\text{M}_{1}(s)$
$\displaystyle=\mathcal{A}\int_{-\infty}^{\infty}e^{sz}e^{\beta
z}\text{K}_{0}(\alpha|z|)dz$ (46)
$\displaystyle=\mathcal{A}\pi\frac{1}{\sqrt{\alpha^{2}-(s+\beta)^{2}}}$

From the normalization condition, $\text{M}_{1}(0)=1$, we find a relation
between $\mathcal{A}$ and $\alpha$, $\beta$. Now to take into account the
factor $\frac{k}{k_{B}T}$ in the Eq. (41), defining
$\alpha_{ws}=\frac{\alpha}{k/k_{B}T}$, $\beta_{ws}=\frac{\beta}{k/k_{B}T}$ and
$\mathcal{A}_{ws}=\frac{\sqrt{\alpha_{ws}^{2}-\beta_{ws}^{2}}}{\pi}$, we
finally obtain the exact expression of the short-time work probability density
function

$\displaystyle
P(W)=\mathcal{A}_{ws}\exp(\beta_{ws}W)\text{K}_{0}(\alpha_{ws}|W|).$ (47)

The related moment generating function will be exactly the same as given in
the Eq. (46) after replacing $\mathcal{A}$ and $\alpha$, $\beta$ by
$\mathcal{A}_{ws}$ and $\alpha_{ws}$, $\beta_{ws}$.

###  IV Cumulative work probability density function

Assuming the time-series of $W$ uncorrelated, we can say $W_{\tau_{m}}$ is
just the summation over the independent points from that time-series of $W$ as
defined in the Eq. (40). Therefore, the corresponding moment generating
function will be the following:

$\text{M}_{n}(s)=\left(\frac{\mathcal{A}_{ws}\pi}{\sqrt{\alpha_{ws}^{2}-(s+\beta_{ws})^{2}}}\right)^{n}.$
(48)

Interestingly, the characteristic function of this moment-generating function
can be written as

$\text{CF}(s)=\left(\frac{\mathcal{A}_{ws}\pi}{\sqrt{\alpha_{ws}^{2}-\beta_{ws}^{2}}}\right)^{n}\left(1-\frac{is}{\alpha_{ws}-\beta_{ws}}\right)^{-\frac{n}{2}}\left(1+\frac{is}{\alpha_{ws}+\beta_{ws}}\right)^{-\frac{n}{2}}.$
(49)

The inverse fourier transform of $\text{CF}(s)$ is the probability density
function of the sum of $n$ number of $W$ points. Now, note that
$\left(1-\frac{is}{\alpha_{ws}-\beta_{ws}}\right)^{-\frac{n}{2}}\left(1+\frac{is}{\alpha_{ws}+\beta_{ws}}\right)^{-\frac{n}{2}}$
is related to the CF of the probability density function of
$\text{y}=\text{y}_{1}-\text{y}_{2}$ where $\text{y}_{1}$ and $\text{y}_{2}$
are two independent numbers drawn from two Gamma distributions
$\Gamma(\frac{n}{2},\alpha-\beta)$ and $\Gamma(\frac{n}{2},\alpha+\beta)$
respectively [103, 102]. Therefore, the desired probability density function
(for the uncorrelated W) is the convolution of two Gamma distributions and is
given by

$\text{P}(W_{\tau_{m}})=\frac{\mathcal{A}_{ws}^{n}\pi^{n}}{\sqrt{\pi}\Gamma(\frac{n}{2})}\frac{1}{(2\alpha_{ws})^{\frac{n-1}{2}}}e^{\beta_{ws}Z}|W_{\tau_{m}}|^{\frac{n-1}{2}}\text{K}_{\frac{n-1}{2}}(\alpha_{ws}|W_{\tau_{m}}|),$
(50)

where $\text{K}_{\frac{n-1}{2}}$ is the modified Bessel function of order
$(n-1)/2$. Further, it can be shown that $\alpha_{ws}$ and $\beta_{ws}$ are
also related to the mean and variance of $W_{\tau_{m}}$ from the moment
generating function Eq. (48) such that

$\begin{split}&\langle
W_{\tau_{m}}\rangle=\frac{d}{ds}[\text{M}_{n}(s)]_{s=0}=\frac{n\beta_{ws}}{\alpha_{ws}^{2}-\beta_{ws}^{2}},\\\
&\text{Var}(\text{W}_{\tau_{m}})=\frac{d^{2}}{ds^{2}}[\text{M}_{n}(s)]_{s=0}=\frac{n(\alpha_{ws}^{2}+\beta_{ws}^{2})}{\alpha_{ws}^{2}-\beta_{ws}^{2}}.\end{split}$
(51)

Using these two relations, $\beta_{ws}$ and $\alpha_{ws}$ can be calculated
as,

$\begin{split}&\beta_{ws}=\frac{n\langle\text{W}_{\tau_{m}}\rangle}{n\text{Var}(\text{W}_{\tau_{m}})-2\langle\text{W}_{\tau_{m}}\rangle^{2}},\\\
&\alpha_{ws}=\left[\beta_{ws}^{2}+\frac{n\beta_{ws}}{\langle\text{W}_{\tau_{m}}\rangle}\right]^{\frac{1}{2}}.\end{split}$
(52)

Note that Eq. (50) would converge to the short-time work probability
distribution function for $n\rightarrow 1$.

Unfortunately, the time-series of $W$ is not uncorrelated, and hence, Eq. (50)
does not describe the experimentally and numerically measured data. However,
we have found an approximate expression of the cumulative work probability
distribution function. Assuming that the correlation will not change the form
of $P(W_{\tau_{m}})$ given in the Eq. (50), we can use the mean and variance
of $W_{\tau_{m}}$ calculated directly from the Langevin equation, i.e., Eqns
(29) and (32) respectively, to calculate the $\alpha$ and $\beta$ parameters.
Notably, Eqns (29) and (32) do not neglect the correlation in $W$. We call
these new parameters $\alpha_{w}$ and $\beta_{w}$, and finally reach the
approximate expression of the cumulative work probability distribution
function which is the following:

$\text{P}(W_{\tau_{m}})=\frac{\mathcal{A}_{w}^{n}\pi^{n}}{\sqrt{\pi}\Gamma(\frac{n}{2})}\frac{1}{(2\alpha_{w})^{\frac{n-1}{2}}}e^{\beta_{w}Z}|W_{\tau_{m}}|^{\frac{n-1}{2}}\text{K}_{\frac{n-1}{2}}(\alpha_{w}|W_{\tau_{m}}|).$
(53)

where, $\mathcal{A}_{w}=\frac{\sqrt{\alpha_{w}^{2}-\beta_{w}^{2}}}{\pi}$. We
have found that this approximate expression of $P(W_{\tau_{m}})$ matches the
experimental and numerical data excellently which is described in the main
text.

##  Appendix B Work fluctuation theorem

Here we show the ratio of probabilities corresponding to the positive and
negative fluctuations of the stochastic work done on a microscopic particle in
the viscoelastic (Fig. 7(a)) and viscous bath (Fig. 7(b)),– estimated from the
numerical data. As discussed in the main text, $R(W_{\tau_{m}})$ for the
viscous bath follows the unit slope line for higher strength of the external
noise than the same in the viscoelastic bath. The slope, however, as expected,
converges to one as $\theta$ tends to zero (Fig. 7(c)).

Figure 7: Ratio of probabilities corresponding to the positive and negative
work fluctuations is plotted for different strengths of external noise in the
viscoelastic (a) and viscous (b) bath, respectively.(c) Plot of $2\beta_{w}$
as a function of $\theta$.

##  Appendix C  Entropy production rate

The entropy production rate of this system can be calculated using the Fokker-
plank based method as described in Ref. [104, 14, 16, 63]. Starting from an
arbitrary initial condition for $x,X$ and $x_{0}$, the system will reach to a
nonequilibrium steady state in the long time with a characteristics
probability distribution ($P_{ss}(\mathbf{Y})$) and current
($\mathbf{J}_{ss}(\mathbf{Y})$) that can be described in terms of the steady
state covariance matrix $\bm{\sigma}$ as shown in Eq.(22).

Now the total entropy production rate (in units of $k_{B}s^{-1}$) of the
system can be obtained as,

$\Pi=\int
d\mathbf{Y}\frac{\mathbf{D}_{f}^{-1}\mathbf{J}_{ss}^{2}(\mathbf{Y})}{P_{ss}(\mathbf{Y})}=\frac{Ak^{2}(\tau+\tau_{e})}{T\tau_{e}(\gamma_{0}\tau+\gamma\tau_{e}+\gamma_{0}\tau_{e}+k\tau_{e}^{2}+k\tau\tau_{e})}.$
(54)

Note that this expression is exactly similar to the mean rate of work done
($\langle W_{\tau_{m}}\rangle/\tau_{m}$) as shown in Eq.(29).

##  Appendix D Method to obtain the rheological properties of the fluid

We deploy a phase-sensitive active microrheological technique to measure the
fluid characteristics of the viscoelastic medium [71]. In this technique, an
optically trapped microscopic particle embedded in the fluid is perturbed by
an external square modulation containing sinusoids of several frequencies, and
the position fluctuations of the particle are recorded. From the recorded
trajectory, the relative phase of the particle’s response to the modulation is
measured as a function of the modulation frequency. The relaxation timescale
and the relative viscosity of the fluid can be directly estimated by fitting
the phase response curve with the theoretical expression calculated from the
Langevin description of the trapped particle in the viscoelastic fluid,
following Jeffrey’s model. The overdamped dynamics of the particle trapped in
an optical potential with stiffness $k$ can be written in the frequency domain
(taking Fourier transform of the overdamped version of Eq. (1)) as:

$i\omega\Gamma(\omega)x(\omega)+kx(\omega)=kx_{0}(\omega)$ (55)

where,

$\displaystyle\Gamma(\omega)=\int_{0}^{\infty}\Gamma(t)\exp(i\omega t)\
dt=\Bigg{[}\gamma_{0}+\frac{\gamma}{-i\omega\tau+1}\Bigg{]}$ (56)
$\displaystyle=6\pi
a_{0}\mu_{s}\Bigg{[}1+\frac{\mu_{r}}{-i\omega\tau+1}\Bigg{]}.$ (57)

The relative viscosity $\mu_{r}$ is the ratio of corresponding viscosities of
the polymer ($\mu_{p}$) and the solvent($\mu_{s}$). $a_{0}$ denotes the radius
of the trapped particle. Now Eq.57 can be substituted in Eq.55 to get the
phase response of the particle as:

$\delta(\omega)=\tan^{-1}\Bigg{[}\frac{\frac{1+\mu_{r}}{\tau^{2}}\omega+\omega^{3}}{\frac{k}{\tau^{2}\gamma_{0}}+(\frac{k}{\gamma_{0}}-\frac{\mu_{r}}{\tau})\omega^{2}}\Bigg{]}.$
(58)

As described in Ref.[71], the trapped microparticle embedded in the
viscoelastic fluid is perturbed with a square wave modulation of frequency $3\
Hz$. The phase response of the particle to the odd harmonics of the square
wave modulation is obtained experimentally and fitted with Eq.58 to estimate
the relaxation timescale ($\tau$), and relative viscosity ($\mu_{r}$) of the
fluid as shown in Fig.1(b) of the main text.

Figure 8: Phase response of the trapped particle for different frequencies and
the corresponding fit for the phase-sensitive active micro-rheology [71].

##  Appendix E Experimental setup

Our Optical tweezers setup (Fig. 9) is developed around a standard inverted
microscope (Olympus IX71) with a high numerical aperture objective lens (100X,
NA = 1.3). A semiconductor laser (Lasever, Max power 500 mW) of wavelength
1064 nm is tightly focused through the objective lens on the sample to trap a
micro-particle dispersed in it. The trapping beam is passed through an
acousto-optic modulator (AOM) placed at a plane conjugate to the objective
lens. We modulate the beam by using the first-order diffracted beam off the
AOM with a computer-generated signal fed to the AOM controller via a data
acquisition card (which acts as a signal generator here). We employ a second
low-power laser beam (stationary and copropagating with the trapping beam) of
wavelength 785 nm to track the position of the particle, which we determine
from the back-scattered light incident on a balanced detection system [105],
constructed using two high gain-bandwidth photo-detectors (Thorlabs PDA100A2)
placed orthogonal to each other. The displacement data of the particle is
recorded into a computer from the balanced detection signal via a data
acquisition card (NI DAQ) at a spatio-temporal resolution of 1 nm-10kHz.

Figure 9: Schematic of our setup. The annotations are as follows: M: Mirror,
DM: Dichroic mirror, EM: Edge mirror, L: Plano-convex lens of different focal
lengths, HWP: Half-wave plate, PD: Photodiode, AOM: Acousto optic modulator,
DAQ: Data acquisition card (National Instruments).

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