# Time-correlation functions of stochastic three-sphere micromachines

Jun Li  Department of Physics, Wenzhou University, Wenzhou, Zhejiang 325035,
China  Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou,
Zhejiang 325001, China  Ziluo Zhang  Wenzhou Institute, University of Chinese
Academy of Sciences, Wenzhou, Zhejiang 325001, China  Institute of Theoretical
Physics, Chinese Academy of Sciences, Beijing 100190, China  Zhanglin Hou
Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou,
Zhejiang 325001, China  Oujiang Laboratory, Wenzhou, Zhejiang 325000, China
Kento Yasuda  Research Institute for Mathematical Sciences, Kyoto University,
Kyoto 606-8502, Japan  Linli He  Corresponding author<EMAIL_ADDRESS>Department of Physics, Wenzhou University, Wenzhou, Zhejiang 325035, China
Shigeyuki Komura  Corresponding author<EMAIL_ADDRESS>Wenzhou Institute,
University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325001, China
Oujiang Laboratory, Wenzhou, Zhejiang 325000, China  Department of Chemistry,
Graduate School of Science, Tokyo Metropolitan University, Tokyo 192-0397,
Japan

###### Abstract

We discuss and compare the statistical properties of two stochastic three-
sphere micromachines, i.e., odd micromachine and thermal micromachine. We
calculate the steady state time-correlation functions for these micromachines
and decompose them into the symmetric and antisymmetric parts. In both cases,
the cross-correlation between the two spring extensions has an antisymmetric
part, which is a direct consequence of the broken time-reversal symmetry. For
the odd micromachine, the antisymmetric part of the correlation function is
proportional to the odd elasticity, whereas it is proportional to the
temperature difference between the two edge spheres for the thermal
micromachine. The entropy production rate and the Green-Kubo relations for the
two micromachines are also obtained. Comparing the results of the two models,
we argue an effective odd elastic constant of the thermal micromachine. The
effective odd elasticity of the thermal micromachine is proportional to the
temperature difference among the spheres, which causes an internal heat flow
and leads to directional locomotion in the presence of hydrodynamic
interactions.

##  I Introduction

Microswimmers are tiny objects moving in fluids, such as sperm cells or motile
bacteria, that swim in a fluid and are expected to be relevant to
microfluidics and microsystems Lauga09a ; LaugaBook ; Hosaka22 . By
transforming chemical energy into mechanical work, microswimmers change their
shapes and can move in viscous environments. According to Purcell’s scallop
theorem, reciprocal body motion cannot be used for locomotion in a Newtonian
fluid Purcell1977 ; Ishimoto12 . As one of the simplest models exhibiting
nonreciprocal body motion, Najafi and Golestanian proposed a model of a three-
sphere microswimmer Najafi04 ; Golestanian08 , in which three in-line spheres
are linked by two arms of varying lengths. Later, such a three-sphere
microswimmer has been experimentally realized Leoni09 ; Grosjean16 ;
Grosjean18 . Various extensions of the original three-sphere microswimmer
model were considered and summarized in Ref. Yasuda23 . Among them, we focus
on the two stochastic microswimmers, i.e., thermal microswimmer in which the
spheres have different temperatures Hosaka17 ; Sou19 ; Sou21  and odd
microswimmer in which the two springs (rather than arms) have odd elasticity
Yasuda21 ; Kobayashi23 .

The concept of odd elasticity was proposed by Scheibner et al. to account for
nonreciprocal interactions in active systems Scheibner20 ; Fruchart23 . They
showed that the odd component of the elastic constant matrix quantifies the
amount of work extracted along quasistatic deformation cycles. Generalized odd
elasticity exists not only in elastic materials but also in generic
micromachines, such as molecular motors and catalytic enzymes that exhibit
nonequilibrium steady state dynamics Yasuda21catalytic ; YasudaOM22 ;
YasudaTCF22 ; KobayashiODD23 ; Ishimoto22 ; Ishimoto23 . Since the thermal
microswimmer model was proposed before the work by Scheibner et al., it is
necessary to understand how it can be quantitatively characterized in terms of
effective odd elasticity and further compare it with the odd microswimmer.

For this purpose, we discuss various quantities obtained from the two
stochastic microswimmer models and particularly focus on their time-
correlation functions. This is because the antisymmetric part of the cross-
correlation function can exist for nonequilibrium micromachines when the time-
reversal symmetry is broken YasudaTCF22 ; KobayashiODD23 . In order to have a
proper comparison between the two models, we generalize the odd microswimmer
model to have different even elastic constants. On the other hand, we mostly
neglect hydrodynamic interactions between the different spheres except when we
calculate the average velocity. In the absence of hydrodynamic interactions,
the considered models do not exhibit directed locomotion but undergo Brownian
motion Sou19 ; Sou21 . Hence, we use the word “micromachine” instead of
“microswimmer” throughout this paper.

After explaining the thermal three-sphere micromachine and the odd three-
sphere micromachine, we show the average velocity, entropy production rate,
extension-extension and velocity-velocity time correlation functions, and the
corresponding Green-Kubo relations. The antisymmetric parts of the cross-
correlation functions are explicitly obtained for the two micromachines,
reflecting the degree of broken time-reversal symmetry. For the odd
micromachines, the time-correlation functions exhibit oscillatory behaviors
when the odd elasticity is large enough. We shown that the effective odd
elasticity of the thermal microswimmer is essentially proportional to the
temperature difference between the two edge spheres. Such a temperature
difference causes a heat flow inside the micromachine, which can be quantified
by the entropy production rate.

In Sec. II, we describe the model of the odd three-sphere micromachine and
show its statistical properties as mentioned above. In Sec. III, we perform a
similar analysis for the thermal three-sphere micromachine. In Sec. IV, we
compare the two stochastic micromachines and discuss the effective odd
elasticity of the thermal micromachine. A summary and some further discussion
are given in Sec. V. The two Appendices list the velocity-velocity correlation
functions for the two models.

##  II Odd three-sphere micromachine

Figure 1:  Two stochastic three-sphere micromachines consisting of three
spheres with a radius $a$ and two springs with a natural length $\ell$. These
micromachines are immersed in a fluid with viscosity $\eta$. The positions of
the spheres are denoted by $x_{i}$ ($i=1,2,3$) in a one-dimensional
coordinate, and the spring extensions with respect to $\ell$ are denoted by
$u_{\alpha}$ ($\alpha={\rm A},{\rm B}$). (a) Odd three-sphere micromachine in
which the two springs have the odd elastic constant $K^{\rm o}$ in addition to
the even elastic constants $K^{\rm e}_{\rm A}$ and $K^{\rm e}_{\rm B}$. The
temperature of all the spheres is $T$. The ratios are defined by
$\kappa=K_{\rm B}^{\rm e}/K_{\rm A}^{\rm e}$ and $\lambda=K^{\rm o}/K_{\rm
A}^{\rm e}$. (b) Thermal three-sphere micromachine in which the two springs
have only the two elastic constants $K^{\rm e}_{\rm A}$ and $K^{\rm e}_{\rm
B}$. However, the three spheres are in equilibrium with independent heat baths
at different temperatures $T_{i}$ ($i=1,2,3$).

###  II.1 Model

We first describe the model of the odd micromachine that is slightly extended
from our previous model Yasuda21 ; Kobayashi23 . As schematically shown in
Fig. 1(a), this model consists of three spheres of radius $a$ positioned along
a one-dimensional coordinate system, denoted by $x_{i}$ ($i=1,2,3$). These
three spheres are connected by two springs that exhibit both even and odd
elasticities. We denote the two spring extensions by $u_{\rm
A}=x_{2}-x_{1}-\ell$ and $u_{\rm B}=x_{3}-x_{2}-\ell$, where $\ell$ is the
constant natural length. Then the forces $F_{\rm A}$ and $F_{\rm B}$ conjugate
to $u_{\rm A}$ and $u_{\rm B}$, respectively, are given by
$F_{\alpha}=-K_{\alpha\beta}u_{\beta}$ ($\alpha,\beta={\rm A},{\rm B}$). For
an odd elastic spring, the elastic constant matrix $K_{\alpha\beta}$ is given
by

$\displaystyle\mathbf{K}=\left(\begin{array}[]{cc}K_{\rm A}^{\rm e}&K^{\rm
o}\vspace{1ex}\\\ -K^{\rm o}&K_{\rm B}^{\rm e}\\\ \end{array}\right).$ (3)

In the two-dimensional configuration space, $K_{\rm A}^{\rm e}$ and $K_{\rm
B}^{\rm e}$ are positive even elastic constants, while $K^{\rm o}$ is an odd
elastic constant that can take both positive and negative values. In our
previous study, we only studied the case of $K_{\rm A}^{\rm e}=K_{\rm B}^{\rm
e}$ Yasuda21 ; Kobayashi23 . Then, the forces $f_{i}$ acting on each sphere
are given by $f_{1}=-F_{\rm A}$, $f_{2}=F_{\rm A}-F_{\rm B}$, and
$f_{3}=F_{\rm B}$. Notice that these forces automatically satisfy the force-
free condition, i.e., $f_{1}+f_{2}+f_{3}=0$ Yasuda17a .

The above odd micromachine is immersed in a fluid of shear viscosity $\eta$
and temperature $T$. The Langevin equation of each sphere is given by DoiBook

$\displaystyle\dot{x}_{i}=M_{ij}f_{j}+\xi_{i},$ (4)

where $\dot{x}_{i}=dx_{i}/dt$ and $M_{ij}$ is hydrodynamic mobility
coefficient matrix Najafi04 ; Golestanian08

$\displaystyle M_{ij}=\begin{cases}1/(6\pi\eta a)&(i=j),\vspace{1ex}\\\
1/(4\pi\eta|x_{i}-x_{j}|)&(i\neq j).\end{cases}$ (5)

In Eq. (4), the Gaussian white-noise sources $\xi_{i}$ have zero mean
$\langle\xi_{i}(t)\rangle=0$, and their correlations satisfy the fluctuation-
dissipation theorem DoiBook

$\displaystyle\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=2k_{\rm
B}TM_{ij}\delta(t-t^{\prime}),$ (6)

where $k_{\rm B}$ is the Boltzmann constant. This condition assures that the
surrounding fluid is in thermal equilibrium. Since the noise amplitudes depend
on the particle positions, we use the Itô interpretation in Eq. (4)
Kobayashi23 .

###  II.2 Average velocity

It is convenient to introduce the characteristic time scale $\tau=6\pi\eta
a/K_{\rm A}^{\rm e}$ describing the spring relaxation time. We define the
ratios between the elastic constants by $\kappa=K_{\rm B}^{\rm e}/K_{\rm
A}^{\rm e}$ and $\lambda=K^{\rm o}/K_{\rm A}^{\rm e}$. In the following
analysis, we assume $u_{\alpha}\ll\ell$ and $a\ll\ell$, and focus on the
leading-order contribution. The instantaneous total velocity of the
micromachine is simply given by $V=(\dot{x}_{1}+\dot{x}_{2}+\dot{x}_{3})/3$.
Using Eqs. (3)-(5) and taking the statistical average, we obtain Yasuda21 ;
Kobayashi23

$\displaystyle\langle V\rangle$
$\displaystyle=\frac{a}{8\ell^{2}\tau}\Big{[}(-1+3\lambda)\langle u_{\rm
A}^{2}\rangle+(\kappa+3\lambda)\langle u_{\rm B}^{2}\rangle$
$\displaystyle+[3(1-\kappa)-2\lambda]\langle u_{\rm A}u_{\rm
B}\rangle\Big{]},$ (7)

where we have used $\langle u_{\alpha}\rangle=0$.

The equal-time correlation functions $\langle u_{\alpha}u_{\beta}\rangle$
appearing in Eq. (7) can be obtained from the reduced Langevin equations for
$\dot{u}_{\rm A}=\dot{x}_{2}-\dot{x}_{1}$ and $\dot{u}_{\rm
B}=\dot{x}_{3}-\dot{x}_{2}$ as

$\displaystyle\dot{u}_{\alpha}=\Gamma_{\alpha\beta}u_{\beta}+\Xi_{\alpha},$
(8)

where the friction matrix $\Gamma_{\alpha\beta}$ and the noise vector
$\Xi_{\alpha}$ are given by

$\displaystyle\mathbf{\Gamma}=-\frac{1}{\tau}\left(\begin{array}[]{cc}2+\lambda&-\kappa+2\lambda\\\
-1-2\lambda&2\kappa-\lambda\\\
\end{array}\right),\quad\mathbf{\Xi}=\left(\begin{array}[]{cc}\xi_{2}-\xi_{1}\\\
\xi_{3}-\xi_{2}\\\ \end{array}\right).$ (13)

Here, under the assumption $u_{\alpha}\ll\ell$, we have neglected the
hydrodynamic interactions acting between different spheres, i.e.,
$M_{ij}\approx 0$ when $i\neq j$ in Eq. (5). It should be noted that the
friction matrix $\Gamma_{\alpha\beta}$ is nonreciprocal, i.e.,
$\Gamma_{\rm{AB}}\neq\Gamma_{\rm{BA}}$ when $\lambda\neq 0$ even if
$\kappa=1$. Such a nonreciprocal interaction between $\langle u_{\rm
A}\rangle$ and $\langle u_{\rm B}\rangle$ results from the odd elasticity
$K^{\rm o}$.

To deal with the above Langevin equations, we introduce the bilateral Fourier
transform of a function $f(t)$ as
$f(\omega)=\int_{-\infty}^{\infty}dt\,f(t)e^{-{\rm i}\omega t}$ and its
inverse transform as
$f(t)=\int_{-\infty}^{\infty}(d\omega/2\pi)\,f(\omega)e^{{\rm i}\omega t}$.
Solving Eq. (8) in the frequency domain, we obtain

$\displaystyle u_{\rm A}(\omega)$
$\displaystyle=\frac{\left[(2\kappa-\lambda)+{\rm
i}\omega\tau\right]\xi_{1}(\omega)-\left[(\kappa+\lambda)+{\rm
i}\omega\tau\right]\xi_{2}(\omega)-(\kappa-2\lambda)\xi_{3}(\omega)}{-3(\kappa+\lambda^{2})-2(1+\kappa){\rm
i}\omega\tau+(\omega\tau)^{2}}\tau,$ (14) $\displaystyle u_{\rm B}(\omega)$
$\displaystyle=\frac{(1+2\lambda)\xi_{1}(\omega)+\left[(1-\lambda)+{\rm
i}\omega\tau\right]\xi_{2}(\omega)-\left[(2+\lambda)+{\rm
i}\omega\tau\right]\xi_{3}(\omega)}{-3(\kappa+\lambda^{2})-2(1+\kappa){\rm
i}\omega\tau+(\omega\tau)^{2}}\tau.$ (15)

Using Eqs. (14) and (15), one can calculate the correlation functions $\langle
u_{\rm\alpha}(\omega)u_{\rm\beta}(\omega^{\prime})\rangle$. Then, the equal-
time correlation functions are obtained as

$\displaystyle\langle u_{\rm A}^{2}\rangle$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi}\langle
u_{\rm A}(\omega)u_{\rm A}(\omega^{\prime})\rangle$
$\displaystyle=\frac{k_{\rm
B}T(\kappa+\kappa^{2}-\kappa\lambda+2\lambda^{2})}{K_{\rm A}^{\rm
e}(1+\kappa)(\kappa+\lambda^{2})},$ (16) $\displaystyle\langle u_{\rm
B}^{2}\rangle$ $\displaystyle=\frac{k_{\rm
B}T(1+\kappa+\lambda+2\lambda^{2})}{K_{\rm A}^{\rm
e}(1+\kappa)(\kappa+\lambda^{2})},$ (17) $\displaystyle\langle u_{\rm A}u_{\rm
B}\rangle$ $\displaystyle=-\frac{k_{\rm
B}T[(1-\kappa)\lambda+\lambda^{2}]}{K_{\rm A}^{\rm
e}(1+\kappa)(\kappa+\lambda^{2})}.$ (18)

Here, we have neglected the cross-correlations of the noise,
$\langle\xi_{i}\xi_{j}\rangle$ with $i\neq j$, because they only contribute to
the higher order terms in $a/\ell$. When $\lambda=0$, the above expressions
reduce to $\langle u_{\rm A}^{2}\rangle=k_{\rm B}T/K_{\rm A}^{\rm e}$,
$\langle u_{\rm B}^{2}\rangle=k_{\rm B}T/K_{\rm B}^{\rm e}$, and $\langle
u_{\rm A}u_{\rm B}\rangle=0$, reproducing the well-known thermal equilibrium
case.

Substituting the equal-time correlations in Eqs. (16)-(18) into Eq. (7), we
obtain the steady state average velocity of the odd micromachine as

$\displaystyle\langle V\rangle=\frac{7k_{\rm
B}T\lambda}{24\pi\eta\ell^{2}(1+\kappa)},$ (19)

which reduces to the previous result when $\kappa=1$ Yasuda21 ; Kobayashi23 .
Recalling $\lambda=K^{\rm o}/K_{\rm A}^{\rm e}$, we see that $\langle
V\rangle$ is proportional to the odd elastic constant $K^{\rm o}$ whose sign
determines the swimming direction. Since $\langle V\rangle$ is also
proportional to $k_{\rm B}T$, thermal fluctuations are responsible for the
locomotion of the odd micromachine.

###  II.3 Entropy production rate

Next, we calculate the steady state average entropy production rate
$\langle\dot{\sigma}\rangle$ of the odd three-sphere micromachine, which is
given by

$\displaystyle\langle\dot{\sigma}\rangle=-\sum_{i=1}^{3}\frac{\langle\dot{Q}_{i}\rangle}{T}.$
(20)

According to the framework of stochastic energetics established by Sekimoto
SekimotoBook , the time-derivative of the heat gained by the $i$-th sphere is

$\displaystyle\dot{Q}_{i}=6\pi\eta a(-\dot{x}_{i}+\xi_{i})\circ\dot{x}_{i},$
(21)

where $\dot{x}_{i}$ and $\xi_{i}$ are given by Eq. (4) and $\circ$ indicates
the Stratonovich product (the summation over $i$ is not taken in this
expression). Then, the lowest-order average heat flows become

$\displaystyle{\langle\dot{Q}_{1}\rangle}$ $\displaystyle=\frac{k_{\rm
B}T\lambda(1-2\lambda)}{\tau(1+\kappa)},$ (22)
$\displaystyle{\langle\dot{Q}_{2}\rangle}$ $\displaystyle=\frac{k_{\rm
B}T\lambda(-1+\kappa-2\lambda)}{\tau(1+\kappa)},$ (23)
$\displaystyle{\langle\dot{Q}_{3}\rangle}$ $\displaystyle=-\frac{k_{\rm
B}T\lambda(\kappa+2\lambda)}{\tau(1+\kappa)},$ (24)

which all vanish when $\lambda=0$. Substituting Eqs. (22)-(24) into Eq. (20),
we obtain the average entropy production rate of the odd micromachine as

$\displaystyle\langle\dot{\sigma}\rangle=\frac{6k_{\rm
B}\lambda^{2}}{\tau(1+\kappa)},$ (25)

which is obviously non-negative, $\langle\dot{\sigma}\rangle\geq 0$, and
vanishes only when $\lambda=0$. This result is consistent with the second law
for nonequilibrium steady states Freitas22 .

In our previous work with $\kappa=1$ Yasuda21 , we obtained the entropy
production rate by using the combination of the friction matrix and the
covariant matrix of the Gaussian distribution function Weiss03 ; Weiss07 . We
also showed that the entropy production rate coincides with the power (work
per unit time) of the odd micromachine. Hence, all the extracted work due to
odd elasticity is converted into entropy production in the steady state.

Figure 2: Plots of the scaled extension-extension correlation functions (a)
$\phi_{\rm{AA}}^{\rm s}$, (b) $\phi_{\rm{BB}}^{\rm s}$, (c)
$\phi_{\rm{AB}}^{\rm s}$, and (d) $\phi_{\rm{AB}}^{\rm a}$ as a function of
$t/\tau$ for $\kappa=1$. The dimensionless odd elasticity is chosen as
$\lambda=0$ (black), $0.25$ (red), and $5$ (green). For these parameters,
$\mu=\sqrt{3\lambda^{2}-(1-\kappa+\kappa^{2})}$ is purely imaginary, and the
correlation functions do not exhibit any oscillatory behaviors.  Figure 3:
Plots of the scaled extension-extension correlation functions (a)
$\phi_{\rm{AA}}^{\rm s}$, (b) $\phi_{\rm{BB}}^{\rm s}$, (c)
$\phi_{\rm{AB}}^{\rm s}$, and (d) $\phi_{\rm{AB}}^{\rm a}$ as a function of
$t/\tau$ for $\kappa=1$. The dimensionless odd elasticity is chosen as
$\lambda=1$ (black), $2$ (red), and $3$ (green). For these parameters,
$\mu=\sqrt{3\lambda^{2}-(1-\kappa+\kappa^{2})}$ is real and the correlation
functions exhibit oscillatory behaviors.

###  II.4 Time-correlation functions

Next, we calculate the time-correlation functions of the odd micromachine in
the steady state. Using Eqs. (14) and (15), we perform inverse Fourier
transform and obtain the extension-extension time-correlation function matrix
$\phi_{\alpha\beta}(t)$ defined by

$\displaystyle\langle u_{\alpha}(t)u_{\beta}(0)\rangle=\frac{k_{\rm
B}T}{K_{\rm A}^{\rm e}}\phi_{\alpha\beta}(t),$ (26)

where $\phi_{\alpha\beta}$ is dimensionless. Generally, one can decompose the
time-correlation functions into the symmetric and antisymmetric parts as
YasudaTCF22 ; KobayashiODD23

$\displaystyle\phi_{\alpha\beta}(t)=\phi_{\alpha\beta}^{\mathrm{s}}(t)+\phi_{\alpha\beta}^{\mathrm{a}}(t),$
(27)

where $\phi_{\alpha\beta}^{\mathrm{s}}(t)=\phi_{\beta\alpha}^{\mathrm{s}}(t)$
and $\phi_{\alpha\beta}^{\mathrm{a}}(t)=-\phi_{\beta\alpha}^{\mathrm{a}}(t)$.
When $\alpha=\beta$ (auto-correlation), only the symmetric part is allowed due
to the time-translational invariance of the steady state and hence the
antisymmetric parts should vanish, i.e., $\phi_{\rm{AA}}^{\rm
a}=\phi_{\rm{BB}}^{\rm a}=0$. When $\alpha\neq\beta$ (cross-correlation) and
if the system is in thermal equilibrium, the antisymmetric part of the cross-
correlation $\phi_{\rm{AB}}^{\rm a}$ should also vanish due to the time-
reversal symmetry DoiBook . In nonequilibrium situations, however,
$\phi_{\rm{AB}}^{\rm a}$ can exist because the time-reversal symmetry is
generally broken YasudaTCF22 ; KobayashiODD23 . In other words,
$\phi_{\rm{AB}}^{\rm a}$ quantifies the degree of nonequilibrium in stochastic
micromachines.

The inverse Fourier transform of the correlation function can be
straightforwardly performed by employing the residue theorem. After some
calculation, we obtain the time-dependent extension-extension correlation
functions as

$\displaystyle\phi_{\rm{AA}}^{\rm s}(t)$
$\displaystyle=\frac{\kappa+\kappa^{2}-\kappa\lambda+2\lambda^{2}}{(1+\kappa)(\kappa+\lambda^{2})}\cos\left(\mu
t/\tau\right)e^{-(1+\kappa)|t|/\tau}$
$\displaystyle-\frac{\kappa(1-\kappa+\lambda)}{(\kappa+\lambda^{2})\mu}\sin\left(\mu|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(28) $\displaystyle\phi_{\rm{BB}}^{\rm s}(t)$
$\displaystyle=\frac{1+\kappa+\lambda+2\lambda^{2}}{(1+\kappa)(\kappa+\lambda^{2})}\cos\left(\mu
t/\tau\right)e^{-(1+\kappa)|t|/\tau}$
$\displaystyle+\frac{1-\kappa+\lambda}{(\kappa+\lambda^{2})\mu}\sin\left(\mu|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(29) $\displaystyle\phi_{\rm{AB}}^{\rm s}(t)$
$\displaystyle=-\frac{(1-\kappa)\lambda+\lambda^{2}}{(1+\kappa)(\kappa+\lambda^{2})}\cos\left(\mu
t/\tau\right)e^{-(1+\kappa)|t|/\tau}$
$\displaystyle+\frac{\kappa-(1-\kappa)\lambda}{(\kappa+\lambda^{2})\mu}\sin\left(\mu|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(30) $\displaystyle\phi_{\rm{AB}}^{\rm a}(t)$
$\displaystyle=-\frac{3\lambda}{(1+\kappa)\mu}\sin\left(\mu
t/\tau\right)e^{-(1+\kappa)|t|/\tau},$ (31)

where we have used the notation
$\mu=\sqrt{3\lambda^{2}-(1-\kappa+\kappa^{2})}$. When
$\lambda^{2}<(1-\kappa+\kappa^{2})/3$, $\mu$ is purely imaginary so that
“$\cos$” and “$\sin$” functions should be regarded as “$\cosh$” and “$\sinh$”
functions, respectively. The equal-time correlations in Eqs. (16)-(18) can be
recovered by setting $t=0$ in the above equations.

As mentioned before, the auto-correlations have only the symmetric parts
$\phi_{\rm{AA}}^{\rm s}$ and $\phi_{\rm{BB}}^{\rm s}$, whereas the cross-
correlation contains both the symmetric part $\phi_{\rm{AB}}^{\rm s}$ and the
antisymmetric part $\phi_{\rm{AB}}^{\rm a}$. However, $\phi_{\rm{AB}}^{\rm a}$
exists only for nonzero $\lambda$ and measures the degree of the broken time-
reversal symmetry. Obviously, the symmetric parts are even functions in time,
and the antisymmetric part is an odd function in time. In Fig. 2, we plot (a)
$\phi_{\rm{AA}}^{\rm s}$, (b) $\phi_{\rm{BB}}^{\rm s}$, (c)
$\phi_{\rm{AB}}^{\rm s}$, and (d) $\phi_{\rm{AB}}^{\rm a}$ as a function of
$t/\tau$ when $\kappa=1$ and $\lambda=0,0.25,0.5$ (when $\mu$ is imaginary).
In Fig. 3, we plot the same functions when $\kappa=1$ and $\lambda=1,2,3$
(when $\mu$ is real). In Figs. 2(a) and (b), the auto-correlation functions
decay monotonically, whereas the symmetric part of the cross-correlation
function exhibits maximum values and even takes negative values in Figs. 2(c).
As we increase the odd elasticity $\lambda$ in Fig. 3, the oscillatory
behaviors of the time-correlation functions become pronounced.

The velocity-velocity correlation function matrix $\psi_{\alpha\beta}(t)$ can
be obtained by taking the second time derivative of the extension-extension
correlation functions as DoiBook

$\displaystyle\langle\dot{u}_{\alpha}(t)\dot{u}_{\beta}(0)\rangle=-\frac{\partial^{2}}{\partial
t^{2}}\langle
u_{\alpha}(t)u_{\beta}(0)\rangle=\frac{D}{\tau}\psi_{\alpha\beta}(t),$ (32)

where we have introduced the diffusion coefficient of a sphere $D=k_{\rm
B}T/(6\pi\eta a)$. Alternatively, one can also use Eq. (8) to calculate
$\langle\dot{u}_{\alpha}(t)\dot{u}_{\beta}(0)\rangle$ directly, as shown in
Appendix A. Similar to Eq. (27), $\psi_{\alpha\beta}(t)$ can be decomposed
into the symmetric and antisymmetric parts as
$\psi_{\alpha\beta}(t)=\psi_{\alpha\beta}^{\mathrm{s}}(t)+\psi_{\alpha\beta}^{\mathrm{a}}(t)$.
Following the same argument as before, we have $\psi_{\rm{AA}}^{\rm
a}=\psi_{\rm{BB}}^{\rm a}=0$ for the auto-correlation functions. However, the
antisymmetric part of the cross-correlation $\psi_{\rm{AB}}^{\rm a}$ exists
when $\lambda$ is nonzero.

The explicit expressions of the velocity-velocity correlation functions and
their plots are given in Appendix A. We note that the symmetric part of the
velocity-velocity correlation functions has a sharp peak at $t=0$, which
arises from the noise term in the Langevin equation. For very short time,
$t\ll\tau$, the velocity-velocity correlation functions are approximated as
DoiBook

$\displaystyle\langle\dot{u}_{\rm A}(t)\dot{u}_{\rm A}(0)\rangle$
$\displaystyle=\langle\dot{u}_{\rm B}(t)\dot{u}_{\rm B}(0)\rangle\approx
4D\delta(t),$ (33) $\displaystyle\langle\dot{u}_{\rm A}(t)\dot{u}_{\rm
B}(0)\rangle$ $\displaystyle=\langle\dot{u}_{\rm B}(t)\dot{u}_{\rm
A}(0)\rangle\approx-2D\delta(t).$ (34)

###  II.5 Green-Kubo relations

At equilibrium, the time integral of the velocity-velocity correlation
function for a free particle gives the diffusion coefficient, known as one of
the Green-Kubo relations Zwanzig . Recently, Han et al. argued that an
equilibrium-like Green-Kubo relation holds near the steady state of an
isotropic active fluid if the activated and fluctuating degrees of freedom are
statistically decoupled Han21 . Epstein and Mandadapu obtained the Green-Kubo
relation for the odd viscosity Avron98 , which explicitly violates the time-
reversal symmetry at the level of the steady state stress fluctuations
Epstein20 ; Hargus20 .

For the odd micromachine, we perform the time integral of the velocity-
velocity correlation functions given in Appendix A. From the auto-correlation
functions in Eqs. (62) and (63), we obtain

$\displaystyle\int_{0}^{\infty}dt\,\langle\dot{u}_{\rm A}(t)\dot{u}_{\rm
A}(0)\rangle=\int_{0}^{\infty}dt\,\langle\dot{u}_{\rm B}(t)\dot{u}_{\rm
B}(0)\rangle=0.$ (35)

Here, the time integral of Eq. (33) gives $2D$ which is canceled by the rest
of the terms in Eqs. (62) and (63). The result that the integrals in Eq. (35)
vanish is reasonable because the particles are connected by the springs
DoiBook .

For the cross-correlation functions in Eqs. (64)-(65), on the other hand, one
obtains

$\displaystyle\int_{0}^{\infty}dt\,\left[\langle\dot{u}_{\rm A}(t)\dot{u}_{\rm
B}(0)\rangle-\langle\dot{u}_{\rm B}(t)\dot{u}_{\rm
A}(0)\rangle\right]=-\frac{6\lambda D}{1+\kappa}.$ (36)

In this calculation, only the antisymmetric part of the correlation function
$\psi_{\rm AB}^{\mathrm{a}}$ remains. Equation (36) explicitly demonstrates
that the broken time-reversal symmetry is caused by the finite odd elasticity
$\lambda$. The above result also suggests that one can estimate the odd
elasticity of a micromachine by measuring the cross-correlation function
KobayashiODD23 .

##  III Thermal three-sphere micromachine

###  III.1 Model

In this section, we discuss another type of stochastic micromachine that is
also purely driven by thermal fluctuation but without any explicit odd
elasticity. Here, the key assumption is that the three spheres are in
equilibrium with independent heat baths having different temperatures Hosaka17
; Sou19 ; Sou21 . In this micromachine, the heat transfer occurs from a hotter
sphere to a colder one, driving the whole system out of equilibrium.

As shown in Fig. 1(b), we consider a three-sphere micromachine in which the
three spheres are in equilibrium with independent heat baths having different
temperatures $T_{i}$ ($i=1,2,3$) Hosaka17 ; Sou19 ; Sou21 . On the other hand,
the two springs have only even elasticity characterized by $K_{\rm A}^{\rm e}$
and $K_{\rm B}^{\rm e}$, whereas an explicit odd elastic constant does not
exist. The equation of motion of each sphere is the same as in Eq. (4),
although the thermal noise $\xi_{i}$ has different statistical properties. For
the thermal micromachine, we assume that Gaussian white-noise sources
$\xi_{i}$ have zero mean $\langle\xi_{i}(t)\rangle_{\rm t}=0$, and their
correlations satisfy

$\displaystyle\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle_{\rm
t}=2D_{ij}\delta(t-t^{\prime}).$ (37)

Here, $D_{ij}$ is the mutual diffusion coefficient matrix given by

$\displaystyle D_{ij}=\begin{cases}k_{\rm B}T_{i}/(6\pi\eta
a)&(i=j),\vspace{1ex}\\\ k_{\rm
B}\Theta(T_{i},T_{j})/(4\pi\eta|x_{i}-x_{j}|)&(i\neq j),\end{cases}$ (38)

where $\Theta(T_{i},T_{j})$ is a function of $T_{i}$ and $T_{j}$. The
effective temperature $\Theta$ can be the mobility-weighted average Grosberg15
, which in the present case is simply given by
$\Theta(T_{i},T_{j})=(T_{i}+T_{j})/2$ because all the spheres have the same
size. However, its explicit functional form is not needed here, and we only
require that $\Theta$ should satisfy an appropriate fluctuation-dissipation
theorem in thermal equilibrium. Such a simplification is justified because we
only consider the limit of $a\ll\ell$.

###  III.2 Average velocity

The average velocity of the thermal micromachine can be obtained similarly as
before by using Eq. (7) with $\lambda=0$ because odd elasticity does not exist
in this model. In our previous paper, the average velocity was obtained as
Hosaka17

$\displaystyle\langle V\rangle_{\rm t}=\frac{k_{\rm
B}T_{2}\left[(2-5\kappa)\theta_{1}-7(1-\kappa)+(5-2\kappa)\theta_{3}\right]}{144\pi\eta\ell^{2}(1+\kappa)},$
(39)

where $\kappa=K_{\rm B}^{\rm e}/K_{\rm A}^{\rm e}$ as before and we have
defined the temperature ratios by $\theta_{1}=T_{1}/T_{2}$ and
$\theta_{3}=T_{3}/T_{2}$. When the three temperatures are identical,
$T_{1}=T_{2}=T_{3}$ or $\theta_{1}=\theta_{3}=1$, the velocity vanishes
identically, $\langle V\rangle_{\rm t}=0$. This means that the temperature
difference among the spheres causes the locomotion of the thermal three-sphere
micromachine.

When the two springs are equivalent and $\kappa=1$, $\langle V\rangle_{\rm t}$
does not depend on $T_{2}$ and is proportional to the temperature difference
$T_{3}-T_{1}$. In this case, the average velocity vanishes when $T_{1}=T_{3}$
even though $T_{1}$ and $T_{3}$ are different from $T_{2}$.

###  III.3 Entropy production rate

The average entropy production rate can also be calculated similarly to the
previous section. For the thermal micromachine, the lowest-order average heat
flows were obtained as Hosaka17

$\displaystyle\langle\dot{Q}_{1}\rangle_{\rm t}$ $\displaystyle=\frac{k_{\rm
B}T_{2}[(3+2\kappa)\theta_{1}-(3+\kappa)-\kappa\theta_{3}]}{6\tau(1+\kappa)},$
(40) $\displaystyle\langle\dot{Q}_{2}\rangle_{\rm t}$
$\displaystyle=\frac{k_{\rm
B}T_{2}[-(3+\kappa)\theta_{1}+(3+2\kappa+3\kappa^{2})-(\kappa+3\kappa^{2})\theta_{3}]}{6\tau(1+\kappa)},$
(41) $\displaystyle\langle\dot{Q}_{3}\rangle_{\rm t}$
$\displaystyle=\frac{k_{\rm
B}T_{2}[-\kappa\theta_{1}-(\kappa+3\kappa^{2})+(2\kappa+3\kappa^{2})\theta_{3}]}{6\tau(1+\kappa)},$
(42)

which all vanish when $T_{1}=T_{2}=T_{3}$ or $\theta_{1}=\theta_{3}=1$. It is
worth to note that the above heat flows satisfy
$\langle\dot{Q}_{1}\rangle_{\rm t}+\langle\dot{Q}_{2}\rangle_{\rm
t}+\langle\dot{Q}_{3}\rangle_{\rm t}=0$ so that the total heat is conserved.

Using these results and keeping in mind the different temperatures of the
spheres, the steady state average entropy production rate is given by

$\displaystyle\langle\dot{\sigma}\rangle_{\rm t}=$
$\displaystyle-\sum_{i=1}^{3}\frac{\langle\dot{Q}_{i}\rangle_{\rm t}}{T_{i}}$
$\displaystyle=$ $\displaystyle\frac{k_{\rm
B}}{6\tau(1+\kappa)}\Biggl{[}(3+\kappa)\left(\theta_{1}+\frac{1}{\theta_{1}}\right)+\kappa\left(\frac{\theta_{1}}{\theta_{3}}+\frac{\theta_{3}}{\theta_{1}}\right)$
$\displaystyle+\kappa(1+3\kappa)\left(\theta_{3}+\frac{1}{\theta_{3}}\right)-6(1+\kappa+\kappa^{2})\Biggr{]}.$
(43)

One can easily confirm the second law, $\langle\dot{\sigma}\rangle_{\rm t}\geq
0$, by using the inequality $A+B\geq 2\sqrt{AB}$ for $A,B>0$ and
$\langle\dot{\sigma}\rangle_{\rm t}$ vanishes when $\theta_{1}=\theta_{3}=1$.
The above entropy production rate reduces to that in Ref. Sou21  when
$\kappa=1$.

Figure 4: Plots of the scaled extension-extension correlation functions (a)
$\Phi_{\rm{AA}}^{\rm s}$, (b) $\Phi_{\rm{BB}}^{\rm s}$, (c)
$\Phi_{\rm{AB}}^{\rm s}$, and (d) $\Phi_{\rm{AB}}^{\rm a}$ as a function of
$t/\tau$ for $\kappa=1$. The dimensionless temperature ratios between the
spheres are chosen as $(\theta_{1},\theta_{3})=(1,1)$ (black), $(2,3)$ (red),
and $(3,5)$ (green).

###  III.4 Time-correlation functions

Next, we calculate the extension-extension time-correlation function matrix
$\Phi_{\alpha\beta}$ for the thermal micromachine and decompose it into the
symmetric and antisymmetric parts as

$\displaystyle\langle u_{\alpha}(t)u_{\beta}(0)\rangle_{\rm t}=\frac{k_{\rm
B}T_{2}}{K_{\rm A}^{\rm e}}\Phi_{\alpha\beta}(t)=\frac{k_{\rm B}T_{2}}{K_{\rm
A}^{\rm
e}}\left[\Phi_{\alpha\beta}^{\mathrm{s}}(t)+\Phi_{\alpha\beta}^{\mathrm{a}}(t)\right].$
(44)

The time-translational invariance requiers $\Phi_{\rm{AA}}^{\rm
a}=\Phi_{\rm{BB}}^{\rm a}=0$ as before. After repeating similar calculations,
we obtain

$\displaystyle\Phi_{\rm AA}^{\mathrm{s}}(t)$
$\displaystyle=\frac{(3+4\kappa)\theta_{1}+(3+\kappa)+\kappa\theta_{3}}{6(1+\kappa)}\cosh\left(\rho
t/\tau\right)e^{-(1+\kappa)|t|/\tau}$
$\displaystyle+\frac{(-3+4\kappa)\theta_{1}-(3-\kappa)+\kappa\theta_{3}}{6\rho}\sinh\left(\rho|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(45) $\displaystyle\Phi_{\rm BB}^{\mathrm{s}}(t)$
$\displaystyle=\frac{\theta_{1}+(1+3\kappa)+(4+3\kappa)\theta_{3}}{6\kappa(1+\kappa)}\cosh\left(\rho
t/\tau\right)e^{-(1+\kappa)|t|/\tau}$
$\displaystyle+\frac{\theta_{1}+(1-3\kappa)+(4-3\kappa)\theta_{3}}{6\kappa\rho}\sinh\left(\rho|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(46) $\displaystyle\Phi_{\rm{AB}}^{\rm s}(t)$
$\displaystyle=\frac{\theta_{1}-2+\theta_{3}}{3(1+\kappa)}\cosh\left(\rho
t/\tau\right)e^{-(1+\kappa)|t|/\tau}$
$\displaystyle+\frac{\theta_{1}+1+\theta_{3}}{3\rho}\sinh\left(\rho|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(47) $\displaystyle\Phi_{\rm{AB}}^{\rm a}(t)$
$\displaystyle=\frac{-\theta_{1}+(1-\kappa)+\kappa\theta_{3}}{2(1+\kappa)\rho}\sinh\left(\rho
t/\tau\right)e^{-(1+\kappa)|t|/\tau},$ (48)

where we have used the notation $\rho=\sqrt{1-\kappa+\kappa^{2}}$. Since
$1-\kappa+\kappa^{2}>0$, $\rho$ is real for any $\kappa$, and all the above
time-correlation functions do not oscillate in time. In Eq. (48), we see that
$\Phi_{\rm{AB}}^{\rm a}(t)$ vanishes when $\theta_{1}=\theta_{3}=1$.

Next, we calculate the velocity-velocity correlation function matrix
$\Psi_{\alpha\beta}(t)$ for the thermal micromachine as

$\displaystyle\langle\dot{u}_{\alpha}(t)\dot{u}_{\beta}(0)\rangle_{\rm
t}=\frac{D_{22}}{\tau}\Psi_{\alpha\beta}(t)=\frac{D_{22}}{\tau}\left[\Psi_{\alpha\beta}^{\mathrm{s}}(t)+\Psi_{\alpha\beta}^{\mathrm{a}}(t)\right],$
(49)

where $D_{22}=k_{\rm B}T_{2}/(6\pi\eta a)$ is the diffusion coefficient
defined by using the temperature $T_{2}$ [see Eq. (38)]. The explicit
expressions of the velocity-velocity correlation functions and their plots are
given in Appendix B. Similar to the odd micromachine, the symmetric parts of
the velocity-velocity correlation functions have a sharp peak at $t=0$. In the
short time limit, $t\ll\tau$, the velocity-velocity correlation functions
become

$\displaystyle\langle\dot{u}_{\rm A}(t)\dot{u}_{\rm A}(0)\rangle_{\rm t}$
$\displaystyle\approx 2(\theta_{1}+1)D_{22}\delta(t),$ (50)
$\displaystyle\langle\dot{u}_{\rm B}(t)\dot{u}_{\rm B}(0)\rangle_{\rm t}$
$\displaystyle\approx 2(1+\theta_{3})D_{22}\delta(t),$ (51)
$\displaystyle\langle\dot{u}_{\rm A}(t)\dot{u}_{\rm B}(0)\rangle_{\rm t}$
$\displaystyle=\langle\dot{u}_{\rm B}(t)\dot{u}_{\rm A}(0)\rangle_{\rm
t}\approx-2D_{22}\delta(t).$ (52)

These results coincide with Eqs. (33) and (34) when $\theta_{1}=\theta_{3}=1$
and if $D_{22}$ can be identified with $D$.

###  III.5 Green-Kubo relations

Having obtained the velocity-velocity correlation functions for the thermal
micromachine, we now calculate the Green-Kubo relations as in the previous
section. From the auto-correlation functions in Eqs. (66) and (67), we have

$\displaystyle\int_{0}^{\infty}dt\,\langle\dot{u}_{\rm A}(t)\dot{u}_{\rm
A}(0)\rangle_{\rm t}=\int_{0}^{\infty}dt\,\langle\dot{u}_{\rm
B}(t)\dot{u}_{\rm B}(0)\rangle_{\rm t}=0,$ (53)

similar to Eq. (35). From the cross-correlation functions in Eqs. (68) and
(69), however, we obtain

$\displaystyle\int_{0}^{\infty}dt\,\left[\langle\dot{u}_{\rm A}(t)\dot{u}_{\rm
B}(0)\rangle_{\rm t}-\langle\dot{u}_{\rm B}(t)\dot{u}_{\rm A}(0)\rangle_{\rm
t}\right]$
$\displaystyle=-\frac{[\theta_{1}-(1-\kappa)-\kappa\theta_{3}]D_{22}}{1+\kappa}.$
(54)

This result implies that the time-reversal symmetry is broken for the thermal
microswimmer when the temperatures are different.

##  IV Effective odd elasticity of the thermal micromachine

So far, we have discussed the statistical properties of the odd and thermal
micromachines, and obtained their time-correlation functions to discuss the
Green-Kubo relations. Although these two stochastic models are different, it
is worthwhile to consider the effective odd elasticity of the thermal
micromachine. As we mentioned earlier, odd elasticity gives a quantitative
measure of the work that can be extracted from active systems Scheibner20 ;
Fruchart23 , and nonequilibrium cyclic dynamics of micromachines can be
quantified by effective odd elasticity Yasuda21catalytic ; YasudaOM22 . Such
an approach is possible for stochastic micromachines by looking at the
violation of the time-reversal symmetry of the cross-correlation functions
YasudaTCF22 ; KobayashiODD23 .

For this purpose, we compare the Green-Kubo relations in Eqs. (36) and (54),
and define an effective odd elasticity $\lambda_{\rm t}$ ratio of the thermal
micromachine as

$\displaystyle\lambda_{\rm
t}=\frac{D_{22}}{6D}[\theta_{1}-(1-\kappa)-\kappa\theta_{3}].$ (55)

If we further assume $D_{22}=D$ and $\kappa=1$, we obtain

$\displaystyle\lambda_{\rm t}=\frac{T_{1}-T_{3}}{6T_{2}}.$ (56)

This result clearly demonstrates that the effective odd elasticity of the
thermal micromachine is purely determined by the temperature difference
between the edge spheres. The above relations further indicate that
$\lambda_{\rm t}$ is inversely proportional to the middle sphere temperature
$T_{2}$.

It should be mentioned, however, that the above argument is not the only way
to obtain the effective odd elasticity of the thermal micromachine. For
example, one can also compare the average velocities of the two micromachines
as given in Eqs. (19) and (39) or the entropy production rates in Eqs. (25)
and (43). Although the comparison of different quantities leads to a slightly
modified numerical factor, the effective odd elasticity is always proportional
to the temperature difference among the spheres, as shown in Eq. (56). This is
because both the odd elasticity and the temperature difference lead to the
violation of the time-reversal symmetry. One of the advantages of using the
Green-Kubo relations to define $\lambda_{\rm t}$ is that they contain
information on the cross-correlation functions over the whole time.

##  V Summary and discussion

In this paper, we have calculated the time-correlation functions of the two
different stochastic three-sphere micromachines, i,e., the odd micromachine
and the thermal micromachine. For both models, the cross-correlation functions
of the two springs contain the antisymmetric part, which is a direct
consequence of the broken time-reversal symmetry of the micromachines. For the
odd micromachine, the antisymmetric part of the correlation function is
proportional to the odd elastic constant [see Eq. (31)], whereas it is
proportional to the temperature difference among the spheres [see Eq. (48)].
We have also obtained the entropy production rate and the Green-Kubo relations
for the two micromachines. A comparison of these results allows us to discuss
the effective odd elasticity of the thermal micromachine [see Eq. (55)], even
though the odd elasticity is not explicitly included in the model. For the
thermal micromachine, the effective odd elasticity is proportional to the
temperature difference between the spheres, which causes an internal heat flow
and leads to directional locomotion.

Evaluation of effective odd elasticity was previously discussed for a model
micromachine driven by catalytic chemical reactions KobayashiODD23 . We
calculated the time-correlation functions of the structural variables and
analyzed them in terms of Langevin dynamics with effective odd elasticity
YasudaTCF22 . It was also shown that the odd elasticity is directly related to
the quantity called nonreciprocality of a micromachine. For a deterministic
micromachine undergoing cyclic motions, the nonreciprocality $R$ is defined by
Yasuda21

$\displaystyle R=\frac{1}{2}\oint dt\,(u_{\rm A}\dot{u}_{\rm B}-\dot{u}_{\rm
A}u_{\rm B}),$ (57)

where the integral is taken over one cycle, and $R$ represents the area
enclosed by the trajectory in the configuration space Shapere89 . For
stochastic micromachines, the average nonreciprocality $\langle R\rangle$ is
essentially given by the time derivative of the antisymmetric part of the
extension-extension correlation function, such as given by Eq. (31) that is
proportional to the odd elasticity. Hence, the effective odd elasticity
introduced in Eq. (56) directly characterizes the nonreciprocality of the
thermal micromachines.

Recently, Hargus et al. discussed odd diffusivity in two-dimensional chiral
active matter and derived the Green-Kubo relation for the odd diffusion
coefficient Hargus20prl . A similar relation was also obtained for the odd
mobility of a passive tracer in a chiral active fluid Poggioli23 . In their
results, both odd diffusivity and odd mobility result from the broken time-
reversal symmetry in active systems and are related to the antisymmetric part
of the cross-correlation functions, which is similar to our result. In our
stochastic micromachines, the integrals of the auto-correlations vanish as in
Eqs. (35) and (53) because the spheres are connected by the springs. Although
the motions of the micromachines are one-dimensional in our models, the
configurational space spanned by $u_{\rm A}$ and $u_{\rm B}$ are two-
dimensional. It is useful to consider an effective odd elasticity in such a
configurational space, and the obtained Green-Kubo relations for the cross-
correlation functions in Eqs. (36) and (54) are directly proportional to the
odd elasticity. We also note that the oscillatory trajectory seen in the
chiral random walk Hargus20prl  is analogous to the oscillatory behaviors of
the time-correlation functions in this paper due to the odd elasticity.

In this work, the elastic constant matrix for the odd micromachine was given
by Eq. (3) in which we took into account the difference between $K_{\rm
A}^{\rm e}$ and $K_{\rm B}^{\rm e}$. However, the most general form of the
elastic constant matrix can take the form

$\displaystyle\mathbf{K}=\left(\begin{array}[]{cc}K_{\rm A}^{\rm e}&K^{\rm
e}_{\rm AB}+K^{\rm o}\vspace{1ex}\\\ K^{\rm e}_{\rm AB}-K^{\rm o}&K_{\rm
B}^{\rm e}\\\ \end{array}\right),$ (60)

where we have introduced another even elastic constant $K^{\rm e}_{\rm AB}$ in
the off-diagonal components KobayashiODD23 . Such an even elasticity can
induce amplitude asymmetry between the two springs Lin23 . Although the time-
correlation functions can be easily calculated in the presence of $K^{\rm
e}_{\rm AB}$, Eq. (3) is sufficient to compare with the thermal micromachine.

To calculate the time-correlation functions, we have neglected hydrodynamic
interactions acting between the spheres. This is justified because
hydrodynamic interactions are higher-order contributions for the correlation
functions, which will be calculated in our future work. It should be mentioned
again, however, that hydrodynamic interactions are necessary for the
locomotion of the three-sphere micromachines.

###### Acknowledgements.

We thank R. García-Millán, L.-S. Lin, and Z. Xiong for the useful discussion.
Z.H. L.H. and S.K. acknowledge the support by the National Natural Science
Foundation of China (Nos. 12104453, 22273067, 12274098, and 12250710127). K.Y.
acknowledges the support by a Grant-in-Aid for JSPS Fellows (Grants No.
22KJ1640) from the JSPS. S.K. acknowledges the startup grant of Wenzhou
Institute, University of Chinese Academy of Sciences (No. WIUCASQD2021041).
K.Y and S.K. acknowledge the support by the Japan Society for the Promotion of
Science (JSPS) Core-to-Core Program “Advanced core-to-core network for the
physics of self-organizing active matter” (No. JPJSCCA20230002). J.L. and Z.Z.
contributed equally to this work.

##  Appendix A Velocity-velocity correlation functions of the odd three-sphere
micromachine

As mentioned in the text, the velocity-velocity correlation functions can be
calculated by using Eq. (8) as

$\displaystyle\langle\dot{u}_{\alpha}(t)\dot{u}_{\beta}(0)\rangle$
$\displaystyle=\langle[\Gamma_{\alpha\gamma}u_{\gamma}(t)+\Xi_{\alpha}(t)][\Gamma_{\beta\delta}u_{\delta}(0)+\Xi_{\beta}(0)]\rangle.$
(61)

The correlations between the displacement and noise, $\langle
u_{\alpha}(t)\Xi_{\beta}(0)\rangle$, can also be calculated straightforwardly.

For the odd micromachine, the expressions of the velocity-velocity correlation
functions defined in Eq. (32) are given as follows:

$\displaystyle\psi_{\rm{AA}}^{\rm s}(t)$
$\displaystyle=4\tau\delta(t)+\frac{6\lambda^{2}-3\kappa\lambda-(4+5\kappa+\kappa^{2})}{1+\kappa}\cos\left(\mu
t/\tau\right)e^{-(1+\kappa)|t|/\tau}-\frac{12\lambda^{2}-3\kappa\lambda-(4-\kappa+\kappa^{2})}{\mu}\sin\left(\mu|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(62) $\displaystyle\psi_{\rm{BB}}^{\rm s}(t)$
$\displaystyle=4\tau\delta(t)+\frac{6\lambda^{2}+3\lambda-(1+5\kappa+4\kappa^{2})}{1+\kappa}\cos\left(\mu
t/\tau\right)e^{-(1+\kappa)|t|/\tau}-\frac{12\lambda^{2}+3\lambda-(1-\kappa+4\kappa^{2})}{\mu}\sin\left(\mu|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(63) $\displaystyle\psi_{\rm{AB}}^{\rm s}(t)$ $\displaystyle=-2\tau\delta(t)$
$\displaystyle-\frac{3\lambda^{2}+3(1-\kappa)\lambda-2(1+2\kappa+\kappa^{2})}{1+\kappa}\cos\left(\mu
t/\tau\right)e^{-(1+\kappa)|t|/\tau}+\frac{6\lambda^{2}+3(1-\kappa)\lambda-(2+\kappa+2\kappa^{2})}{\mu}\sin\left(\mu|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(64) $\displaystyle\psi_{\rm{AB}}^{\rm a}(t)$
$\displaystyle=\begin{cases}-6\lambda\cos\left(\mu
t/\tau\right)e^{-(1+\kappa)t/\tau}-\dfrac{3\lambda[3\lambda^{2}-(2+\kappa+2\kappa^{2})]}{(1+\kappa)\mu}\sin\left(\mu
t/\tau\right)e^{-(1+\kappa)t/\tau}&(t>0),\vspace{1ex}\\\ 6\lambda\cos\left(\mu
t/\tau\right)e^{(1+\kappa)t/\tau}-\dfrac{3\lambda[3\lambda^{2}-(2+\kappa+2\kappa^{2})]}{(1+\kappa)\mu}\sin\left(\mu
t/\tau\right)e^{(1+\kappa)t/\tau}&(t<0),\end{cases}$ (65)

where $\mu=\sqrt{3\lambda^{2}-(1-\kappa+\kappa^{2})}$.

##  Appendix B Velocity-velocity correlation functions of the thermal three-
sphere micromachine

Figure 5: Plots of the scaled velocity-velocity correlation functions (a)
$\psi_{\rm{AA}}^{\rm s}$, (b) $\psi_{\rm{BB}}^{\rm s}$, (c)
$\psi_{\rm{AB}}^{\rm s}$, and (d) $\psi_{\rm{AB}}^{\rm a}$ as a function of
$t/\tau$ for $\kappa=1$. The dimensionless odd elasticity is chosen as
$\lambda=0$ (black), $1$ (red), and $2$ (green).  Figure 6: Plots of the
scaled velocity-velocity correlation functions (a) $\Psi_{\rm{AA}}^{\rm s}$,
(b) $\Psi_{\rm{BB}}^{\rm s}$, (c) $\Psi_{\rm{AB}}^{\rm s}$, and (d)
$\Psi_{\rm{AB}}^{\rm a}$ as a function of $t/\tau$ for $\kappa=1$. The
dimensionless temperature ratios between the spheres are chosen as
$(\theta_{1},\theta_{3})=(1,1)$ (black), $(2,3)$ (red), and $(3,5)$ (green).

For the thermal micromachine, the expressions of the velocity-velocity
correlation functions defined in Eq. (49) are given as follows:

$\displaystyle\Psi_{\rm{AA}}^{\rm s}(t)$
$\displaystyle=2(\theta_{1}+1)\tau\delta(t)$
$\displaystyle-\frac{(4+5\kappa)\theta_{1}+(4+5\kappa+3\kappa^{2})-\kappa^{2}\theta_{3}}{2(1+\kappa)}\cosh\left(\rho
t/\tau\right)e^{-(1+\kappa)|t|/\tau}-\frac{(\kappa-4)\theta_{1}-(4-\kappa+3\kappa^{2})+\kappa^{2}\theta_{3}}{2\rho}\sinh\left(\rho|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(66) $\displaystyle\Psi_{\rm{BB}}^{\rm s}(t)$
$\displaystyle=2(1+\theta_{3})\tau\delta(t)$
$\displaystyle-\frac{-\theta_{1}+(3+5\kappa+4\kappa^{2})+\kappa(5+4\kappa)\theta_{3}}{2(1+\kappa)}\cosh\left(\rho
t/\tau\right)e^{-(1+\kappa)|t|/\tau}-\frac{\theta_{1}-(3-\kappa+4\kappa^{2})+\kappa(1-4\kappa)\theta_{3}}{2\rho}\sinh\left(\rho|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(67) $\displaystyle\Psi^{\rm s}_{\rm{AB}}(t)$
$\displaystyle=-2\tau\delta(t)+\frac{\kappa\theta_{1}+2(1+\kappa+\kappa^{2})+\kappa\theta_{3}}{1+\kappa}\cosh\left(\rho
t/\tau\right)e^{-(1+\kappa)|t|/\tau}-\frac{\kappa\theta_{1}+(2-\kappa+2\kappa^{2})+\kappa\theta_{3}}{\rho}\sinh\left(\rho|t|/\tau\right)e^{-(1+\kappa)|t|/\tau},$
(68) $\displaystyle\Psi^{\rm a}_{\rm{AB}}(t)$
$\displaystyle=\begin{cases}-[\theta_{1}-(1-\kappa)-\kappa\theta_{3}]\cosh\left(\rho
t/\tau\right)e^{-(1+\kappa)t/\tau}+\dfrac{(2+\kappa+2\kappa^{2})[\theta_{1}-(1-\kappa)-\kappa\theta_{3}]}{2(1+\kappa)\rho}\sinh\left(\rho
t/\tau\right)e^{-(1+\kappa)t/\tau}&(t>0),\vspace{1ex}\\\
[\theta_{1}-(1-\kappa)-\kappa\theta_{3}]\cosh\left(\rho
t/\tau\right)e^{(1+\kappa)t/\tau}+\dfrac{(2+\kappa+2\kappa^{2})[\theta_{1}-(1-\kappa)-\kappa\theta_{3}]}{2(1+\kappa)\rho}\sinh\left(\rho
t/\tau\right)e^{(1+\kappa)t/\tau}&(t<0),\end{cases}$ (69)

where $\rho=\sqrt{1-\kappa+\kappa^{2}}$.

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