J. Phys. Soc. Jpn.
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Short Notes

Journal of the Physical Society of Japan 90, 075001 (2021)
https://doi.org/10.7566/JPSJ.90.075001

Odd Microswimmer
Kento Yasuda+, Yuto Hosaka,
Isamu Sou, and Shigeyuki Komura

Fig. 1. (Color online) Odd microswimmer in a ﬂuid with a viscosity η andtemperature T. Three spheres of radius a are connected by two springs with anatural length ‘. Each spring has both even elastic constant K e and odd elastic
constant K o . The positions of the spheres are denoted as xi (i ¼ 1; 2; 3), andthe spring extensions with respect to ‘ are denoted as uA and uB .

Department of Chemistry, Graduate School of Science,
Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan(Received March 25, 2021; accepted May 7, 2021;
published online June 7, 2021)
We propose a model for a thermally driven microswimmer in which threespheres are connected by two springs with odd elasticity. We demonstratethat the presence of odd elasticity leads to the directional locomotion of thestochastic microswimmer.

Although micromachines such as proteins and enzymes
experience the inﬂuence of strong thermal ﬂuctuations, theyoften exhibit directional locomotion under nonequilibrium
conditions.1) To elucidate this type of phenomena, wepreviously proposed a thermally driven elastic microswimmer
consisting of three spheres.2) In this model, the three spheres
were assumed to be in equilibrium with independent heatbaths characterized by diﬀerent temperatures.
Recently, Scheibner et al. introduced the concept of “oddelasticity”, which can arise from active and nonreciprocalinteractions.3) Importantly, the odd part of the elastic constanttensor quantiﬁes the amount of work extracted along
quasistatic deformation cycles. In this paper, we propose anovel type of thermally driven microswimmer in which thethree spheres are connected with springs having not only evenelasticity,4) but also odd elasticity.3) We explicitly demonstrate that the proposed stochastic “odd microswimmer” canexhibit a directional locomotion as a result of odd elasticity.
Additionally, we provide a simple physical interpretation of
the average velocity within the nonequilibrium statisticalphysics.
Consider a three-sphere microswimmer in which thepositions of the three spheres of radius a are given by xi(i ¼ 1; 2; 3) in a one-dimensional coordinate system (seeFig. 1).5) These three spheres are connected by two springs
that exhibit both even and odd elasticity. We denote the twospring extensions as uA ¼ x2  x1  ‘ and uB ¼ x3  x2  ‘,
where ‘ is the natural length. Then, the forces FA and FB
conjugate to uA and uB , respectively, are given by F ¼K u (;  ¼ A; B). For an odd spring, the elastic constantK is given by3)
K ¼ K e  þ K o  ;ð1Þ

where K e and K o are the even and odd elastic constants,
respectively, in the 2D conﬁguration space spanned by uA anduB (unlike the real 2D space in Ref. 3),  is the Kronecker
delta, and  is the 2D Levi-Civita tensor with AA ¼
BB ¼ 0 and AB ¼ BA ¼ 1. The presence of odd elasticity
K o in Eq. (1) reﬂects the nonreciprocal interaction between
the two springs such that uA and uB inﬂuence each other in adiﬀerent manner.6) The forces fi acting on each sphere are
given by f1 ¼ FA , f2 ¼ FA  FB , and f3 ¼ FB . These forces
satisfy the force-free condition, i.e., f1 þ f2 þ f3 ¼ 0.
The odd microswimmer described above is immersed in aﬂuid with a shear viscosity of η and temperature T. Then theequations of motion for each sphere are given by2,4,5)

ð2Þ
x_i ¼ Mij fj þ i ;where x_i ¼ dxi =dt and Mij are the hydrodynamic mobility
coeﬃcients5)(1=ð6aÞ(i ¼ j):ð3ÞMij ¼
1=ð4jxi  xj jÞ (i ≠ j)
In Eq. (2), the Gaussian white-noise sources i have zeromean hi ðtÞi ¼ 0, and their correlations satisfy the following
ﬂuctuation–dissipation theorem:hi ðtÞj ðt 0 Þi ¼ 2kB TMij ðt  t 0 Þ;

ð4Þ

where kB is the Boltzmann constant.
It is convenient to introduce the characteristic time scale¼ 6a=K e and the ratio between the two spring constants¼ K o =K e . In the following analysis, we assume uA ; uB ‘ and a  ‘, and focus solely on the leading-ordercontribution. The total velocity of the microswimmer isgiven by V ¼ ðx_1 þ x_2 þ x_3 Þ=3. After taking the statistical
average and using Eqs. (1)–(3), we obtain2)
ahVi ¼ 2 ½hu2B i  hu2A i þ ð3hu2B i þ 3hu2A i  2huA uB iÞ8‘ð5Þþ O½ða=‘Þ2 ; ðu=‘Þ3 ;
where we use huA i ¼ huB i ¼ 0.
The equal-time correlation functions appearing in Eq. (5)can be obtained from the reduced Langevin equations foru_A ¼ x_2  x_1 and u_B ¼ x_3  x_2 asu_ ¼  u þ  þ O½a=‘;
where  and  are!
2þ
1 þ 21
¼;
1  22

¼

ð6Þ
2  13  2!
:

ð7Þ

Notice that  is nonreciprocal, i.e., AB ≠ BA when ≠ 0.
By solving Eq. (6) in the Fourier domain and using Eq. (4),
we obtain the following equal-time correlation functions:2)
kB T2huA i ¼ e 1 þ O½a=‘;ð8Þ
K2ð1 þ 2 ÞkB T
þ O½a=‘;ð9Þhu2B i ¼ e 1 þK2ð1 þ 2 Þ2kB T
þ O½a=‘:ð10Þ
huA uB i ¼  eK 2ð1 þ 2 ÞHere, we neglect the cross-correlations hi j i with i ≠ jbecause they only contribute to higher orders in a=‘. When
¼ 0, the above expressions reduce to hu2A i ¼ hu2B i ¼
kB T=K e and huA uB i ¼ 0, reproducing the thermal equilibrium situation. We have hu2A i < hu2B i when > 0, because
the eﬀective elastic constant of spring A is greater than thatof spring B.

075001-1©2021 The Physical Society of Japan

J. Phys. Soc. Jpn.
Downloaded from journals.jps.jp by 182.168.196.10 on 06/07/21

Short Notes

J. Phys. Soc. Jpn. 90, 075001 (2021)

K. Yasuda et al.

By substituting Eqs. (8)–(10) into Eq. (5), we obtain theaverage velocity as
hVi ¼

7akB Tþ O½ða=‘Þ2 ; ðu=‘Þ3 :
8‘2 K eð11ÞHere, hVi is proportional to the odd elastic constant K o that
can take either positive or negative value. Because hVi is also
proportional to kB T, thermal ﬂuctuations are responsible forthe locomotion of the odd microswimmer. Therefore, ourmodel provides a novel type of Brownian ratchet.
Next, we discuss the nonequilibrium statistical propertiesof the odd microswimmer.7,8) For the time-dependent
probability distribution function pðuA ; uB ; tÞ, the Fokker–Planck equation corresponding to Eq. (6) can be written asp_ ¼ @ j , where @ ¼ @=ð@u Þ and j is the probability ﬂux
given by7)j ¼  u p  D @ p:Here, D is the diﬀusion matrix
D¼kB T
6a2

1

12

ð12Þ!
;

Fig. 2. (Color online) Steady-state scaled probability distribution function
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
p~ ¼ pkB T=K e and steady-state scaled probability ﬂux ~j ¼ j kB T=K e(arrows) in the conﬁguration space spanned by uA and uB when ¼
K o =K e ¼ 1.

ð13Þwhich satisﬁes the relationship h ðtÞ ðt 0 Þi ¼ 2D ðt  t 0 Þ
according to Eq. (4).
Owing to the reproductive property of Gaussian distributions, the steady-state probability distribution function thatsatisﬁes p_ ¼ 0 is given by a Gaussian function7)
11 1pðuA ; uB Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp  ðC Þ u u :ð14Þ22 det C
Here, C ¼ hu u i is the covariance matrix obtained from
Eqs. (8)–(10) as!
1  =2 þ 2
 2 =2
kB T 1C¼ e; ð15Þ
K 1þ 21 þ =2 þ 2
 2 =2and ðC1 Þ is the inverse matrix of C . For our purposes,
we explicitly show thatkB T 2 4 þ 7 2 þ 3 4
det C ¼:
ð16ÞKe4ð1 þ 2 Þ2In Fig. 2, we plot the steady-state probability distributionfunction in Eq. (14) and corresponding probability ﬂux inEq. (12) when ¼ 1. The probability distribution function is
distorted by the negative correlation (CAB ¼ CBA   2 =2)
between uA and uB . One can see a counter-clockwise loopof the probability ﬂux. Such a probability ﬂux becomesclockwise for < 0 and vanishes when ¼ 0. The existenceof a probability ﬂux loop indicates that the detailed balance isbroken in the nonequilibrium steady state.
The steady-state probability ﬂux can be conveniently
expressed in terms of a frequency matrix  asj ¼  u p.7) For the proposed odd microswimmer, thefrequency matrix is given by!
 2
2 þ  2 23
¼; ð17Þ
2ð4 þ 3 2 Þ 2 þ þ 2 2which is traceless. Then, the two eigenvalues of  aregiven bypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
3¼ i
ð18Þ
4 þ 7 2 þ 3 4:2
ð4 þ 3 Þ

Because these eigenvalues are purely imaginary, theprobability current in the conﬁguration space is rotational.
Comparing Eq. (11) with Eqs. (16) and (18), we obtain thefollowing simple expression for the average velocity:
jhVij ¼

7a pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
det Cj j:
12‘2

ð19Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Here, 7a=ð12‘2 Þ is the geometrical factor,5)det C kB T=K e is the explored area in the conﬁguration space, andj j  1 is the speed of the rotational probability ﬂux.7)
Finally, we consider the work that can be extracted when
odd elasticity exsits.3) For the stochastic odd microswimmer,_ ¼ K hu_ u i,the average power
can be evaluated as hWi
R
where W ¼ du F . From Eq. (6), we obtain hu_A uB i ¼
hu_B uA i ¼ 3kB T =ð2K e Þ and hu_A uA i ¼ hu_B uB i ¼ 0. Byusing these results, we can estimate the power of the odd_ ¼ 3kB T 2 = . We have conﬁrmed thatmicroswimmer as hWi
this power coincides with the average entropy production rate
obtained by the expression h _i ¼ Tr½ðCD1 þ IÞ,8)
where I is the identity matrix. Therefore, all the extracted
work due to odd elasticity is converted into the entropy
production. It is also useful to note that the average velocity
can be alternatively written as hVi ¼ 7a=ð12‘2 Þhu_B uA i.
Acknowledgment K.Y. and Y.H. acknowledge support by a Grant-in-Aidfor JSPS Fellows (Grants Nos. 18J21231 and 19J20271) from the JSPS. S.K.
acknowledges support by a Grant-in-Aid for Scientiﬁc Research (C) (GrantsNos. 18K03567 and 19K03765) from the JSPS, and support by a Grant-in-Aid for
Scientiﬁc Research on Innovative Areas “Information Physics of Living Matters”(Grant No. 20H05538) from the MEXT of Japan.
+

Present address: Research Institute for Mathematical Sciences, Kyoto
University, Kyoto 606-8502, Japan1) H. Yuan, X. Liu, L. Wang, and X. Ma, Bioactive Mater. 6, 1727 (2021).
2) Y. Hosaka, K. Yasuda, I. Sou, R. Okamoto, and S. Komura, J. Phys. Soc.
Jpn. 86, 113801 (2017).
3) C. Scheibner, A. Souslov, D. Banerjee, P. Surówka, W. T. M. Irvine, andV. Vitelli, Nat. Phys. 16, 475 (2020).
4) K. Yasuda, Y. Hosaka, M. Kuroda, R. Okamoto, and S. Komura, J. Phys.
Soc. Jpn. 86, 093801 (2017).
5) R. Golestanian and A. Ajdari, Phys. Rev. E 77, 036308 (2008).
6) K. Era, Y. Koyano, Y. Hosaka, K. Yasuda, H. Kitahata, and S. Komura,Europhys. Lett. 133, 34001 (2021).
7) I. Sou, Y. Hosaka, K. Yasuda, and S. Komura, Phys. Rev. E 100, 022607(2019).
8) I. Sou, Y. Hosaka, K. Yasuda, and S. Komura, Physica A 562, 125277(2021).

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