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Oscillatory active nematics represent nonequilibrium suspensions of microscopic objects, such as natural or artificial molecular machines, that cyclically change their shapes and thus operate as oscillating force dipoles. In this mini-review, hydrodynamic collective effects in such active nematics are discussed. Microscopic stirring at low Reynolds numbers induces non-thermal fluctuating flows and passive particles become advected by them. Similar to advection of particles in macroscopic turbulent flows, this enhances diffusion of tracer particles. Furthermore, their drift and accumulation in regions with stronger activity or higher concentration of force dipoles take place. Analytical investigations and numerical simulations both for 2D and 3D systems were performed.

Nematics are formed by elongated molecules. When temperature is decreased, they undergo a phase transition to a liquid-crystalline state where the molecules are orientationally ordered, but the translational order is absent and the system remains fluid.^{1}^{)} For classical nematics, the shapes of the molecules are fixed (or only subject to equilibrium thermal fluctuations). There are however important applications where active nematics with the elements that cyclically change their shape have to be analyzed.

First of all, this situation is encountered when the elements represent molecular machines, either natural or artificial.^{2}^{,}^{3}^{)} The characteristic property of a machine is that, in each operation cycle, it repeatedly changes its conformation, powered by the energy that is externally supplied. Protein machines, acting as motors, enzymes, ion pumps, or performing operations with other biomolecules, constitute a fundamental component of the living cell; typically they are driven by the energy released in the hydrolysis of ATP. On the other hand, it becomes also possible to design and synthesize non-biological molecules that behave as machines, with the energy supplied by illumination or in a chemical way.^{4}^{,}^{5}^{)} Investigations of artificial molecular machines is a rapidly developing field. Molecular machines can be incorporated into liquid crystals (see, e.g., the experiments^{6}^{)}), so that their orientational ordering is induced. Moreover, under crowded conditions that are characteristic for biological cells, they can undergo an orientational ordering transition and produce liquid crystals themselves. Note that, in such a liquid crystal, the molecules are active and cyclically elongate or contract.

A different class of oscillatory active nematics is constituted by bacteria and other microorganisms. It is well known that a biological cell may actively change its shape and this can result in the swimming effect.^{7}^{)} However, one can also imagine microorganisms that do not propel themselves, but just repeatedly change their shapes in a reciprocal way. Again, such microorganisms may undergo an orientational ordering transition, either as a result of crowding or external condition.

Moreover, these active nematics can be manufactured too. Indeed, one can prepare oscillating dumbbells where the distance between two beads is cyclically changed through application of an external periodic field or due to other effects. Such dumbbells can form orientationally ordered states as well.

Whereas the nature of the considered active nematics may vary, there are common properties shared by all of them. From the hydrodynamical point of view, almost any object immersed into a fluid and changing its shape represents a force dipole.^{8}^{)} Thus, one needs to consider hydrodynamic effects in large populations of active force dipoles.

Previously, such hydrodynamical effects have been analyzed assuming that the orientational ordering of active force dipoles was absent.^{9}^{)} It has been shown that, even in this case, diffusion of passive tracer particles becomes enhanced and, if concentration or activity gradients for force dipoles are created, directed drift of tracer particles takes place.^{9}^{,}^{10}^{)} Detailed investigations were performed for two-dimensional systems, such as biomembranes with active protein inclusions,^{11}^{)} and the effects in viscoelastic three-dimensional media were also analyzed.^{12}^{)} In this article, we construct a general theory for hydrodynamic effects of orientationally ordered active force dipoles and illustrate it by numerical simulations for several selected set-ups. We also briefly review the previous results for disordered force dipoles. Our approach is general and therefore the analysis is applicable not only for biological systems, but also for the synthetic and artificial machines.

The system that we consider in this study represents a population of microscopic objects that cyclically change their shape and are immersed into a viscous fluid. We assume that hydrodynamic flows induced by such oscillating objects are characterized by low Reynolds numbers, so that inertia is absent and the Navier–Stokes equations can be linearized. Our aim is to determine the intensity and statistical properties of non-thermal flow velocity fluctuations caused by the active elements.

It is known that, at a distance much longer than its size, any object that asymmetrically changes its shape can generally be described as a force dipole. The force dipole *i* is characterized by its position ^{9}^{)} the magnitude of its force dipole is *F* is the force acting between them. Note that the magnitude *m* of the force dipole can be positive or negative, depending on whether there are repulsive or attractive forces between the beads.

Because inertia is absent on the considered length and time scales, the flows are instantaneously following changes in the force dipoles. The flow velocity *μ* being the fluid viscosity. Note that *μ* is the bulk viscosity for

The expression (1) is invariant with respect to the reflection

To avoid the ambiguity involved in the choice of vectors *i* can be introduced,

If there are *N* force dipoles whose positions, orientations and magnitudes vary with time, and we have

Certain assumptions about statistical properties of active force dipoles will be made: The mean magnitude of the force dipole is zero,

In this study, we will consider two limiting cases: when the nematic order is absent and force dipoles are randomly oriented, and when all force dipoles are perfectly aligned. The intermediate situation, where both the nematic order and orientational fluctuations are present, should be a subject for further research.

When orientational order is absent,

In the case of perfect nematic orientational order, *λ* represents the nematic order parameter.

Spatial positions *t* (*R* from its initial position at *D* is the diffusion coefficient and

Finally, Eq. (6) can be also written in the form

In absence of orientational order, the autocorrelation function of this noise is *c* is the concentration of force dipoles at the reference point

If the orientational order is present, we have

Equation (16), that includes the active noise

A small tracer particle immersed into a fluid will always move, at low Reynolds numbers, at the same velocity as the local flow velocity of the fluid. Hence, the particle will be advected by the fluctuating flow field. If

Here, an important distinction has to be made. If we keep fixed the observation point

Thus, we expect that advection will have two effects. First, diffusion of a tracer particle will be enhanced and, second, under certain conditions the drift of a tracer particle will be induced.

Suppose that the nematic order is absent. Then, if deviations

In the linear order in the noise intensity, the displacement *T* is

An important role in the last equation is played by the product

Analyzing Eq. (27), one can notice that mean-square displacements within time *T* behave as *T*,

The diffusion coefficients are defined by equations

Additionally, the mean drift velocity *T* for the particle starting its motion at location *T* and we finally obtain

The same derivations can be performed in the case of orientationally ordered force dipoles. As a result, Eqs. (37) and (38) become derived, but, in these equations, the tensor

Since diffusion coefficients and the drift velocity are known, the Fokker–Planck equation for the probability density ^{13}^{)} Because the particles do not interact each with another, the same equation holds for their concentration

Thus, generally, the activity of force dipoles induces not only the diffusion of passive particles, but also their drift. Both result from advection of particles in fluctuating hydrodynamic fields of active force dipoles and represent therefore an analog of turbulent diffusion effects at high Reynolds numbers.

In absence of nematic order, general Eqs. (31) and (36) for diffusion coefficients and the drift velocity can be cast into a simpler form. Using explicit expressions (2) and (3) for tensor functions ^{11}^{)}

These integrals have singularities at *c* of force dipoles (or their activity

Further transformation can be also performed for the normalized drift velocity

A similar derivation could be performed^{11}^{)} for

Suppose that the system is uniform, i.e., ^{9}^{)} that diffusion enhancement is isotropic,

If the gradients of *χ* and *c* are sufficiently small, the local approximation (50) can be used and the evolution Eq. (41) for the distribution of passive particle takes the form

The drift is counterbalanced by the concentration gradient and finally a stationary distribution of passive particles becomes established. In the local approximation, it is given by^{10}^{)}

The force dipoles correspond to physical objects, such as molecular machines, that are also immersed into the fluid and are free to diffuse. These microscopic objects, like passive tracers, will be also advected in fluctuating flow fields and their diffusion will be enhanced, as described by Eq. (50). Moreover, their drift described by Eq. (48) will also take place. The evolution of the concentration distribution of such objects will be governed by Eq. (51) where *n* should be replaced by *c*, so that we obtain^{10}^{)}

The above results were obtained in the local approximation. To discuss how strong nonlocal effects are, we consider a situation when force dipoles occupy only a spherical region of radius *R*. Depending on the experimental setup, this can be a region where active bacteria are concentrated or an intracellular domain rich with active proteins. Moreover, we assume that, within this region, their activity and concentration are fixed,

We find that nonlocal effects significantly modify the prediction (50) of the local approximation only within a layer with the width about the cutoff length

Figure 1(a) shows accumulation of passive particles inside the region occupied by active force dipoles. In this simulation, force dipoles have the concentration distribution ^{14}^{)} We see that gradual aggregation of particles inside the sphere takes place through an “adsorption” process: the particles reaching the boundary are dragged inside the sphere and their concentration near the boundary starts to increase. This process continues until a stationary distribution is achieved where adsorption and evaporation compensate each another.

Figure 1. (Color online) Accumulation of passive particles inside (a) a sphere and (b) a disk occupied by active force dipoles. Consequent snapshots of the concentration distribution are shown. The parameters are

The local approximation does not hold at ^{11}^{)} The results of the analysis hold however also in the general case.

Inside the disk (for

Outside of the disk (for *r*.

Similar to 3D media, passive particles will accumulate inside the disk. In the final stationary state, the particles are uniformly distributed with concentrations ^{11}^{)}

Numerical simulations for evolution of the concentration of passive particles have been performed.^{11}^{)} Figure 1(b) shows snapshots of the distribution at different subsequent times. In this simulation, force dipoles were distributed as in Eq. (56).^{15}^{)} As seen in Fig. 1(b), the disk filled with active dipoles is sucking the particles from the region around it. Therefore, the concentration of passive particles starts to increase near the border of the disk. Later on, the particles become uniformly distributed within it.

If oscillating force dipoles are orientationally ordered, flows of passive particles, induced by dipole activity, persist even in the steady state. Although principal effects are similar, the descriptions are different for 2D and 3D media and we consider separately these two cases.

The nematic order tensor is *λ* is the nematic order parameter and *M* is a diagonal matrix,

Similar to the orientationally disordered 3D case, local approximation can be used again and we obtain

The equation for concentration of passive particles is

The existence of circulating fluxes in the steady state can be demonstrated by considering a situation where force dipoles occupy a sphere of radius *R* and are absent outside of it,

Numerical simulations were performed assuming that concentration distribution of force dipoles is smooth and given by Eq. (56) and that ^{16}^{)} Figures 2(a) and 2(b) show the distribution

Figure 2. (Color online) Distribution of passive particles (a, b) and their fluxes (c, d) in the steady state of a 3D system with orientationally ordered force dipoles that occupy a sphere in the center (

The nematic tensor in 2D systems is

As an example, we consider a situation when the dipoles are uniformly distributed inside a disk of radius *R* and are absent outside of it; we also assume

Inside the disk (

Outside of the disk, diffusion is anisotropic and, moreover, cross diffusion takes place. The components of the diffusion tensor outside of the disk (

Furthermore, temporal evolution of the distribution of passive particles and this distribution in the final steady state can be obtained by integrating Eq. (40). We have performed the numerical integration assuming that the distribution of force dipoles is fixed and given by Eq. (56). We also assume that ^{17}^{)}

Figure 3 presents the results for the final distribution of passive particles and for the flux. We see that there are significant differences from the three-dimensional case (cf. Fig. 2). While the 3D distributions had a two-fold symmetry in the median cross-section, the symmetry is four-fold in 2D. Now the particles are leaving the disk both along the orientation line of force dipoles and along the line orthogonal to it.

Figure 3. (Color online) Distribution of passive particles (a, b) and their fluxes (c, d) in the steady state of a 2D system with orientationally ordered force dipoles that occupy a circle in the center (

In oscillatory active nematics, a variety of hydrodynamic effects can be observed. The activity of force dipoles, corresponding to the elements that cyclically change their shapes, leads to stirring of the fluid and generation of non-thermal fluctuating flows within it. Passive particles are advected by such fluctuating flows and, as a result, their diffusion is enhanced. This phenomenon is analogous to turbulent diffusion, but takes place at low Reynolds numbers where inertial effects and nonlinearities are absent. Furthermore, directed drift of tracer particles is induced if the spatial distribution of force dipoles (or of their activity) is non-uniform. The effects are local in 3D systems and nonlocal in 2D. Depending on a system, orientational ordering in oscillatory active nematics can arise. As we have seen, hydrodynamic behavior is sensitive to the nematic order in populations of force dipoles.

An important consequence is that passive particles are driven by fluctuating flows into the regions with the high concentration or activity of force dipoles. This leads to a non-equilibrium redistribution of passive particles in the medium, maintained only as long as the activity of force dipoles persists. This provides a possibility to control spatial distributions of particles by purely hydrodynamic means, without applying any force fields.

In original publications,^{9}^{,}^{10}^{)} hydrodynamical effects in oscillatory active nematics were studied under an assumption that orientations and positions of active force dipoles did not change significantly within a cycle, so that the dipoles could be treated as immobile. In the present article, we allowed for orientational fluctuations in the disordered state. Furthermore, systems with the orientational nematic order were also analyzed.

The considered phenomena are important for biological cells. It was shown by in vivo experiments that active non-thermal noise dominates transport phenomena in the cytoplasm.^{18}^{)} Such non-thermal noise can be due to molecular motors operating on the cytoskeleton^{18}^{,}^{19}^{)} or be caused by the metabolic activity inside the cell.^{20}^{)} To analyze such effects, viscoelastic properties of the cytoplasm have to be taken into account.^{12}^{)} Moreover, biological membranes typically include a large number of proteins and many such inclusions operate as molecular machines, cyclically changing their shapes. Active proteins are typically confined within protein rafts. On the length scales shorter than about a micrometer, lipid bilayers behave as 2D fluids^{21}^{,}^{22}^{)} and, therefore, fluctuating lipid flows are induced when inclusions change their shapes. Their activity enhances diffusion and induces redistribution of particles over the membrane.^{11}^{)}

Experiments with suspensions of synthetic molecular machines^{4}^{)} or with the oscillating dumbbells can be performed too. Moreover, similar effects are expected for populations of microorganisms that, while not acting as swimmers, reciprocally change their shapes. With this respect, it should be noted that in bacterial layers formed by swimming microorganisms, diffusion can be enhanced by a factor up to 100;^{23}^{)} these effects have been theoretically investigated.^{24}^{–}^{26}^{)}

Our analysis was based on a number of simplifications. Energetic interactions between the particles were not taken into account, although they might become important when accumulation of the particles in some regions occurs. We have also not taken into account fluctuation effects, even though they may become essential on short length scales. Moreover, only the situations with a perfect orientational order or without any order were considered by us. The respective theory extensions are the task for some future work.

## Acknowledgments

We are grateful to R. Kapral for stimulating discussions. This work was supported by JSPS KAKENHI Grant Nos. JP25103008, JP26520205, and JP15K05199. This work was supported in part by the Core-to-Core Program “Nonequilibrium dynamics of soft matter and information” (Grant No. 23002) to Y.K. and H.K., and by the EU program “Nonequilibrium dynamics of soft and active matter” (Grant No. PIRSES-GA-2011-295243) to A.M.

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## Author Biographies

**Alexander S. Mikhailov** was born in Russia in 1950. He obtained his Diploma (1973), Ph.D. (1976) and Dr.Sc. (1984) degrees from the Lomonosov Moscow State University. He works in Germany since 1990, first as a guest professor in the University of Stuttgart (1990–1991). Starting from 1995, he is a staff scientist (professor) and a group leader in the Department of Physical Chemistry of the Fritz Haber Institute in Berlin. He has been a guest professor in the Hokkaido, Kyoto and Hiroshima universities; and was awarded the International Solvay Chair in Chemistry in 2009. He is an editor of *Progress of Theoretical and Experimental Physics*. He has been working on a broad range of topics in theoretical chemical and biological physics, nonlinear dynamics and network science.

**Yuki Koyano** was born in Chiba, Japan in 1991. She obtained her B.Sci. (2013) and her M.Sci. (2015) from Chiba University. She is an doctor course student (2015–) at Department of Physics, Chiba University. She has been working on nonlinear physics, especially on self-propelled systems.

**Hiroyuki Kitahata** was born in Osaka, Japan in 1978. He obtained his B.Sci. (2001), his M.Sci. (2003), and his D.Sci. (2006) from Kyoto University. He was an assistant professor (2004–2008) at Department of Physics, Kyoto University, and a lecturer (2008–2010) and an associate professor (2011–) at Department of Physics, Chiba University. He is a head editor of the *Journal of the Physical Society of Japan*. He has been working on nonlinear physics, especially on active matter and spatio-temporal self-organization.