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Onsager's variational principle provides us with a systematic way to derive dynamical equations for various soft matter and active matter. By reformulating the Onsager-Machlup variational principle (OMVP), which is a time-global principle, we propose a new method to incorporate thermal fluctuations. To demonstrate the utility of the statistical formulation of OMVP, we obtain the diffusion constant of a Brownian particle embedded in a viscous fluid by maximizing the modified Onsager-Machlup integral for the surrounding fluid. We also apply our formulation to a Brownian particle in a steady shear flow, which is a typical nonequilibrium system. Possible extensions of our formulation to internally driven active systems are also discussed.

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 symmetric and antisymmetric parts. In both models, 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. We find that it 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.

We discuss hydrodynamic forces acting on a two-dimensional liquid domain that moves laterally within a supported fluid membrane in the presence of odd viscosity. Since active rotating proteins can accumulate inside the domain, we focus on the difference in odd viscosity between the inside and outside of the domain. Taking into account the momentum leakage from a two-dimensional incompressible fluid to the underlying substrate, we analytically obtain the fluid flow induced by the lateral domain motion and calculate the drag and lift forces acting on the moving liquid domain. In contrast to the passive case without odd viscosity, the lateral lift arises in the active case only when the in and out odd viscosities are different. The in-out contrast in the odd viscosity leads to nonreciprocal hydrodynamic responses of an active liquid domain.

We propose a model that describes cyclic state transitions of a micromachine driven by a catalytic chemical reaction. We consider a mechanochemical coupling of variables representing the degree of a chemical reaction and the internal state of a micromachine. The total free energy consists of a tilted periodic potential and a mechanochemical coupling energy. We assume that the reaction variable obeys a deterministic stepwise dynamics characterized by two typical timescales, i.e., the mean first passage time and the mean first transition path time. To estimate the functionality of a micromachine, we focus on the quantity called "nonreciprocality" and further discuss its dependence on the properties of catalytic reaction. For example, we show that the nonreciprocality is proportional to the square of the mean first transition path time. The explicit calculation of the two timescales within the decoupling approximation model reveals that the nonreciprocality is inversely proportional to the square of the energy barrier of catalytic reaction.

We discuss the linear hydrodynamic response of a two-dimensional active chiral compressible fluid with odd viscosity. The viscosity coefficient represents broken time-reversal and parity symmetries in the 2D fluid and characterizes the deviation of the system from a passive fluid. Taking into account the hydrodynamic coupling to the underlying bulk fluid, we obtain the odd viscosity-dependent mobility tensor, which is responsible for the nonreciprocal hydrodynamic response to a point force. Furthermore, we consider a finite-size disk moving laterally in the 2D fluid and demonstrate that the disk experiences a nondissipative lift force in addition to the dissipative drag one.

We study the Brownian motion of a charged colloid, confined between two charged walls, for small separation between the colloid and the walls. The system is embedded in an ionic solution. The combined effect of electrostatic repulsion and reduced diffusion due to hydrodynamic forces results in a specific motion in the direction perpendicular to the confining walls. The apparent diffusion coefficient at short times as well as the diffusion characteristic time are shown to follow a sigmoid curve as a function of a dimensionless parameter. This parameter depends on the electrostatic properties and can be controlled by tuning the solution ionic strength. At low ionic strength, the colloid moves faster and is localized, while at high ionic strength it moves slower and explores a wider region between the walls, resulting in a larger diffusion characteristic time.

The dynamics of a membrane coupled to an active fluid on top of a substrate is considered theoretically. It is assumed that the director field of the active fluid has rotational symmetry in the membrane plane. This situation is likely to be relevant for in vitro reconstructed actomyosin-membrane system. Different from a membrane coupled to a polar active fluid, this model predicts that only when the viscosity of the fluid above the membrane is sufficiently large, a contractile active fluid is able to slow down the relaxation of the membrane for perturbations with wavelength comparable to the thickness of the active fluid. Hence, our model predicts a finite-wavelength instability in the limit of strong contractility, which is different from a membrane coupled to a polar active fluid. However, a membrane coupled to an extensile active fluid is always unstable against long-wavelength perturbations due to active extensile stress enhanced membrane undulation.

We discuss the shear viscosity of a Newtonian solution of catalytic enzymes and substrate molecules. The enzyme is modeled as a two-state dimer consisting of two spherical domains connected with an elastic spring. The enzymatic conformational dynamics is induced by the substrate binding and such a process is represented by an additional elastic spring. Employing the Boltzmann distribution weighted by the waiting times of enzymatic species in each catalytic cycle, we obtain the shear viscosity of dilute enzyme solutions as a function of substrate concentration and its physical properties. The substrate affinity distinguishes between fast and slow enzymes, and the corresponding viscosity expressions are obtained. Furthermore, we connect the obtained viscosity with the diffusion coefficient of a tracer particle in enzyme solutions.

We discuss the nonequilibrium statistical mechanics of a thermally driven micromachine consisting of three spheres and two harmonic springs [Y. Hosaka et al., J. Phys. Soc. Jpn. 86, 113801 (2017)JUPSAU0031-901510.7566/JPSJ.86.113801]. We obtain the nonequilibrium steady state probability distribution function of such a micromachine and calculate its probability flux in the corresponding configuration space. The resulting probability flux can be expressed in terms of a frequency matrix that is used to distinguish between a nonequilibrium steady state and a thermal equilibrium state satisfying detailed balance. The frequency matrix is shown to be proportional to the temperature difference between the spheres. We obtain a linear relation between the eigenvalue of the frequency matrix and the average velocity of a thermally driven micromachine that can undergo a directed motion in a viscous fluid. This relation is consistent with the scallop theorem for a deterministic three-sphere microswimmer.

We study pattern formation of skin cancers by means of numerical simulation of a binary system consisting of cancer and healthy cells. We extend the conventional model H for macrophase separations by considering a logistic growth of cancer cells and also a mechanical friction between dermis and epidermis. Importantly, our model exhibits a microphase separation due to the proliferation of cancer cells. By numerically solving the time evolution equations of the cancer composition and its velocity, we show that the phase separation kinetics strongly depends on the cell proliferation rate as well as on the strength of hydrodynamic interactions. A steady-state diagram of cancer patterns is established in terms of these two dynamical parameters and some of the patterns correspond to clinically observed cancer patterns. Furthermore, we examine in detail the time evolution of the average composition of cancer cells and the characteristic length of the microstructures. Our results demonstrate that different sequence of cancer patterns can be obtained by changing the proliferation rate and/or hydrodynamic interactions.

A model of three-disk micromachine swimming in a quasi-two-dimensional supported membrane is proposed. We calculate the average swimming velocity as a function of the disk size and the arm length. Due to the presence of the hydrodynamic screening length in the quasi-two-dimensional fluid, the geometric factor appearing in the average velocity exhibits three different asymptotic behaviors depending on the microswimmer size and the hydrodynamic screening length. This is in sharp contrast with a microswimmer in a three-dimensional bulk fluid that shows only a single scaling behavior. We also find that the maximum velocity is obtained when the disks are equal-sized, whereas it is minimized when the average arm lengths are identical. The intrinsic drag of the disks on the substrate does not alter the scaling behaviors of the geometric factor.

We theoretically investigate the dynamics of a floating lipid bilayer membrane coupled with a two-dimensional cytoskeleton network, taking into account explicitly the intermonolayer friction, the discrete lattice structure of the cytoskeleton, and its prestress. The lattice structure breaks lateral continuous translational symmetry and couples Fourier modes with different wave vectors. It is shown that within a short time interval a long-wavelength deformation excites a collection of modes with wavelengths shorter than the lattice spacing. These modes relax slowly with a common renormalized rate originating from the long-wavelength mode. As a result, and because of the prestress, the slowest relaxation is governed by the intermonolayer friction. Conversely, and most interestingly, forces applied at the scale of the cytoskeleton for a sufficiently long time can cooperatively excite large-scale modes.

We discuss the hydrodynamic collective effects due to active protein molecules that are immersed in lipid bilayer membranes and modeled as stochastic force dipoles. We specifically take into account the presence of the bulk solvent that surrounds the two-dimensional fluid membrane. Two membrane geometries are considered: the free membrane case and the confined membrane case. Using the generalized membrane mobility tensors, we estimate the active diffusion coefficient and the drift velocity as a function of the size of a diffusing object. The hydrodynamic screening lengths distinguish the two asymptotic regimes of these quantities. Furthermore, the competition between the thermal and nonthermal contributions in the total diffusion coefficient is characterized by two length scales corresponding to the two membrane geometries. These characteristic lengths describe the crossover between different asymptotic behaviors when they are larger than the hydrodynamic screening lengths.

With the use of the "two-fluid model," we discuss anomalous diffusion induced by active force dipoles in viscoelastic media. Active force dipoles, such as proteins and bacteria, generate nonthermal fluctuating flows that lead to a substantial increment of the diffusion. Using the partial Green's function of the two-fluid model, we first obtain passive (thermal) two-point correlation functions such as the displacement cross-correlation function between the two-point particles separated by a finite distance. We then calculate active (nonthermal) one-point and two-point correlation functions due to active force dipoles. The time correlation of a force dipole is assumed to decay exponentially with a characteristic time scale. We show that the active component of the displacement cross-correlation function exhibits various crossovers from super-diffusive to subdiffusive behaviors depending on the characteristic time scales and the particle separation. Our theoretical results are intimately related to the microrheology technique to detect fluctuations in nonequilibrium environment.

We study the relaxation dynamics of a compressible bilayer vesicle with an asymmetry in the viscosity of the inner and outer fluid medium. First we explore the stability of the vesicle free energy which includes a coupling between the membrane curvature and the local density difference between the two monolayers. Two types of instabilities are identified: a small wavelength instability and a larger wavelength instability. Considering the bulk fluid viscosity and the inter-monolayer friction as the dissipation sources, we next employ Onsager's variational principle to derive the coupled equations both for the membrane and the bulk fluid. The three relaxation modes are coupled to each other due to the bilayer and the spherical structure of the vesicle. Most importantly, a higher fluid viscosity inside the vesicle shifts the crossover mode between the bending and the slipping to a larger value. As the vesicle parameters approach the unstable regions, the relaxation dynamics is dramatically slowed down, and the corresponding mode structure changes significantly. In some limiting cases, our general result reduces to the previously obtained relaxation rates.

We propose a model that accounts for the budding transition of asymmetric two-component lipid domains, where the two monolayers (leaflets) have different average compositions controlled by independent chemical potentials. Assuming a coupling between the local curvature and local lipid composition in each of the leaflets, we discuss the morphology and thermodynamic behavior of asymmetric lipid domains. The membrane free-energy contains three contributions: the bending energy, the line tension, and a Landau free-energy for a lateral phase separation. Within a mean-field treatment, we obtain various phase diagrams containing fully budded, dimpled, and flat states as a function of the two leaflet compositions. The global phase behavior is analyzed, and depending on system parameters, the phase diagrams include one-phase, two-phase, and three-phase regions. In particular, we predict various phase coexistence regions between different morphologies of domains, which may be observed in multicomponent membranes or vesicles.

Active motions of a biological membrane can be induced by nonthermal fluctuations that occur in the outer environment of the membrane. We discuss the dynamics of a membrane interacting hydrodynamically with an active wall that exerts random velocities on the ambient fluid. Solving the hydrodynamic equations of a bound membrane, we first derive a dynamic equation for the membrane fluctuation amplitude in the presence of different types of walls. Membrane two-point correlation functions are calculated for three different cases: (i) a static wall, (ii) an active wall, and (iii) an active wall with an intrinsic time scale. We focus on the mean squared displacement (MSD) of a tagged membrane describing the Brownian motion of a membrane segment. For the static wall case, there are two asymptotic regimes of MSD (∼t^{2/3} and ∼t^{1/3}) when the hydrodynamic decay rate changes monotonically. In the case of an active wall, the MSD grows linearly in time (∼t) in the early stage, which is unusual for a membrane segment. This linear-growth region of the MSD is further extended when the active wall has a finite intrinsic time scale.

We propose a model that accounts for the budding behavior of domains in lipid bilayers, where each of the bilayer leaflets has a coupling between its local curvature and the local lipid composition. The compositional asymmetry between the two monolayers leads to an overall spontaneous curvature. The membrane free energy contains three contributions: the bending energy, the line tension, and a Landau free energy for a lateral phase separation. Within a mean-field treatment, we obtain various phase diagrams which contain fully budded, dimpled, and flat states. In particular, for some range of membrane parameters, the phase diagrams exhibit a tricritical behavior as well as a three-phase coexistence region. The global phase diagrams can be divided into three types and are analyzed in terms of the curvature-composition coupling parameter and domain size.

We study diffusion coefficients of liquid domains by explicitly taking into account the two-layered structure called leaflets of the bilayer membrane. In general, the velocity fields associated with each leaflet are different and the layers sliding past each other cause frictional coupling. We obtain analytical results of diffusion coefficients for a circular liquid domain in a leaflet, and quantitatively study their dependence on the interleaflet friction. We also show that the diffusion coefficients diverge in the absence of coupling between the bilayer and solvents, even when the interleaflet friction is taken into account. In order to corroborate our theory, the effect of the interleaflet friction on the correlated diffusion is examined.

The dynamic shape relaxation of the two-layer vesicle is calculated. In addition to the undulation relaxation where the two bilayers move in the same direction, the squeezing mode appears when the gap between the two bilayers is small. At a large gap, the inner vesicle relaxes much faster, whereas the slow mode is mainly due to the outer-layer relaxation. We have calculated the viscoelasticity of the dilute two-layer-vesicle suspension. It is found that for a small gap, the applied shear drives the undulation mode strongly while the slow squeezing mode is not much excited. In this limit, the complex viscosity is dominated by the fast-mode contribution. On the other hand, the slow mode is strongly driven by shear for a larger gap. We have determined the crossover gap, which depends on the interaction between the two bilayers. For a series of samples where the gap is changed systematically, it is possible to observe the two amplitude switchings.

We consider the formation of finite-size domains in lipid bilayers consisting of saturated and hybrid lipids. First, we describe a monolayer model that includes a coupling between a compositional scalar field and a two-dimensional vectorial order parameter. Such a coupling yields an effective two-dimensional microemulsion free energy for the lipid monolayer, and its characteristic length of compositional modulations can be considered as the origin of finite-size domains in biological membranes. Next, we consider a coupled bilayer composed of two modulated monolayers and discuss the static and dynamic properties of concentration fluctuations above the transition temperature. We also investigate the micro-phase separation below the transition temperature and compare the micro-phase separated structures with statics and dynamics of concentration fluctuations above the transition.

We consider the phase separation in an asymmetrically charged lipid bilayer membrane consisting of neutral and negatively charged lipids that are in contact with in and out ionic solutions having different ionic strengths. The two asymmetric leaflets are coupled through electrostatic interactions. Based on a free-energy approach, the critical point and phase diagrams are calculated for different ionic strengths of the two solutions and coupling parameter. An increase of the coupling constant or asymmetry in the salt concentration between the in and out solutions yields a higher phase-separation temperature because of electrostatic interactions. As a consequence, the phase-coexistence region increases for strong screening (small Debye length). Finally, possible three-phase coexistence regions in the phase diagram are predicted.

The diffusion coefficient of an inclusion in a liquid membrane is investigated by taking into account the interaction between membranes and bulk solvents of arbitrary thickness. As illustrative examples, the diffusion coefficients of two types of inclusions, a circular domain composed of fluid with the same viscosity as the host membrane and that of a polymer chain embedded in the membrane, are studied. The diffusion coefficients are expressed in terms of the hydrodynamic screening lengths, which vary according to the solvent thickness. When the membrane fluid is dragged by the solvent of finite thickness, via stick boundary conditions, multiple hydrodynamic screening lengths together with the weight factors to the diffusion coefficients are obtained from the characteristic equation. The conditions for which the diffusion coefficients can be approximated by the expression including only a single hydrodynamic screening length are also shown.

We performed small-angle neutron scattering and neutron spin echo experiments on a ternary microemulsion composed of ionic surfactant AOT, water, and decane. Thermal fluctuations of monolayers have been investigated as a function of temperature and pressure. The amphiphilic monolayers become more flexible with increasing temperature and more rigid with increasing pressure. These results are consistent with the microscopic picture that the head-head repulsion of the AOT molecules is enhanced at high temperature while an attractive interaction between the hydrophobic tails of the AOT molecules increases at high pressure.

Dynamical fluctuations of the bicontinuous microemulsion and lamellar structures in ternary C12E5-water-n-octane amphiphilic system are studied by means of neutron spin echo (NSE) spectrometry. The decay rates of the time correlation of the concentration were analyzed in terms of three theories: (1) A. G. Zilman and R. Granek, Phys. Rev. Lett. 77, 4788 (1996), (2) M. Nonomura and T. Ohta, J. Chem. Phys. 110, 7516 (1999), and (3) R. Granek and M. E. Cates, Phys. Rev. A 46, 3319 (1992), in the first of which a Langevin equation for membrane plaquettes and in the latter two of which time-dependent Ginzburg-Landau equations for the order parameters are considered. The result shows that the intermediate correlation functions I(q,t) for the ranges of 0