Nonreciprocity and odd viscosity in chiral active fluids

Significance

Active materials are composed of many individually driven entities and span many length-scales and many disciplines: from living systems such as cells or birds to metamaterials and swarms of robots. In recent years, chiral active materials have received considerable interest because of their wide applicability in biology and the new nonreciprocal physics they entail. Perhaps the simplest example of chiral active matter is a collection of spinning noninteracting molecules. We show that odd (Hall) viscosity is generic in these systems, which are nevertheless naturally nonreciprocal, breaking Onsager reciprocity. We also find evidence of a nonreciprocal phase transition when interactions between spinning molecules are present. Our results open a broad study avenue with vast applications to bio and bioinspired materials.

Abstract

Odd viscosity couples stress to strain rate in a dissipationless way. It has been studied in plasmas under magnetic fields, superfluid ${\mathrm{He}}^3$, quantum-Hall fluids, and recently in the context of chiral active matter. In most of these studies, odd terms in the viscosity obey Onsager reciprocal relations. Although this is expected in equilibrium systems, it is not obvious that Onsager relations hold in active materials. By directly coarse-graining the kinetic energy and independently using both the Poisson-bracket formalism and a kinetic theory derivation, we find that the appearance of a nonvanishing angular momentum density, which is a hallmark of chiral active materials, necessarily breaks Onsager reciprocal relations. This leads to a non-Hermitian dynamical matrix for the total hydrodynamic momentum and to the appearance of odd viscosity and other nondissipative contributions to the viscosity. Furthermore, by accounting for both the angular momentum density and interactions that lead to odd viscosity, we find regions in the parameter space in which 3D odd mechanical waves propagate and regions in which they are mechanically unstable. The lines separating these regions are continuous lines of exceptional points, suggesting a possible nonreciprocal phase transition.

Chiral active materials are composed of complex molecules that break both parity and time-reversal symmetry (TRS) at the microscale. This is generally a result of continuous injection of energy and angular momentum through local torques. Realizations of such fluids are found in a variety of systems across length scales, from nanoscale biomolecular motors (1–3), actomyosin networks (4), and microscale active colloids (5–7), to macroscale-driven chiral grains (8–10). An important consequence of the breaking of parity and TRS is the possible appearance of odd viscosity. Unlike the regular viscosity that dissipates energy, odd viscosity is “reactive” (11, 12) and can be obtained using a Hamiltonian theory with no need of adding dissipation (13). Importantly, Onsager reciprocal relations^* (14–18) predict that such odd viscosity only appears when TRS is broken (19–21), be it due to an external magnetic field as was studied in gases (22), plasmas (23, 24), and in the context of quantum-Hall fluids (25–29), or due to local torques injected at the particle level (12, 15, 30–32).

By construction, the stress only enters the dynamics via $\mathrm{\nabla }·\mathit{\sigma }$, where $\mathit{\sigma }$ is the stress tensor. As a result, the dynamics is not modified by the transformation $\mathit{\sigma }\to \mathit{\sigma }+\mathrm{\nabla }\times \mathbf{B}$ (33–35) ($\mathbf{B}$ is an arbitrary second rank tensor), hence $\mathit{\sigma }$ is not unique. Among the different stress tensors are that associated with center-of-mass (CM) momentum alone and that associated with spin angular momentum (SAM) as well. We will refer to the latter momentum as the total hydrodynamic momentum (or hydrodynamic momentum for shortness), with density $\mathbf{g}$. This momentum differs from the real total momentum as it does not include motion of rapidly decaying nonhydrodynamic molecular modes.^† In passive systems in which the SAM density of rotating molecules (or complex particles) relaxes more rapidly than linear momentum density, there is no difference between the two stress tensors (in the hydrodynamic limit). However, in chiral active materials in which SAM density is driven, a significantly different behavior may arise. We argue that the experimentally accessible surface forces are those related to the total hydrodynamic momentum.

In all studies we are aware of, odd terms in the viscosity always obey Onsager reciprocal relations, mainly due to the fact that their origin is usually associated with reciprocal interparticle collisions, even though these may break parity (33). Even when the possible existence of such nonreciprocal odd terms is assumed phenomenologically from symmetry arguments (13, 18, 30), molecular dynamic simulations accounting for the interparticle interactions (30) have supported reciprocity. This is quite surprising considering the fact that in active materials, reciprocity need not be obeyed, and nonreciprocity is quite common (36, 37). Of course, microscopic interactions obey Newton’s third law and are reciprocal, but effective interactions mediated by an “active” nonequilibrium medium are not necessarily so, and they are often responsible for the frequent occurrence of nonreciprocity (38–40). The observation that active materials do not obey Onsager reciprocity is not new (36, 38, 41, 42), but in this work, we find a quite remarkable and simple instance of this—the mere existence of nonvanishing SAM breaks Onsager reciprocal relations. In passive systems, this has no consequence as angular momentum relaxes rapidly and does not affect the hydrodynamic equations. However, as we show below, in the presence of active torques, this is no longer the case, and reciprocity is generically broken.

In ref. 12, we coarse-grained the total hydrodynamic momentum density $\mathbf{g}$ and found that it obeys the Belinfante–Rosenfeld relation (43, 44), which for classical fields means that the total hydrodynamic momentum density equals the CM momentum density plus half of the curl of the internal angular momentum density, $\mathit{\ell }$ (35). Then, by writing the kinetic energy in the common form $H_k=\int \text{d}\mathbf{r}{\mathbf{g}}^2/(2\rho )$, we found that odd-viscosity, which obeys Onsager reciprocal relations, emerges from the Poisson-bracket (PB) formalism (45, 46). The resultant excitation spectrum exhibits 3D odd mechanical waves, even in noninteracting systems.

In what follows, we treat the molecules as rigid bodies that exhibit translational and rotational motion, ignoring finite-frequency nonhydrodynamic molecular modes. We then coarse-grain the microscopic kinetic energy directly to obtain the well-known continuum kinetic energy of a collection of rigid molecules (47) with additional boundary terms. Surprisingly, we find that these two coarse-graining (CG) methods are not equivalent. We believe that the CG of the Hamiltonian presented here is correct, and we have supported this finding using a kinetic theory derivation (Materials and Methods) that does not rely on CG of the Hamiltonian itself.

The CG of this work gives the familiar dynamics of CM and angular momenta, with no odd viscosity (16, 48). However, when writing the total hydrodynamic momentum dynamics, odd viscosity naturally emerges, and more “odd” terms appear in the viscosity, including one that couples vorticity and pressure [odd pressure (13)]. These new viscosity terms exactly cancel terms in the dynamics of the total hydrodynamic momentum and do not allow for propagation of 3D odd mechanical waves. Instead, they require a longitudinal wave to always be accompanied by a transverse wave but not vice versa. Importantly, this means that the dynamics are nonreciprocal, and, indeed, the dynamical matrix for long-wavelength excitations is non-Hermitian. In the presence of an additional odd viscosity that can originate in fluctuations or interparticle interactions (30), we find instability lines of exceptional points (49), which suggests the existence of a nonreciprocal phase transition (37).

In what follows, we consider a fluid of dumbbells, which are diatomic molecules composed of two point masses, $m$, separated by distance $2a$ as depicted in Fig. 1. We define the CM position of dumbbell $\alpha$ to be ${\mathbf{r}}^{\alpha }$ while ${\mathit{\nu }}^{\alpha }$ is a unit vector pointing from mass “2” to mass “1.” It is important to note that we only consider a fluid of dumbbells for clearer presentation. The derivation below also applies to general complex rigid molecules (index $\alpha$) that are composed of multiple subparticles (atoms) with mass $m^{\alpha \mu }$ and momentum ${\mathbf{p}}^{\alpha \mu }$ located at ${\mathbf{r}}^{\alpha \mu }$. We show this in SI Appendix, section III.

Fig. 1.

Illustration of our model system that is composed of many diatomic molecules. Each of these molecules is composed of two equal point masses, $m$, separated by distance $2a$. The vector $\mathit{\nu }$ points from mass 2 to mass 1, and ${\mathbf{r}}^{\alpha }$ points to the CM of the $\alpha$ molecule. Diatomic molecules are the minimal molecules that have internal angular momentum, and therefore this is the minimal system that will exhibit “kinetic” odd viscosity solely due to molecules spinning, which could be a result of external field or internal active torques. The existence of angular momentum also breaks Onsager reciprocal relations.

Total Hydrodynamic Momentum

We start by introducing the concept of total hydrodynamic momentum, a density field that captures the momentum of all atoms (not only the molecules CM) in the hydrodynamic limit, i.e., in the limit in which all “fast modes” are neglected. In practice, all finite-frequency modes are discarded and the CG only accounts for the molecules’ zero modes. For simple molecules with no states of self-stress (50, 51) these zero modes are the rigid body translations and rotations (see Materials and Methods for more discussion). Hereafter, we will refer to this momentum as the hydrodynamic momentum. As we show below, the hydrodynamic momentum flux (or the hydrodynamic momentum stress tensor) is very different from the one associated with the CM.

To model a fluid of dumbbells, we write the momentum of the two point masses in terms of the CM momentum, ${\mathbf{p}}^{\alpha }$, and ${\mathit{\nu }}^{\alpha }$ as ${\mathbf{p}}_{1,2}^{\alpha }=\frac{1}{2}{\mathbf{p}}^{\alpha }\pm ma{\dot{\mathit{\nu }}}^{\alpha }$ (Here $\dot{X}={\partial }_{t}X$), such that the total momentum density for the diatomic fluid, ${\widehat{g}}_i(\mathbf{r})=\sum _{\alpha \mu }{p}_i^{\alpha \mu }\delta \left(\mathbf{r}-{\mathbf{r}}^{\alpha \mu }\right)$, can be split into the CM momentum density, ${\widehat{\mathbf{g}}}^c\equiv \sum _{\alpha }{\mathbf{p}}^{\alpha }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha })$, and a spin-like momentum density, $\widehat{\mathbf{g}}={\widehat{\mathbf{g}}}^c+{\widehat{\mathbf{g}}}^s$, with

$$\begin{array}{c}\hfill {\widehat{g}}_i^s(\mathbf{r})=\frac{1}{2}\mathrm{\nabla }\times \widehat{\mathit{\ell }}-\frac{1}{2}\mathrm{\nabla }·\widehat{\mathbf{A}},\end{array}$$ [1]

where $\widehat{\mathit{\ell }}(\mathbf{r})=\sum _{\alpha }{\mathit{\ell }}^{\alpha }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha })$ and $\widehat{\mathbf{A}}(\mathbf{r})=I\sum _{\alpha }{\dot{\mathbf{Q}}}^{\alpha }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha })$. Here, $Q_{\mathit{ij}}^{\alpha }={\nu }_i^{\alpha }{\nu }_j^{\alpha }-{\delta }_{\mathit{ij}}/d$ is the alignment tensor of molecule $\alpha$ ($d$ is the spatial dimension) and ${\widehat{Q}}_{\mathit{ij}}(\mathbf{r})=\sum _{\alpha }{Q}_{\mathit{ij}}^{\alpha }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha })$. The angular momentum of each molecule is related to its angular velocity, ${\mathrm{\Omega }}^{\alpha }\equiv {\mathit{\nu }}^{\alpha }\times {\dot{\mathit{\nu }}}^{\alpha }$, by ${\mathit{\ell }}^{\alpha }=I{\mathbf{\Omega }}^{\alpha }$. The moment of inertia of each molecule is $I=M{a}^2$ with $M=2m$ being the molecule mass. Note that these expressions are written in the long wavelength limit (SI Appendix, section I) indicating that we already performed some kind of CG in which the atoms that form the molecule are not considered explicitly. Instead, each rigid molecule is described using its CM and angular momenta, which are the generalized momenta associated with the molecule normal zero-frequency modes (47). Therefore, $\widehat{\mathbf{g}}={\widehat{\mathbf{g}}}^c+{\widehat{\mathbf{g}}}^s$ is the total hydrodynamic momentum and not the real total momentum. In principle, one can coarse-grain the total momentum directly (without taking the hydrodynamic limit at this stage), but then the constraints between the atoms within each molecule will have to be considered explicitly (see Materials and Methods for further discussion).

CG Eq. 1 gives ${\mathbf{g}}^s(\mathbf{r})=\frac{1}{2}\mathrm{\nabla }\times \mathit{\ell }-\frac{1}{2}\mathrm{\nabla }·\mathbf{A}$. Throughout this work, we use a CG method (52) that is described in Materials and Methods following ideas of P. C. Martin. Assuming the system is in its disordered phase, $\mathbf{A}=\mathbf{Q}=0$, the well-known form of the hydrodynamic momentum density is recovered (12, 35).

$$\begin{array}{c}\hfill \mathbf{g}(\mathbf{r})={\mathbf{g}}^c+\frac{1}{2}\mathbf{\nabla }\times \mathit{\ell },\end{array}$$ [2]

where ${\mathbf{g}}^c=\rho {\mathbf{v}}^c$ is the CM momentum density, $\rho$ the mass density, ${\mathbf{v}}^c$ the CM velocity, and $\mathit{\ell }(\mathbf{r})=\rho (\mathbf{r})\tilde{I}\mathrm{\Omega }(\mathbf{r})$ is the SAM density, where $\tilde{I}=I/M=a^2$ is the moment of inertia per unit mass and $\mathbf{\Omega }$ is the rotation-rate vector. The relation of (Eq. 2) is well known for classical fields (35) and is referred to in the quantum physics literature as the Belinfante–Rosenfeld relation in which case the CM momentum is the canonical momentum (derived from Noether theorem) and the angular momentum is the molecular spin (44).

In the classical fields context, Martin et al. (35) showed that for passive fluids the difference between the total momentum and the CM stress tensors is a microscopic quantity that has no hydrodynamic effects. Our analysis below agrees with this conclusion. However, we find that this is not the case in chiral active fluids, in which case the difference between the stress tensors has hydrodynamic effects and is the source of odd viscosity.

Importantly, the macroscopic stress, which is the one accessible experimentally, is the one related to the low-frequency part of the momentum of all atoms. This is by definition what we call the hydrodynamic momentum stress and not just the CM momentum stress. As stated above, for passive fluids this distinction is rather academic. Remarkably, in chiral active fluids, this difference is no longer microscopic—it results in the appearance of odd viscosity and nonreciprocity in the hydrodynamic momentum dynamics. We show this in detail below. Fig. 2 illustrates how such difference may arise.

Fig. 2.

Cartoon of a fluid of rotating dumbbells. The dumbbells rotate around their CM but their CM do not move. In this specific setting, one of the dumbbells hit the container wall thus exerting a force on it. This is perhaps the simplest example of how the CM momentum does not capture all the stress, but the hydrodynamic momentum does.

We remark that the real total momentum takes into account the motion of all atoms, which includes all molecules translations, rotations, and internal modes. The CM momentum has a special role as it is also conserved on its own, and as is well known, is sufficient to describe the hydrodynamics of passive fluids, in which all other degrees of freedom relax in microscopic times. However, when other, nonhydrodynamic zero modes are active (or driven) they should be accounted for in the CG process, while finite-frequency modes can usually be neglected (see SI Appendix, section II for more details).

Dynamics of the Hydrodynamic Momentum

Let us start by restating the well-known dynamics of the CM and angular momenta of isotropic fluids. This can be a simple Newtonian fluid or a complex fluid, but with no order (e.g., no nematic or polar order). Because we are interested in chiral active fluids, we first consider the dynamics of the SAM density (12, 15, 16, 31, 46, 53):

$$\begin{array}{c}\hfill {\dot{\ell }}_i(\mathbf{r})+{\mathrm{\nabla }}_j\left({\ell }_i{v}_j^c\right)=-\mathrm{\Gamma }\left({\mathrm{\Omega }}_i-{\omega }_i^c\right)+{\tau }_i,\end{array}$$ [3]

where $\mathit{\tau }$ is an external torque density, ${\mathit{\omega }}^c=\frac{1}{2}\mathrm{\nabla }\times {\mathbf{v}}^c$ is the rotation vector, and the dissipative term $-\mathrm{\Gamma }(\mathbf{\Omega }-{\mathit{\omega }}^c)$ is the one that provides preference for a dissipation-free steady-state in which $\mathbf{\Omega }={\mathit{\omega }}^c$, such that the fluid rotates as a rigid body. The continuity and Navier–Stokes equations are (16, 48):

$$\begin{array}{cc}& \dot{\rho }+\mathrm{\nabla }·{\mathbf{g}}^c=0,\hfill \\ \hfill & {\dot{g}}_i^c+{\mathrm{\nabla }}_j\left(v_j^c{g}_i^c\right)={\mathrm{\nabla }}_j[-P{\delta }_{\mathit{ij}}+{\eta }_{\mathit{ijkl}}^e{\mathrm{\nabla }}_l{v}_k^c\hfill \end{array}$$ [4a]

$$\begin{array}{c}+\frac{\mathrm{\Gamma }}{2}{\epsilon }_{\mathit{ijk}}\left({\mathrm{\Omega }}_k-{\omega }_k^c\right)]+f_i,\hfill \end{array}$$ [4b]

where $\mathit{f}$ is an external force density and ${\eta }_{\mathit{ijkl}}^e=\lambda {\delta }_{\mathit{ij}}{\delta }_{\mathit{kl}}+\eta \left({\delta }_{\mathit{ik}}{\delta }_{\mathit{jl}}+{\delta }_{\mathit{il}}{\delta }_{\mathit{jk}}\right)$ is the usual viscosity tensor for an isotropic fluid, with $\lambda$ and $\eta$ being constants. We also include a dissipative antisymmetric stress $\sim \left(\mathbf{\Omega }-{\mathit{\omega }}^c\right)$ with $\mathrm{\Gamma }$ being a rotational viscosity (16, 45), which is required due to the presence of a similar term in (Eq. 3) and conservation of total angular momentum (15). Here, $P=\rho \frac{\delta F}{\delta \rho }-\mathcal{F}$ is the thermodynamic pressure and $F[\rho ]=\int \text{d}\mathbf{r}\mathcal{F}$ is the free energy.

These well-known dynamic equations have been used extensively in the literature and clearly show no sign of odd viscosity. Odd viscosity is only revealed upon the realization that the CM momentum density does not account for all of the hydrodynamic momentum. Taking the time derivative of (Eq. 2) and substituting the dynamics of ${\mathbf{g}}^c$ and $\mathit{\ell }$ from Eqs. 3 and 4 we get the hydrodynamic momentum dynamics (see SI Appendix, section IV for more details):

$$\begin{array}{cc}& {\dot{g}}_i+{\mathrm{\nabla }}_j(g_i{v}_{v_j})={\mathrm{\nabla }}_j{\sigma }_{\mathit{ij}}+f_i,\hfill \end{array}$$ [5a]

$$\begin{array}{cc}\hfill & {\sigma }_{\mathit{ij}}=-P{\delta }_{\mathit{ij}}+\left({\eta }_{\mathit{ijkl}}^o+{\eta }_{\mathit{ijkl}}^e\right){\mathrm{\nabla }}_l{v}_k+\frac{1}{2}{\epsilon }_{\mathit{ijk}}{\tau }_k,\hfill \end{array}$$ [5b]

$$\begin{array}{cc}\hfill & {\eta }_{\mathit{ijkl}}^o=-\frac{{\ell }_n}{4}\left({\gamma }_{ijkl;n}^o+{\epsilon }_{\mathit{ilk}}{\delta }_{\mathit{nj}}+{\epsilon }_{\mathit{jlk}}{\delta }_{\mathit{ni}}-2{\epsilon }_{\mathit{lkn}}{\delta }_{\mathit{ij}}\right),\hfill \end{array}$$ [5c]

with the usual (12, 24, 31, 54) odd viscosity tensor

$$\begin{array}{c}\hfill {\gamma }_{ijkl;n}^o={\epsilon }_{\mathit{iln}}{\delta }_{\mathit{jk}}+{\epsilon }_{\mathit{ikn}}{\delta }_{\mathit{jl}}+{\epsilon }_{\mathit{jkn}}{\delta }_{\mathit{il}}+{\epsilon }_{\mathit{jln}}{\delta }_{\mathit{ik}}.\end{array}$$ [6]

In deriving the above equations we divided the velocity gradient tensor into its symmetric and antisymmetric parts and dropped a nonhydrodynamic term $\sim \mathrm{\nabla }{\left(\mathrm{\nabla }\times \mathit{\ell }\right)}^2$ (SI Appendix, section IV). Importantly, the dissipative term $\sim \left(\mathbf{\Omega }-{\mathit{\omega }}^c\right)$ is canceled in the derivation of (Eq. 5) such that one can obtain odd viscosity without the need of introducing dissipation at all. Indeed, the dynamics of the hydrodynamic momentum can be derived using PBs directly and the odd terms in the hydrodynamic momentum appear as reactive terms (Materials and Methods).

Note that $\mathit{\ell }$ in (Eq. 5) is not yet a viscosity, rather it is a dynamical variable that obeys the dynamics of (Eq. 3) (which also shows that the SAM relaxes in finite time and is therefore not hydrodynamic). Moreover, it seems that our simple manipulation, of using the hydrodynamic momentum, leads to the appearance of “odd” terms whenever the angular momentum density does not vanish, hence, one may think it appears also in passive fluids. This is of course not the case, and $\mathit{\ell }$ only contributes to the hydrodynamic equation of the hydrodynamic momentum in the presence of active (or external) torques.

This can be understood as follows. Both TRS and parity are broken by $\mathit{\ell }$, whereas the presence of $\mathit{\tau }$, which creates activity and gives both $\mathbf{\Omega }\simeq {\mathbf{\Omega }}^0=\mathit{\tau }/\mathrm{\Gamma }$ and $\mathit{\ell }\simeq {\mathit{\ell }}^0=\underset{\_}{\underset{\_}{\mathbf{I}}}·\mathit{\tau }/\mathrm{\Gamma }$ steady-state values in the long-wavelength (hydrodynamic) limit (SI Appendix, section V), is responsible for the existence of odd terms in (Eq. 5b). [Note that the external torque may include surface friction, e.g., $\mathit{\tau }=\tilde{\mathit{\tau }}-{\mathrm{\Gamma }}^{\text{ex}}\mathbf{\Omega }$ (12).] In the absence of activity, there is no breaking of the continuous rotational symmetry, and $\mathbf{\Omega }$ relaxes in microscopic times to $\mathit{\omega }$. Then, $\mathit{\ell }$ in (Eq. 5b) is $\sim \mathit{\omega }\mathrm{\nabla }\mathbf{v}\sim {\mathrm{\nabla }}^2$, hence, it is not hydrodynamic and must be omitted from the hydrodynamic equations, Eqs. 5a–5c. Therefore, odd viscosity only appears in (Eq. 5) after relaxation of SAM and in the presence of activity. For the same reasons, we expect to find odd viscosity in magnetic fluids, such as ferrofluids (55) and ferronematics (56), in which the magnetization breaks continuous rotational symmetry similarly to the SAM in the chiral active fluid we consider.

Consequently, in the passive case, one cannot distinguish between the total momentum and the CM momentum densities as was also pointed out in ref. 35. Remarkably, in chiral active materials, this statement is no longer valid, and as we show here (and in ref. 12), the hydrodynamic momentum stress is not the same as the CM stress. Then, the stress tensor that is related to the experimentally measured surface forces is the momentum flux of the hydrodynamic momentum, which includes the odd terms.

The hydrodynamic momentum stress of (Eq. 5b) (after SAM relaxation, $\mathit{\ell }\to {\mathit{\ell }}^0$) contains the odd viscosity, the so-called odd pressure (13) that couples vorticity (or rotations) to pressure, and two more terms that couple vorticity to shears that involve the direction of $\mathit{\ell }$ (57), which do not appear in 2D. Importantly, the existence of these nondissipative terms does not depend on the free-energy $F$, hence, these terms will appear even in a noninteracting system, e.g., a dilute gas of spinning particles with an average common rotation axis.

In Materials and Methods, we have derived the dynamics of Eqs. 4 and 5 using the PB formalism, which depends on the coarse-grained Hamiltonian. Because we find that the direct CG of the kinetic energy gives a different Hamiltonian from that used in ref. 12, and therefore leads to different dynamics, we support our current findings using another, unrelated derivation of the dynamics. We employed a kinetic theory that does not rely on the CG of the Hamiltonian to derive the hydrodynamic equations (the kinetic theory derivation is also detailed in Materials and Methods).

Excitation Spectrum

We continue by calculating the excitation spectrum in the presence of odd viscosity. For simplicity, we assume that $\mathit{\tau }$ is constant (e.g., as a result of an external field) so that $\mathit{\ell }\simeq {\mathit{\ell }}^0$ is also constant. We also set $\mathbf{f}=0$ such that the dynamics of the hydrodynamic momentum of (Eq. 5) obeys

$$\begin{array}{c}\hfill {\dot{g}}_i+{\mathrm{\nabla }}_j\left(v_j{g}_i\right)=-{\mathrm{\nabla }}_i\tilde{P}+\eta {\mathrm{\nabla }}^2{v}_i-\frac{1}{2}{\epsilon }_{\mathit{ijn}}{\ell }_n^0{\mathrm{\nabla }}_j\mathrm{\nabla }·\mathbf{v},\end{array}$$ [7]

with $\tilde{P}\equiv P-(\lambda +\eta )\mathrm{\nabla }·\mathbf{v}$ being the mechanical pressure (diagonal part of the stress).^‡ (Note that the term $\mathrm{\nabla }\times \mathit{\tau }/2$, which should appear in the right-hand-side of (Eq. 7), vanishes in the case of constant $\mathit{\tau }$.) To find the excitation spectrum we linearize (Eq. 7) and use the Fourier-transform $\left[\mathbf{v},\delta \rho \right]=\int \frac{\text{d}\mathbf{k}}{{(2\pi )}^3}\left[\tilde{\mathbf{v}},\delta \tilde{\rho }\right]{\text{e}}^{i\left(\mathbf{k}·\mathbf{r}-st\right)}$, leading to:

$$\begin{array}{c}\hfill \left(\begin{array}{cccc}s+i\nu k^2& 0& 2i{\ell }_{r}k{k}_{\perp }& 0\\ 0& s+i\nu k^2& 0& 0\\ 0& 0& s+i{\nu }_L{k}^2& -kc\\ 0& 0& -kc& s\end{array}\right)\left(\begin{array}{c}{\tilde{v}}_1\\ {\tilde{v}}_2\\ {\tilde{v}}_L\\ \tilde{h}\end{array}\right)=0.\end{array}$$ [8]

Here, $\nu \equiv \eta /{\rho }_0$, ${\nu }_L\equiv 2\nu +\lambda /{\rho }_0$, ${\ell }_r\equiv {\ell }^0/(4{\rho }_0)$, ${\mathit{\ell }}^0={\ell }^0\widehat{z}$, and $h=\delta \rho /(c{\rho }_0)$ where $\rho ={\rho }_0+\delta \rho$ and $c$ is the speed of sound ($c^2=\partial P/\partial \rho$). We further define $v_{1,2}\equiv \mathbf{v}·{\widehat{\mathbf{e}}}_{1,2}$ and $v_L\equiv \mathbf{v}·\widehat{\mathbf{k}}$, where $\widehat{\mathbf{k}}=(k_x,k_y,k_z)/k$, ${\widehat{\mathbf{e}}}_1=(-k_y,k_x,0)/k_{\perp }$, and ${\widehat{\mathbf{e}}}_2=(-k_x{k}_z,-k_y{k}_z,k_{\perp }^2)/(k{k}_{\perp })$ are a set of orthonormal vectors with $k=\sqrt{{\mathbf{k}}^2}$, $k_{\perp }=\sqrt{k_x^2+k_y^2}$, and $k_{\Vert }=\mathbf{k}·{\mathit{\ell }}^0/{\ell }^0$.

This matrix is not Hermitian, but its eigenvalues can easily be found using the determinant:

$$\begin{array}{c}\hfill {\left(s+i\nu k^2\right)}^2\left(s^2-c^2{k}^2+is{\nu }_L{k}^2\right)=0.\end{array}$$ [9]

The dispersion relation is thus the one found for simple fluids (48), where there are two dissipative transverse modes with $s_T=-i\nu k^2$ and two longitudinal decaying waves that obey the usual sound waves relation $s_L=\frac{1}{2}k\left[-i{\nu }_{L}k\pm \sqrt{4c^2-{\nu }_L^2{k}^2}\right]$. The transverse eigenvectors [$\overrightarrow{v}=(v_1,v_2,v_L,h)$] are trivial [${\overrightarrow{v}}_T=(1,0,0,0);(0,1,0,0)$] while the longitudinal eigenvectors to lowest order in $g=2{\ell }_r{k}_{\perp }/c$ and $g_{\nu }=k{\nu }_L/c$ are ${\overrightarrow{v}}_L=(\mp ig,0,1,\pm 1+i{g}_{\nu }/2)$.

Interestingly, this means that although the presence of nonvanishing SAM density does not affect the dispersion relation, it creates (a nonreciprocal) coupling, which is evident in the eigenfunctions of (Eq. 8), between longitudinal and transverse modes such that a longitudinal wave is always accompanied by a transverse wave (but not vice versa).

The result of this work stands in contrast to ref. 12 that predicted the presence of SAM density itself is sufficient to produce a modified type of transverse “odd” mechanical waves. It is, nevertheless, expected that spin–spin interactions (that were neglected in this work so far) will lead to odd viscosity (that obeys Onsager reciprocal relations) (30) such that (Eq. 5c) will have an additional term $({\eta }_n^o/4){\gamma }_{ijkl;n}^o$. Because this “interaction odd viscosity” is a result of spin–spin interaction (30), which together with $\mathit{\ell }$ defines the broken symmetry direction of the system, we expect that ${\mathit{\eta }}^o\Vert \mathit{\ell }$.^§ Then, assuming that $\mathit{\tau }$, $\mathit{\ell }$, and ${\mathit{\eta }}^o$ are constants (and $\mathbf{f}=0$), the linearized equations in the Fourier space, (Eq. 8), become

$$\begin{array}{c}\hfill \left(\begin{array}{cccc}s+i\nu k^2& -i{\nu }^{o}k{k}_{\Vert }& 2ik{k}_{\perp }({\ell }_r+{\nu }^o)& 0\\ i{\nu }^{o}k{k}_{\Vert }& s+i\nu k^2& 0& 0\\ -2i{\nu }^{o}k{k}_{\perp }& 0& s+i{\nu }_L{k}^2& -kc\\ 0& 0& -kc& s\end{array}\right)\left(\begin{array}{c}{\tilde{v}}_1\\ {\tilde{v}}_2\\ {\tilde{v}}_L\\ \tilde{h}\end{array}\right)=0,\end{array}$$ [10]

where ${\nu }^o=\left({\mathit{\eta }}^o·\widehat{z}\right)/(4{\rho }_0)$. This matrix is again non-Hermitian, and when dissipation is negligible, $\nu ={\nu }_L=0$, the dimensionless eigenvalue equation is

$$\begin{array}{c}\hfill x^4-x^2\left[1+{({\tilde{\eta }}_{\Vert }^o)}^2+4{\tilde{\eta }}_{\perp }^o\left({\tilde{\eta }}_{\perp }^o+{\tilde{\ell }}_{\perp }\right)\right]+{({\tilde{\eta }}_{\Vert }^o)}^2=0,\end{array}$$ [11]

where we define $x=s/(kc)$, ${\tilde{\eta }}_{\Vert }^o={\nu }^o{k}_{\Vert }/c$, ${\tilde{\eta }}_{\perp }^o={\nu }^o{k}_{\perp }/c$, and ${\tilde{\ell }}_{\perp }={\ell }_r{k}_{\perp }/c$. The solutions for this equation can be written as $x^2=\left[C\pm \sqrt{D}\right]/2$ where $C=1+{({\tilde{\eta }}_{\Vert }^o)}^2+4{\tilde{\eta }}_{\perp }^o\left({\tilde{\eta }}_{\perp }^o+{\tilde{\ell }}_{\perp }\right)$ and $D=C^2-4{({\tilde{\eta }}_{\Vert }^o)}^2$. If the discriminant $D\ge 0$ then $x^2$ is real. If also $C>0$ then $x^2>0$ and there are four modes of nondecaying waves. When $D\ge 0$ but $C<0$, we have $x^2<0$ and $x$ is pure imaginary, such that there are two modes of nondecaying waves and two unstable modes. When the discriminant is negative $x^2$ becomes a complex number and $x=\mathfrak{R}\{x\}+i\mathfrak{I}\{x\}$ with $\mathfrak{R}\{x\}=\pm \frac{1}{2}{\left[\left(\sqrt{|D|+C^2}+C\right)\right]}^{1/2}$ and $\mathfrak{I}\{x\}=\pm \frac{1}{2}{\left[\sqrt{|D|+C^2}-C\right]}^{1/2}$, which means that there are two unstable modes with positive imaginary part. Therefore, if $D<0$ or $C<0$ the system is linearly unstable, hence the onset of instability is given by $D=0$. A special case is when ${\tilde{\eta }}_{\Vert }^o=0$, for which there are only two modes of nondecaying waves (the regular sound waves) and two zero modes.

The various instability regions are plotted in Fig. 3 in the $({\tilde{\ell }}_{\perp },{\tilde{\eta }}_{\perp }^o)$ [for (A) and (D)], $({\tilde{\ell }}_{\perp },{\tilde{\eta }}_{\Vert }^o)$ [for (B) and (E)], and $({\tilde{\eta }}_{\perp }^o,{\tilde{\eta }}_{\Vert }^o)$ [for (C) and (F)] phase-spaces. A necessary condition for instability is ${\tilde{\ell }}_{\perp }{\tilde{\eta }}_{\perp }^o<0$, such that there are two instability branches as shown in Fig. 3. In the stable regions, there are four propagating modes. When ${\tilde{\ell }}_{\perp }=0$, we recover the results of ref. 12 in which odd waves always propagate and reciprocity is restored.

Fig. 3.

Regions of instability in the $({\tilde{\ell }}_{\perp },{\tilde{\eta }}_{\perp }^o)$ [for (A) and (D)], $({\tilde{\ell }}_{\perp },{\tilde{\eta }}_{\Vert }^o)$ [for (B) and (E)], and $({\tilde{\eta }}_{\perp }^o,{\tilde{\eta }}_{\Vert }^o)$ [for (C) and (F)] phase-spaces. Instability regions are the dark gray areas in (A–C). Lines of exceptional points at which $D=0$ marks the transition, suggesting a nonreciprocal phase transition (37). A necessary condition for instability is ${\tilde{\ell }}_{\perp }{\tilde{\eta }}_{\perp }^o<0$, such that there are two instability branches. Parameters used are as follows. Top row: (A) ${\tilde{\eta }}_{\Vert }^o=-0.5$, (B) ${\tilde{\eta }}_{\perp }^o=0.5$ and (C) $\left|{\tilde{\ell }}_{\perp }\right|=0.5$; Bottom row: The lines (solid blue, dotted red, dashed black, magenta dash-dotted) correspond to (D) ${\tilde{\eta }}_{\Vert }^o=(-0.5,1,2,3)$, (E) ${\tilde{\eta }}_{\perp }^o=(-0.5,-0.1,0.1,0.5)$, and (F) ${\tilde{\ell }}_{\perp }=(0.5,0.7,1,1.2)$.

Remarkably, for any nonvanishing ${\tilde{\ell }}_{\perp }$, the lines of $D=0$ that separate stable and unstable regions are lines of exceptional points (49). Along these lines, instead of having four propagating modes (two pairs of $\pm$ eigenvalues), each pair of eigenvectors (and eigenvalues) collapse into one, leaving only two propagating modes. Recently, it has been suggested that such exceptional lines mark a new type of phase transition denoted as nonreciprocal phase transition (37). However, our system does not precisely fall within the class of theories examined in ref. 37 as here there is no spontaneous symmetry breaking and the exceptional points have nonzero eigenvalues. It is worth noting that in these exceptional points, there are two other modes, but they are not of the form of regular normal modes (see SI Appendix, section VII for details); instead, they have the following form: $\left(\mathbf{W}+(t+c)\mathbf{V}\right){\text{e}}^{i\left(\mathbf{k}·\mathbf{r}-st\right)}$, where $\mathbf{V}$ is an eigenvector of the dynamical matrix and $\mathbf{W}$ is a generalized eigenvector^¶ of the dynamical matrix in Eq. 10. These solutions grow slowly (algebraically) with time and seem unstable, but may be stabilized in the presence of viscosity such that the solution grows at short times and decays at long times, making the system stable along the exceptional lines.

Conclusions

The study of chiral active matter has increased interest in odd viscosity dramatically in recent years, from pure theoretical work (12, 13, 31, 59) to molecular dynamic simulations (30) and experimental work (7). So far, and quite surprisingly, there is no evidence that odd terms in the viscosity break Onsager relations. In our recent work (12), we proposed a microscopic model for odd viscosity in active matter, which reproduced an odd viscosity that obeys Onsager relations in the hydrodynamic momentum stress tensor. For that purpose, we have used the coarse-grained hydrodynamic momentum density, which accounts for both the CM linear momentum density and the SAM density.

In this paper, we directly coarse-grain the kinetic energy which results in somewhat different results. The reactive part of the CM stress tensor does not contain any trace of the SAM density and therefore has no odd viscosity. It trivially obeys Onsager reciprocal relations. However, the CM momentum density does affect the reactive part of the dynamics of the angular momentum density. This, by itself is a manifestation of nonreciprocity. Indeed, when writing the reactive part of the dynamics of the hydrodynamic momentum density, which includes both CM linear momentum density and SAM density, odd viscosity appears together with an odd pressure and two other terms that couple vorticity to shears involving the direction of $\mathit{\ell }$. We, therefore, conclude that the mere existence of a nonvanishing SAM density breaks Onsager reciprocal relations and gives rise to odd viscosity. Unlike our previous work (12), using our direct CG approach we find that there are no new excitations in a 3D fluid (or gas) of noninteracting spinning constituents. Instead, a longitudinal wave will always be accompanied by a transverse wave, but not vice versa, thus breaking Onsager reciprocity. To verify the discrepancy between our two CG approaches we have verified our current results using a kinetic theory.

It is expected that in sufficiently dense chiral fluids, interactions will play an increasing role. Although we did not propose a microscopic model for such interactions, recent work (30, 33) suggests that these exist and give rise to another odd viscosity that obeys Onsager reciprocal relations. Considering such an effect together with the “kinetic odd viscosity” gives rise to a non-Hermitian dynamical matrix. Analyzing the excitation spectrum of this matrix in 3D reveals regions in which waves propagate (as we found in ref. 12) and regions that are linearly unstable, suggesting an emergence of a new inhomogeneous phase. Surprisingly, the boundaries that separate these regions are densely packed surfaces of exceptional points, which may indicate the existence of a nonreciprocal phase transition (37). As there is no spontaneous symmetry breaking in our system and also because $\mathit{\ell }$ breaks both chirality and reciprocity simultaneously, our system does not seem to precisely fit within the framework of ref. 37. To further study the transition and classify it, one must go beyond the linearization scheme we used to obtain the excitation spectrum. We intend to investigate this in future work.

From a broader perspective, our results highlight the significance of using the hydrodynamic momentum density in the study of chiral active materials where angular momentum density does not vanish. In passive fluids, the CM and hydrodynamic momentum densities are essentially identical; in chiral active materials, they differ considerably. As we have shown, it is the hydrodynamic momentum that is related to actual forces. Without accounting for the stress caused by the spinning of the molecules one cannot resolve the true forces that acts on the system boundaries.

Materials and Methods

PBs for Fields.

In the main text, we show how to derive the hydrodynamic momentum dynamics from the well-known dynamics of the CM and angular momenta. Here, we detail the derivation of these equations using the PB formalism for fields (12, 45, 48, 60). We are interested in the dynamics of the coarse-grained total momentum, CM momentum, and angular momentum densities. The microscopic fields ${\widehat{\mathrm{\Phi }}}_{\mu }\left(\{{\mathbf{q}}_i^{\alpha }\},\{{\mathit{\pi }}_i^{\alpha }\},t\right)$, with $\{{\mathbf{q}}_i^{\alpha }\}$ and $\{{\mathit{\pi }}_i^{\alpha }\}$ the generalized coordinates and conjugate momenta, are then coarse-grained to give mesoscopic fields ${\mathrm{\Phi }}_{\mu }(\mathbf{r},t)={[{\widehat{\mathrm{\Phi }}}_{\mu }\left(\{{\mathbf{q}}_i^{\alpha }\},\{{\mathit{\pi }}_i^{\alpha }\},t\right)]}_c$ (the definition of ${[\widehat{\mathcal{O}}]}_c$ can be found in another section below), and the statistical mechanics of the latter is determined by the coarse-grained Hamiltonian $H[\{{\mathrm{\Phi }}_{\mu }\}]$. The reactive (nondissipative) part of the dynamics of the coarse-grained fields is found by using (12, 45, 46, 61)

$$\begin{array}{c}\hfill \frac{\partial {\mathrm{\Phi }}_{\mu }(\mathbf{r},t)}{\partial t}{|}_{\mathrm{reactive}}=-\int \text{d}{\mathbf{r}}^{\prime }\{{\mathrm{\Phi }}_{\mu }(\mathbf{r}),{\mathrm{\Phi }}_{\nu }({\mathbf{r}}^{\prime })\}\frac{\delta H}{\delta {\mathrm{\Phi }}_{\nu }({\mathbf{r}}^{\prime })},\end{array}$$ [12]

where $\{{\mathrm{\Phi }}_{\mu }(\mathbf{r}),{\mathrm{\Phi }}_{\nu }({\mathbf{r}}^{\prime })\}={[\{{\widehat{\mathrm{\Phi }}}_{\mu }(\mathbf{r}),{\widehat{\mathrm{\Phi }}}_{\nu }({\mathbf{r}}^{\prime })\}]}_c$ and

$$\begin{array}{c}\hfill \{{\widehat{\mathrm{\Phi }}}_{\mu }(\mathbf{r}),{\widehat{\mathrm{\Phi }}}_{\nu }({\mathbf{r}}^{\prime })\}={\displaystyle \sum _{\alpha i}}[\frac{\partial {\widehat{\mathrm{\Phi }}}_{\mu }(\mathbf{r})}{\partial {\pi }_i^{\alpha }}\frac{\partial {\widehat{\mathrm{\Phi }}}_{\nu }({\mathbf{r}}^{\prime })}{\partial q_i^{\alpha }}-\frac{\partial {\widehat{\mathrm{\Phi }}}_{\mu }(\mathbf{r})}{\partial q_i^{\alpha }}\frac{\partial {\widehat{\mathrm{\Phi }}}_{\nu }({\mathbf{r}}^{\prime })}{\partial {\pi }_i^{\alpha }}].\end{array}$$ [13]

Unlike the microscopic dynamics, PBs do not produce the complete mesoscopic dynamics. The CG procedure neglects many degrees of freedom, which appear as an additional dissipative term in the coarse-grained dynamics

$$\begin{array}{c}\hfill \frac{\partial {\mathrm{\Phi }}_{\mu }(\mathbf{r},t)}{\partial t}{|}_{\mathrm{dissipative}}=-\int \text{d}{\mathbf{r}}^{\prime }{\mathrm{\Gamma }}_{\mu \nu }(\mathbf{r},{\mathbf{r}}^{\prime })\frac{\delta H}{\delta {\mathrm{\Phi }}_{\nu }({\mathbf{r}}^{\prime })}.\end{array}$$ [14]

The dissipative tensor $\mathbf{\Gamma }$ is symmetric and positive semidefinite. It is generally a function of all fields $\{{\mathrm{\Phi }}_{\mu }\}$, and following Curie’s symmetry principle, it must obey the system symmetries. Moreover, ${\mathrm{\Phi }}_{\mu }$ is dissipatively coupled to ${\mathrm{\Phi }}_{\nu }$ only if they have the same sign under time reversal. This is the signature of dissipation where the flux $\partial {\mathrm{\Phi }}_{\mu }/\partial t$ has the opposite sign under time reversal from its conjugate force $\delta H/\delta {\mathrm{\Phi }}_{\nu }$.

CG the Kinetic Energy of Complex Molecules.

A necessary ingredient in the PB formalism is the knowledge of the coarse-grained Hamiltonian. In this section, we use the CG procedure that is detailed in the next section to directly coarse-grain the kinetic energy of a fluid of dumbbells (complete derivation, including for a general complex molecule is deferred to SI Appendix, section III). The microscopic kinetic Hamiltonian is

$$\begin{array}{c}\hfill H_k={\displaystyle \sum _{\alpha \mu }}\frac{{\left({\mathbf{p}}^{\alpha \mu }\right)}^2}{2m^{\alpha \mu }},\end{array}$$ [15]

where the atoms $\mu$ of molecule $\alpha$ are subjected to constraints that force them to move as a rigid body (one can consider also the vibrational motion of the atoms within the molecule, but these finite-frequency modes will not contribute to the hydrodynamics, see SI Appendix, section II). The first step before CG is to write the kinetic energy density ${\mathcal{H}}_k$ such that $H_k=\int \text{d}\mathbf{r}{\mathcal{H}}_k$:

$$\begin{array}{c}\hfill {\mathcal{H}}_k(\mathbf{r})={\displaystyle \sum _{\alpha \mu }}\frac{{\left({\mathbf{p}}^{\alpha \mu }\right)}^2}{2m^{\alpha \mu }}\delta \left(\mathbf{r}-{\mathbf{r}}^{\alpha \mu }\right).\end{array}$$ [16]

There are (at least) three ways of CG the kinetic energy: i) To ignore all rigid body constraints and use the CG method described in the next section to obtain ${\mathcal{H}}_k^{(\mathrm{i})}(\mathbf{r})={\left[\mathbf{g}(\mathbf{r})\right]}^2/(2\rho (\mathbf{r}))$, where $\mathbf{g}$ is the hydrodynamic momentum. This is what we did in ref. 12. (ii) Writing the kinetic energy of each rigid molecule using its normal zero-frequency modes and their associated generalized momenta, which are simply the CM and angular momenta (47):

$$\begin{array}{c}\hfill {\mathcal{H}}_k^{(\mathrm{ii})}(\mathbf{r})={\displaystyle \sum _{\alpha }}\left[{\left({\mathbf{p}}^{\alpha }\right)}^2/\left(2M\right)+{\left({\mathit{\ell }}^{\alpha }\right)}^2/(2I)\right]\delta \left(\mathbf{r}-{\mathbf{r}}^{\alpha }\right),\end{array}$$ [17]

where the sum is now over the molecules, ${\mathbf{p}}^{\alpha }$ is the molecule CM momentum with $M$ the molecular mass (for a fluid of dumbbells it is $M=2m$), ${\mathit{\ell }}^{\alpha }$ is the molecule SAM, and $I$ the molecular moment of inertia (for a dumbbell it is constant, $I=M{a}^2$). When coarse-grained this gives ${\mathcal{H}}_k^{(\mathrm{ii})}(\mathbf{r})={\left[{\mathbf{g}}^c(\mathbf{r})\right]}^2/(2\rho (\mathbf{r}))+{\left[\mathit{\ell }(\mathbf{r})\right]}^2/(2I)$ as was used in refs. 15, 46, and 62. iii) The third way, which is what we use hereafter, is to use the same CG that gives the hydrodynamic momentum equation, (Eq. 2), which is just taking the long wavelength limit and then using the CG method of the next section. The kinetic energy then reads (for complete derivation see SI Appendix, section I)

$$\begin{array}{c}\hfill H_k=\int \text{d}\mathbf{r}\left[\frac{{\left({\mathbf{g}}^c\right)}^2}{2\rho }+\frac{{\mathit{\ell }}^2}{2I}+\frac{1}{2}\mathrm{\nabla }·\left(\mathit{\ell }\times {\mathbf{v}}^c-{\mathbf{v}}^c·\mathbf{A}\right)\right].\end{array}$$ [18]

Importantly, the last term in (Eq. 18) is a boundary term and can therefore be ignored in the bulk. Its origin is the same as the $\mathrm{\nabla }\times \mathit{\ell }$ term in (Eq. 2); it comes from molecules that are only partially within the CG volume (SI Appendix, section I). Then, the Hamiltonian in terms of the CM momentum assumes the same form as in CG (ii) in the bulk, which is consistent with classical mechanics textbooks (47).

The CG procedure described in the next section loses information about internal correlations within the CG volume. Therefore, if such correlations do not vanish (on average) such that they cannot be treated as noise in a fluctuating hydrodynamic theory (63), the CG becomes inaccurate. This is precisely what is happening when CG the kinetic energy using method (i) as described above, which gives ${\mathcal{H}}_k^{(\mathrm{i})}={\mathbf{g}}^2/(2\rho )$. The other CGs, (ii) and (iii), only differ in boundary terms, and both consider explicitly the constraints of a rigid body by using the appropriate generalized momenta. Crucially, the difference between the various CG methods is unimportant in passive fluids, because the neglected correlations, which are related to the SAM, relax in microscopic times. However, when the SAM is driven as in chiral active materials, these correlations never decay and the distinction between the various CG methods becomes apparent.

Using the PB formalism with the Hamiltonian of (Eq. 18) gives Eqs. 3 and 4 with no odd viscosity (in the CM dynamics). Indeed, as we have shown in the main text, odd viscosity only appears in the hydrodynamic momentum stress. Intuitively this can be understood by writing the kinetic Hamiltonian in terms of the hydrodynamic momentum with the help of Eqs. 2 and 18:

$$\begin{array}{c}\hfill H_k=\int \text{d}\mathbf{r}[\frac{{\mathbf{g}}^2}{2\rho }+\frac{{\mathit{\ell }}^2}{2I}-\mathit{\ell }·\mathit{\omega }+\frac{1}{2}\mathrm{\nabla }·\left(\mathit{\ell }\times \mathbf{v}-\mathbf{v}·\mathbf{A}\right)],\end{array}$$ [19]

where $\mathit{\omega }=\frac{1}{2}\mathrm{\nabla }\times \mathbf{v}$ is the rotation vector and terms $\sim {(\mathrm{\nabla }\mathit{\ell })}^2$ were neglected. Here, the third term $\sim \mathit{\ell }·\mathit{\omega }$ is the one that was responsible for the appearance of odd viscosity in refs. 12 and 31. (The last term remains a boundary term.) This result differs from the coarse-grained Hamiltonian we used in ref. 12 in which the third term is absent. One can use directly the PBs of the hydrodynamic momentum and this Hamiltonian to derive Eqs. 5a–5c. Importantly, this Hamiltonian is written in terms of the hydrodynamic momentum and the SAM, which are dependent via (Eq. 2). The two independent fields here are the CM momentum and the SAM, which are coarse-grained from the independent molecular degrees of freedom.

Note that although the last term in Eqs. 18 and 19 can be ignored in the bulk it will affect the boundary conditions and may play a significant role in the study of surface states in fluids of rotating particles (7, 59, 64).

CG According to Paul C. Martin.

In this section, we explain in detail the CG procedure we refer to throughout this paper. Since PBs are strictly mechanistic, they do not require any ensemble average or the introduction of temperature. Here, we instead perform a strictly mechanical average. In most of the literature, smoothing $\widehat{\mathcal{O}}=\sum _{\alpha }{\mathcal{O}}^{\alpha }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha })$ is done as follows (see, e.g., ref. 65):

$$\begin{array}{c}\hfill \mathcal{O}(\mathbf{r})\equiv {\left[\widehat{\mathcal{O}}(\mathbf{r})\right]}_c=\int \text{d}{\mathbf{r}}^{\prime }W(\mathbf{r}-{\mathbf{r}}^{\prime })\widehat{\mathcal{O}}({\mathbf{r}}^{\prime })={\displaystyle \sum _{\alpha }}{\mathcal{O}}^{\alpha }W(\mathbf{r}-{\mathbf{r}}^{\alpha }),\end{array}$$ [20]

where $W$ is a smooth function such as a Gaussian or a Heaviside function that obeys $\int \text{d}\mathbf{r}W(\mathbf{r})=1$.

We, however, are not interested in this homogenization procedure (which we refer to as usual), but rather in performing spatial averages within cells of volume $\mathrm{\Delta }V$ (defined by the width of $W$); see Fig. 4. In this process of averaging, we disregard any other intracell information. This extra intracell information is the cause of (thermal) fluctuations, which can be added to our hydrodynamic equations (63). We discuss this briefly in the end of the kinetic theory derivation below. Our aim is to approximate, within such CG volume, functions of the form:

$$\begin{array}{c}\hfill \widehat{\mathcal{O}}(\mathbf{r})={\displaystyle \sum _{\alpha }}{\mathit{A}}^{\alpha }{\mathit{B}}^{\alpha }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha }),\end{array}$$ [21]

Fig. 4.

Illustration of CG the momentum density at point $\mathbf{r}$ in time $t$. Small arrows are the momentum of various molecules in the system. The large thick arrow shows the average momentum within the CG volume, which is marked by the red dashed circle.

using the coarse-grained fields $\mathit{A}(\mathbf{r})=\sum _{\alpha }{\mathit{A}}^{\alpha }W(\mathbf{r}-{\mathbf{r}}^{\alpha })$ and $\mathit{B}(\mathbf{r})=\sum _{\alpha }{\mathit{B}}^{\alpha }W(\mathbf{r}-{\mathbf{r}}^{\alpha })$. To do so, we follow Paul C. Martin ideas and replace ${\mathit{B}}^{\alpha }$ (or ${\mathit{A}}^{\alpha }$) with its average within the CG volume, $\overline{\mathit{B}}(\mathbf{r})=\mathit{B}(\mathbf{r})/n(\mathbf{r})$, where $n(\mathbf{r})\equiv \sum _{\alpha }W(\mathbf{r}-{\mathbf{r}}^{\alpha })$. With this we can write,

$$\begin{array}{c}\hfill \mathcal{O}(\mathbf{r})\simeq \overline{\mathit{B}}(\mathbf{r}){\displaystyle \sum _{\alpha }}{\mathit{A}}^{\alpha }W(\mathbf{r}-{\mathbf{r}}^{\alpha })=\frac{\mathit{A}(\mathbf{r})\mathit{B}(\mathbf{r})}{n(\mathbf{r})}.\end{array}$$ [22]

In SI Appendix, section VI, we provide simple examples and some extensions of this method.

Interestingly, if one chooses cells that are of the size of a dumbbell (or a rigid molecule in the general case), such that only one dumbbell can be within each cell, an exact relation for the microscopic fields $\widehat{\mathcal{O}}(\mathbf{r})$ can be obtained (52):

$$\begin{array}{cc}\hfill \widehat{\mathcal{O}}(\mathbf{r})& ={\displaystyle \sum _{\alpha }}{\mathit{A}}^{\alpha }{\mathit{B}}^{\alpha }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha })\frac{{\sum }_{\beta }\delta (\mathbf{r}-{\mathbf{r}}^{\beta })}{{\sum }_{\gamma }\delta (\mathbf{r}-{\mathbf{r}}^{\gamma })}\hfill \\ \hfill & =\frac{1}{\widehat{n}(\mathbf{r})}{\displaystyle \sum _{\alpha ,\beta }}{\mathit{A}}^{\alpha }{\mathit{B}}^{\beta }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha })\delta (\mathbf{r}-{\mathbf{r}}^{\beta })\hfill \\ \hfill & =\frac{\widehat{A}(r)\widehat{B}(r)}{\widehat{n}(r)},\hfill \end{array}$$ [23]

where $\widehat{\mathcal{O}}(\mathbf{r})$ denotes a microscopic field. The second line of (Eq. 23) is exact because no two molecules occupy the same cell. In fact, using this equation, we can write the Hamiltonian of (Eq. 18) in terms of the microscopic fields and calculate the microscopic PBs, thus getting the reactive part of the dynamics in terms of the microscopic fields. These microscopic dynamics assume exactly the same form as Eqs. 3–5 (but with the microscopic fields $\{{\widehat{\mathcal{O}}}_i\}$), such that odd viscosity is present even in this microscopic formulation. In the microscopic formulation, there is, however, no dissipation (in contrast to (Eq. 5a)) and $F[\rho ]$ of (Eq. 24) can no longer be thought of as free energy and it must be written in terms of the microscopic fields (see, e.g., $F$ in the kinetic theory derivation below).

Note that in ref. 52, a single atom is allowed within a cell, while here a single complex molecule (dumbbell in the simplest case) can be within a cell, which leads to the appearance of the odd terms that are related to the molecule angular momentum. If one insists on CG using cells that allow occupation of a single atom, the rigid body constraints create correlated motion between near coarse-grained volumes. This is an undesired artifact that requires the addition of forces that will maintain such constraints. Treating this is much more complicated than considering the molecules as rigid bodies and coarse-grain using cells containing single molecules as we described above.

In this paper, we are dealing with CG of a fluid of complex molecules, where we find that there is a difference between the stress associated with the total hydrodynamic momentum density and the one associated with the CM momentum density. Obviously, the hydrodynamic momentum coincides with the CM momentum for point-like particles, which is related to the fact that point-like particles cannot rotate and thus do not possess angular momentum. A more complex molecule or “shaped particle” is needed to support angular momentum. The simplest type of such molecule is a dumbbell, which is what we focus on in the main text. As noted above, in the process of CG, we disregard any deviation from the average of any field within the coarse-grained volume. When considering complex molecules, this includes any vibrational modes of the molecules, such that only the molecules’ zero modes contribute to the coarse-grained fields. Intuitively, the culling of finite-frequency modes is a result of averaging over length-scales much larger than the vibration amplitude (long wave-length) and over time-scales much longer than the vibration period (low frequency limit). Each of these averages on its own will remove any finite-frequency modes (in SI Appendix, sections I–II, we show both of these processes independently). It is, therefore, important to identify the zero modes of the molecules before CG.

Dynamics of the Hydrodynamic Momentum.

In terms of the CM momentum, the kinetic Hamiltonian of (Eq. 18) is quite standard (disregarding the boundary term) and the complete Hamiltonian is (12, 45):

$$\begin{array}{c}\hfill H=\int \text{d}\mathbf{r}\left(\frac{{\left({\mathbf{g}}^c\right)}^2}{2\rho }+\frac{{\mathit{\ell }}^2}{2I}+F[\rho (\mathbf{r})]\right),\end{array}$$ [24]

where $F[\rho ]=\int \text{d}\mathbf{r}\mathcal{F}\left(\rho ,\mathrm{\nabla }\rho \right)$ is the free energy, which in the isotropic case is only a functional of $\rho (\mathbf{r})$. The latter statement assumes that interactions are only central-force, i.e., they only depend on the CM positions of the molecules. When friction between molecules is considered, this might not be the case and other types of interactions that depend on $\mathit{\ell }$ may appear (30)—we ignore such interactions for now.

To derive the dynamics of the coarse-grained fields, we use the PBs formalism for fields (12, 45, 61) (see section above). For our specific case, we use the following fundamental PBs (all other PBs vanish):

$$\begin{array}{cc}& \{{\ell }_i(\mathbf{r}),{\ell }_j({\mathbf{r}}^{\prime })\}=-{\epsilon }_{\mathit{ijk}}{\ell }_k(\mathbf{r})\delta (\mathbf{r}-{\mathbf{r}}^{\prime }),\hfill \end{array}$$ [25a]

$$\begin{array}{cc}\hfill & \{{\ell }_i(\mathbf{r}),g_j^c({\mathbf{r}}^{\prime })\}={\ell }_i({\mathbf{r}}^{\prime }){\mathrm{\nabla }}_j\delta (\mathbf{r}-{\mathbf{r}}^{\prime }),\hfill \end{array}$$ [25b]

$$\begin{array}{cc}\hfill & \{g_i^c(\mathbf{r}),\rho ({\mathbf{r}}^{\prime })\}=\rho (\mathbf{r}){\mathrm{\nabla }}_i\delta ({\mathbf{r}}^{\prime }-\mathbf{r}),\hfill \end{array}$$ [25c]

$$\begin{array}{cc}\hfill & \{g_i^c(\mathbf{r}),g_j^c({\mathbf{r}}^{\prime })\}=g_i^c({\mathbf{r}}^{\prime }){\mathrm{\nabla }}_j\delta (\mathbf{r}-{\mathbf{r}}^{\prime })-{\mathrm{\nabla }}_i^{\prime }\left[g_j^c({\mathbf{r}}^{\prime })\delta (\mathbf{r}-{\mathbf{r}}^{\prime })\right].\hfill \end{array}$$ [25d]

Substituting these PBs into (Eq. 12) and using the Hamiltonian of (Eq. 24), we derive the reactive part of the dynamics for ${\mathbf{g}}^c$, $\mathit{\ell }$, and $\rho$:

$$\begin{array}{cc}& \dot{\rho }+\mathrm{\nabla }·{\mathbf{g}}^c=0,\hfill \end{array}$$ [26a]

$$\begin{array}{cc}\hfill & {\dot{g}}_i^c+{\mathrm{\nabla }}_j\left(v_j^c{g}_i^c\right)=-{\mathrm{\nabla }}_{i}P+f_i,\hfill \end{array}$$ [26b]

$$\begin{array}{cc}\hfill & {\dot{\ell }}_i+{\mathrm{\nabla }}_j\left(v_j^c{\ell }_i\right)={\tau }_i,\hfill \end{array}$$ [26a]

where $\mathit{f}$ and $\mathit{\tau }$ are, respectively, external force and torque densities that were added to the dynamics of linear and angular momentum. Here, $P=\rho \frac{\delta F}{\delta \rho }-\mathcal{F}$ is the thermodynamic pressure and $F=\int \text{d}\mathbf{r}\mathcal{F}$. This result is not surprising or interesting, we simply recovered the continuity equation, the reactive parts of the Navier–Stokes equation and the angular momentum density dynamics. When adding the usual dissipative terms for isotropic fluids, we obtain Eqs. 3 and 4. It is also possible to derive directly (Eq. 5) using the PB formalism and the Hamiltonian of (Eq. 19).

Kinetic Theory Derivation.

The kinetic derivation is essentially the same as the Irving–Kirkwood derivation of the equations of hydrodynamics (66), which uses the Liouville equation for the probability density and ensemble average. The only difference between the approaches lies in the type of averaging. In the Irving–Kirkwood approach an ensemble average is used, while in the kinetic theory derivation (65), an average over some small volume (CG) is used. If the coarse-grained volume contains a large number of molecules, these two averages coincide.

We start from the microscopic equation for the hydrodynamic momentum density ((Eq. 1)) and ignore the $\dot{\mathit{Q}}$ term that will vanish once the equation is coarse-grained:

$$\begin{array}{c}\hfill \widehat{\mathbf{g}}(\mathbf{r})\simeq {\displaystyle \sum _{\alpha }}\left[{\mathbf{p}}^{\alpha }+\frac{1}{2}\mathrm{\nabla }\times {\mathit{\ell }}^{\alpha }\right]\delta (\mathbf{r}-{\mathbf{r}}^{\alpha }),\end{array}$$ [27]

where as before ${\mathbf{p}}^{\alpha }=\sum _{\mu }{\mathbf{p}}^{\alpha \mu }$ and ${\mathit{\ell }}^{\alpha }=I{\mathit{\nu }}^{\alpha }\times {\dot{\mathit{\nu }}}^{\alpha }$. We remind the reader that, similarly to the main text, by writing (Eq. 27), we coarse-grain the atoms forming the molecule and treat every molecule as a point-like particle with SAM. Assuming the molecules only interact via two-body central forces [this is of course not general—we expect interactions involving mutual torques will lead to another odd viscosity term on their own (30)], we have ${\dot{\mathbf{p}}}^{c,\alpha }=-{\mathrm{\nabla }}_{\alpha }U\left(\{{\mathbf{r}}^{\beta }\}\right)+{\mathit{f}}^{\alpha }$ and ${\dot{\mathit{\ell }}}^{\alpha }={\mathit{T}}^{\alpha }$ such that

$$\begin{array}{cc}\hfill {\partial }_t{\widehat{g}}_i=& {\displaystyle \sum _{\alpha }}[f_i^{\alpha }-{\mathrm{\nabla }}_{\alpha }U\left(\{{\mathbf{r}}^{\beta }\}\right)-p_i^{\alpha }{v}_j^{\alpha }{\mathrm{\nabla }}_j\hfill \\ \hfill & -\frac{1}{2}{\epsilon }_{\mathit{ikn}}{\ell }_n^{\alpha }{v}_j^{\alpha }{\mathrm{\nabla }}_j{\mathrm{\nabla }}_k+\frac{1}{2}{\epsilon }_{\mathit{ijk}}{T}_k^{\alpha }{\mathrm{\nabla }}_j]\delta (\mathbf{r}-{\mathbf{r}}^{\alpha }).\hfill \end{array}$$ [28]

Here, $U\left(\{{\mathbf{r}}^{\beta }\}\right)=\frac{1}{2}\sum _{\alpha \ne \beta }\varphi \left(\left|{\mathbf{r}}^{\alpha }-{\mathbf{r}}^{\beta }\right|\right)$ is a general two-body potential. CG by employing (Eq. 22) and using (Eq. 2), we get

$$\begin{array}{cc}\hfill {\dot{g}}_i& +{\mathrm{\nabla }}_j(g_i{v}_j)=f_i-{\mathrm{\nabla }}_{i}P\hfill \\ \hfill & +{\mathrm{\nabla }}_j\left[-\frac{1}{2}{\ell }_n\left({\epsilon }_{\mathit{iln}}{\delta }_{\mathit{jk}}+{\epsilon }_{\mathit{jln}}{\delta }_{\mathit{ik}}\right){\mathrm{\nabla }}_l{v}_k+\frac{1}{2}{\epsilon }_{\mathit{ijk}}{\tau }_k\right],\hfill \end{array}$$ [29]

with $\mathbf{f}(\mathbf{r})\equiv \sum _{\alpha }{\mathit{f}}^{\alpha }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha })$ and $\mathit{\tau }(\mathbf{r})\equiv \sum _{\alpha }{\mathit{T}}^{\alpha }\delta (\mathbf{r}-{\mathbf{r}}^{\alpha })$ being the external force and torque densities, respectively, and $F=\int \text{d}\mathbf{r}\text{d}{\mathbf{r}}^{\prime }\left[\rho (\mathbf{r})\varphi \left(\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|\right)\rho ({\mathbf{r}}^{\prime })\right]$. The pressure is as in the main text, $P=\rho \frac{\delta F}{\delta \rho }-\mathcal{F}$, and we have dropped a nonhydrodynamic term $\sim \mathrm{\nabla }{\left(\mathrm{\nabla }\times \mathit{\ell }\right)}^2$. This result can be shown to be equivalent to that obtained in the main text from the PB approach for the standard kinetic energy (SI Appendix, section IV).

Note that within the kinetic derivation above, the thermal pressure term is absent (the pressure above is a result of molecule interactions). In some literature (e.g., ref. 65), the kinetic part of the stress (which is the thermal pressure) is written in a way similar to that of the Irving–Kirkwood kinetic stress (66), ${\sigma }_{\mathit{ij}}^K=-\sum _{\alpha }{m}^{\alpha }{\left(\frac{{\mathbf{p}}^{\alpha }}{m^{\alpha }}-\mathbf{v}\right)}_i{\left(\frac{{\mathbf{p}}^{\alpha }}{m^{\alpha }}-\mathbf{v}\right)}_{j}W(\mathbf{r}-{\mathbf{r}}^{\alpha })$. We use the fluctuating hydrodynamics approach (60, 63) in which any deviation from the average motion appears as thermal noise. In such an approach, ${\mathit{\sigma }}^K$ is absent before any ensemble average. It does appear when considering the ensemble average of the dynamics. By writing $\mathbf{v}=⟨\mathbf{v}⟩+\delta \mathbf{v}$ and $\rho =⟨\rho ⟩+\delta \rho$ (where $⟨...⟩$ denotes average over noise realizations), the streaming term becomes ${\mathrm{\nabla }}_j(g_i{v}_j)={\mathrm{\nabla }}_j\left(⟨\rho ⟩⟨v_i⟩⟨v_j⟩+⟨\rho ⟩⟨\delta v_i\delta v_j⟩\right)$, where the second term gives the ideal gas pressure as desired. Note also that by adding the ideal gas entropy $-T\int \text{d}\mathbf{r}\left[nlogn-n\right]$ to $F$ (thus making it a free-energy rather then just internal energy), the dynamics obtained (without the addition of noise) are the same as after the thermal averaging discussed above.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.