# The Flux Hypothesis for Odd Transport Phenomena

Cory Hargus  Department of Chemical and Biomolecular Engineering, University
of California, Berkeley, California 94720, USA  Université Paris Cité,
Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS, F-75205, 75205
Paris, France  Alhad Deshpande  Department of Chemical and Biomolecular
Engineering, University of California, Berkeley, California 94720, USA  Ahmad
K. Omar  Department of Materials Science and Engineering, University of
California, Berkeley, California 94720, USA  Materials Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Kranthi K. Mandadapu  Department of Chemical and Biomolecular Engineering,
University of California, Berkeley, California 94720, USA  Chemical Sciences
Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA

###### Abstract

Onsager’s regression hypothesis makes a fundamental connection between
macroscopic transport phenomena and the average relaxation of spontaneous
microscopic fluctuations. This relaxation, however, is agnostic to odd
transport phenomena, in which fluxes run orthogonal to the gradients driving
them. To account for odd transport, we generalize the regression hypothesis,
postulating that macroscopic linear constitutive laws are, on average, obeyed
by microscopic fluctuations, whether they contribute to relaxation or not.
From this “flux hypothesis,” Green-Kubo and reciprocal relations follow,
elucidating the separate roles of broken time-reversal and parity symmetries
underlying various odd transport coefficients. As an application, we derive
and verify the Green-Kubo relation for odd collective diffusion in chiral
active matter, first in an analytically-tractable model and subsequently
through molecular dynamics simulations of concentrated active spinners.

Introduction. Ongoing efforts in statistical physics have sought to generalize
the canonical theories of phase transitions, rare events, and transport
processes to include phenomena which are unique to nonequilibrium systems. A
radical example is odd transport phenomena—such as odd viscosity [1, 2, 3, 4,
5, 6, 7, 8, 9], odd diffusion [10, 11, 12], and odd thermal conduction [13]—in
which the gradient of a conserved quantity (momentum, mass, and energy,
respectively), drives a corresponding flux in the orthogonal direction.
Recently, odd transport has been shown to exist in a broad class of systems
known as “chiral active matter.” These systems, which may be biological [14,
15, 16, 17, 18] or synthetic [19, 20, 21, 22, 23], are characterized by
microscopic dynamics breaking time-reversibility and parity symmetry.

The connection between microscopic dynamics and macroscopic transport
phenomena was developed in the pioneering works of Onsager, Kubo, Casimir, and
Zwanzig, among others [24, 25, 26, 27, 28, 29, 30]. By assuming that
microscopic fluctuations about equilibrium tend to relax according to the same
conservation laws governing macroscopic transport (the regression hypothesis),
Onsager derived his celebrated reciprocal relations, showing that time-
reversal symmetry of the microscopic dynamics ensures certain macroscopic
symmetries between transport coefficients [24, 25]. Building upon Onsager’s
regression hypothesis, Kubo, Yokota, and Nakajima derived Green-Kubo relations
[27, 28] giving a quantitative prediction of these transport coefficients from
equilibrium correlations.

Recently, the regression hypothesis was applied beyond its original
equilibrium context to derive Green-Kubo relations for odd viscosity in chiral
active matter [5, 6]. Others have used nonequilibrium linear response theory
[31, 7, 32] to study the odd response of active systems to external driving.
In general, however, odd fluxes driven by spatial gradients do not fit neatly
into either approach: if they do not affect the relaxation of gradients, these
fluxes elude the regression hypothesis, yet beyond equilibrium they need not
maintain a connection (e.g. Einstein relation) to those driven by external
perturbations. Thus, certain fundamental questions remain. Namely, how do odd
transport phenomena generically arise out of chiral microscopic fluctuations,
and when is the breaking of time-reversal symmetry a necessary condition for
odd transport?

In this article we develop a general theoretical framework illuminating the
microscopic origins of odd transport phenomena and, moreover, revealing cases
where odd transport may be possible without breaking time-reversal symmetry.
By introducing the flux hypothesis, a reformulation of Onsager’s regression
hypothesis, we develop general Green-Kubo relations and reciprocal relations,
which we apply and validate in the context of odd collective diffusion of
chiral active spinners.

Linear constitutive behavior and Onsager’s regression hypothesis. Transport
phenomena occur when a system is perturbed from its resting state, which may
be thermodynamic equilibrium or, more generally, a nonequilibrium steady
state. In this work we consider linear constitutive behavior in which a flux
$\bm{J}_{i}$ is driven by spatial gradients of conserved quantities $A_{j}$:

$\bm{J}_{i}=-\sum_{j=1}^{m}\mathbf{M}_{ij}\cdot\bm{\nabla}A_{j}\,.$ (1)

Here, subscripts index over the $m$ quantities being transported (mass,
momentum, heat, etc.) such that $\mathbf{M}_{ij}\in\mathbb{R}^{d\times d}$ is
one of $m^{2}$ matrices of constant transport coefficients in $d$ dimensions.
Equation (1) encapsulates the constitutive laws of Fick, Fourier, and Newton
[33, 34, 35] as terms with $i=j$. Conversely, $i\neq j$ corresponds to cross-
couplings, for instance the Soret and Dufour effects [33], in which
temperature gradients drive mass flux and mass gradients drive heat flux,
respectively.

In the Fourier representation defined by $y^{\bm{q}}(t)=\int d\bm{r}\
e^{-\text{i}\bm{q}\cdot\bm{r}}y(\bm{r},t)$, Eq. (1) becomes

$\bm{J}_{i}^{\bm{q}}=-\sum_{j=1}^{m}\mathbf{M}_{ij}\cdot\text{i}\bm{q}A_{j}^{\bm{q}}\,,$
(2)

with wave vector $\bm{q}$. In the absence of sources or sinks,
$A_{i}^{\bm{q}}$ changes value only as it is transported by the flux
$\bm{J}_{i}^{\bm{q}}$, and thus obeys the conservation law

$\frac{d}{dt}A_{i}^{\bm{q}}=-\text{i}\bm{q}\cdot\bm{J}_{i}^{\bm{q}}\,.$ (3)

Combining Eqs. (2) and (3) yields the relaxation equation

$\frac{d}{dt}A_{i}^{\bm{q}}=-\sum_{j=1}^{m}(\bm{q}\cdot\mathbf{M}_{ij}^{\mathrm{even}}\cdot\bm{q})A_{j}^{\bm{q}}\,.$
(4)

We have explicitly labeled the even part of the transport matrix, defined here
as
$\mathbf{M}_{ij}^{\mathrm{even}}=\frac{1}{2}(\mathbf{M}_{ij}+\mathbf{M}_{ij}^{\intercal})$,
where the transpose $(\cdot)^{\intercal}$ acts on the spatial elements but not
the transport couplings (i.e. subscripts). In contrast, the odd part
$\mathbf{M}^{\mathrm{odd}}_{ij}=\frac{1}{2}(\mathbf{M}_{ij}-\mathbf{M}_{ij}^{\intercal})$
drives fluxes perpendicular to the gradient in Eq. (1), and consequently does
not enter111In Newtonian fluids, the stress $\bm{\sigma}$ and strain
$\nabla\bm{v}$ are connected by a rank-four viscosity tensor $\eta_{ijkl}$ as
$\sigma_{ij}=\eta_{ijkl}\partial_{l}v_{k}$, where all subscripts index spatial
dimensions. The odd viscosity is usually defined as
$\eta_{ijkl}^{\mathrm{odd}}=\frac{1}{2}(\eta_{ijkl}-\eta_{klij})$, and thus
only the part for which $i=k$ generates non-relaxational (i.e. divergence-
free) fluxes, while the remaining parts may affect bulk relaxation. See
Appendix E for application of the flux hypothesis to odd viscosity.  into Eq.
(4). Nevertheless, $\mathbf{M}^{\mathrm{odd}}_{ij}$ can affect the evolution
of $A_{i}^{\bm{q}}(t)$ through boundary conditions involving the fluxes
$\bm{J}_{i}^{\bm{q}}$, with significant and sometimes unexpected consequences
[36, 10, 37, 38, 39, 40].

Figure 1:  Odd collective diffusion. (a) An excited density mode
$\rho^{\bm{q}}$ of wavelength $1/|\bm{q}|$ relaxes on the timescale
$\tau_{r}\sim 1/(D^{\mathrm{c}}_{\parallel}q^{2})$. The ORH states that the
macroscopic relaxation equation for the density (solid box) is obeyed on
average by microscopic fluctuations (dotted box). (b) Underlying the
relaxation process, the density is transported by fluxes. These may include an
odd part which eludes the bulk relaxation (vertical component of the light
blue arrows). The FH states that the macroscopic constitutive law for the flux
(solid box), is obeyed on average by microscopic fluctuations (dotted box),
for both the even and odd components. (c) Typical trajectories of the chiral
Langevin model system described by Eq. (23). (d) The odd diffusivity
$D_{\perp}^{\mathrm{c}}$ of this system (dashed pink line) is obtained on
intermediate timescales at small $\bm{q}$ from the analytical solution to the
Green-Kubo relations in Eq. (25). (e) Rescaling by $\lambda$ collapses all
curves and recovers Markovian behavior as $\lambda\rightarrow 0$, motivating
Eq. (15).

To connect $\mathbf{M}^{\mathrm{even}}_{ij}$ to microscopic dynamics, Onsager
[24, 25] proposed that Eq. (4) applies not only to macroscopic transport, but
also to the average behavior of spontaneous microscopic fluctuations about a
resting state, as illustrated in Fig. 1(a). This is the Onsager Regression
Hypothesis (ORH), expressed mathematically as

$\displaystyle\begin{split}\frac{\langle\hat{A}_{i}^{\bm{q}}(\Delta
t)\rangle_{\\{\hat{A}_{j}^{\bm{q}}(0)=a_{j}^{\bm{q}}\\}}-a_{i}^{\bm{q}}}{\Delta
t}=-\sum_{j=1}^{m}(\bm{q}\cdot\mathbf{M}_{ij}^{\mathrm{even}}\cdot\bm{q})a_{j}^{\bm{q}}\,,\end{split}$
(5)

where $\hat{A}_{i}^{\bm{q}}$ indicates a fluctuating microscopic quantity
corresponding to $A_{i}^{\bm{q}}$, as obtained for example through coarse-
graining of the microscopic dynamics [41, 42]. The notation
$\langle\cdot\rangle_{\\{\hat{A}_{j}^{\bm{q}}(0)=a_{j}^{\bm{q}}\\}}$ indicates
an average over the subensemble compatible with the prescribed initial values
$\\{a_{j}^{\bm{q}}\\}$. A more practical statement of the ORH involving
correlation functions is obtained by multiplying Eq. (5) by $a_{k}^{-\bm{q}}$
and averaging $\\{a_{i}^{\bm{q}}\\}$ over the steady-state distribution of
$\\{\hat{A}_{i}^{\bm{q}}\\}$, yielding

$\displaystyle\begin{split}\frac{1}{\Delta
t}\big{[}\langle\hat{A}_{i}^{\bm{q}}(\Delta
t)\hat{A}_{k}^{-\bm{q}}(0)\rangle-\langle\hat{A}_{i}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(0)\rangle\big{]}\\\
=-\sum_{j=1}^{m}(\bm{q}\cdot\mathbf{M}_{ij}^{\mathrm{even}}\cdot\bm{q})\langle\hat{A}_{j}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(0)\rangle\,.\end{split}$
(6)

The left-hand sides of Eqs. (5) and (6) are formulated as finite-difference
time derivatives, valid for $\Delta t$ much shorter than the fastest
macroscopic relaxation time
$\tau_{r}\sim\min_{ij}1/|\bm{q}\cdot\mathbf{M}_{ij}^{\mathrm{even}}\cdot\bm{q}|$.
For Eq. (6) to reproduce the Markovian behavior of Eq. (4), however, $\Delta
t$ must also be much longer than the correlation time $\tau_{c}$, above which
the increments $\hat{A}_{i}^{\bm{q}}(t+\Delta t)-\hat{A}_{i}^{\bm{q}}(t)$ and
$\hat{A}_{j}^{\bm{q}}(t)-\hat{A}_{j}^{\bm{q}}(t-\Delta t)$ can be considered
statistically independent for all $i$ and $j$. Thus $\Delta t$ must satisfy
the separation of timescales

$\tau_{c}\ll\Delta t\ll\tau_{r}\,,$ (7)

with the upper inequality satisfied in the macroscopic limit
$\bm{q}\rightarrow\bm{0}$. With these considerations, Kubo, Yokota, and
Nakajima [28] derived from Eq. (6) the Green-Kubo relations

$\int_{0}^{\Delta t}dt\
\big{\langle}\hat{\bm{J}}_{i}^{\bm{q}}(t)\hat{\bm{J}}^{-\bm{q}}_{k}(0)\big{\rangle}^{\mathrm{even}}=\sum_{j=1}^{m}\mathbf{M}_{ij}^{\mathrm{even}}\langle\hat{A}_{j}^{\bm{q}}(0)\hat{A}^{-\bm{q}}_{k}(0)\rangle\,,$
(8)

where $\hat{\bm{J}}_{i}^{\bm{q}}$ is the microscopic flux transporting
$\hat{A}_{i}^{\bm{q}}$.

Flux hypothesis. As seen in the previous section, the ORH cannot, by
construction, yield any microscopic information about
$\mathbf{M}_{ij}^{\mathrm{odd}}$. Note, however, that the core idea behind the
ORH can in fact be decomposed into two separate assertions. Firstly,
fluctuations $\hat{A}_{i}^{\bm{q}}$ are conserved and thus obey
$\frac{d}{dt}\hat{A}_{i}^{\bm{q}}=-\text{i}\bm{q}\cdot\hat{\bm{J}}_{i}^{\bm{q}}$
exactly for any realization of the fluctuating dynamics, just as for the
macroscopic fields in Eq. (3). Secondly, fluctuations on average reproduce
macroscopic constitutive behavior. This suggests that, to access
$\mathbf{M}_{ij}^{\mathrm{odd}}$, the latter statement should be posed
independently of the former. Namely, we postulate a flux hypothesis (FH)
stating that Eq. (2) applies not only to macroscopic fluxes arising in
response to externally-driven gradients, but also to the average behavior of
microscopic fluxes arising in response to the spontaneous fluctuations of
$\hat{A}_{i}^{\bm{q}}$. More precisely,

$~{}\langle\hat{\bm{J}}^{\bm{q}}_{i}(\Delta
t)\rangle_{\\{\hat{A}_{j}^{\bm{q}}(0)=a_{j}^{\bm{q}}\\}}=-\sum_{j=1}^{m}\mathbf{M}_{ij}\cdot\text{i}\bm{q}\,a_{j}^{\bm{q}}\,,$
(9)

as illustrated in Fig. 1(b). Again, $\Delta t$ must satisfy the separation of
timescales in Eq. (7): it should be long enough that constitutive behavior is
Markovian, yet short enough that no relaxation of $\hat{A}_{j}^{\bm{q}}$ has
occurred. Multiplying by $a_{k}^{-\bm{q}}$ and averaging, as in Eq. (6),
yields

$\langle\hat{\bm{J}}_{i}^{\bm{q}}(\Delta
t)\hat{A}_{k}^{-\bm{q}}(0)\rangle=-\sum_{j=1}^{m}\mathbf{M}_{ij}\cdot\text{i}\bm{q}\langle\hat{A}_{j}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(0)\rangle\,.$
(10)

Note that contracting both sides of this equation with $\text{i}\bm{q}$ and
invoking the conservation law recovers Eq. (6). The ORH is therefore a
corollary of the FH.

Assuming stationarity, Eq. (10) then implies

$\int_{0}^{\Delta t}dt\
\langle\hat{\bm{J}}_{i}^{\bm{q}}(t)\hat{\bm{J}}_{k}^{-\bm{q}}(0)\rangle=\sum_{j=1}^{m}\mathbf{M}_{ij}\langle\hat{A}_{j}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(0)\rangle\,,$
(11)

as obtained in Appendix A. Note that Eq. (11) is identical to Eq. (8), except
that it holds for all components of $\mathbf{M}_{ij}$, both even and odd.
Finally, defining

$\displaystyle\big{[}\mathbf{g}^{-1}(\bm{q})\big{]}_{ij}$
$\displaystyle=\langle\hat{A}_{i}^{\bm{q}}(0)\hat{A}_{j}^{-\bm{q}}(0)\rangle\,,$
(12) $\displaystyle\mathbf{L}_{ij}(\bm{q},\Delta t)$
$\displaystyle=\int_{0}^{\Delta t}dt\
\langle\hat{\bm{J}}_{i}^{\bm{q}}(t)\hat{\bm{J}}_{j}^{-\bm{q}}(0)\rangle\,,$
(13)

and rearranging Eq. (11) yields the Green-Kubo relations

$\mathbf{M}_{ij}(\bm{q},\Delta t)=\sum_{k=1}^{m}\mathbf{L}_{ik}(\bm{q},\Delta
t)g_{kj}(\bm{q})\,,$ (14)

where $\mathbf{L}_{ik}$ encodes the dynamical response and $g_{kj}$ sets the
scale of static fluctuations.

Separation of timescales and self-similarity. For both even and odd transport,
there remains the ambiguity of identifying an intermediate timescale $\Delta
t$ satisfying the inequalities in Eq. (7). One possibility is to take first
the macroscopic limit $\bm{q}\rightarrow\bm{0}$ and then send $\Delta
t\rightarrow\infty$ [43, 44]. Moreover, for gradient-driven transport, self-
similarity of the time correlation functions suggests rescaling space and time
as $\bm{q}\mapsto\lambda\bm{q}$ and $\Delta t\mapsto\Delta t/\lambda^{2}$ in
$\mathbf{M}_{ij}(\bm{q},\Delta t)$, as illustrated in Figs. 1(d,e). In this
way, the macroscopic and Markovian characteristics of Eq. (2) are both
recovered at small values of the scaling parameter $\lambda$, in line with
work by Zwanzig [29]. The Green-Kubo relations in Eq. (14) may then be
evaluated as

$\mathbf{M}_{ij}=\lim_{\Delta t\rightarrow 0}\lim_{\lambda\rightarrow
0}\sum_{k=1}^{m}\mathbf{L}_{ik}(\lambda\bm{q},\Delta
t/\lambda^{2})g_{kj}(\lambda\bm{q})\,,$ (15)

noting that the limits do not commute. The inner limit recovers the Markovian,
macroscopic phenomenology of Eq. (1). The outer limit then ensures that
correlations are instantaneous with respect to the relaxation time.

Reciprocal relations. By connecting fluctuations to transport phenomena, the
Green-Kubo relations (15) reveal the microscopic symmetries involved in odd
transport. Most fundamentally, odd transport requires antisymmetry at the
level of the time correlation functions in Eq. (13). For dynamics obeying
time-reversal symmetry (TRS), this implies the reciprocal relations

$\mathbf{L}_{ij}\stackrel{{\scriptstyle\text{TRS}}}{{=}}h_{i}h_{j}\mathbf{L}_{ji}^{\intercal}\
,$ (16)

where $h_{i}\in\\{-1,1\\}$ indicates whether the flux
$\hat{\bm{J}}_{i}^{\bm{q}}$ is an even ($h_{i}=1$) or odd ($h_{i}=-1$)
function of the particle velocities. For instance, a fluid stress will be even
while a diffusive flux is odd. Onsager’s regression hypothesis leads to
$\mathbf{L}_{ij}^{\mathrm{even}}\stackrel{{\scriptstyle\text{TRS}}}{{=}}h_{i}h_{j}\mathbf{L}_{ji}^{\mathrm{even}}$,
while the flux hypothesis yields the additional odd reciprocal relations

$\mathbf{L}_{ij}^{\mathrm{odd}}\stackrel{{\scriptstyle\text{TRS}}}{{=}}-h_{i}h_{j}\mathbf{L}_{ji}^{\mathrm{odd}}\,.$
(17)

When $i=j$, we find
$\mathbf{L}_{ii}^{\mathrm{odd}}\stackrel{{\scriptstyle\text{TRS}}}{{=}}-\mathbf{L}_{ii}^{\mathrm{odd}}=0$,
indicating that TRS-breaking is indeed a requirement for self-induced odd
transport. Interestingly, odd cross terms ($i\neq j$) are in principal
possible without breaking TRS, requiring only parity symmetry breaking. This
suggests the possibility of equilibrium (i.e. passive) matter exhibiting odd
cross couplings, for instance between viscous and diffusive transport.

Application to odd collective diffusion.

Figure 2:  Collective diffusion of concentrated active spinners. (a, c) Green-
Kubo estimates of $D_{\parallel}^{\mathrm{c}}$ and $D_{\perp}^{\mathrm{c}}$
from Eq. (21) (error bars for 95 % confidence intervals) and NEMD direct
measurements from Eq. (28) (solid lines, error bars smaller than the symbol
size). In the dilute limit, $D_{\parallel}^{\mathrm{c}}=\frac{1}{2}$ and
$D_{\perp}^{\mathrm{c}}=0$, marked by black stars. (b, d). Contributions of
$\mathbf{D}^{\mathrm{self}}$ (dotted) and $\mathbf{D}^{\mathrm{distinct}}$
(solid), showing the latter to dominate in concentrated solutions
($\bar{\rho}\gtrsim 0.1$).

We now apply the preceding theoretical framework in a concrete and novel
setting involving odd collective diffusion. Application to other odd transport
phenomena proceeds analogously, as shown in Appendix E for odd thermal
conductivity and odd viscosity222Note also that while the ORH recovers a set
of six Green-Kubo relations for the eight viscosity coefficients involved in
isotropic, compressible two-dimensional chiral fluids [5], the FH offers the
advantage of recovering eight independent relations for each of the viscosity
coefficients.. Collective (or “down-gradient”) diffusion in concentrated
solutions, whether passive or active, involves the flux $\bm{J}$ of solute
relative to a stationary medium driven by gradients in its density $\rho$,
with the linear constitutive law

$\bm{J}(\bm{r},t)=-\mathbf{D}^{\mathrm{c}}\big{(}\rho\big{)}\cdot\bm{\nabla}\rho(\bm{r},t)\,,$
(18)

as depicted in Fig. 1. Unlike self-diffusion in dilute solutions, whose odd
part has been described previously [10, 11, 12], the collective diffusivity
$\mathbf{D}^{\mathrm{c}}$ depends on the solute density due to interactions
between solute particles. In two-dimensional isotropic systems,
$\mathbf{D}^{\mathrm{c}}=D^{\mathrm{c}}_{\parallel}\bm{\delta}-D^{\mathrm{c}}_{\perp}\bm{\epsilon}$,
where $\bm{\delta}=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$ and
$\bm{\epsilon}=\begin{bmatrix}0&1\\\ \text{-}1&0\end{bmatrix}$ and
$D^{\mathrm{c}}_{\parallel}$ and $D^{\mathrm{c}}_{\perp}$ are the even and odd
collective diffusivity, respectively. For small perturbations about a
homogeneous density, $\rho(\bm{r},t)=\bar{\rho}+\delta\rho(\bm{r},t)$, Eq.
(18) assumes the linear form

$\bm{J}(\bm{r},t)=-\mathbf{D}^{\mathrm{c}}(\bar{\rho})\cdot\bm{\nabla}\delta\rho(\bm{r},t)\,,$
(19)

matching Eq. (1), with $\mathbf{M}=\mathbf{D}^{\mathrm{c}}(\bar{\rho})$ and
$A=\delta\rho$.

The Green-Kubo relations in Eq. (14) are then

$\displaystyle\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t\,;\,\bar{\rho})=\frac{\int_{0}^{\Delta t}dt\
\langle\hat{\bm{J}}^{\bm{q}}(t)\hat{\bm{J}}^{-\bm{q}}(0)\rangle}{\langle\hat{\rho}^{\bm{q}}\hat{\rho}^{-\bm{q}}\rangle}\,,$
(20)

identifying
$g^{-1}(\bm{q})=\langle\hat{\rho}^{\bm{q}}\hat{\rho}^{-\bm{q}}\rangle$ with
the static structure factor and $\mathbf{L}(\bm{q},\Delta t)=\int_{0}^{\Delta
t}dt\ \langle\hat{\bm{J}}^{\bm{q}}(t)\hat{\bm{J}}^{-\bm{q}}(0)\rangle$ as the
solute flux autocorrelation function (see Appendix B.1). For $N$ solute
particles at positions $\bm{r}_{\alpha}$ with velocities $\bm{v}_{\alpha}$, we
define the microscopic fields
$\hat{\rho}(\bm{r},t)=\sum_{\alpha=1}^{N}\delta\big{(}\bm{r}-\bm{r}_{\alpha}(t)\big{)}$,
where $\langle\hat{\rho}\rangle=\bar{\rho}$, and
$\hat{\bm{J}}(\bm{r},t)=\sum_{\alpha=1}^{N}\bm{v}_{\alpha}(t)\delta\big{(}\bm{r}-\bm{r}_{\alpha}(t)\big{)}$.
Equation (20) is then

$\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t\,;\,\bar{\rho})=\frac{\sum\limits_{\alpha\beta}\int_{0}^{\Delta
t}dt\langle\bm{v}_{\alpha}(t)\bm{v}_{\beta}(0)\cos\big(\bm{q}\cdot\bm{\Delta
r}_{\alpha\beta}(t)\big{missing})\rangle}{\sum\limits_{\alpha\beta}\langle\cos\big(\bm{q}\cdot\bm{\Delta
r}_{\alpha\beta}(0)\big{missing})\rangle}\,,$ (21)

having defined $\bm{\Delta
r}_{\alpha\beta}(t):=\bm{r}_{\alpha}(t)-\bm{r}_{\beta}(0)$.

In the dilute limit, correlations vanish for $\alpha\neq\beta$. For identical
particles, Eq. (21) then simplifies to the Green-Kubo relation for self-
diffusion of a single particle

$\displaystyle\mathbf{D}^{\mathrm{self}}$ $\displaystyle=\lim_{\Delta
t\rightarrow 0}\lim_{\lambda\rightarrow 0}\int_{0}^{\Delta
t/\lambda^{2}}dt\langle\bm{v}(t)\bm{v}(0)\cos\big(\lambda\bm{q}\cdot\bm{\Delta
r}(t)\big{missing})\rangle$ $\displaystyle=\int_{0}^{\infty}dt\
\langle\bm{v}(t)\bm{v}(0)\rangle\,,$ (22)

consistent with Ref. [10], showing $D_{\perp}^{\mathrm{self}}$ to arise from
chiral random motion, as reviewed in Appendix B.2.

Chiral Langevin diffusion. From Eq. (17), it is clear that any system
exhibiting odd diffusion must break TRS. A straightforward model system
meeting this requirement consists of non-interacting particles described by
the Langevin equation

$\displaystyle\begin{split}\dot{\bm{r}}_{\alpha}(t)&=\bm{v}_{\alpha}(t)\\\
\dot{\bm{v}}_{\alpha}(t)&=-\bm{\gamma}\bm{v}_{\alpha}(t)+\bm{\eta}_{\alpha}(t)\,,\end{split}$
(23)

with the asymmetric friction
$\bm{\gamma}=\gamma_{\parallel}\bm{\delta}+\gamma_{\perp}\bm{\epsilon}$ and
thermal noise obeying
${\langle\bm{\eta}_{\alpha}(t)\bm{\eta}_{\beta}(t^{\prime})\rangle=2k_{\mathrm{B}}T\bm{\gamma}\delta_{\alpha\beta}\delta(t-t^{\prime})}$.
The odd friction $\gamma_{\perp}$ breaks TRS by inducing chiral motion (see
Fig. 1(c)), playing an identical role to the Lorentz force affecting particles
carrying charge $q$ and moving in the plane normal to a magnetic field $B$, in
which case $|\gamma_{\perp}|=|qB|$. More generally, odd friction can be a
consequence of chiral dynamics intrinsic to the solute or arising out of
interactions with the solvent [45, 32], and serves as a useful model for
detailed-balance-violating currents frequently encountered in living systems
[46, 47, 48].

Given the non-interacting nature of the particles, the self-diffusivity then
follows directly from Eq. (The Flux Hypothesis for Odd Transport Phenomena) as
$\mathbf{D}^{\mathrm{self}}=k_{\mathrm{B}}T\bm{\gamma}^{-1}$, with even and
odd components

$\displaystyle{D_{\parallel}^{\mathrm{self}}}=k_{\mathrm{B}}T\frac{\gamma_{\parallel}}{\gamma_{\parallel}^{2}+\gamma_{\perp}^{2}}\,,\hskip
7.22743pt{D_{\perp}^{\mathrm{self}}}=k_{\mathrm{B}}T\frac{\gamma_{\perp}}{\gamma_{\parallel}^{2}+\gamma_{\perp}^{2}}\,.$
(24)

Independently, the collective diffusivity
$\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta t)$ can be solved exactly (see Appendix
C, [49, 50]) yielding

$\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t)=k_{\mathrm{B}}T\bm{\gamma}^{-1}(\bm{\delta}-e^{-\bm{\gamma}\Delta
t})e^{-\frac{1}{4}q^{2}\langle|\bm{\Delta r}(\Delta t)|^{2}\rangle}\,.$ (25)

Note that $\lim_{\Delta t\rightarrow 0}\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t)=\lim_{\Delta t\rightarrow\infty}\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta t)=0$
for any finite $\bm{q}$. The correct value of $\mathbf{D}^{\mathrm{c}}$ is
attained only on intermediate timescales, as plotted in Fig. 1(d), or upon
rescaling (Fig. 1(e)) and applying the limits of Eq. (15) in the correct
order, resulting in

$\displaystyle\mathbf{D}^{\mathrm{c}}$ $\displaystyle=\lim_{\Delta
t\rightarrow 0}\lim_{\lambda\rightarrow
0}\mathbf{D}^{\mathrm{c}}(\lambda\bm{q},\Delta t/\lambda^{2})$ (26)
$\displaystyle=\lim_{\Delta t\rightarrow
0}k_{\mathrm{B}}T\bm{\gamma}^{-1}e^{-q^{2}D_{\parallel}^{\mathrm{self}}\Delta
t}=k_{\mathrm{B}}T\bm{\gamma}^{-1}\,,$

confirming that $\mathbf{D}^{\mathrm{c}}=\mathbf{D}^{\text{self}}$ for both
the even and odd components, due to the non-interacting setting.

Collective diffusion of concentrated active spinners. Odd collective diffusion
can also arise out of collisions between spinning solute particles, as
encountered in a range of physical contexts [14, 15, 16, 17, 18, 19, 20, 22,
51, 23]. Consider a system of chiral active spinners with positions
$\bm{r}_{\alpha}$ and orientations $\theta_{\alpha}$, evolving as

$\displaystyle\begin{split}\dot{\bm{r}}_{\alpha}&=\bm{v}_{\alpha}\,,\hskip
17.07164pt\dot{\theta}_{\alpha}=\Omega_{\alpha}\\\
m\dot{\bm{v}}_{\alpha}&=-\gamma\bm{v}_{\alpha}-\bm{\nabla}_{\bm{r}_{\alpha}}(V^{\mathrm{int}}+V^{\mathrm{ext}})+\bm{\eta}_{\alpha}\\\
I\dot{\Omega}_{\alpha}&=-\gamma_{\theta}\Omega_{\alpha}+\Gamma^{a}-\partial_{\theta_{\alpha}}(V^{\mathrm{int}}+V^{\mathrm{ext}})+\xi_{\alpha}\,.\end{split}$
(27)

Here, $m\bm{v}_{\alpha}(t)$ and $I\Omega_{\alpha}(t)$ are the linear and
angular momenta, with mass $m$, moment of inertia $I$, friction coefficients
$\gamma$ and $\gamma_{\theta}$, and noise obeying the fluctuation-dissipation
relations
${\langle\bm{\eta}_{\alpha}(t)\bm{\eta}_{\beta}(t^{\prime})\rangle=2k_{\mathrm{B}}T\gamma\delta_{\alpha\beta}\delta(t-t^{\prime})\bm{\delta}}$
and
${\langle\xi_{\alpha}(t)\xi_{\beta}(t^{\prime})\rangle=2k_{\mathrm{B}}T\gamma_{\theta}\delta_{\alpha\beta}\delta(t-t^{\prime})}$.
Pairwise interactions depend on the relative position and orientation of the
particles, i.e.
$V^{\mathrm{int}}=\frac{1}{2}\sum_{\alpha\neq\beta}V_{\alpha\beta}(|\bm{r}_{\alpha}-\bm{r}_{\beta}|,|\theta_{\alpha}-\theta_{\beta}|)$.
In practice, we simulate rigid dumbbells using LAMMPS software [52] (see
Appendix D for simulation details).

Chirality is imposed on the rotational motion via the active torque
$\Gamma^{a}$, whose magnitude relative to thermal forces is quantified by the
Péclet number $\mathrm{Pe}=\Gamma^{a}/k_{\mathrm{B}}T$. Chirality then enters
the translational motion through $V^{\mathrm{int}}$. In other words,
$\Gamma^{a}$ biases the interaction statistics, such that the average
collision path of two spinners is parity-asymmetric, though $V^{\mathrm{int}}$
contains no inherent asymmetry [4, 13]. Thus, in contrast to the previous
section, odd diffusion is expected to vanish in the dilute limit.

The results of computing $D_{\parallel}^{\mathrm{c}}$ and
$D_{\perp}^{\mathrm{c}}$ from the Green-Kubo relations in Eq. (20) are shown
in Figs. 2(a,c). Note that the zero-wave vector solute flux
$\hat{\bm{J}}^{\bm{q}=\mathbf{0}}=\sum_{\alpha=1}^{N}\bm{v}_{\alpha}(t)=N\bar{\bm{v}}(t)$,
where the center-of-mass velocity $\bar{\bm{v}}(t)$, loses all information
about pairwise interactions $V^{\mathrm{int}}$, while density fluctuations
also vanish at $\bm{q}=\bm{0}$. Thus, $\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t)$ can only be evaluated at small but finite $\bm{q}$, i.e. in the limit form
of Eq. (15). To validate the Green-Kubo estimates, we independently calculate
$\mathbf{D}^{\mathrm{c}}$ using nonequilibrium molecular dynamics (NEMD),
imposing a small perturbation $V^{\mathrm{ext}}(x)=-\delta V\cos(qx)$ and
measuring the steady-state density and flux, which admit the solution (see
Appendix D)

$\displaystyle\begin{split}\rho(x)&=\bar{\rho}\bigg{(}1+\frac{\delta
V}{D_{\parallel}^{\mathrm{c}}\gamma}\cos(qx)\bigg{)}\,,\\\
J_{y}&=-D_{\perp}^{\mathrm{c}}\partial_{x}\rho=\frac{D_{\perp}^{\mathrm{c}}\bar{\rho}q\delta
V}{D_{\parallel}^{\mathrm{c}}\gamma}\sin(qx)\,.\end{split}$ (28)

Note that the odd flux $J_{y}$ persists even at steady state. Figures 2(a,c)
show these values to be in good agreement with Green-Kubo predictions from Eq.
(21) across a range of densities $\bar{\rho}$ and active torques
$\mathrm{Pe}$.

The Green-Kubo formulas in Eq. (21) further allow decomposition of the
diffusion tensor as
$\mathbf{D}^{\mathrm{c}}=\mathbf{D}^{\mathrm{self}}+\mathbf{D}^{\mathrm{distinct}}$,
where the “distinct” part comprises the $\alpha\neq\beta$ terms in the
summation. These are plotted in Figs. 2(b,d), where the distinct contributions
are seen to dominate $\mathbf{D}^{\mathrm{c}}$ for $\bar{\rho}\gtrsim 0.1$.
The self-diffusivity $\mathbf{D}^{\mathrm{self}}$ diminishes at high density,
where the motion of individual spinners is increasingly impeded by collisions.
As anticipated, ${D}^{\mathrm{self}}_{\perp}$ also vanishes in the dilute
limit, thereby attaining a maximum at intermediate density.

Conclusion. Macroscopic transport coefficients are encoded in microscopic,
steady-state fluctuations, as formulated in Onsager’s regression hypothesis.
Yet the existence of odd transport phenomena, as observed in an ever-widening
range of active systems, necessitates making this connection directly at the
level of constitutive behavior, rather than bulk relaxation. The result is the
flux hypothesis, which we have invoked to obtain Green-Kubo relations, which
we subsequently validated in an application to odd collective diffusion.
Further, the ensuing reciprocal relations reveal that time-reversal and parity
symmetry breaking are requirements for odd self-induced transport, while only
the latter symmetry must be broken for odd cross-couplings. Future directions
include further validation of the flux hypothesis in nonequilibrium settings,
and applications to odd cross-couplings, in both passive and active systems.

Acknowledgements. We are grateful to Lila Sarfati, Frédéric van Wijland,
Julien Tailleur, and Karthik Shekhar for stimulating discussions. C.H. and
A.D. are supported by the National Science Foundation Graduate Research
Fellowship Program under Grant Nos. DGE 1752814 and 2146752, respectively.
C.H. also acknowledges the support of ANR grant THEMA. A.K.O. is supported by
the Laboratory Directed Research and Development Program of Lawrence Berkeley
National Laboratory under U.S. Department of Energy Contract No. DE-
AC02-05CH11231. K.K.M is supported by Director, Office of Science, Office of
Basic Energy Sciences, of the U.S. Department of Energy under contract No.
DEAC02-05CH11231. This research used resources of the National Energy Research
Scientific Computing Center (NERSC), a U.S. Department of Energy Office of
Science User Facility located at Lawrence Berkeley National Laboratory, using
NERSC award BES-ERCAP0023682.

## References

  * Avron  _et al._ [1995] J. Avron, R. Seiler, and P. G. Zograf, Viscosity of quantum hall fluids, Physical Review Letters 75, 697 (1995).
  * Avron [1998] J. Avron, Odd viscosity, Journal of Statistical Physics 92, 543 (1998).
  * Banerjee  _et al._ [2017] D. Banerjee, A. Souslov, A. G. Abanov, and V. Vitelli, Odd viscosity in chiral active fluids, Nature Communications 8, 1573 (2017).
  * Liao  _et al._ [2019] Z. Liao, M. Han, M. Fruchart, V. Vitelli, and S. Vaikuntanathan, A mechanism for anomalous transport in chiral active liquids, Journal of Chemical Physics 151, 194108 (2019).
  * Epstein and Mandadapu [2020] J. M. Epstein and K. K. Mandadapu, Time-reversal symmetry breaking in two-dimensional nonequilibrium viscous fluids, Physical Review E 101, 52614 (2020).
  * Hargus  _et al._ [2020] C. Hargus, K. Klymko, J. M. Epstein, and K. K. Mandadapu, Time reversal symmetry breaking and odd viscosity in active fluids: Green-Kubo and NEMD results, Journal of Chemical Physics 152, 201102 (2020).
  * Han  _et al._ [2021] M. Han, M. Fruchart, C. Scheibner, S. Vaikuntanathan, J. J. D. Pablo, and V. Vitelli, Fluctuating hydrodynamics of chiral active fluids, Nature Physics 17, 1260 (2021).
  * Fruchart  _et al._ [2023] M. Fruchart, C. Scheibner, and V. Vitelli, Odd viscosity and odd elasticity, Annual Review of Condensed Matter Physics 14, 471 (2023).
  * Hosaka  _et al._ [2023] Y. Hosaka, R. Golestanian, and A. Vilfan, Lorentz reciprocal theorem in fluids with odd viscosity, Physical Review Letters 131, 178303 (2023).
  * Hargus  _et al._ [2021] C. Hargus, J. M. Epstein, and K. K. Mandadapu, Odd diffusivity of chiral random motion, Physical Review Letters 127, 178001 (2021).
  * Kalz  _et al._ [2022] E. Kalz, H. D. Vuijk, I. Abdoli, J.-U. Sommer, H. Löwen, and A. Sharma, Collisions enhance self-diffusion in odd-diffusive systems, Physical Review Letters 129, 090601 (2022).
  * Muzzeddu  _et al._ [2022] P. L. Muzzeddu, H. D. Vuijk, H. Löwen, J. U. Sommer, and A. Sharma, Active chiral molecules in activity gradients, Journal of Chemical Physics 157, 134902 (2022).
  * Fruchart  _et al._ [2022] M. Fruchart, M. Han, C. Scheibner, and V. Vitelli, The odd ideal gas: Hall viscosity and thermal conductivity from non-hermitian kinetic theory, arXiv:2202.02037 (2022).
  * Diluzio  _et al._ [2005] W. R. Diluzio, L. Turner, M. Mayer, P. Garstecki, D. B. Weibel, H. C. Berg, and G. M. Whitesides, Escherichia coli swim on the right-hand side, Nature 435, 1271 (2005).
  * Friedrich and Jülicher [2007] B. M. Friedrich and F. Jülicher, Chemotaxis of sperm cells, Proceedings of the National Academy of Sciences 104, 13256 (2007).
  * Drescher  _et al._ [2009] K. Drescher, K. C. Leptos, I. Tuval, T. Ishikawa, T. J. Pedley, and R. E. Goldstein, Dancing volvox: hydrodynamic bound states of swimming algae, Physical Review Letters 102, 1 (2009).
  * Petroff  _et al._ [2015] A. P. Petroff, X.-L. Wu, and A. Libchaber, Fast-moving bacteria self-organize into active two-dimensional crystals of rotating cells, Physical Review Letters 114, 158102 (2015).
  * Tan  _et al._ [2022] T. H. Tan, A. Mietke, J. Li, Y. Chen, H. Higinbotham, P. J. Foster, S. Gokhale, J. Dunkel, and N. Fakhri, Odd dynamics of living chiral crystals, Nature 607, 287 (2022).
  * Tsai  _et al._ [2005] J.-C. Tsai, F. Ye, J. Rodriguez, J. P. Gollub, and T. Lubensky, A chiral granular gas, Physical Review Letters 94, 214301 (2005).
  * Kümmel  _et al._ [2013] F. Kümmel, B. Ten Hagen, R. Wittkowski, I. Buttinoni, R. Eichhorn, G. Volpe, H. Löwen, and C. Bechinger, Circular motion of asymmetric self-propelling particles, Physical Review Letters 110, 1 (2013).
  * Nourhani  _et al._ [2016] A. Nourhani, S. J. Ebbens, J. G. Gibbs, and P. E. Lammert, Spiral diffusion of rotating self-propellers with stochastic perturbation, Physical Review E 94, 1 (2016).
  * Soni  _et al._ [2019] V. Soni, E. S. Bililign, S. Magkiriadou, S. Sacanna, D. Bartolo, M. J. Shelley, and W. T. M. Irvine, The odd free surface flows of a colloidal chiral fluid, Nature Physics 15, 1188 (2019).
  * Ángel López-Castaño  _et al._ [2023] M. Ángel López-Castaño, A. M. Seco, A. M. Seco, Álvaro Rodríguez-Rivas, and F. V. Reyes, Spin and velocity correlations in a confined two-dimensional fluid of disk-shaped active rotors, Physics of Fluids 35, 033321 (2023).
  * Onsager [1931a] L. Onsager, Reciprocal relations in irreversible processes I., Physical Review 37, 405 (1931a).
  * Onsager [1931b] L. Onsager, Reciprocal relations in irreversible processes II., Physical Review 38, 2265 (1931b).
  * Casimir [1945] H. B. Casimir, On Onsager’s principle of microscopic reversibility, Reviews of Modern Physics 17, 343 (1945).
  * Kubo [1957] R. Kubo, Statistical-mechanical theory of irreversible processes 1. General theory and simple applications to magnetic and conduction problems, Journal of Physical Society of Japan 12, 570 (1957).
  * Kubo  _et al._ [1957] R. Kubo, M. Yokota, and S. Nakajima, Statistical-mechanical theory of irreversible processes II. Response to thermal disturbance, Journal of Physical Society of Japan 12, 1203 (1957).
  * Zwanzig [1964] R. Zwanzig, Elementary derivations of time-correlation formulas for transport coefficients, Journal of Chemical Physics 40, 2527 (1964).
  * Zwanzig [1965] R. Zwanzig, Time-correlation functions and transport coefficients in statistical mechanics, Annual Review of Physical Chemistry 16, 67 (1965).
  * Baiesi and Maes [2013] M. Baiesi and C. Maes, An update on the nonequilibrium linear response, New Journal of Physics 15, 013004 (2013).
  * Poggioli and Limmer [2023] A. R. Poggioli and D. T. Limmer, Odd mobility of a passive tracer in a chiral active fluid, Physical Review Letters 130, 158201 (2023).
  * de Groot and Mazur [1984] S. R. de Groot and P. Mazur,  _Non-Equilibrium Thermodynamics_ (Dover, 1984).
  * Deen [1998] W. M. Deen,  _Analysis of Transport Phenomena_ (Oxford University Press, 1998).
  * Bird  _et al._ [2007] R. B. Bird, W. E. Stewart, and E. N. Lightfoot,  _Transport Phenomena_ , 2nd ed. (Wiley, 2007).
  * Ganeshan and Abanov [2017] S. Ganeshan and A. G. Abanov, Odd viscosity in two-dimensional incompressible fluids, Physical Review Fluids 2, 094101 (2017).
  * Kirkinis [2023] E. Kirkinis, Null-divergence nature of the odd viscous stress for an incompressible liquid, Physical Review Fluids 8, 014104 (2023).
  * Batton and Rotskoff [2024] C. H. Batton and G. M. Rotskoff, Microscopic origin of tunable assembly forces in chiral active environments, arXiv:2310.17763 (2024).
  * Lou  _et al._ [2022] X. Lou, Q. Yang, Y. Ding, P. Liu, K. Chen, X. Zhou, F. Ye, M. Yang, and R. Podgornik, Odd viscosity-induced Hall-like transport of an active chiral fluid, Proceedings of the National Academy of Sciences 119, e2201279119 (2022).
  * Ghimenti  _et al._ [2023] F. Ghimenti, L. Berthier, G. Szamel, and F. van Wijland, Sampling efficiency of transverse forces in dense liquids, Physical Review Letters 131, 257101 (2023).
  * Irving and Kirkwood [1950] J. H. Irving and J. G. Kirkwood, The statistical mechanical theory of transport processes IV: The equations of hydrodynamics, Journal of Chemical Physics 18, 817 (1950).
  * Hardy [1982] R. J. Hardy, Formulas for determining local properties in molecular-dynamics simulations: Shock waves, Journal of Chemical Physics 76, 622 (1982).
  * Kubo  _et al._ [1991] R. Kubo, M. Toda, and N. Hashitsume,  _Statistical Physics II_, edited by M. Cardona, P. Fulde, K. Von Klitzing, H.-J. Queisser, and H. K. V. Lotsch, Springer Series in Solid-State Sciences, Vol. 31 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1991).
  * Evans and Morriss [2008] D. J. Evans and G. P. Morriss,  _Statistical Mechanics of Nonequilibrium Liquids_ (Cambridge, 2008).
  * Reichhardt and Reichhardt [2019] C. Reichhardt and C. J. O. Reichhardt, Active microrheology, Hall effect, and jamming in chiral fluids, Physical review E 100, 1 (2019).
  * Dinis  _et al._ [2012] L. Dinis, P. Martin, J. Barral, J. Prost, and J. F. Joanny, Fluctuation-response theorem for the active noisy oscillator of the hair-cell bundle, Physical Review Letters 109, 1 (2012).
  * Gnesotto  _et al._ [2018] F. S. Gnesotto, F. Mura, J. Gladrow, and C. P. Broedersz, Broken detailed balance and non-equilibrium dynamics in living systems: A review, Reports on Progress in Physics 81, 066601 (2018).
  * O’Byrne  _et al._ [2022] J. O’Byrne, Y. Kafri, J. Tailleur, and F. van Wijland, Time-(ir)reversibility in active matter: From micro to macro, Nature Reviews Physics 4, 167 (2022).
  * Chandrasekhar [1943] S. Chandrasekhar, Stochastic problems in physics and astronomy, Reviews of Modern Physics 15, 1 (1943).
  * Kampen [1981] N. G. V. Kampen,  _Stochastic Processes in Physics and Chemistry_, Vol. 36 (Elsevier, 1981).
  * Reyes  _et al._ [2022] F. V. Reyes, M. A. López-Castaño, and Álvaro Rodríguez-Rivas, Diffusive regimes in a two-dimensional chiral fluid, Communications Physics 5, 1 (2022).
  * Plimpton [1995] S. J. Plimpton, Fast parallel algorithms for short-range molecular dynamics, Journal of Computational Physics 117, 1 (1995).

Appendices

##  A Green-Kubo Relations from the Flux Hypothesis

In this Appendix we provide the details of obtaining the Green-Kubo relations
(Eq. (14) of the main text) from the Flux Hypothesis, starting with Eq. (10)
of the main text:

$\langle\hat{\bm{J}}_{i}^{\bm{q}}(\Delta
t)\hat{A}_{k}^{-\bm{q}}(0)\rangle=-\sum_{j=1}^{m}\mathbf{M}_{ij}\cdot\text{i}\bm{q}\langle\hat{A}_{j}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(0)\rangle\,.$
(A.1)

Note that the right-hand side of Eq. (A.1) preserves all information about
$\mathbf{M}_{ij}^{\text{odd}}$, since the equality must hold for all
orientations of the wave vector $\bm{q}$. The left-hand side may be expanded
as

$\langle\hat{\bm{J}}_{i}^{\bm{q}}(\Delta
t)\hat{A}_{k}^{-\bm{q}}(0)\rangle=\langle\hat{\bm{J}}_{i}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(0)\rangle+\int_{0}^{\Delta
t}\frac{d}{dt}\langle\hat{\bm{J}}_{i}^{\bm{q}}(t)\hat{A}_{k}^{-\bm{q}}(0)\rangle
dt\,.$ (A.2)

In what follows, we assume for simplicity that the static correlator
$\langle\hat{\bm{J}}_{i}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(0)\rangle=0$. The
remaining correlation function can be simplified as follows,

$\displaystyle~{}\begin{split}\frac{d}{dt}\langle\hat{\bm{J}}_{i}^{\bm{q}}(t)\hat{A}_{k}^{-\bm{q}}(0)\rangle&=\frac{d}{dt}\langle\hat{\bm{J}}_{i}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(-t)\rangle\\\
=&-\frac{d}{dt^{\prime}}\langle\hat{\bm{J}}_{i}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(t^{\prime})\rangle\big{|}_{t^{\prime}=-t}\\\
=&-\langle\hat{\bm{J}}_{i}^{\bm{q}}(0)\hat{\bm{J}}_{k}^{-\bm{q}}(-t)\rangle\cdot\text{i}\bm{q}\\\
=&-\langle\hat{\bm{J}}_{i}^{\bm{q}}(t)\hat{\bm{J}}_{k}^{-\bm{q}}(0)\rangle\cdot\text{i}\bm{q}\,,\end{split}$
(A.3)

where we have used the stationarity of the correlation function to transfer
the time derivative to the state variable $\hat{A}_{k}^{-\bm{q}}$ and applied
the conservation equation

$~{}\frac{d}{dt}\hat{A}_{i}^{\bm{q}}(t)=-\text{i}\bm{q}\cdot\hat{\bm{J}}^{\bm{q}}_{i}(t)\,.$
(A.4)

Note that Eq. (A.4) must hold not only on average but also identically for any
realization of the microscopic fluctuating variables [42]. Inputting Eq. (A.3)
into Eq. (A.1) yields

$\int_{0}^{\Delta t}dt\
\langle\hat{\bm{J}}_{i}^{\bm{q}}(t)\hat{\bm{J}}_{k}^{-\bm{q}}(0)\rangle\cdot\text{i}\bm{q}=\sum_{j=1}^{m}\mathbf{M}_{ij}\langle\hat{A}_{j}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(0)\rangle\cdot\text{i}\bm{q}\,.$
(A.5)

The fact that this equality holds for any orientation of $\bm{q}$ implies that

$\int_{0}^{\Delta t}dt\
\langle\hat{\bm{J}}_{i}^{\bm{q}}(t)\hat{\bm{J}}_{k}^{-\bm{q}}(0)\rangle=\sum_{j=1}^{m}\mathbf{M}_{ij}\langle\hat{A}_{j}^{\bm{q}}(0)\hat{A}_{k}^{-\bm{q}}(0)\rangle\,.$
(A.6)

Equation (A.6) exactly mirrors the Green-Kubo relations obtained from the
Onsager regression hypothesis (8), except that we have retained all
information about $\mathbf{M}_{ij}^{\text{odd}}$. Defining

$\big{[}\mathbf{g}^{-1}(\bm{q})\big{]}_{ij}=\langle\hat{A}_{i}^{\bm{q}}(0)\hat{A}_{j}^{-\bm{q}}(0)\rangle$
(A.7)

and

$\mathbf{L}_{ij}(\bm{q},\Delta t)=\int_{0}^{\Delta t}dt\
\langle\hat{\bm{J}}_{i}^{\bm{q}}(t)\hat{\bm{J}}_{j}^{-\bm{q}}(0)\rangle\,,$
(A.8)

where we note that $\mathbf{g}(\bm{q})$ is invertible as state variables must
be orthogonal, Eq. (A.6) then yields a closed form expression for
$\mathbf{M}_{ij}(\bm{q},\Delta t)$:

$\mathbf{M}_{ij}(\bm{q},\Delta t)=\sum_{k=1}^{m}\mathbf{L}_{ik}(\bm{q},\Delta
t)g_{kj}(\bm{q})\,,$ (A.9)

recovering Eq. (14) of the main text. Obtaining the macroscopic transport
coefficient $\mathbf{M}_{ij}$ amounts to taking the macroscopic limit
$\bm{q}\rightarrow\bm{0}$ and choosing an appropriate correlation time $\Delta
t$ of Eq. (A.9). In what follows, we work with both Eqs. (A.1) and (A.9) when
applying the flux hypothesis to collective diffusion.

##  B Green-Kubo Relations for Odd Diffusion

###  B.1 Collective Diffusion

The linear constitutive equation governing the flux of a species $\bm{J}$ in
response to density gradients $\rho$ is Fick’s Law

$\bm{J}(\bm{r},t)=-\mathbf{D}^{\mathrm{c}}(\rho)\bm{\nabla}\rho(\bm{r},t)\,,$
(B.10)

where $\rho(\bm{r},t)\in\mathbb{R}$ is the density field,
$\bm{J}(\bm{r},t)\in\mathbb{R}^{d}$ is the flux and
$\mathbf{D}^{\mathrm{c}}\in\mathbb{R}^{d\times d}$ is the diffusivity. Non-
linearity in this constitutive relation is accounted for through the density
dependence of $\mathbf{D}^{\mathrm{c}}(\rho)$. For small perturbations
$\delta\rho(\bm{r},t)$ about the constant and homogeneous density
$\bar{\rho}$, Eq. (B.10) is linearized as

$\bm{J}(\bm{r},t)=-\mathbf{D}^{\mathrm{c}}(\bar{\rho})\bm{\nabla}\delta\rho(\bm{r},t)\,.$
(B.11)

Introducing the Fourier transform $y^{\bm{q}}(t)=\int d\bm{r}\
e^{-\text{i}\bm{q}\cdot\bm{r}}y(\bm{r},t)$, Eq. (B.11) can be expressed as

$\bm{J}^{\bm{q}}(t)=-\mathbf{D}^{\mathrm{c}}(\bar{\rho})\cdot\text{i}\bm{q}\delta\rho^{\bm{q}}(t)\,,$
(B.12)

where the density modes are
$\delta\rho^{\bm{q}}=\rho^{\bm{q}}-\bar{\rho}\delta(\bm{q})$. Since we are
only interested in Fourier modes with finite $\bm{q}$, we can ignore the
$\bar{\rho}\delta(\bm{q})$ contribution, and replace $\delta\rho^{\bm{q}}(t)$
with $\rho^{\bm{q}}(t)$.

The flux hypothesis for collective diffusion, as stated from Eq. (10) of the
main text, is

$\mathbf{D}^{\mathrm{c}}\cdot\text{i}\bm{q}\langle\hat{\rho}^{\bm{q}}\hat{\rho}^{-\bm{q}}\rangle=-\langle\hat{\bm{J}}^{\bm{q}}(\Delta
t)\hat{\rho}^{-\bm{q}}(0)\rangle\,.$ (B.13)

Invoking the conservation law
$\frac{d}{dt}\hat{\rho}^{\bm{q}}=-\text{i}\bm{q}\cdot\hat{\bm{J}}^{\bm{q}}$
together with stationarity, as in Appendix A, leads directly to the Green-Kubo
relations

$\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t)=\langle\delta\hat{\rho}^{\bm{q}}\delta\hat{\rho}^{-\bm{q}}\rangle^{-1}\int_{0}^{\Delta
t}dt\langle\hat{\bm{J}}^{\bm{q}}(t)\hat{\bm{J}}^{-\bm{q}}(0)\rangle\,,$ (B.14)

mirroring Eq. (14) of the main text.

For an $N$-particle system with positions $\\{\bm{r}_{\alpha}\\}$ and
velocities $\\{\bm{v}_{\alpha}\\}$, the flux and density fields are defined
through coarse-graining [41, 42] as

$\displaystyle\hat{\rho}(\bm{r},t)$
$\displaystyle:=\sum_{\alpha=1}^{N}\delta(\bm{r}-\bm{r}_{\alpha})\,,$ (B.15)
$\displaystyle\hat{\bm{J}}(\bm{r},t)$
$\displaystyle:=\sum_{\alpha=1}^{N}\bm{v}_{\alpha}\delta(\bm{r}-\bm{r}_{\alpha})\,,$
(B.16)

with corresponding Fourier modes given by

$\displaystyle\hat{\rho}^{\bm{q}}(t)$
$\displaystyle=\sum_{\alpha=1}^{N}e^{-\text{i}\bm{q}\cdot\bm{r}_{\alpha}(t)}\,,$
(B.17) $\displaystyle\hat{\bm{J}}^{\bm{q}}(t)$
$\displaystyle=\sum_{\alpha=1}^{N}\bm{v}_{\alpha}(t)e^{-\text{i}\bm{q}\cdot\bm{r}_{\alpha}(t)}\,.$
(B.18)

Equation (B.13) is then

$\mathbf{D}^{\mathrm{c}}\cdot\text{i}\bm{q}\sum_{\alpha\beta}\langle
e^{-\text{i}\bm{q}\cdot(\bm{r}_{\alpha}-\bm{r}_{\beta})}\rangle=-\sum_{\alpha\beta}\big{\langle}\bm{v}_{\alpha}(\Delta
t)e^{-\text{i}\bm{q}\cdot(\bm{r}_{\alpha}(\Delta
t)-\bm{r}_{\beta}(0))}\big{\rangle}\,.$ (B.19)

Recalling that $\mathbf{D}^{\mathrm{c}}$ is real, Eq. (B.13) may be
reexpressed as

$\displaystyle\begin{split}\mathbf{D}^{\mathrm{c}}\cdot\bm{q}&=-\imaginary[\langle\hat{\rho}^{\bm{q}}\hat{\rho}^{-\bm{q}}\rangle^{-1}\langle\hat{\bm{J}}^{\bm{q}}(\Delta
t)\hat{\rho}^{-\bm{q}}(0)\rangle\big{]}\\\
&=\frac{\sum_{\alpha\beta}\langle\bm{v}_{\alpha}(\Delta
t)\sin\big(\bm{q}\cdot[\bm{r}_{\alpha}(\Delta
t)-\bm{r}_{\beta}(0)]\big{missing})\rangle}{\sum_{\alpha\beta}\langle\cos\big(\bm{q}\cdot[\bm{r}_{\alpha}-\bm{r}_{\beta}]\big{missing})\rangle}\,,\end{split}$
(B.20)

where $\imaginary[\bm{y}]$ denotes the imaginary part of
$\bm{y}\in\mathbb{C}^{d}$. Here we have used the fact that the density-density
correlation $\langle\rho^{\bm{q}}\rho^{-\bm{q}}\rangle$ is real, as all
particles are indistinguishable and thus the system is identical under
exchange of particles $\alpha$ and $\beta$. With these arguments, the Green-
Kubo relation in Eq. (B.14) leads to

$\mathbf{D}^{\mathrm{c}}=\frac{\sum_{\alpha\beta}\int_{0}^{\Delta t}dt\
\langle\bm{v}_{\alpha}(t)\bm{v}_{\beta}(0)\cos\big(\bm{q}\cdot[\bm{r}_{\alpha}(t)-\bm{r}_{\beta}(0)]\big{missing})\rangle}{\sum_{\alpha\beta}\langle\cos\big(\bm{q}\cdot[\bm{r}_{\alpha}-\bm{r}_{\beta}]\big{missing})\rangle}\,.$
(B.21)

In the dilute limit, the terms in the sums for which $\alpha\neq\beta$ vanish,
and Eq. (B.19) reduces to

$\mathbf{D}^{\mathrm{c}}\cdot\text{i}\bm{q}\stackrel{{\scriptstyle\text{dilute}}}{{=}}-\langle\bm{v}(\Delta
t)e^{-\text{i}\bm{q}\cdot\bm{\Delta r}(\Delta t)}\rangle\,,$ (B.22)

or, taking the imaginary part of both sides,

$\mathbf{D}^{\mathrm{c}}\cdot\bm{q}\stackrel{{\scriptstyle\text{dilute}}}{{=}}\langle\bm{v}(\Delta
t)\sin\big(\bm{q}\cdot\bm{\Delta r}(\Delta t)\big{missing})\rangle\,.$ (B.23)

Here, $\bm{\Delta r}_{\alpha}(\Delta t)=\bm{r}_{\alpha}(\Delta
t)-\bm{r}_{\alpha}(0)$, and where we have assumed for simplicity that all
particles are identical, thus removing the summations. The corresponding
Green-Kubo relations, obtained by differentiating and then integrating in
time, are

$\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t)\stackrel{{\scriptstyle\text{dilute}}}{{=}}\int_{0}^{\Delta t}dt\
\bigg{\langle}\bm{v}(t)\otimes\bm{v}(0)\cos\big(\bm{q}\cdot\bm{\Delta
r}(t)\big{missing})\bigg{\rangle}\,,$ (B.24)

Using the self-similar form of Eq. (15), we find the macroscopic collective
diffusion to be

$\displaystyle\begin{split}\mathbf{D}^{\mathrm{c}}&\stackrel{{\scriptstyle\text{dilute}}}{{=}}\lim_{\Delta
t\rightarrow 0}\lim_{\lambda\rightarrow 0}\int_{0}^{\Delta t/\lambda^{2}}dt\
\bigg{\langle}\bm{v}(t)\bm{v}(0)\cos\big(\lambda\bm{q}\cdot\bm{\Delta
r}(t)\big{missing})\bigg{\rangle}\\\ &=\lim_{\Delta t\rightarrow
0}\lim_{\lambda\rightarrow 0}\int_{0}^{\Delta t/\lambda^{2}}dt\
\bigg{\langle}\bm{v}(t)\bm{v}(0)\big{[}1-\frac{1}{2}\lambda^{2}\bm{q}\cdot\bm{\Delta
r}(t)\bm{\Delta
r}(t)\cdot\bm{q}+\mathcal{O}(\lambda^{4})\big{]}\bigg{\rangle}\\\
=\int_{0}^{\infty}dt\
\langle\bm{v}(t)\bm{v}(0)\rangle&-\frac{1}{2}\lim_{\Delta t\rightarrow
0}\lim_{\lambda\rightarrow 0}\bigg{[}\lambda^{2}\int_{0}^{\tau_{c}}dt\
\bigg{\langle}\bm{v}(t)\bm{v}(0)\bm{q}\cdot\bm{\Delta r}(t)\bm{\Delta
r}(t)\cdot\bm{q}\bigg{\rangle}\\\ &\hskip
51.21504pt+\lambda^{2}\int_{\tau_{c}}^{\Delta t/\lambda^{2}}dt\
\big{\langle}\bm{v}(t)\bm{v}(0)\big{\rangle}\bm{q}\cdot\big{\langle}\bm{\Delta
r}(t)\bm{\Delta r}(t)\big{\rangle}\cdot\bm{q}\bigg{]}\\\
&=\int_{0}^{\infty}dt\ \langle\bm{v}(t)\bm{v}(0)\rangle\,,\end{split}$ (B.25)

recovering the Green-Kubo relation for self-diffusion derived in our previous
work [10]. In deriving Eq. (B.25), on the second line, the cosine term has
been expanded in $\lambda$. On the following line, the zeroth order term in
the expansion yields the expected Green-Kubo relation, with the upper limit of
integration going to $\infty$ upon applying the limit $\lambda\rightarrow 0$.
The remaining integral is then separated into an integral for $t\leq\tau_{c}$,
and another for $t>\tau_{c}$. The former integral is finite, and thus vanishes
under the limit $\lambda\rightarrow 0$. In the latter, where $t>\tau_{c}$, the
velocities become uncorrelated with the squared displacement. With the
assumption that $\langle\bm{v}(t)\bm{v}(0)\rangle=0$ for $t>\tau_{c}$, this
integral also vanishes. More precisely, because the mean squared displacement
$\langle\bm{\Delta r}(t)\bm{\Delta r}(t)\rangle$ grows linearly with $t$, the
velocity correlation $\langle\bm{v}(t)\bm{v}(0)\rangle=0$ must decay faster
than $1/t$ as $t\rightarrow\infty$, which is already the criterion for
integrability of the velocity autocorrelation function.

###  B.2 Self-Diffusion

Macroscopic transport phenomena typically arise out of complex, many-body
microscopic dynamics, which generally do not admit an exact solution, but can
instead be understood in terms of statistical averages and fluctuations. In
this way, the flux hypothesis provides a necessary connection between
microscopic dynamics and macroscopic constitutive phenomena, providing a
starting place for deriving Green-Kubo relations, just as Onsager’s regression
hypothesis has done for the subset of transport phenomena excluding odd
transport.

Dilute self-diffusion, however, can be understood in terms of the stochastic
dynamics of a single solute particle. Accordingly, as previously shown in
[10], Green-Kubo relations for dilute self-diffusivity (including odd
diffusivity) can be derived without the flux hypothesis, by assuming only
separation of timescales, as in Eq. (7). In this appendix, we provide a self-
contained derivation of the Green-Kubo relation for the dilute odd
diffusivity, following Ref. [10], which does not require invoking the flux
hypothesis. Our motivation is to highlight parallels between this derivation
and the more general one which uses the flux hypothesis, as demonstrated in
the following appendix for odd collective diffusion.

To set the stage, we first show the canonical derivation of the mean-squared
displacement relation for the diffusivity, which neglects the odd part. This
canonical derivation considers the time evolution of the probability density
$P(\bm{r},t|\bm{r}_{0})$ of the particle being at position $\bm{r}$ at time
$t$ conditioned on having started at $\bm{r}_{0}$ at time $0$. This
probability is then expanded by considering all sources of incoming
probability from a distance $\bm{\Delta r}$ away at a time $\Delta t$ in the
past

$P(\bm{r},t+\Delta t|\bm{r}_{0})=\int d\bm{\Delta r}P(\bm{r}-\bm{\Delta
r},t|\bm{r}_{0})\Pi(\bm{\Delta r};\Delta t)\,,$ (B.26)

where $\Pi(\bm{\Delta r};\Delta t)$ is the transition probability, i.e. the
probability of the particle undergoing a displacement $\bm{\Delta r}$ in the
time interval $\Delta t$. The Markovian form of Eq. (B.26) requires $\Delta
t\gg\tau_{c}$ where $\tau_{c}$ is the timescale on which increments of the
particle motion are correlated. Assuming now that $\Delta t$ also satisfies
$\Delta t\ll\tau_{r}$, as in Eq. (7), we can expand the LHS of Eq. (B.26) to
first order in time and the RHS to second order in space, yielding

$\displaystyle\begin{split}&P(\bm{r},t|\bm{r}_{0})+\Delta
t\partial_{t}P(\bm{r},t|\bm{r}_{0})\\\ &=\int d\bm{\Delta
r}\left[P(\bm{r},t|\bm{r}_{0})-\bm{\Delta
r}\cdot\bm{\nabla}P(\bm{r},t|\bm{r}_{0})+\frac{1}{2}\bm{\Delta
r}\cdot\bm{\nabla}\bm{\nabla}P(\bm{r},t|\bm{r}_{0})\cdot\bm{\Delta
r}\right]\Pi(\bm{\Delta r};\Delta t)+\mathcal{O}(\bm{\Delta r}^{3},\Delta
t^{2})\\\ &=P(\bm{r},t|\bm{r}_{0})-\langle\bm{\Delta r}(\Delta
t)\rangle\cdot\bm{\nabla}P(\bm{r},t|\bm{r}_{0})+\frac{1}{2}\langle\bm{\Delta
r}(\Delta t)\bm{\Delta r}(\Delta
t)\rangle:\bm{\nabla}\bm{\nabla}P(\bm{r},t|\bm{r}_{0})+\mathcal{O}(\bm{\Delta
r}^{3},\Delta t^{2})\,.\end{split}$ (B.27)

Dividing both sides by $\Delta t$ and neglecting the higher order terms then
yields the relaxation equation

$\partial_{t}P(\bm{r},t|\bm{r}_{0})=-\bar{\bm{v}}\cdot\bm{\nabla}P(\bm{r},t|\bm{r}_{0})+\bm{\nabla}\cdot\mathbf{D}^{\mathrm{self,\,even}}\cdot\bm{\nabla}P(\bm{r},t|\bm{r}_{0})\,,$
(B.28)

where $\bar{\bm{v}}=\langle\bm{\Delta r}(\Delta t)\rangle/\Delta t$ is the
drift velocity and the symmetric (i.e. even) part of the diffusivity tensor
obeys

$\displaystyle\begin{split}\mathbf{D}^{\mathrm{self,\,even}}&=\lim_{\Delta
t\rightarrow\infty}\frac{1}{2\Delta t}\langle\bm{\Delta r}(\Delta
t)\otimes\bm{\Delta r}(\Delta t)\rangle^{\mathrm{even}}\\\
&=\int_{0}^{\infty}dt\
\langle\bm{v}(t)\otimes\bm{v}(0)\rangle^{\mathrm{even}}\,,\end{split}$ (B.29)

upon sending $\Delta t\rightarrow\infty$. The first equality is the well-known
mean-squared displacement formula [50]. In the final equality, defining
$\bm{\Delta r}=\int_{0}^{\Delta t}dt\ \bm{v}(t)$ obtains the corresponding
Green-Kubo formula for the even part of the diffusivity.

The time evolution of $P(\bm{r},t|\bm{r}_{0})$, however, is agnostic to odd
diffusion. To derive the Green-Kubo relation for the odd self-diffusivity, we
must instead consider the fluxes—i.e. the constitutive behavior—directly, as
in [10]. We now consider the joint probability density
$P(\bm{r},\bm{v},t|\bm{r}_{0})$, of the particle being at $\bm{r}$ with
velocity $\bm{v}$ at time $t$, conditioned on having been at $\bm{r}_{0}$ at
time $0$. The evolution of the density $\rho(\bm{r},t|\bm{r}_{0})$ and the
flux $\bm{J}(\bm{r},t|\bm{r}_{0})$, given that the particle started at
$\bm{r}_{0}$, are then the zeroth and first moments of the joint probability
density with respect to the velocity:

$\displaystyle\rho(\bm{r},t|\bm{r}_{0})$ $\displaystyle=\int d\bm{v}\
P(\bm{r},\bm{v},t|\bm{r}_{0})\,,$ (B.30)
$\displaystyle\bm{J}(\bm{r},t|\bm{r}_{0})$ $\displaystyle=\int d\bm{v}\
\bm{v}P(\bm{r},\bm{v},t|\bm{r}_{0})\,.$ (B.31)

We now introduce into the joint probability the variable $\bm{r}^{\prime}$
representing the position of the particle at time $t-\Delta t$, and
marginalize it back out; that is, ${P(\bm{r},\bm{v},t|\bm{r}_{0})=\int
d\bm{r}^{\prime}\ P(\bm{r},\bm{v},t;\bm{r}^{\prime},t-\Delta t|\bm{r}_{0})}$.
The expression for the flux is then

$\displaystyle\begin{split}\bm{J}(\bm{r},t|\bm{r}_{0})&=\iint d\bm{v}\
d\bm{r}^{\prime}\ \bm{v}P(\bm{r},\bm{v},t;\bm{r}^{\prime},t-\Delta
t|\bm{r}_{0})\\\ &=\iint d\bm{v}\ d\bm{r}^{\prime}\
\bm{v}\rho(\bm{r}^{\prime},{t-\Delta
t}|\bm{r}_{0})P(\bm{r},\bm{v},t|\bm{r}^{\prime},t-\Delta
t;\bm{r}_{0})\,,\end{split}$ (B.32)

where the second equality uses the factorization property of the conditional
probability and the third replaces $P(\bm{r}^{\prime},t-\Delta t|\bm{r}_{0})$
with $\rho(\bm{r}^{\prime},{t-\Delta t}|\bm{r}_{0})$ from Eq. (B.30). So far,
no assumptions have been made on the dynamics of the particle. We now require
$\Delta t\gg\tau_{c}$ (where $\tau_{c}$ is the timescale on which the random
motion is correlated with itself) such that the dynamics are Markovian. That
is,

$P(\bm{r},\bm{v},t|\bm{r}^{\prime},t-\Delta
t;\bm{r}_{0})=P(\bm{r},\bm{v},t|\bm{r}^{\prime},t-\Delta t)=:\Pi(\bm{\Delta
r},\bm{v};\Delta t)$ (B.33)

is the transition probability of the particle undergoing a displacement
$\bm{\Delta r}=\bm{r}-\bm{r}^{\prime}$ and ending with velocity $\bm{v}$ in a
time interval $\Delta t$, and is independent of $\bm{r}_{0}$. Making the
change of variables $\bm{\Delta r}=\bm{r}-\bm{r}^{\prime}$, this leaves

$\displaystyle\begin{split}\bm{J}(\bm{r},t|\bm{r}_{0})&=\iint d\bm{v}\
d\bm{\Delta r}\ \bm{v}\rho(\bm{r}-\bm{\Delta r},t-\Delta
t|\bm{r}_{0})\Pi(\bm{\Delta r},\bm{v};\Delta t)\\\ &=\iint d\bm{v}\
d\bm{\Delta r}\ \bm{v}\big{[}\rho(\bm{r},t|\bm{r}_{0})-\bm{\Delta
r}\cdot\bm{\nabla}\rho(\bm{r},t|\bm{r}_{0})\big{]}\Pi(\bm{\Delta
r},\bm{v};\Delta t)+\mathcal{O}(\bm{\Delta r}^{2},\Delta t)\\\
&=\langle\bm{v}\rangle\rho(\bm{r},t|\bm{r}_{0})-\langle\bm{v}\otimes\bm{\Delta
r}\rangle\cdot\bm{\nabla}\rho(\bm{r},t|\bm{r}_{0})+\mathcal{O}(\bm{\Delta
r}^{2},\Delta t)\\\
&=\bar{\bm{v}}\rho(\bm{r},t|\bm{r}_{0})-\mathbf{D}^{\mathrm{self}}\cdot\bm{\nabla}\rho(\bm{r},t|\bm{r}_{0})+\mathcal{O}(\bm{\Delta
r}^{2},\Delta t)\,.\end{split}$ (B.34)

In the second equality, $\rho(\bm{r}-\bm{\Delta r},t-\Delta t|\bm{r}_{0})$ has
been expanded to first order in the position and zeroth order in time, under
the assumption that $\Delta t\ll\tau_{r}$, thus requiring the separation of
timescales as in Eq. (7). The final line reveals a drift term
$\bar{\bm{v}}=\langle\bm{v}\rangle$ and the diffusivity

$\displaystyle\begin{split}\mathbf{D}^{\mathrm{self}}&=\lim_{t\rightarrow\infty}\langle\bm{v}(t)\otimes\bm{\Delta
r}(t)\rangle\\\ &=\lim_{t\rightarrow\infty}\int_{0}^{t}dt^{\prime}\
\langle\bm{v}(t)\otimes\bm{v}(t^{\prime})\rangle\\\
&=\lim_{t\rightarrow\infty}\int_{0}^{t}dt^{\prime}\
\langle\bm{v}(t-t^{\prime})\otimes\bm{v}(0)\rangle\\\ &=\int_{0}^{\infty}dt\
\langle\bm{v}(t)\otimes\bm{v}(0)\rangle\,.\end{split}$ (B.35)

The final equality gives the desired Green-Kubo relation, which follows from
invoking stationarity in the third equality and performing a change of
variables in the final equality. Note that Eq. (B.35) holds for both the
antisymmetric (odd) as well as the symmetric (even) part of $\mathbf{D}$.
Considering a two-dimensional isotropic system where the diffusivity takes the
form $\mathbf{D}=D_{\parallel}\bm{\delta}-D_{\perp}\bm{\epsilon}$, we arrive
at the Green-Kubo relations

$\displaystyle D^{\mathrm{self}}_{\parallel}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}dt\
\langle\bm{v}(t)\cdot\bm{v}(0)\rangle=\lim_{t\rightarrow\infty}\frac{1}{4t}\langle|\bm{\Delta
r}(t)|^{2}\rangle\,,$ (B.36) $\displaystyle D^{\mathrm{self}}_{\perp}$
$\displaystyle=-\frac{1}{2}\int_{0}^{\infty}dt\
\langle\bm{v}(t)\bm{v}(0)\rangle:\bm{\epsilon}$ (B.37)
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}dt\
\langle\bm{v}_{y}(t)\bm{v}_{x}(0)\rangle-\langle\bm{v}_{x}(t)\bm{v}_{y}(0)\rangle\,,$

Equation (B.37) reveals the odd diffusivity to be nonzero only for dynamics
breaking time-reversal and parity symmetries, i.e. chiral random motion.
Finally, note that while $D^{\mathrm{self}}_{\parallel}$ can be expressed in
terms of the mean squared displacement, as in Eq. (B.29), no such relationship
exists for the odd diffusivity $D^{\mathrm{self}}_{\perp}$ due to its inherent
antisymmetry.

##  C Dilute Inertial Langevin Dynamics

###  C.1 One Dimension

In this appendix, we shall derive the exact form of the correlation function
in Eq. (B.22) for dilute Langevin diffusion. We then show that carrying out
the limits in Eq. (15) yields the standard Green-Kubo expression for self-
diffusion. Before proceeding to the two-dimensional case where odd collective
diffusion is possible, we briefly solve for the correlation function in Eq.
(B.22) in a one-dimensional system of particles subject to the underdamped
Langevin dynamics

$\displaystyle\dot{x}_{\alpha}(t)$ $\displaystyle=v_{\alpha}(t)\,,$ (C.38)
$\displaystyle\dot{v}_{\alpha}(t)$ $\displaystyle=-\gamma
v_{\alpha}(t)+\eta_{\alpha}(t)\,,$ (C.39)

where $\gamma$ is the friction coefficient and $\eta_{\alpha}(t)$ represents
Gaussian white noise satisfying $\langle\eta_{\alpha}(t)\rangle=0$ and
$\langle\eta_{\alpha}(t)\eta_{\beta}(t^{\prime})\rangle=2\gamma
k_{\mathrm{B}}T\delta(t-t^{\prime})\delta_{\alpha\beta}$ [49, 50].

Solving Eqs. (C.38) and (C.39) for the particle displacement yields

$\displaystyle\begin{split}\Delta x(\Delta t)&=\int_{0}^{\Delta t}dt\ v(t)\\\
&=\int_{0}^{\Delta t}dt\ \bigg{[}e^{-\gamma t}v(0)+\int_{0}^{t}dt^{\prime}\
e^{-\gamma(t-t^{\prime})}\eta(t^{\prime})\bigg{]}\\\
&=\big{(}1-e^{-\gamma\Delta t}\big{)}\frac{v(0)}{\gamma}+\int_{0}^{\Delta
t}dt\ \frac{1}{\gamma}\big{(}1-e^{-\gamma(\Delta
t-t)}\big{)}\eta(t)\,.\end{split}$ (C.40)

where for clarity we have removed particle index $\alpha$, as the particles
are identical and non-interacting. The final equality above is obtained by
changing the order of integration. Now, using $\langle\eta(t)v(0)\rangle=0$
and the equilibrium equipartition identity
$\langle|v(0)|^{2}\rangle=k_{\mathrm{B}}T$, the mean squared displacement is

$\displaystyle\begin{split}\langle|\Delta x(\Delta
t)|^{2}\rangle&=\frac{k_{\mathrm{B}}T}{\gamma^{2}}(1-e^{-\gamma\Delta
t})^{2}+\int_{0}^{\Delta t}dt\int_{0}^{\Delta t}dt^{\prime}\
\frac{1}{\gamma^{2}}(1-e^{-\gamma(\Delta t-t)})(1-e^{-\gamma(\Delta
t-t^{\prime})})\langle\eta(t)\eta(t^{\prime})\rangle\\\
&=\frac{k_{\mathrm{B}}T}{\gamma^{2}}(1-e^{-\gamma\Delta
t})^{2}+\frac{2k_{\mathrm{B}}T}{\gamma}\int_{0}^{\Delta t}dt\
(1-e^{-\gamma(\Delta t-t)})^{2}\\\
&=\frac{2k_{\mathrm{B}}T}{\gamma^{2}}\big{[}\gamma\Delta t+e^{-\gamma\Delta
t}-1\big{]}\,,\end{split}$ (C.41)

where the self-diffusivity is given by

$D^{\mathrm{self}}:=\lim_{\Delta t\rightarrow\infty}\frac{1}{2\Delta
t}\langle|\Delta x(\Delta t)|^{2}\rangle=\frac{k_{\mathrm{B}}T}{\gamma}\,.$
(C.42)

The correlation function in Eq. (B.22) can be expressed as the time derivative
of the moment generating function

$D^{\mathrm{c}}(q,\Delta t)=-\frac{1}{\text{i}q}\langle v(\Delta
t)e^{-\text{i}q\Delta x(\Delta
t)}\rangle=-\frac{1}{q^{2}}\frac{\partial}{\partial\Delta t}\langle
e^{-\text{i}q\Delta x(\Delta t)}\rangle\,.$ (C.43)

A cumulant expansion yields

$\langle e^{-\text{i}q\Delta x(\Delta
t)}\rangle=\exp\\{-\frac{1}{2}q^{2}\big{\langle}|\Delta x(\Delta
t)|^{2}\big{\rangle}\bigg{\\}}\,,$ (C.44)

where only the second cumulant remains due to the Gaussian nature of the noise
$\eta(t)$. Putting together Eqs. (C.41), (C.43) and (C.44) results in

$\displaystyle\begin{split}D^{\mathrm{c}}(q,\Delta
t)&=-\frac{1}{q^{2}}\frac{\partial}{\partial\Delta
t}\exp\\{-\frac{1}{2}q^{2}\langle|\Delta x(\Delta t)|^{2}\rangle\bigg{\\}}\\\
&=\frac{k_{\mathrm{B}}T}{\gamma}[1-e^{-\gamma\Delta
t}]\exp\\{-q^{2}\frac{k_{\mathrm{B}}T}{\gamma^{2}}\big{[}\gamma\Delta
t+e^{-\gamma\Delta t}-1\big{]}\bigg{\\}}\,.\end{split}$ (C.45)

Rescaling by $\lambda$ and evaluating the limit $\lambda\rightarrow 0$
recovers Markovian constitutive behavior

$\displaystyle\begin{split}\lim_{\lambda\rightarrow 0}D^{\mathrm{c}}(\lambda
q,\Delta t/\lambda^{2})&=\lim_{\lambda\rightarrow
0}\frac{k_{\mathrm{B}}T}{\gamma}[1-e^{-\gamma\Delta
t/\lambda^{2}}]\exp\\{-\lambda^{2}q^{2}\frac{k_{\mathrm{B}}T}{\gamma^{2}}\big{[}\gamma\Delta
t/\lambda^{2}+e^{-\gamma\Delta t/\lambda^{2}}-1\big{]}\bigg{\\}}\\\
&=\frac{k_{\mathrm{B}}T}{\gamma}\exp\\{-\frac{k_{\mathrm{B}}T}{\gamma}q^{2}\Delta
t\bigg{\\}}\,.\end{split}$ (C.46)

Evaluating the limit $\Delta t\rightarrow 0$ then yields

$\displaystyle\begin{split}D^{\mathrm{c}}=\lim_{\Delta t\rightarrow
0}\lim_{\lambda\rightarrow 0}D^{\mathrm{c}}(\lambda q,\Delta
t/\lambda^{2})=\lim_{\Delta t\rightarrow
0}\frac{k_{\mathrm{B}}T}{\gamma}\exp\\{-\frac{k_{\mathrm{B}}T}{\gamma}q^{2}\Delta
t\bigg{\\}}=\frac{k_{\mathrm{B}}T}{\gamma}\,,\end{split}$ (C.47)

which agrees with the self-diffusion result in Eq. (C.42) due to the non-
interacting setting.

###  C.2 Two Dimensions with Odd Friction

We now proceed to the two dimensional case, where additional attention is
needed to obtain the odd (antisymmetric) part of the diffusivity. In two
dimensions, the Eqs. (C.38) and (C.39) for inertial Langevin dynamics
generalize to

$\displaystyle\dot{\bm{r}}_{\alpha}(t)$ $\displaystyle=\bm{v}_{\alpha}(t)\,,$
(C.48) $\displaystyle\dot{\bm{v}}_{\alpha}(t)$
$\displaystyle=-\bm{\gamma}\cdot\bm{v}_{\alpha}(t)+\bm{\eta}_{\alpha}(t)\,,$
(C.49)

where $\bm{\gamma}=\gamma_{\parallel}\bm{\delta}+\gamma_{\perp}\bm{\epsilon}$
is the isotropic friction on the particle due to interactions with its
environment. The odd friction $\gamma_{\perp}$ causes the particle motion to
break time-reversal and parity symmetries. In practice, this term could be the
consequence of hydrodynamic asymmetry causing the particle to deflect from its
current direction with a systematic bias [20, 21], or the consequence of a
charged particle moving through the plane normal to a magnetic field. We
assume the noise $\bm{\eta}_{\alpha}$ is related to the friction through a
fluctuation dissipation relation
$\langle\bm{\eta}_{\alpha}(t)\bm{\eta}_{\beta}(t^{\prime})\rangle=2k_{\mathrm{B}}T\bm{\gamma}\delta_{\alpha\beta}\delta(t-t^{\prime})$.

Solving for the particle displacement from Eqs. (C.48) and (C.49) yields

$\displaystyle\begin{split}\bm{\Delta r}(\Delta t)&=\int_{0}^{\Delta t}dt\
\bm{v}(t)\\\ &=\int_{0}^{\Delta t}dt\
\bigg{[}e^{-\bm{\gamma}t}\cdot\bm{v}(0)+\int_{0}^{t}dt^{\prime}\
e^{-\bm{\gamma}(t-t^{\prime})}\cdot\bm{\eta}(t^{\prime})\bigg{]}\\\
&=\bm{\gamma}^{-1}\big{(}\bm{\delta}-e^{-\bm{\gamma}\Delta
t}\big{)}\cdot\bm{v}(0)+\bm{\gamma}^{-1}\int_{0}^{\Delta t}dt\
\big{(}\bm{\delta}-e^{-\bm{\gamma}(\Delta t-t)}\big{)}\cdot\bm{\eta}(t)\\\
&=\mathbf{G}(\Delta t)\cdot\bm{v}(0)+\int_{0}^{\Delta t}dt\ \mathbf{G}(\Delta
t-t)\cdot\bm{\eta}(t)\,,\end{split}$ (C.50)

where the particle index $\alpha$ has been ignored due to particles being
identical and non-interacting. The final line defines the Green’s function
$\mathbf{G}(\Delta t)=\bm{\gamma}^{-1}\big{(}\bm{\delta}-e^{-\bm{\gamma}\Delta
t}\big{)}$. It then follows that the first and second moments of the
displacement, conditioned on an initial particle velocity $\bm{v}(0)$, are

$\displaystyle\langle\bm{\Delta r}(\Delta t)\rangle_{\bm{v}(0)}$
$\displaystyle=\mathbf{G}(\Delta t)\cdot\bm{v}(0)\,,$ (C.51)
$\displaystyle\langle\bm{\Delta r}(\Delta t)\bm{\Delta r}(\Delta
t)\rangle_{\bm{v}(0)}$ $\displaystyle=\mathbf{G}(\Delta
t)\big{(}\bm{v}(0)\bm{v}(0)\big{)}\mathbf{G}^{\intercal}(\Delta t)$
$\displaystyle+2k_{\mathrm{B}}T{\bm{\gamma}^{\intercal}}^{-1}\bigg{[}\Delta
t-\Tr\mathbf{G}(\Delta
t)+\frac{1}{2\gamma_{\parallel}}\big{(}1-e^{-2\gamma_{\parallel}\Delta
t}\big{)}\bigg{]}\,.$ (C.52)

The initial velocity $\bm{v}(0)$ is distributed in the steady state
corresponding to Eqs. (C.48) and (C.49) by the Maxwell-Boltzmann distribution

$W_{ss}(\bm{v}_{0})=\frac{1}{2\pi
k_{\mathrm{B}}T}\exp\left[-\frac{|\bm{v}_{0}|^{2}}{2k_{\mathrm{B}}T}\right]\,.$
(C.53)

From isotropy of the dynamics, the diffusivity tensor will be of the form
$\mathbf{D}^{\mathrm{c}}=D^{\mathrm{c}}_{\parallel}\bm{\delta}-D^{\mathrm{c}}_{\perp}\bm{\epsilon}$,
where the presence of the odd diffusivity $D^{\mathrm{c}}_{\perp}$ renders the
situation more complicated than the one-dimensional case solved in the
previous section. Under time-reversal of the dynamics,
$D^{\mathrm{c}}_{\parallel}$ remains unchanged while $D^{\mathrm{c}}_{\perp}$
changes sign, as can be seen from the Green-Kubo relation (20). Thus, we shall
consider the time-reversed diffusivity
${\mathbf{D}^{\mathrm{c}}}^{\intercal}=D^{\mathrm{c}}_{\parallel}\bm{\delta}+D^{\mathrm{c}}_{\perp}\bm{\epsilon}$,
taking advantage of the fact that it can be readily computed from the time-
reversed form of Eq. (B.22), i.e.,

$\displaystyle\begin{split}&\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t)\cdot\text{i}\bm{q}=-\langle\bm{v}(\Delta
t)e^{-\text{i}\bm{q}\cdot\bm{\Delta r}(\Delta t))}\rangle\\\
\xRightarrow{\mathcal{T}}&{\mathbf{D}^{\mathrm{c}}}^{\intercal}(\bm{q},\Delta
t)\cdot\text{i}\bm{q}=\langle\bm{v}(0)e^{\text{i}\bm{q}\cdot\bm{\Delta
r}(\Delta t))}\rangle=\int d\bm{v}_{0}\ W_{ss}(\bm{v}_{0})\bm{v}_{0}\langle
e^{\text{i}\bm{q}\cdot\bm{\Delta r}(\Delta
t))}\rangle_{\bm{v}_{0}}\,,\end{split}$ (C.54)

where $\xRightarrow{\mathcal{T}}$ indicates the time-reversal operation and
$\langle\cdot\rangle_{\bm{v}_{0}}$ is the expectation conditioned on the
initial velocity $\bm{v}(0)=\bm{v}_{0}$. It is important to note that, while
in the one-dimensional case described in the previous section it was possible
to obtain the $D^{\mathrm{c}}$ simply by time differentiation of the moment
generating function, in the two-dimensional case such an approach would yield
an expression only for the even part of $\mathbf{D}^{\mathrm{c}}$ and not the
odd part. Nevertheless, one may expand the conditional moment generating
function to obtain

$\langle e^{\text{i}\bm{q}\cdot\bm{\Delta r}(\Delta
t)}\rangle_{\bm{v}_{0}}=\exp\\{\text{i}\bm{q}\cdot\langle\bm{\Delta r}(\Delta
t)\rangle_{\bm{v}_{0}}\bigg{\\}}\exp\\{-\frac{1}{2}\bm{q}\cdot\big{\langle}\big{(}\bm{\Delta
r}(\Delta t)\bm{\Delta r}(\Delta t)-\langle\bm{\Delta r}(\Delta
t)\rangle_{\bm{v}_{0}}\langle\bm{\Delta r}(\Delta
t)\rangle_{\bm{v}_{0}}\big{)}\big{\rangle}_{\bm{v}_{0}}\cdot\bm{q}\bigg{\\}}\,.$
(C.55)

As in the one-dimensional case, all higher cumulants vanish due to the
Gaussian nature of the noise. Due to the initial velocity condition
$\bm{v}(0)=\bm{v}_{0}$, however, the first cumulant $\langle\bm{\Delta
r}(\Delta t)\rangle_{\bm{v}_{0}}$ does not vanish. Using the identities in
Eqs. (C.51)-(C.2), the second cumulant simplifies as

$\displaystyle\begin{split}\bm{q}\cdot\bigg{\langle}\big{(}\bm{\Delta
r}(\Delta t)\bm{\Delta r}(\Delta t)-\langle\bm{\Delta r}(\Delta
t)\rangle_{\bm{v}_{0}}\langle\bm{\Delta r}(\Delta
t)\rangle_{\bm{v}_{0}}\big{)}\bigg{\rangle}_{\bm{v}_{0}}\cdot\bm{q}&=\frac{1}{2}q^{2}\big{\langle}|\bm{\Delta
r}(\Delta t)|^{2}\big{\rangle}-\bm{q}\cdot\mathbf{G}(\Delta
t)\langle\bm{v}_{0}\bm{v}_{0}\rangle\mathbf{G}^{\intercal}(\Delta
t)\cdot\bm{q}\\\ &=\frac{1}{2}q^{2}\big{\langle}|\bm{\Delta r}(\Delta
t)|^{2}\big{\rangle}-k_{\mathrm{B}}T\bm{q}\cdot\mathbf{G}(\Delta
t)\mathbf{G}^{\intercal}(\Delta t)\cdot\bm{q}\,.\end{split}$ (C.56)

In the first equality, by recognizing from Eq. (C.2) that the second cumulant
does not depend on the initial velocity, we average each term individually
over the initial velocity according to the distribution $W_{ss}(\bm{v}_{0})$,
from which $\langle\bm{v}(0)\bm{v}(0)\rangle=k_{\mathrm{B}}T\bm{\delta}$. With
Eq. (C.56), Eq. (C.54) is then a Gaussian integral which evaluates to

$\displaystyle\begin{split}{\mathbf{D}^{\mathrm{c}}}^{\intercal}(\bm{q},\Delta
t)\cdot\text{i}\bm{q}&=\int d\bm{v}_{0}\ W_{ss}(\bm{v}_{0})\
\bm{v}_{0}\exp\\{\text{i}\bm{q}\cdot\mathbf{G}(\Delta
t)\cdot\bm{v}_{0}\bigg{\\}}\exp\\{-\frac{1}{4}q^{2}\big{\langle}|\bm{\Delta
r}(\Delta
t)|^{2}\big{\rangle}+\frac{k_{\mathrm{B}}T}{2}\bm{q}\cdot\mathbf{G}(\Delta
t)\mathbf{G}^{\intercal}(\Delta t)\cdot\bm{q}\bigg{\\}}\\\
&=e^{-\frac{1}{4}q^{2}\langle|\bm{\Delta r}(\Delta
t)|^{2}\rangle}\frac{1}{2\pi k_{\mathrm{B}}T}\int d\bm{v}_{0}\
\bm{v}_{0}\exp\\{-\frac{1}{2k_{\mathrm{B}}T}|\bm{v}_{0}|^{2}+\text{i}\bm{q}\cdot\mathbf{G}(\Delta
t)\cdot\bm{v}_{0}+\frac{k_{\mathrm{B}}T}{2}\bm{q}\cdot\mathbf{G}(\Delta
t)\mathbf{G}^{\intercal}(\Delta t)\cdot\bm{q}\bigg{\\}}\\\
&=e^{-\frac{1}{4}q^{2}\langle|\bm{\Delta r}(\Delta
t)|^{2}\rangle}\frac{1}{2\pi k_{\mathrm{B}}T}\int d\bm{v}_{0}\
\bm{v}_{0}\exp\\{-\frac{1}{2k_{\mathrm{B}}T}\big{|}\bm{v}_{0}-k_{\mathrm{B}}T\mathbf{G}^{\intercal}(\Delta
t)\cdot\text{i}\bm{q}\big{|}^{2}\bigg{\\}}\\\
&=k_{\mathrm{B}}T\mathbf{G}^{\intercal}(\Delta
t)\cdot\text{i}\bm{q}\,e^{-\frac{1}{4}q^{2}\langle|\bm{\Delta r}(\Delta
t)|^{2}\rangle}\,.\end{split}$ (C.57)

From isotropy, the orientation of $\bm{q}$ may be chosen arbitrarily and thus

$\boxed{\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t)=k_{\mathrm{B}}T\mathbf{G}(\Delta t)e^{-\frac{1}{4}q^{2}\langle|\bm{\Delta
r}(\Delta t)|^{2}\rangle}}\,.$ (C.58)

From Eq. (C.2) together with equipartition identity
$\langle\bm{v}(0)\bm{v}(0)\rangle=k_{\mathrm{B}}T\bm{\delta}$, the mean-
squared displacement is

$\langle|\bm{\Delta r}(\Delta t)|^{2}\rangle=k_{\mathrm{B}}T\mathbf{G}(\Delta
t):\mathbf{G}(\Delta
t)+4k_{\mathrm{B}}T\frac{\gamma_{\parallel}}{\gamma_{\parallel}^{2}+\gamma_{\perp}^{2}}\bigg{[}\Delta
t-\Tr\mathbf{G}(\Delta
t)+\frac{1}{2\gamma_{\parallel}}\big{(}1-e^{-2\gamma_{\parallel}\Delta
t}\big{)}\bigg{]}$ (C.59)

The odd diffusivity $D^{\mathrm{c}}_{\perp}(\bm{q},\Delta
t)=-\frac{1}{2}\bm{\epsilon}:\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta t)$ is
plotted from the expression in Eq. (C.58) in Fig. 1(d) of the main text. Note
that $\lim_{\Delta t\rightarrow 0}\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta
t)=\lim_{\Delta t\rightarrow\infty}\mathbf{D}^{\mathrm{c}}(\bm{q},\Delta t)=0$
and it is thus only on intermediate timescales $\tau_{c}\ll\Delta
t\ll\tau_{r}$ the collective diffusivity can be obtained. As indicated in the
main text, this can be achieved by rescaling by $\lambda$ and evaluating the
limit

$\lim_{\lambda\rightarrow 0}\mathbf{D}^{\mathrm{c}}(\lambda\bm{q},\Delta
t/\lambda^{2})=\exp\\{-k_{\mathrm{B}}T\frac{\gamma_{\parallel}}{\gamma_{\parallel}^{2}+\gamma_{\perp}^{2}}q^{2}\Delta
t\big{\\}}k_{\mathrm{B}}T\bm{\gamma}^{-1}\,,$ (C.60)

recovering Markovian behavior consistent with the macroscopic phenomenology.
Then, taking the short-time limit

$\mathbf{D}^{\mathrm{c}}=\lim_{\Delta t\rightarrow 0}\lim_{\lambda\rightarrow
0}\mathbf{D}^{\mathrm{c}}(\lambda\bm{q},\Delta t/\lambda^{2})=\lim_{\Delta
t\rightarrow
0}\exp\\{-k_{\mathrm{B}}T\frac{\gamma_{\parallel}}{\gamma_{\parallel}^{2}+\gamma_{\perp}^{2}}q^{2}\Delta
t\big{\\}}k_{\mathrm{B}}T\bm{\gamma}^{-1}=k_{\mathrm{B}}T\bm{\gamma}^{-1}\,.$
(C.61)

Contracting with $\bm{\delta}$ and $\bm{\epsilon}$ to obtain the even and odd
components, respectively, results in

$\displaystyle D^{\mathrm{c}}_{\parallel}$
$\displaystyle={k_{\mathrm{B}}T}\frac{\gamma_{\parallel}}{\gamma_{\parallel}^{2}+\gamma_{\perp}^{2}}\,,$
(C.62) $\displaystyle D^{\mathrm{c}}_{\perp}$
$\displaystyle={k_{\mathrm{B}}T}\frac{\gamma_{\perp}}{\gamma_{\parallel}^{2}+\gamma_{\perp}^{2}}\,.$
(C.63)

This can be compared to the direct solution of the self-diffusivity from Eq.
(The Flux Hypothesis for Odd Transport Phenomena) or the expression in Eq.
(B.35) derived without the flux hypothesis, where

$\displaystyle\begin{split}\mathbf{D}^{\mathrm{self}}&=\int_{0}^{\infty}dt\
\langle\bm{v}(t)\bm{v}(0)\rangle\\\ &=\int_{0}^{\infty}dt\
e^{-\bm{\gamma}t}\langle\bm{v}(0)\bm{v}(0)\rangle+\int_{0}^{t}dt^{\prime}\
e^{-\bm{\gamma}(t-t^{\prime})}\langle\bm{\eta}(t^{\prime})\bm{v}(0)\rangle\\\
&=k_{\mathrm{B}}T\bm{\gamma}^{-1}\,,\end{split}$ (C.64)

with components

$\displaystyle D^{\mathrm{self}}_{\parallel}$
$\displaystyle={k_{\mathrm{B}}T}\frac{\gamma_{\parallel}}{\gamma_{\parallel}^{2}+\gamma_{\perp}^{2}}\,,$
(C.65) $\displaystyle D^{\mathrm{self}}_{\perp}$
$\displaystyle={k_{\mathrm{B}}T}\frac{\gamma_{\perp}}{\gamma_{\parallel}^{2}+\gamma_{\perp}^{2}}\,.$
(C.66)

As expected due to the dilute (i.e. non-interacting) setting, the collective
diffusivity recovers the self diffusivity when evaluated correctly in the
macroscopic, intermediate-timescale regime. Note that Fig. 1(c) plots several
typical trajectories of Eqs. (C.48)-(C.49) with
$k_{\mathrm{B}}T=\gamma_{\parallel}=\gamma_{\perp}=1$. These same parameters
are used in Figs. 1(d) and 1(e), where $t\in[0,12]$ and $t\in[0,4]$,
respectively, and where colorbars are scaled logarithmically with $\bm{q}$ and
$\lambda\in[0.01,1.5]$.

##  D Simulation Details

Molecular dynamics simulations were carried out using the LAMMPS simulation
software [52] with custom modifications111Available at
https://github.com/mandadapu-group/active-matter.. As a minimal model of the
interacting active spinners described by Eq. (27), we simulated rigid dumbbell
particles constructed from two beads of width $\sigma$ held together by a
rigid bond of length $\ell$. These beads interact with one another through a
Weeks-Chandler-Andersen potential, defined by

$V^{\mathrm{WCA}}(r)=\begin{cases}4\epsilon\bigg{[}\big{(}\sigma/r\big{)}^{12}-\big{(}\sigma/r\big{)}^{6}\bigg{]}+\epsilon&r<2^{1/6}\sigma\\\
0&r\geq 2^{1/6}\sigma\,,\\\ \end{cases}$ (D.67)

where $r=|\bm{r}_{\alpha}^{a}-\bm{r}_{\beta}^{b}|$ is the distance between
bead $a$ in dumbbell $\alpha$ and bead $b$ in dumbbell $\beta$. The position
of each dumbbell is defined as its center of mass
$\bm{r}_{\alpha}=\frac{1}{2}(\bm{r}_{\alpha}^{1}+\bm{r}_{\alpha}^{2})$, and
the angle $\theta_{\alpha}$ of the dumbbell relative to the $x$-axis is
defined such that $\bm{\Delta
r}_{\alpha}=\bm{r}_{\alpha}^{2}-\bm{r}_{\alpha}^{1}=\ell\begin{bmatrix}\cos\theta_{\alpha}\\\
\sin\theta_{\alpha}\end{bmatrix}$. Conversely, the bead positions are related
to the new coordinates by
$\bm{r}_{\alpha}^{1}=\bm{r}_{\alpha}-\frac{1}{2}\bm{\Delta r}_{\alpha}$ and
$\bm{r}_{\alpha}^{2}=\bm{r}_{\alpha}+\frac{1}{2}\bm{\Delta r}_{\alpha}$. The
force on the dumbbell center of mass is then

$-\frac{\partial
V^{\mathrm{WCA}}}{\partial\bm{r}_{\alpha}}=-\bigg{(}\frac{\partial\bm{r}_{\alpha}^{1}}{\partial\bm{r}_{\alpha}}\frac{\partial}{\partial\bm{r}_{\alpha}^{1}}+\frac{\partial\bm{r}_{\alpha}^{2}}{\partial\bm{r}_{\alpha}}\frac{\partial}{\partial\bm{r}_{\alpha}^{2}}\bigg{)}V^{\mathrm{WCA}}=-\bigg{(}\frac{\partial}{\partial\bm{r}_{\alpha}^{1}}+\frac{\partial}{\partial\bm{r}_{\alpha}^{2}}\bigg{)}V^{\mathrm{WCA}}\,,$
(D.68)

and the torque is

$\displaystyle\begin{split}-\frac{\partial
V^{\mathrm{WCA}}}{\partial\theta_{\alpha}}&=-\frac{\partial\bm{\Delta
r}_{\alpha}}{\partial\theta_{\alpha}}\cdot\bigg{(}\frac{\partial\bm{r}_{\alpha}^{1}}{\partial\bm{\Delta
r}_{\alpha}}\frac{\partial
V^{\mathrm{WCA}}}{\partial\bm{r}_{\alpha}^{1}}+\frac{\partial\bm{r}_{\alpha}^{2}}{\partial\bm{\Delta
r}_{\alpha}}\frac{\partial
V^{\mathrm{WCA}}}{\partial\bm{r}_{\alpha}^{2}}\bigg{)}\\\
&=\frac{\ell}{2}\bigg{[}\sin\theta\bigg{(}\frac{\partial
V^{\mathrm{WCA}}}{\partial x_{\alpha}^{2}}-\frac{\partial
V^{\mathrm{WCA}}}{\partial
x_{\alpha}^{1}}\bigg{)}-\cos\theta\bigg{(}\frac{\partial
V^{\mathrm{WCA}}}{\partial y_{\alpha}^{2}}-\frac{\partial
V^{\mathrm{WCA}}}{\partial y_{\alpha}^{1}}\bigg{)}\bigg{]}\,.\end{split}$
(D.69)

The Green-Kubo estimates for $\mathbf{D}^{\mathrm{c}}$ are then computed from
Eq. (21) in a periodic simulation box of size $L=120\sigma$ for the wave
vector $|\bm{q}|=\frac{2\pi}{L}$. The Green-Kubo values reported in Fig. 2 are
computed as the average over $60$ independent replicas, while 95% confidence
intervals (i.e. error bars) are computed as twice the standard error over
these replicas. All simulations use the settings $\gamma=2,\
\gamma_{\theta}=\frac{1}{2},\ m=2,\ I=\frac{1}{2},\ \ell=1,\
k_{\mathrm{B}}T=1,\ \sigma=1,$ and $\epsilon=1$. The translational and angular
velocity evolution in (27) was discretized with Euler-Maruyama method and a
time step of $\Delta t=0.005$.

As described in the main text, the Green-Kubo results are compared against
direct “non-equilibrium molecular dynamics” (NEMD) measurements of
$\mathbf{D}^{\mathrm{c}}$ made by perturbing the system with a small
sinusoidal potential and measuring the response. Specifically, an external
potential $V^{\mathrm{ext}}=-\delta V\cos(qx)$ is imposed with $\delta
V=0.2\bar{\rho}$ and $q=\frac{2\pi}{L}$. This potential drives a flux of the
spinners relative to the substrate, independent of the flux driven by density
gradients. The result is a drift-diffusion equation for the density evolution
of this single component system is then

$\bm{J}=-\mathbf{D}^{\mathrm{c}}\bm{\nabla}\rho+\gamma^{-1}\rho\mathbf{F}^{\mathrm{ext}}\,,$
(D.70)

where $\mathbf{F}^{\mathrm{ext}}=-\bm{\nabla}V^{\mathrm{ext}}=-q\delta
V\sin(qx)\hat{\bm{e}}_{x}$. For small $\delta V$, the density can be
linearized as $\rho(\bm{r})=\bar{\rho}+\delta\rho(\bm{r})$, such that the
external force acts by $\bar{\rho}$ to leading order. In the steady state,
$\dot{\rho}=-\bm{\nabla}\cdot\bm{J}=0$, requiring $J_{x}=0$. Solving Eq.
(D.70) for the steady state then yields

$\displaystyle\rho(x)$ $\displaystyle=\bar{\rho}\bigg{(}1+\frac{\delta
V}{D_{\parallel}^{\mathrm{c}}\gamma}\cos(qx)\bigg{)}\,,$ (D.71) $\displaystyle
J_{x}$ $\displaystyle=0\,,$ (D.72) $\displaystyle J_{y}$
$\displaystyle=-D_{\perp}^{\mathrm{c}}\partial_{x}\rho=\frac{D_{\perp}^{\mathrm{c}}\bar{\rho}q\delta
V}{D_{\parallel}^{\mathrm{c}}\gamma}\sin(qx)\,.$ (D.73)

Finally, given $|\bm{q}|=\frac{2\pi}{L}$ and the sinusoidal nature of the
applied potential $V^{\text{ext}}$, the simulation box can be divided into
intervals $x\in(0,L/4]$, $x\in(L/4,L/2]$, $x\in(L/2,3L/4]$ and $x\in(3L/4,L)$
to improve sampling statistics. This approach yields

$\frac{\bar{\rho}L\delta V}{2\pi
D_{\parallel}^{\mathrm{c}}\gamma}=\frac{1}{4}\bigg{[}\int_{0}^{L/4}\delta\rho(x)dx\
-\int_{L/4}^{L/2}\delta\rho(x)dx\ -\int_{L/2}^{3L/4}\delta\rho(x)dx\
+\int_{3L/4}^{L}\delta\rho(x)dx\ \bigg{]}$ (D.74)

and

$\frac{D_{\perp}^{\mathrm{c}}\bar{\rho}qL\delta V}{2\pi
D_{\parallel}^{\mathrm{c}}\gamma}=\frac{1}{4}\bigg{[}\int_{0}^{L/4}J_{y}(x)dx\
+\int_{L/4}^{3L/4}J_{y}(x)dx\ -\int_{L/2}^{3L/4}J_{y}(x)dx\
-\int_{3L/4}^{L}J_{y}(x)dx\ \bigg{]}$ (D.75)

where $\delta\rho(x)$ is the steady state value measured in simulation after
averaging in time and $y$. The NEMD results reported in Fig. 2 are averaged
over 60 replicas, with error bars corresponding to 95% confidence intervals
smaller than the line widths and consequently omitted.

##  E Green-Kubo Relations for 2d Momentum and Energy Transport

In this Appendix we show the utililty of the flux hypothesis to recover Green-
Kubo relations for the transport coefficients governing two-dimensional (2d)
momentum and energy transport, including odd viscous and odd thermal
transport. In subsection E.1, we consider the viscous behaviors of systems
coupling the stress tensor (momentum flux) $\bm{\sigma}$ to a velocity
gradient $\bm{\nabla}\bm{v}$ via the fourth order viscosity tensor
$\bm{\eta}^{(4)}$. In subsection E.2 we consider viscous behaviors of systems
with internal spin, by allowing for the additional coupling of the stress
tensor to an internal spin angular momentum field $m$. In subsection E.3, we
consider energy transport of systems with odd heat conduction, coupling the
heat flux vector $\bm{Q}$ to a temperature gradient $\bm{\nabla}T$ via the
second order thermal conductivity tensor $\bm{\kappa}$.

For higher order transport tensors such as the viscosity $\bm{\eta}^{(4)}$, we
can apply the flux hypothesis using two approaches. In the first approach
(used in subsection E.1), we consider the momentum in each coordinate as a
separate quantity. Thus in 2d, we decompose the linear momentum into $\rho
v_{x}$ and $\rho v_{y}$ and the momentum flux into $\sigma_{xi}$ and
$\sigma_{yi}$, $i\in\\{x,y\\}$. Here, Latin indices now refer to spatial
degrees of freedom and Einstein summation notation is used. The fourth order
viscosity tensor $\eta_{ijkl}$ is accordingly unraveled into the four matrices
$\eta_{xixj},\eta_{yiyj},\eta_{xiyj},$ $\eta_{yixj}$, and the result of Eq.
(14) may then be directly applied. However, in this unraveled state we are
unable to apply tensor representation theorems222A tensor representation
theorem reduces the number of transport coefficients by constructing a basis
set corresponding to the material symmetry. For example, in two-dimensional
isotropic systems, the fourth order viscosity tensor is reduced from 16 to 6
coefficients (see SI-I of Ref. [5] for a representation theorem for isotropic
tensors of order $n$ in $d$ dimensions). to simplify the viscosity tensor into
independent coefficients (e.g. the shear and odd viscosities [5]), as each
coefficient may have contributions from multiple unraveled matrices. To use
the representation theorems, we require reassembly of the Green-Kubo relations
for the four matrices $\eta_{xixj},\eta_{yiyj},\eta_{xiyj},$ $\eta_{yixj}$,
into a Green-Kubo relation for the full fourth order tensor $\eta_{ijkl}$.

In the second approach (used in subsection E.2), we consider the momentum
$\rho\bm{v}$ as a vectorial quantity, enabling for the use of tensor
representation theorems. We apply the flux hypothesis directly on the fourth
order tensor $\eta_{ijkl}$ to recover Green-Kubo relations.

###  E.1 Viscosity for Systems with No Internal Spin

We now apply the flux hypothesis to recover Green-Kubo relations for viscosity
$\bm{\eta}^{(4)}$ in 2d isotropic systems as presented in [5]. We first
present Green-Kubo relations for the case of systems with no internal spin. In
this case, the stress tensor $\bm{\sigma}$ is linearly dependent on the
velocity gradient $\bm{\nabla}\bm{v}$ alone as

$\sigma_{ij}=\eta_{ijkl}\partial_{l}v_{k}\,,$ (E.76)

with the corresponding balance law of linear momentum,

$\rho\dot{v}_{i}=\partial_{j}\sigma_{ij}\,.$ (E.77)

This balance law holds in both the macroscopic sense and instantaneously for
any microscopic realization.

Considering microscopic fluctuations from a spatially homogeneous steady state
with density $\rho_{0}$ and no average bulk velocity, we may write the
linearized Fourier-transformed balance law as

$\rho_{0}\partial_{t}\hat{v}_{i}^{\bm{q}}=\mathrm{i}q_{j}\hat{\sigma}_{ij}^{\bm{q}}\,.$
(E.78)

Using matrix-vector products, the constitutive law (E.76) may be translated to
its Fourier counterpart and then put into the form of Eq. (2) as

$\bm{t}_{x}^{\bm{q}}:=\begin{bmatrix}\sigma_{xx}^{\bm{q}}\\\
\sigma_{xy}^{\bm{q}}\end{bmatrix}=\begin{bmatrix}\eta_{xxxx}&\eta_{xxxy}\\\
\eta_{xyxx}&\eta_{xyxy}\end{bmatrix}\begin{bmatrix}\mathrm{i}q_{x}\\\
\mathrm{i}q_{y}\end{bmatrix}v_{x}^{\bm{q}}+\begin{bmatrix}\eta_{xxyx}&\eta_{xxyy}\\\
\eta_{xyyx}&\eta_{xyyy}\end{bmatrix}\begin{bmatrix}\mathrm{i}q_{x}\\\
\mathrm{i}q_{y}\end{bmatrix}v_{y}^{\bm{q}}\,,$ (E.79)

$\bm{t}_{y}^{\bm{q}}:=\begin{bmatrix}\sigma_{yx}^{\bm{q}}\\\
\sigma_{yy}^{\bm{q}}\end{bmatrix}=\begin{bmatrix}\eta_{yxxx}&\eta_{yxxy}\\\
\eta_{yyxx}&\eta_{yyxy}\end{bmatrix}\begin{bmatrix}\mathrm{i}q_{x}\\\
\mathrm{i}q_{y}\end{bmatrix}v_{x}^{\bm{q}}+\begin{bmatrix}\eta_{yxyx}&\eta_{yxyy}\\\
\eta_{yyyx}&\eta_{yyyy}\end{bmatrix}\begin{bmatrix}\mathrm{i}q_{x}\\\
\mathrm{i}q_{y}\end{bmatrix}v_{y}^{\bm{q}}\,.$ (E.80)

Here, we define the fluxes $\bm{t}_{i}:=[\sigma_{ix},\sigma_{iy}]^{\intercal}$
and define an unraveled matrix-analog to the 4th order viscosity tensor,

$\mathbf{H}_{ij}:=\begin{bmatrix}\eta_{ixjx}&\eta_{ixjy}\\\
\eta_{iyjx}&\eta_{iyjy}\end{bmatrix}\,.$ (E.81)

Thus, mirroring Eq. (2) we obtain

$\bm{t}_{i}^{\bm{q}}=\sum_{k\in\\{x,y\\}}\mathbf{H}_{ik}\cdot\text{i}\bm{q}v^{\bm{q}}_{k}\,.$
(E.82)

We see that the quantities $\bm{t}_{i}$, $\mathbf{H}_{ij}$, and
$v_{i}^{\bm{q}}$ are in direct analogy to $\bm{J}_{i}$, $\mathbf{M}_{ij}$, and
$A_{i}^{\bm{q}}$, respectively, in the main text. Accordingly, we may apply
the flux hypothesis to obtain Green-Kubo relations for $\mathbf{H}_{ij}$,
which from Eq. (14) are

$\mathbf{H}_{ij}(\bm{q},\Delta
t)=\sum_{k\in\\{x,y\\}}\mathbf{K}_{ik}(\bm{q},\Delta t)s_{kj}(\bm{q})\ ,$
(E.83)

where $\mathbf{K}_{ij}$ and $s_{ij}$ are defined in direct analogy to
$\mathbf{L}_{ij}$ and $g_{ij}$, respectively, in the main text as follows:

$\mathbf{K}_{ij}(\bm{q},\Delta t):=\int_{0}^{\Delta t}dt\
\langle\hat{\bm{t}}^{\bm{q}}_{i}(t)\hat{\bm{t}}^{-\bm{q}}_{j}(0)\rangle=\int_{0}^{\Delta
t}dt\begin{bmatrix}\langle\hat{\sigma}_{ix}^{\bm{q}}(t)\hat{\sigma}_{jx}^{-\bm{q}}(0)\rangle&\langle\hat{\sigma}_{ix}^{\bm{q}}(t)\hat{\sigma}_{jy}^{-\bm{q}}(0)\rangle\\\
\langle\hat{\sigma}_{iy}^{\bm{q}}(t)\hat{\sigma}_{jx}^{-\bm{q}}(0)\rangle&\langle\hat{\sigma}_{iy}^{\bm{q}}(t)\hat{\sigma}_{jy}^{-\bm{q}}(0)\rangle\end{bmatrix}\,,$
(E.84)
$[\mathbf{s}^{-1}(\bm{q})]_{ij}:=\rho_{0}\langle\hat{v}_{i}^{\bm{q}}\hat{v}_{j}^{-\bm{q}}\rangle\,.$
(E.85)

Here, we note that $\mathbf{s}$ possesses an extra factor of the average mass
density $\rho_{0}$ when compared to $\mathbf{g}$.

For isotropic systems, the static velocity correlator is diagonal to leading
order (see SI-IV of Ref. [5])

$\lim_{\bm{q}\rightarrow\bm{0}}[\mathbf{s}^{-1}(\bm{q})]_{ij}=\rho_{0}\mu\delta_{ij}\
,$ (E.86)

where we have defined

$\mu\delta_{ij}:=\lim_{\bm{q}\rightarrow\bm{0}}\langle\hat{v}_{i}^{\bm{q}}\hat{v}_{j}^{-\bm{q}}\rangle\,.$
(E.87)

Substituting the definitions of $\mathbf{H}_{ij}$, $\mathbf{K}_{ij}$, and
$\mathbf{s}$ into Eq. (E.83), we obtain the Green-Kubo relations for the
viscosity tensor:

$\begin{bmatrix}\eta_{ixjx}&\eta_{ixjy}\\\
\eta_{iyjx}&\eta_{iyjy}\end{bmatrix}=\lim_{\bm{q}\rightarrow\bm{0}}\sum_{k\in\\{x,y\\}}\int_{0}^{\Delta
t}dt\begin{bmatrix}\langle\hat{\sigma}_{ix}^{\bm{q}}(t)\hat{\sigma}_{kx}^{-\bm{q}}(0)\rangle&\langle\hat{\sigma}_{ix}^{\bm{q}}(t)\hat{\sigma}_{ky}^{-\bm{q}}(0)\rangle\\\
\langle\hat{\sigma}_{iy}^{\bm{q}}(t)\hat{\sigma}_{kx}^{-\bm{q}}(0)\rangle&\langle\hat{\sigma}_{iy}^{\bm{q}}(t)\hat{\sigma}_{ky}^{-\bm{q}}(0)\rangle\end{bmatrix}\frac{1}{\rho_{0}\mu}\delta_{kj}\,.$
(E.88)

Equation (E.88) can be reduced to

$\begin{bmatrix}\eta_{ixjx}&\eta_{ixjy}\\\
\eta_{iyjx}&\eta_{iyjy}\end{bmatrix}=\frac{1}{\rho_{0}\mu}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\begin{bmatrix}\langle\hat{\sigma}_{ix}^{\bm{q}}(t)\hat{\sigma}_{jx}^{-\bm{q}}(0)\rangle&\langle\hat{\sigma}_{ix}^{\bm{q}}(t)\hat{\sigma}_{jy}^{-\bm{q}}(0)\rangle\\\
\langle\hat{\sigma}_{iy}^{\bm{q}}(t)\hat{\sigma}_{jx}^{-\bm{q}}(0)\rangle&\langle\hat{\sigma}_{iy}^{\bm{q}}(t)\hat{\sigma}_{jy}^{-\bm{q}}(0)\rangle\end{bmatrix}\,.$
(E.89)

Finally, reorganizing $\mathbf{H}_{ij}$ into a 4th-order tensor $\eta_{ijkl}$,
we write the Green-Kubo relations for the full viscosity tensors as

$\eta_{ijkl}=\frac{1}{\rho_{0}\mu}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle\,,$
(E.90)

with $\rho_{0}\mu=Vk_{\text{B}}T_{\text{eff}}$ playing the role of effective
temperature for non-equilibrium steady states.

For 2d isotropic systems, the viscosity tensor can be further reduced from 16
to 6 coefficients through the orthogonal basis $s_{ijkl}^{(\alpha)}$ as
outlined in Eq. (8) and Table I of Ref. [5], where

$\eta_{ijkl}=\sum_{\alpha=1}^{6}\lambda_{\alpha}s_{ijkl}^{(\alpha)}=\lambda_{1}\delta_{ij}\delta_{kl}+\lambda_{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl})+\lambda_{3}\epsilon_{ij}\epsilon_{kl}+\lambda_{4}(\epsilon_{ik}\delta_{jl}+\epsilon_{jl}\delta_{ik})+2\lambda_{5}\epsilon_{ij}\delta_{kl}-2\lambda_{6}\delta_{ij}\epsilon_{kl}\,,$
(E.91)

and $\\{\lambda_{\alpha}\\}$ are the relevant transport coefficients. The
first three viscosities are even under parity symmetry: $\lambda_{1}$ and
$\lambda_{2}$ are respectively the typical bulk and shear viscosities, and
$\lambda_{3}$ is the rotational viscosity. The latter three viscosities are
odd under parity symmetry and may arise in chiral systems; $\lambda_{4}$ is
the “odd” viscosity coupling shear along one axis with normal stress along an
orthogonal axis, $\lambda_{5}$ creates a torque in response to compression,
and lastly $\lambda_{6}$ creates a normal stress in response to vorticity.

Using Eq. (E.91) in conjunction with Eq. (E.90), we obtain the following
Green-Kubo relations for each of the six viscosities in 2d:

$\lambda_{1}=\frac{1}{4\rho_{0}\mu}\delta_{ij}\delta_{kl}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle\,,$
(E.92)
$\lambda_{2}=\frac{1}{8\rho_{0}\mu}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl})\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle\,,$
(E.93)
$\lambda_{3}=\frac{1}{4\rho_{0}\mu}\epsilon_{ij}\epsilon_{kl}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle\,,$
(E.94)
$\lambda_{4}=\frac{1}{8\rho_{0}\mu}(\epsilon_{ik}\delta_{jl}+\epsilon_{jl}\delta_{ik})\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle\,,$
(E.95)
$\lambda_{5}=\frac{1}{8\rho_{0}\mu}\epsilon_{ij}\delta_{kl}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle\,,$
(E.96)
$\lambda_{6}=-\frac{1}{8\rho_{0}\mu}\delta_{ij}\epsilon_{kl}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle\,.$
(E.97)

Unlike previous work applying the Onsager regression hypothesis to this
context [5], the flux hypothesis holds the utility of isolating an explicit
independent expression for each viscosity coefficient.

###  E.2 Viscosity for Systems with Internal Spin

We now apply the flux hypothesis to develop Green-Kubo relations for
viscosities in 2d isotropic systems with internal spin. Here, we showcase how
to apply the flux hypothesis directly on higher order tensors. Allowing the
stress tensor $\sigma_{ij}$ to linearly depend on the velocity gradient
$\partial_{j}v_{i}$ and the spin field $m$ results in the constitutive law
(see Eq. (6) in Ref. [5]),

$\sigma_{ij}=\eta_{ijkl}\partial_{l}v_{k}+\gamma_{ij}m\ ,$ (E.98)

where $\eta_{ijkl}$ is the standard viscosity tensor and $\gamma_{ij}$ is
another viscous transport tensor characterizing the stress response to the
spin angular momentum. In the Fourier representation, Eq. (E.98) becomes

$\sigma_{ij}^{\bm{q}}=\eta_{ijkl}\mathrm{i}q_{l}v_{k}^{\bm{q}}+\gamma_{ij}m^{\bm{q}}\,.$
(E.99)

While the linear momentum balance remains the same as Eq. (E.77), the spin
angular momentum $m$ follows the balance law

$\rho\dot{m}=\partial_{i}C_{i}-\epsilon_{ij}\sigma_{ij}\,,$ (E.100)

with couple stress tensor $C_{i}$ [5]. We may once again consider microscopic
fluctuations from a spatially homogeneous steady state with density $\rho_{0}$
and no average bulk velocity to obtain the linearized Fourier-transformed
balance laws as Eq. (E.78) and

$\rho_{0}\partial_{t}\hat{m}^{\bm{q}}=\mathrm{i}q_{i}\hat{C}^{\bm{q}}_{i}-\epsilon_{ij}\hat{\sigma}^{\bm{q}}_{ij}\,.$
(E.101)

Given the multiplicity of transport coefficients entering via tensors of
different order, it is convenient to apply the flux hypothesis directly on the
constitutive law via Eq. (9). To that end, given Eq. (E.99), the flux
hypothesis states the fluctuations in stress $\hat{\sigma}_{ij}^{\bm{q}}$,
velocity $\tilde{v}_{i}^{\bm{q}}$, and internal spin $\tilde{m}^{\bm{q}}$ are
related by

$\langle\hat{\sigma}_{ij}^{\bm{q}}(\Delta
t)\rangle_{\\{\hat{v}_{k}^{\bm{q}}(0)=\tilde{v}_{k}^{\bm{q}},\hat{m}^{\bm{q}}(0)=\tilde{m}^{\bm{q}}\\}}=\eta_{ijkl}\mathrm{i}q_{l}\tilde{v}^{\bm{q}}_{k}+\gamma_{ij}\tilde{m}^{\bm{q}}\,.$
(E.102)

We now multiply by the initial velocity fluctuation $\tilde{v}_{m}^{-\bm{q}}$,
assume $\\{\tilde{v}_{j}^{\bm{q}}\\}$ and $\\{\tilde{m}^{\bm{q}}\\}$
respectively follow the steady-state distribution of the fluctuating variables
$\\{\hat{v}_{j}^{\bm{q}}\\}$ and $\\{\hat{m}^{\bm{q}}\\}$, and then take a
full ensemble average to obtain

$\langle\hat{\sigma}_{ij}^{\bm{q}}(\Delta
t)\hat{v}^{-\bm{q}}_{m}(0)\rangle=\eta_{ijkl}\mathrm{i}q_{l}\langle\hat{v}^{\bm{q}}_{k}(0)\hat{v}^{-\bm{q}}_{m}(0)\rangle+\gamma_{ij}\langle\hat{m}^{\bm{q}}(0)\hat{v}^{-\bm{q}}_{m}(0)\rangle\,.$
(E.103)

Assuming there exist no static correlations between the velocity fluctuations
and the stress fluctuations, the left-hand side term may be written using
stationarity as

$\displaystyle\begin{split}\langle\hat{\sigma}_{ij}^{\bm{q}}(\Delta
t)\hat{v}^{-\bm{q}}_{m}(0)\rangle&=\langle\hat{\sigma}_{ij}^{\bm{q}}(0)\hat{v}^{-\bm{q}}_{m}(0)\rangle+\int_{0}^{\Delta
t}\frac{\partial}{\partial
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{v}^{-\bm{q}}_{m}(0)\rangle dt\\\
&=\int_{0}^{\Delta t}\frac{\partial}{\partial
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{v}^{-\bm{q}}_{m}(0)\rangle dt\\\
&=\int_{0}^{\Delta t}\frac{\partial}{\partial
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(0)\hat{v}^{-\bm{q}}_{m}(-t)\rangle dt\\\
&=-\int_{0}^{\Delta
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(0)\frac{\partial}{\partial
t^{\prime}}\hat{v}^{-\bm{q}}_{m}(t^{\prime})\rangle\bigg{|}_{t^{\prime}=-t}dt\\\
&=-\int_{0}^{\Delta
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\frac{\partial}{\partial
t^{\prime}}\hat{v}^{-\bm{q}}_{m}(t^{\prime})\rangle\bigg{|}_{t^{\prime}=0}dt\,.\end{split}$
(E.104)

Utilizing the balance of linear momentum in Eq. (E.78) with average steady-
state mass density $\rho_{0}$, we obtain from Eqs. (E.103) and (E.104)

$\frac{\mathrm{i}q_{l}}{\rho_{0}}\int_{0}^{\Delta t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{ml}^{-\bm{q}}(0)\rangle=\eta_{ijkl}\mathrm{i}q_{l}\langle\hat{v}^{\bm{q}}_{k}\hat{v}^{-\bm{q}}_{m}\rangle+\gamma_{ij}\langle\hat{m}^{\bm{q}}\hat{v}^{-\bm{q}}_{m}\rangle\,.$
(E.105)

It is convenient to decouple $\gamma_{ij}$ from $\eta_{ijkl}$ in Eq. (E.105).
To this end, we may also multiply Eq. (E.102) by the initial spin fluctuation
$\tilde{m}^{-\bm{q}}$ and take a full ensemble average to yield

$\langle\hat{\sigma}_{ij}^{\bm{q}}(\Delta
t)\hat{m}^{-\bm{q}}(0)\rangle=\eta_{ijkl}\mathrm{i}q_{l}\langle\hat{v}^{\bm{q}}_{k}(0)\hat{m}^{-\bm{q}}(0)\rangle+\gamma_{ij}\langle\hat{m}^{\bm{q}}(0)\hat{m}^{-\bm{q}}(0)\rangle\,.$
(E.106)

Again, assuming there exist no static correlations in stress fluctuations and
spin field fluctuations, the left-hand side of Eq. (E.106) can be expressed
using stationarity as

$\displaystyle\begin{split}\left<\hat{\sigma}_{ij}^{\bm{q}}(\Delta
t)\hat{m}^{-\bm{q}}(0)\right>&=\langle\hat{\sigma}_{ij}^{\bm{q}}(0)\hat{m}^{-\bm{q}}(0)\rangle+\int_{0}^{\Delta
t}\frac{\partial}{\partial
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{m}^{-\bm{q}}(0)\rangle dt\\\
&=\int_{0}^{\Delta t}\frac{\partial}{\partial
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{m}^{-\bm{q}}(0)\rangle dt\\\
&=\int_{0}^{\Delta t}\frac{\partial}{\partial
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(0)\hat{m}^{-\bm{q}}(-t)\rangle dt\\\
&=-\int_{0}^{\Delta
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(0)\frac{\partial}{\partial
t^{\prime}}\hat{m}^{-\bm{q}}(t^{\prime})\rangle\bigg{|}_{t^{\prime}=-t}dt\\\
&=-\int_{0}^{\Delta
t}\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\frac{\partial}{\partial
t^{\prime}}\hat{m}^{-\bm{q}}(t^{\prime})\rangle\bigg{|}_{t^{\prime}=0}dt\,.\end{split}$
(E.107)

Now, utilizing the balance of spin in Eq. (E.101), we obtain from Eqs. (E.106)
and (E.107)

$\frac{\mathrm{i}q_{l}}{\rho_{0}}\int_{0}^{\Delta t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{C}_{l}^{-\bm{q}}(0)\rangle+\frac{\epsilon_{kl}}{\rho_{0}}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle=\eta_{ijkl}\mathrm{i}q_{l}\langle\hat{v}^{\bm{q}}_{k}\hat{m}^{-\bm{q}}\rangle+\gamma_{ij}\langle\hat{m}^{\bm{q}}\hat{m}^{-\bm{q}}\rangle\,.$
(E.108)

Equations (E.105) and (E.108) alone may be sufficient to recover Green-Kubo
relations for the tensors $\eta_{ijkl}$ and $\gamma_{ij}$ irrespective of
material symmetry. In what follows, we derive Green-Kubo relations for
isotropic systems, which puts constraints on the nature of transport
coefficient tensors and static correlation tensors. For isotropic systems, the
static correlation tensors
$\langle\hat{v}_{i}^{\bm{q}}\hat{v}_{j}^{-\bm{q}}\rangle,\langle\hat{m}^{\bm{q}}\hat{v}_{i}^{-\bm{q}}\rangle,$
and $\langle\hat{m}^{\bm{q}}\hat{m}^{-\bm{q}}\rangle$ must themselves be
isotropic in the limit $\bm{q}\rightarrow\bm{0}$ as argued in SI-IV of Ref.
[5]. Thus, $\langle\hat{v}^{\bm{q}}_{i}\hat{v}^{-\bm{q}}_{j}\rangle$ must be a
multiple of the Kronecker delta $\delta_{ij}$ to zeroth order in $\bm{q}$, and
the vector $\langle\hat{m}^{\bm{q}}\hat{v}^{-\bm{q}}_{m}\rangle$ must be first
order in $\bm{q}$. Accordingly, we define

$\mu\delta_{ij}:=\lim_{\bm{q}\rightarrow\bm{0}}\langle\hat{v}_{i}^{\bm{q}}\hat{v}_{j}^{-\bm{q}}\rangle\,,$
(E.109)
$\Omega_{ij}:=-\lim_{\bm{q}\rightarrow\bm{0}}\frac{\partial}{\partial\mathrm{i}q_{i}}\langle\hat{m}^{\bm{q}}\hat{v}_{j}^{-\bm{q}}\rangle\,,$
(E.110) $\pi:=\delta_{ij}\Omega_{ij}\,,$ (E.111)
$\tau:=\epsilon_{ij}\Omega_{ij}\,,$ (E.112)
$\nu:=\lim_{\bm{q}\rightarrow\bm{0}}\langle\hat{m}^{\bm{q}}\hat{m}^{-\bm{q}}\rangle\,.$
(E.113)

As the static correlators may be discontinuous at $\bm{q}=\bm{0}$, they are
considered at the limit $\bm{q}\rightarrow\bm{0}$, which may or may not be
strictly equal to their evaluation at $\bm{q}=\bm{0}$. Thus, expanding around
$\bm{q}=\bm{0}^{+}$,
$\langle\hat{v}_{i}^{\bm{q}}\hat{v}_{j}^{-\bm{q}}\rangle=\mu\delta_{ij}+\mathcal{O}(q)$,
$\langle\hat{m}^{\bm{q}}\hat{v}_{i}^{-\bm{q}}\rangle=-\langle\hat{v}_{i}^{\bm{q}}\hat{m}^{-\bm{q}}\rangle=-\mathrm{i}q_{l}\Omega_{li}+\mathcal{O}(q^{2})$,
and $\langle\hat{m}^{\bm{q}}\hat{m}^{-\bm{q}}\rangle=\nu+\mathcal{O}(q^{2})$.

Now, considering the isotropy of the static correlators in Eq. (E.105) and
noting the arbitrary orientation of $\bm{q}$, we obtain the following reduced
Green-Kubo relations:

$\rho_{0}\mu\eta_{ijkl}-\rho_{0}\Omega_{lk}\gamma_{ij}=\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle\,.$
(E.114)

Applying the same assumptions to Eq. (E.108), while also grouping order
$\mathcal{O}(1)$ terms in the limit $\bm{q}\rightarrow\bm{0}$ yields

$\rho_{0}\nu\gamma_{ij}=\lim_{\bm{q}\rightarrow\bm{0}}\epsilon_{kl}\int_{0}^{\Delta
t}dt\
\langle\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\rangle\,.$
(E.115)

Thus far, we have exploited the isotropy of the static correlation functions.
Now, we additionally utilize the isotropy of the transport tensors
$\eta_{ijkl}$ and $\gamma_{ij}$, expanding them with the bases outlined in
Eqs. (7) and (8) and Table I of Ref. [5]. The basis for $\eta_{ijkl}$ is
specified in Eq. (E.91) and the 2d isotropic basis for $\gamma_{ij}$ is
specified as

$\gamma_{ij}=\sum_{\alpha=1}^{2}\gamma_{\alpha}s_{ij}^{(\alpha)}=\gamma_{1}\delta_{ij}+\gamma_{2}\epsilon_{ij}\,.$
(E.116)

The transport coefficients $\gamma_{1}$ and $\gamma_{2}$ couple the spin field
to a normal stress and an antisymmetric stress, respectively. Using these
bases in conjunction with Eqs. (E.114) and (E.115), we obtain the final Green-
Kubo relations for each transport coefficient relating the stress tensor to
the velocity gradient and spin field,

$\lambda_{1}=\frac{1}{4\rho_{0}\mu}\left(\delta_{ij}\delta_{kl}+\frac{\pi}{\nu}\delta_{ij}\epsilon_{kl}\right)\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\left<\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\right>\,,$
(E.117)
$\lambda_{2}=\frac{1}{8\rho_{0}\mu}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl})\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\left<\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\right>\,,$
(E.118)
$\lambda_{3}=\frac{1}{4\rho_{0}\mu}\left(1-\frac{\tau}{\nu}\right)\epsilon_{ij}\epsilon_{kl}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\left<\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\right>\,,$
(E.119)
$\lambda_{4}=\frac{1}{8\rho_{0}\mu}(\epsilon_{ik}\delta_{jl}+\epsilon_{jl}\delta_{ik})\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\left<\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\right>\,,$
(E.120)
$\lambda_{5}=\frac{1}{8\rho_{0}\mu}\left(\epsilon_{ij}\delta_{kl}+\frac{\pi}{\nu}\epsilon_{ij}\epsilon_{kl}\right)\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\left<\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\right>\,,$
(E.121)
$\lambda_{6}=-\frac{1}{8\rho_{0}\mu}\left(1-\frac{\tau}{\nu}\right)\delta_{ij}\epsilon_{kl}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\left<\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\right>\,,$
(E.122)
$\gamma_{1}=\frac{1}{2\rho_{0}\nu}\delta_{ij}\epsilon_{kl}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\left<\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\right>\,,$
(E.123)
$\gamma_{2}=\frac{1}{2\rho_{0}\nu}\epsilon_{ij}\epsilon_{kl}\lim_{\bm{q}\rightarrow\bm{0}}\int_{0}^{\Delta
t}dt\left<\hat{\sigma}_{ij}^{\bm{q}}(t)\hat{\sigma}_{kl}^{-\bm{q}}(0)\right>\,.$
(E.124)

Once again, using the flux hypothesis, we isolate an independent expression
for each transport coefficient, which would be unattainable using the ORH
without additional assumptions.

The above fully separated Green-Kubo relations reveal equivalencies between
two pairs of transport coefficients. Namely, the antisymmetric stress
originating from vorticity (governed by $\lambda_{3}$) is related to the
antisymmetric stress originating from the spin field (governed by
$\gamma_{2}$). Furthermore, the normal stress originating from vorticity
(governed by $\lambda_{6}$) is related to the normal stress originating from
the spin field (governed by $\gamma_{1}$). These findings can be expressed in
the following equalities,

$-\frac{4\lambda_{6}}{\gamma_{1}}=\frac{2\lambda_{3}}{\gamma_{2}}=\frac{(\nu-\tau)}{\mu}\,.$
(E.125)

While the second equality has been shown in [5] via the regression hypothesis,
the flux hypothesis reveals an additional connection between $\lambda_{6}$ and
$\gamma_{1}$. Note in equilibrium due to equipartioning, there exist no static
cross-correlations between spin and velocity, making $\tau=0$. Furthermore, in
equilibrium $\nu/\mu=I/\rho_{0}$, with moment of inertia density $I$. Thus in
equilibrium systems, equivalencies can be made between transport coefficients
relating a stress to vorticity and with those relating that same stress to the
spin field. Stresses then stem from the difference of the vorticity and
intrinsic spin field.

In summary, we demonstrated how to apply the flux hypothesis to higher-order
tensors directly, which leads to Green-Kubo relations for individual transport
coefficients. This is in contrast to the work presented in [5], which utilizes
the Onsager regression hypothesis and yields fewer equations, making it
impractical and perhaps impossible to generally isolate expressions for each
transport coefficient without additional assumptions, thus showing the utility
of the flux hypothesis. The flux hypothesis allows for all information
pertinent to a transport coefficient to be utilized, whether that coefficient
affects temporal relaxation or not.

###  E.3 Thermal Conductivity

We now apply the flux hypothesis to derive a Green-Kubo relation for the
thermal conductivity tensor for 2d isotropic systems, including both the even
and odd thermal conductivities [13]. Neglecting body heating and viscous
dissipation, the balance law for the specific internal energy $u$ is

$\rho\dot{u}=-\bm{\nabla}\cdot\bm{Q}\,,$ (E.126)

which holds for each microscopic realization with $\bm{Q}$ being the
conductive heat flux vector. Considering microscopic fluctuations from a
spatially homogeneous steady state with no average bulk velocity, we may write
the linearized Fourier-transformed balance law as

$\rho_{0}\partial_{t}\hat{u}^{\bm{q}}=-\text{i}\bm{q}\cdot\hat{\bm{Q}}^{\bm{q}}\,.$
(E.127)

To close this differential equation, we necessitate a constitutive law
relating the heat flux to a field variable. Assuming Fourier’s Law, which
states that the heat flux $\bm{Q}$ is linearly related to the temperature
gradient $\bm{\nabla}T$, we write

$\bm{Q}=-\bm{\kappa}\cdot\bm{\nabla}T\,,$ (E.128)

with $\bm{\kappa}$ being the second order thermal conductivity tensor. Further
assuming that temperature and energy are related, we may express Fourier’s law
as

$\bm{Q}=-\bm{\kappa}\cdot\frac{dT}{du}\bm{\nabla}u\,,$ (E.129)

with $\frac{dT}{du}$ being the inverse specific heat capacity, which we will
take to be constant.

The quantities $\bm{Q}$, $u$, and $\bm{\kappa}$ are in direct analogy to the
quantities $\bm{J}_{i}$, $A_{i}$, and $\mathbf{M}_{ij}$ in the main text.
Thus, mirroring Eq. (14), we have

$\bm{\kappa}=\mathbf{G}(\bm{q},\Delta t)c(\bm{q})\ ,$ (E.130)

where $\mathbf{G}$ and $c$ are

$\mathbf{G}(\bm{q},\Delta t)=\int_{0}^{\Delta
t}dt\langle\hat{\bm{Q}}^{\bm{q}}(t)\hat{\bm{Q}}^{-\bm{q}}(0)\rangle\,,$
(E.131)
$c^{-1}(\bm{q})=\rho_{0}\frac{dT}{du}\langle\hat{u}^{\bm{q}}\hat{u}^{-\bm{q}}\rangle\,.$
(E.132)

Thus, the Green-Kubo relation for the thermal conductivity tensor becomes

$\bm{\kappa}=\lim_{\bm{q}\rightarrow\bm{0}}\frac{1}{\rho_{0}\frac{dT}{du}\left<\hat{u}^{\bm{q}}\hat{u}^{-\bm{q}}\right>}\int_{0}^{\Delta
t}dt\langle\hat{\bm{Q}}^{\bm{q}}(t)\hat{\bm{Q}}^{-\bm{q}}(0)\rangle\,.$
(E.133)

For 2d isotropic systems, we may expand the thermal conductivity tensor using
the representation theorem in SI-I of Ref. [5] as

$\kappa_{ij}=\kappa_{\parallel}\delta_{ij}+\kappa_{\perp}\epsilon_{ij}\ ,$
(E.134)

where $\kappa_{\parallel}$ and $\kappa_{\perp}$ refer to the even and odd
thermal conductivity, respectively. A non-zero odd thermal conductivity
$\kappa_{\perp}$ induces heat fluxes orthogonal to temperature gradients. By
contracting $\delta_{ij}$ or $\epsilon_{ij}$ with Eq. (E.133), we obtain

$\kappa_{\parallel}=\lim_{\bm{q}\rightarrow\bm{0}}\frac{1}{2\rho_{0}\frac{dT}{du}\left<\hat{u}^{\bm{q}}\hat{u}^{-\bm{q}}\right>}\int_{0}^{\Delta
t}dt\langle\hat{Q}_{i}^{\bm{q}}(t)\hat{Q}_{j}^{-\bm{q}}(0)\rangle\delta_{ij}\,,$
(E.135)
$\kappa_{\perp}=\lim_{\bm{q}\rightarrow\bm{0}}\frac{1}{2\rho_{0}\frac{dT}{du}\left<\hat{u}^{\bm{q}}\hat{u}^{-\bm{q}}\right>}\int_{0}^{\Delta
t}dt\langle\hat{Q}_{i}^{\bm{q}}(t)\hat{Q}_{j}^{-\bm{q}}(0)\rangle\epsilon_{ij}\,.$
(E.136)

Note with the substitution
$\frac{du}{dT}=\rho_{0}V\langle\hat{u}^{2}\rangle/(k_{\text{B}}T^{2})$, we
recover the usual pre-factor of $V/(k_{\text{B}}T^{2})$. Equation (E.135) is
the standard Green-Kubo relation for the symmetric part of the thermal
conductivity tensor. However, as can be seen from combining Eqs. (E.126),
(E.129) and (E.134), there exist no contributions of $\kappa_{\perp}$ to
energy relaxation due to the divergence free nature of odd fluxes. Therefore,
there exists no possibility of obtaining the Green-Kubo relations for
$\kappa_{\perp}$ from the Onsager regression hypothesis. The flux hypothesis,
which instead acts on the level of the constitutive law (E.129), is able to
yield a Green-Kubo relation for $\kappa_{\perp}$, showing that both time-
reversal symmetry and parity symmetry breaking is necessary to observe odd
thermal fluxes.