# Supplemental material for “Evaluation of transition rates from
nonequilibrium instantons”

Eric R. Heller<EMAIL_ADDRESS>David T. Limmer<EMAIL_ADDRESS>Department of Chemistry, University of California, Berkeley, 94720 Berkeley,
USA

###### Abstract

In this supplemental material, we derive an alternative, formally equivalent
version of our nonequilibrium instanton rate equation in the van-Vleck or
trajectory formalism, which lends itself to numerically efficient rate
calculations in large, complex systems that require a large number of beads.
Furthermore, we elaborate on the continuous time-translation symmetry of the
instanton, the efficient instanton optimization and the connection of
instanton theory to Kramers–Langer theory at equilibrium for explicit-particle
models and stochastic field theories.

##  I Time-translation symmetry

As described in the main text, the instanton is an infinite-time trajectory.
The typical “double kink” shape of an optimal transition path between reactant
and product is illustrated in Fig. SI.1, where the path length is shown as a
function of time.

Figure SI.1: Plot of the path length as a function of time along an instanton
in the long-time limit. The comparatively fast transitions between the fixed
points of the force are separated by long dwell phases at the reactant,
product and TS configurations, giving rise to the typical double kink shape.
The activation path reaches the TS at time $t_{\text{f}}$. The red dashed
paths indicate equivalent activation paths related through time translation.

It can be seen that the instanton may dwell for an arbitrary, quasi-infinite
time at the reactant, product or TS configurations (the fixed points of the
force) without changing its action because force, velocity and canonical
momentum go to zero at these points [Eq. (5)]. Hence, there exists an infinite
number of instantons related through time translation, as indicated in Fig.
SI.1.

The rate-determining activation path, connecting reactant and TS, corresponds
to the first kink in Fig. SI.1. In numerical calculations, the long-time limit
is reached when the activation process is fully resolved, which is fast
compared to the infinitely long dwell phases. The time-translation symmetry of
the activation path manifests itself as a zero eigenvalue in the second-
derivative matrix of the action, rendering a “naïve” Laplace approximation of
the rate divergent. We address this issue in the main text by transforming the
integral over the zero mode to a collective time integral, as encountered in
the evaluation of flux–flux correlation functions, which must be solved
separately.

##  II Instanton optimization

Locating an instanton consists in the minimization of the discretized
Onsager–Machlup action in Eq. (8), where the endpoints are fixed to the
reactant and TS configurations. The TS can be located either by optimizing a
saddle point of the force directly or by locating the full minimum-action
pathway between reactant and product. A whole host of established minimization
routines may be put to use for the instanton optimization. In this work, we
employ conjugate-gradient and Newton–Raphson algorithms that minimize the
magnitude of the gradient vector of the discretized action with respect to all
intermediate bead positions. For the latter, a small shift must be added to
the diagonal of the second-derivative matrix to avoid problems with zero
eigenvalues [1]. The banded, symmetric properties of the second-derivative
matrix of the action may be leveraged to significantly improve the memory and
time efficiency of Newton–Raphson optimizers for large systems. Furthermore,
we update the second-derivative matrix during the optimization [2], requiring
only a single calculation of the second-derivative matrix for the whole
minimization. Alternatively, specialized instanton optimization routines have
been suggested [3, 4].

The instanton rate must be converged with respect to both the number of beads,
$N$, and time, $t_{\text{f}}$. For the latter, a plateauing plot of the path
length, as in Fig. SI.1, indicates that the long-time limit is reached.
Additionally, the numerical value of the “zero eigenvalue” from time
translation can be monitored because the time-translation symmetry is only
approximate for long but finite time. As $t_{\text{f}}$ is increased, the
ratio between the zero eigenvalue and the remaining eigenvalues decreases. A
sufficiently small ratio thus indicates convergence. However, the most
reliable metric for convergence remains the comparison between actions and
rates at different values of $t_{\text{f}}$ and $N$. We have found it is often
useful to first converge a minimum-action path with a relatively short time
based on a simple initial guess, e.g., equidistant beads along a straight line
connecting the endpoints. The result can then be used as an initial guess for
subsequent optimizations with gradually increasing $t_{\text{f}}$ and $N$.

##  III Computational details

The instanton rates for the particle model in Fig. (2) are calculated with
$N=800$ beads and a propagation time of $t_{\text{f}}=4$, which corresponds to
convergence to within less than $1\%$. An example of a convergence table is
shown in Table SI.1. As expected, for longer propagation times, more beads are
required to converge the rate. In order to judge whether the rate is
converged, it is particularly useful to look at the “diagonal convergence” in
Table SI.1, i.e., comparing rates computed at the same time step
$\tau=t_{\text{f}}/N$. Thus, we find that the rate at $t_{\text{f}}=4$ and
$N=800$ is in close agreement with the rate at $t_{\text{f}}=8$ and $N=1600$,
indicating convergence.

Table SI.1: Convergence of the rate constant $k$ with respect to propagation time $t_{\text{f}}$ and number of beads $N$ for the system defined in Eq. (10) of the main text with $\nu=4$ and $\epsilon=1$.  $N$ | $t_{\text{f}}$
---|---
1/2 | 1 | 2 | 4 | 8
50 | 0.1660 | 0.1442 | 0.1155 | 0.1119 | 0.3771
100 | 0.1649 | 0.1421 | 0.1178 | 0.1114 | 0.1119
200 | 0.1644 | 0.1400 | 0.1190 | 0.1135 | 0.1114
400 | 0.1641 | 0.1404 | 0.1196 | 0.1150 | 0.1135
800 | 0.1640 | 0.1401 | 0.1199 | 0.1159 | 0.1150
1600 | 0.1639 | 0.1399 | 0.1201 | 0.1163 | 0.1159

Note that a very good approximation to the reaction rate with less than $5\%$
deviation from the fully converged result can already be obtained with only
$50$ beads for propagation times of $2$ or $4$.

The DVR results in Fig. 2 have been generated on a grid of lengths 9 and 6
along $x_{1}$ and $x_{2}$, discretized by 150 points in each direction [5, 6].
The first non-zero eigenvalue of the Fokker–Planck operator of a bistable
system corresponds to the sum of the forward and backward rate constants [7].
Due to the symmetry of the system defined in Eq. (10), the forward and
backward rate constants are identical, so that $k$ is obtained as half of the
smallest non-zero eigenvalue.

The convergence of the instanton rates for the active field theory defined in
the main text is somewhat more challenging because the motion along the
reaction coordinate is orders of magnitude slower than the motion along some
of the other degrees of freedom in the system. Hence, relatively long
propagation times of up to $t_{\text{f}}=60$ for $h=-0.1$ and $\nu=0.25$ are
required to reach the long-time limit while a small time step must be used to
correctly describe the fast modes. In order to converge the instanton rates to
within less than $5\%$, we therefore used 10000–30000 beads. Due to the
efficiency of the method, even calculations with this many beads take less
than one day on 32 CPUs. However, it may be possible to reduce the number of
required beads by employing different discretization schemes.

##  IV Trajectory formalism

The direct calculation of the instanton rate from a discretized path integral,
as presented in the main text, offers high numerical stability but requires
diagonalizing a $Nf\times Nf$ matrix, scaling as $\mathcal{O}(N^{3})$ with the
number of beads. Utilizing the banded structure of
$\nabla_{\text{s}}^{2}S_{N}$ reduces the scaling to $\mathcal{O}(N^{2})$.
However, for certain large systems requiring many beads for convergence, the
rate calculation may still become the bottleneck. Here, we present an
alternative, computationally more efficient approach to calculating the
instanton rate within the “trajectory formalism”, building upon the work of
Althorpe and Richardson for Euclidean actions [8, 9].

The key quantity in the definition of the rate constant is the joint
probability density [Eqs. (3),(4)], which we approximate by Laplace’s method

$\displaystyle\rho(\mathsf{x}_{\text{f}},t_{\text{f}};\mathsf{x}_{\text{i}},0)\sim[\det(\mathbf{D})]^{-Nf/2}\sqrt{\frac{[\det(\mathbf{J})]^{-1}}{(4\pi\epsilon\,\tau)^{f}}}\,\mathrm{e}^{-S_{N}/\epsilon},$
(SI.1)

where all quantities are evaluated at the instanton, $\bar{\mathsf{x}}(t)$,
connecting $\mathsf{x}_{\text{i}}$ to $\mathsf{x}_{\text{f}}$ in time
$t_{\text{f}}$. The instanton satisfies $\nabla_{\text{r}}S_{N}=0$, where
$\nabla_{\text{r}}=(\partial/\partial\mathsf{x}_{1},...,\partial/\partial\mathsf{x}_{N-1})$
defines the derivative vector with respect to all intermediate beads. The
fluctuations around the instanton are encoded in
$\mathbf{J}=2\tau\nabla_{\text{r}}^{2}S_{N}$. Alternatively, the probability
density in Eq. (SI.1) may be written in analogy to the van-Vleck approximation
of the quantum-mechanical propagator [10] 111If a system exhibits more than
one (dominant) minimum-action pathway, Eqs. (SI.1) and (SI.2) consist of a sum
over all instantons.

$\displaystyle\rho(\mathsf{x}_{\text{f}},t_{\text{f}};\mathsf{x}_{\text{i}},0)\sim\sqrt{\frac{C}{(2\pi\epsilon)^{f}}}\,\mathrm{e}^{-S(\mathsf{x}_{\text{i}},\mathsf{x}_{\text{f}};t_{\text{f}})/\epsilon},$
(SI.2)

where the information about the fluctuations around the optimal path is
contained in the prefactor

$\displaystyle
C=\det\left(-\frac{\partial^{2}S(\mathsf{x}_{\text{i}},\mathsf{x}_{\text{f}};t_{\text{f}})}{\partial\mathsf{x}_{\text{i}}\partial\mathsf{x}_{\text{f}}}\right)\mathrm{e}^{-\int_{0}^{t_{\text{f}}}\nabla\cdot\mathbf{\mu}\mathsf{F}[\bar{\mathsf{x}}(t)]\mathrm{d}t}.$
(SI.3)

In order for Eqs. (SI.1) and (SI.2) to be equivalent, the prefactors must be
related through

$\displaystyle[\det(\mathbf{D})]^{Nf}(2\tau)^{f}C=[\det(\mathbf{J})]^{-1}.$
(SI.4)

We refer to this formalism as the “trajectory formulation”, because the
locations of the $N-1$ intermediate beads are constrained by requiring that
they minimize the action (and thus form a classical trajectory) for the given
endpoints and time,

$\displaystyle\mathsf{x}_{n}$ $\displaystyle=\bar{\mathsf{x}}_{n}\quad\forall
n\in[1,N-1],$ (SI.5)

where $\bar{\mathsf{x}}_{n}$ are the positions that minimize the discretized
action so that $\nabla_{\text{r}}S_{N}=0$. Hence, the intermediate beads can
be integrated out using Laplace’s method, so that the action
$S(\mathsf{x}_{\text{i}},\mathsf{x}_{\text{f}};t_{\text{f}})$ depends
explicitly only on the two endpoints,
$\mathsf{x}_{\text{i}}\equiv\mathsf{x}_{0}$ and
$\mathsf{x}_{\text{f}}\equiv\mathsf{x}_{N}$ [8]. Therefore, we have
$\frac{\partial\mathsf{x}_{\text{i}}}{\partial\mathsf{x}_{\text{i}}}=\frac{\partial\mathsf{x}_{\text{f}}}{\partial\mathsf{x}_{\text{f}}}=1$,
$\frac{\partial\mathsf{x}_{\text{i}}}{\partial\mathsf{x}_{\text{f}}}=\frac{\partial\mathsf{x}_{\text{f}}}{\partial\mathsf{x}_{\text{i}}}=0$
and non-trivial derivatives of the intermediate beads with respect to the
endpoints.

We now deal with the remaining question of how to evaluate $C$ without
calculating $\det(\mathbf{J})$ [Eq. (SI.4)]. Given the discretized action [Eq.
(8)], a derivative with respect to an intermediate bead is given by

$\displaystyle\frac{\partial S_{N}}{\partial\mathsf{x}_{n}}$
$\displaystyle=-\frac{\tau}{2}\left(\frac{1}{\tau}+\nabla\mathbf{\mu}\mathsf{F}_{n}\right)^{\\!\\!\text{T}}\mathbf{D}^{-1}\left(\frac{\bar{\mathsf{x}}_{n+1}-\bar{\mathsf{x}}_{n}}{\tau}-\mathbf{\mu}\mathsf{F}_{n}\right)$
$\displaystyle+\frac{1}{2}\mathbf{D}^{-1}\left(\frac{\bar{\mathsf{x}}_{n}-\bar{\mathsf{x}}_{n-1}}{\tau}-\mathbf{\mu}\mathsf{F}_{n-1}\right)\quad\forall
n\in[1,N-1],$ (SI.6)

where $\mathsf{F}_{n}\equiv\mathsf{F}(\bar{\mathsf{x}}_{n})$, and we define
$(\nabla\mathbf{\mu}\mathsf{F})_{jk}=\mu_{jm}\frac{\partial F_{m}}{\partial
x_{k}}$, where repeated indices are summed over. Taking a second derivative
with respect to either of the two endpoints leads to the following system of
linear equations

$\displaystyle\sum_{n^{\prime}=1}^{N-1}\sum_{j^{\prime}=1}^{f}J_{nj,n^{\prime}j^{\prime}}\frac{\partial\bar{x}_{n^{\prime}j^{\prime}}}{\partial
x_{\text{i},k}}$
$\displaystyle=[\mathbf{D}^{-1}(\mathbb{I}+\tau\nabla\mathbf{\mu}\mathsf{F}_{0})]\mathsf{e}_{k}\delta_{n1},$
(SI.7a)
$\displaystyle\sum_{n^{\prime}=1}^{N-1}\sum_{j^{\prime}=1}^{f}J_{nj,n^{\prime}j^{\prime}}\frac{\partial\bar{x}_{n^{\prime}j^{\prime}}}{\partial
x_{\text{f},k}}$
$\displaystyle=[\mathbf{D}^{-1}(\mathbb{I}+\tau\nabla\mathbf{\mu}\mathsf{F}_{N-1})]^{\text{T}}\mathsf{e}_{k}\delta_{nN-1},$
(SI.7b)

where $J_{nj,n^{\prime}j^{\prime}}$ are elements of $\mathbf{J}$ with bead
indices $n,n^{\prime}\in[1,N-1]$ and $j,j^{\prime}\in[1,f]$ indexing the
degrees of freedom. For each degree of freedom $k$ with corresponding unit
vector $\mathsf{e}_{k}$, the system of equations is solved for all
$\frac{\partial\bar{\mathsf{x}}_{n}}{\partial x_{\text{i/f},k}}$, vectors
containing derivatives of the coordinates of an intermediate bead with respect
to the $k\text{th}$ component of endpoint $\mathsf{x}_{\text{i/f}}$.

Taking second derivatives of the discretized action with respect to both
endpoints under the constraint of Eq. (SI.5) results in the relations (see
also Refs. 8 and 9)

$\displaystyle\frac{\partial^{2}S}{\partial\mathsf{x}_{\text{i}}\mathsf{x}_{\text{i}}}$
$\displaystyle=-\frac{1}{2\tau}\bigg{[}\tau[\nabla^{2}\mathbf{\mu}\mathsf{F}_{0}][\mathbf{D}^{-1}(\bar{\mathsf{x}}_{1}-\bar{\mathsf{x}}_{0}-\tau\mathsf{F}_{0})]$
$\displaystyle+\left(\mathbb{I}+\tau\nabla\mathbf{\mu}\mathsf{F}_{0}\right)^{\text{T}}\mathbf{D}^{-1}\left(\frac{\partial\bar{\mathsf{x}}_{1}}{\partial\mathsf{x}_{\text{i}}}-\mathbb{I}-\tau\nabla\mathbf{\mu}\mathsf{F}_{0}\right)\bigg{]},$
(SI.8a)
$\displaystyle\frac{\partial^{2}S}{\partial\mathsf{x}_{\text{f}}\mathsf{x}_{\text{f}}}$
$\displaystyle=\frac{1}{2\tau}\mathbf{D}^{-1}\left(\mathbb{I}-(\mathbb{I}+\tau\nabla\mathbf{\mu}\mathsf{F}_{N-1})\frac{\partial\bar{\mathsf{x}}_{N-1}}{\partial\mathsf{x}_{\text{f}}}\right),$
(SI.8b)
$\displaystyle\frac{\partial^{2}S}{\partial\mathsf{x}_{\text{i}}\mathsf{x}_{\text{f}}}$
$\displaystyle=-\frac{1}{2\tau}\frac{\partial\bar{\mathsf{x}}_{N-1}}{\partial\mathsf{x}_{\text{i}}}\left(\mathbb{I}+\tau\nabla\mathbf{\mu}\mathsf{F}_{N-1}\right)\mathbf{D}^{-1}$
(SI.8c)

where each column of the matrix
$\partial\bar{\mathsf{x}}_{n}/\partial\mathsf{x}_{\text{f}}$ is determined by
solving Eqs. (SI.7) for a particular $k$, so that
$(\frac{\partial\bar{x}_{n}}{\partial
x_{\text{f}}})_{jk}=\frac{\partial\bar{x}_{n,j}}{\partial x_{\text{f},k}}$ and
equivalent for derivatives with respect to $\mathsf{x}_{\text{i}}$. The rank 3
tensor in Eqs. (SI.8) is defined as
$(\nabla^{2}\mathbf{\mu}\mathsf{F})_{jkl}=\mu_{jm}\frac{\partial^{2}F_{m}}{\partial
x_{k}\partial x_{l}}$, and
$[\nabla^{2}\mathbf{\mu}\mathsf{F}][\mathbf{D}^{-1}\mathsf{x}]=(\nabla^{2}\mathbf{\mu}\mathsf{F})_{jkl}(\mathbf{D}^{-1}\,\mathsf{x})_{j}$.
Together, Eqs. (SI.7) and (SI.8) enable us to compute the prefactor $C$ [Eq.
(SI.3)]. Notably, the matrices in Eqs. (SI.8) have dimension $f\times f$, as
opposed to $(N-1)f\times(N-1)f$ for $\mathbf{J}$. We thus transformed the
bottleneck of the prefactor calculation from the diagonalization of a large
matrix [Eq. (SI.1)] to the solution of a system of linear equations [Eq.
(SI.7)]. The symmetric, banded, positive-definite properties of $\mathbf{J}$
permit the use of efficient solution algorithms, reducing the scaling of the
computational costs to $\mathcal{O}(N)$.

The trajectory formalism developed above can be used to approximate
probability densities and related quantities of interest. However, as laid out
in Sec. I, the fluctuation matrix obtained in the calculation of rate
constants is ill-defined because of the time-translation symmetry of the
instanton pathway. In the discretized path-integral formalism [Eq. (SI.1)],
this symmetry manifests itself as a zero eigenvalue of $\mathbf{J}$, which can
be readily removed. The trajectory formalism lacks a similar solution.
However, we can break time-translation symmetry by splitting the instanton
into two shorter segments at an arbitrary point $\mathsf{x}_{\text{m}}$ along
the trajectory [1, 12], using the composition property
$\rho(\mathsf{x}_{\text{f}},t_{\text{f}};\mathsf{x}_{\text{i}},0)=\int\rho(\mathsf{x}_{\text{f}},t_{\text{f}};\mathsf{x}_{\text{m}},t_{\text{m}})\rho(\mathsf{x}_{\text{m}},t_{\text{m}};\mathsf{x}_{\text{i}},0)\,\mathrm{d}\mathsf{x}_{\text{m}}$.
The integral to reconnect the probabilities can again be taken by Laplace’s
method, so that the prefactor of the total probability density is given by

$C=C_{\text{a}}C_{\text{b}}\\\
\times\left[\det\left(\frac{\partial^{2}S(\mathsf{x}_{\text{i}},\mathsf{x}_{\text{m}};t_{\text{m}})}{\partial\mathsf{x}_{\text{m}}^{2}}+\frac{\partial^{2}S(\mathsf{x}_{\text{m}},\mathsf{x}_{\text{f}};t_{\text{f}}-t_{\text{m}})}{\partial\mathsf{x}_{\text{m}}^{2}}\right)\right]^{-1},$
(SI.9)

where the prefactors of the shorter segments are

$\displaystyle C_{\text{a}}$
$\displaystyle=\det\left(-\frac{\partial^{2}S(\mathsf{x}_{\text{i}},\mathsf{x}_{\text{m}};t_{\text{m}})}{\partial\mathsf{x}_{\text{i}}\partial\mathsf{x}_{\text{m}}}\right)\mathrm{e}^{-\int_{0}^{t_{\text{m}}}\nabla\cdot\mathbf{\mu}\mathsf{F}[\mathsf{x}(t)]\mathrm{d}t},$
(SI.10) $\displaystyle C_{\text{b}}$
$\displaystyle=\det\left(-\frac{\partial^{2}S(\mathsf{x}_{\text{m}},\mathsf{x}_{\text{f}};t_{\text{f}}-t_{\text{m}})}{\partial\mathsf{x}_{\text{m}}\partial\mathsf{x}_{\text{f}}}\right)\mathrm{e}^{-\int_{t_{\text{m}}}^{t_{\text{f}}}\nabla\cdot\mathbf{\mu}\mathsf{F}[\mathsf{x}(t)]\mathrm{d}t},$
(SI.11)

and the connection piece can also be evaluated with Eqs. (SI.8) and (SI.7).

However, fully reconnecting the segments would reintroduce the problematic
time-translation mode. Instead, we exploit the fact that this mode aligns with
the instanton velocity along the path. Therefore, we can separate the
coordinates of the splitting point $\mathsf{x}_{\text{m}}$ into the mode
parallel to the instanton $q_{\text{m}}$ and the modes orthogonal to it
$\mathsf{Q}_{\text{m}}$. Integration is then performed only over the latter.
Applying this approach to Eq. (2) yields the following equation for the
instanton rate constant

$k\sim\lim_{\mathcal{T}\rightarrow\infty}\frac{\sqrt{C_{\text{a}}C_{\text{b}}}}{(2\pi\epsilon)^{f}}\,\frac{\dot{\bar{q}}_{\text{m}}}{\bar{p}_{\text{f}}}\,\sqrt{\frac{(2\pi\epsilon)^{f-1}}{\Omega_{\text{m}}}}\,\sqrt{\frac{(2\pi\epsilon)^{f-1}}{\Omega^{\ddagger}}}\\\
\times\int_{0}^{\mathcal{T}}\bar{\mathsf{p}}^{\text{T}}_{\text{f}}\mathbf{D}\,\bar{\mathsf{p}}_{\text{f}}\,\mathrm{e}^{-S(\mathsf{x}_{\text{R}},\mathsf{x}^{\ddagger},t_{\text{f}})/\epsilon}\,\mathrm{d}t_{\text{m}},$
(SI.12)

where
$\Omega_{\text{m}}=\det\left(\frac{\partial^{2}S(\mathsf{x}_{\text{R}},\mathsf{x}_{\text{m}};t_{\text{m}})}{\partial\mathsf{Q}_{\text{m}}^{2}}+\frac{\partial^{2}S(\mathsf{x}_{\text{m}},\mathsf{x}^{\ddagger};t_{\text{f}}-t_{\text{m}})}{\partial\mathsf{Q}_{\text{m}}^{2}}\right)$
originates from the composition integral over the orthogonal modes. The
Laplace integration over the dividing surface leads to the term
$\Omega^{\ddagger}=\det\left(\frac{\partial^{2}S(\mathsf{x}_{\text{R}},\mathsf{x}^{\ddagger};t_{\text{f}})}{\partial\mathsf{s}^{2}}\right)$,
where the reaction coordinate along $\bar{\mathsf{p}}_{\text{f}}$ has been
projected out and the remaining $f-1$ coordinates parameterizing the dividing
surface are combined in $\mathsf{s}$.

The velocity along the instanton at the splitting point,
$\dot{\bar{q}}_{\text{m}}$, appears because we transformed the integral over
$q_{\text{m}}$ to a collective time integral. In analogy to the prefactor in
the discretized path-integral formulation, we find that
$C_{\text{b}}\propto\bar{\mathsf{p}}_{\text{f}}^{2}$. Consequently, the term
$C_{\text{b}}/\bar{\mathsf{p}}_{\text{f}}^{2}$ can be factored out of the
integral and evaluated at the instanton because it varies slowly compared to
the exponential even though $\bar{\mathsf{p}}_{\text{f}}\rightarrow 0$ in the
infinite-time limit [13]. By noting that the prefactor in the integrand of Eq.
(SI.12) is the instanton energy [Eq. (6)] and recognizing that the action of a
path connecting distinct points in zero time is infinite, the remaining
collective time integral can be solved by substitution. We thus arrive at the
nonequilibrium instanton formula for the rate constant within the trajectory
formalism

$\displaystyle
k\sim\frac{1}{2\pi}\,\frac{\dot{\bar{q}}_{\text{m}}}{\bar{p}_{\text{f}}}\,\sqrt{\frac{C_{\text{a}}C_{\text{b}}}{\Omega_{\text{m}}\Omega^{\ddagger}}}\,\mathrm{e}^{-S/\epsilon},$
(SI.13)

whose numerical evaluation scales as $\mathcal{O}(N)$.

In practice, achieving a desired numerical accuracy for $C$, calculated from
derivatives with respect to opposite ends of the path, often requires more
beads than the convergence of the connection pieces, obtained from derivatives
with respect to the same end [Eq. (SI.8)]. Therefore, computing $C$ from
$\det(\mathbf{J})$, as shown in Eq. (SI.4), may proof advantageous. The
efficiency of the method can be further enhanced by splitting up the paths
into smaller segments, thus iteratively applying Eq. (SI.9) in analogy to Ref.
14. In this way, the diagonal parts of Eqs. (SI.8) can be used to reconnect
the paths while the prefactors $C_{\text{a}},C_{\text{b}},...$ are obtained by
diagonalizing the smaller $\mathbf{J}$ matrices of the shorter segments. The
independent calculations for individual segments are trivially parallelizable,
leading to an approach that is both efficient and numerically robust.
Alternatively, the components of the prefactor in Eq. (SI.13) can be computed
by numerically solving matrix Riccati or Gelfand–Yaglom equations, as
suggested for large-deviation functions [15, 16, 17, 18, 19]. This highlights
the broader applicability of our methods beyond rate theory.

The extension of the instanton rate in the trajectory formalism [Eq. (SI.13)]
to field-theoretical descriptions, analogous to Eq. (13), is straightforward.
The only differences to Eq. (SI.13) are that: (i) the $f$ zero modes arising
from the translation symmetry at the TS must be excluded from the integral
over the dividing surface and thus projected out of $\Omega^{\ddagger}$ using
the corresponding normal modes at the TS, and (ii) the volume term
$\left(\frac{B_{x}}{2\pi\epsilon\kappa}\right)^{\\!f/2}$ from the integration
over these translation modes must be included.

##  V Connection to Kramers–Langer theory at equilibrium

The transition-state theory estimate of the rate constant at equilibrium is
given by Kramers–Langer theory (KLT) [20, 21, 22, 23]

Figure SI.2: KLT (solid lines) and instanton (dots) rate constants for the
potential defined in Eq. (SI.15) over a range of biases, $b$, and diffusion
anisotropies with $\beta=\epsilon=1$.  Figure SI.3: Color map of the potential
defined below Eq. (10) of the main text. The diffusion is anisotropic because
$\mathbf{D}=\frac{1}{2}\begin{pmatrix}2&1\\\ 1&8\end{pmatrix}\neq\mathbb{I}$.
The stream plot depicts $\mathbf{\mu}\mathsf{F}$, where $\mathsf{F}$ is
conservative ($\nu=0$). The instanton is shown by the green line along with
the direction of the canonical momentum $\bar{\mathsf{p}}_{\text{f}}$ at the
TS and the eigenvectors $\mathsf{u}_{+},\mathsf{v}_{+}$ associated with the
only positive eigenvalues of
$\nabla\mathsf{F}(\mathsf{x}^{\ddagger})\mathbf{\mu}$ and
$\mathbf{\mu}\nabla\mathsf{F}(\mathsf{x}^{\ddagger})$, respectively.

$\displaystyle k_{\text{KLT}}$
$\displaystyle=\frac{\lvert\lambda_{\text{b}}\rvert}{2\pi}\,\sqrt{\frac{\det(\nabla^{2}V_{\text{R}})}{\lvert\det(\nabla^{2}V^{\ddagger})\rvert}}\,\mathrm{e}^{-\beta
V^{\ddagger}},$ (SI.14)

where $V^{\ddagger}=V(\mathsf{x}^{\ddagger})-V(\mathsf{x}_{\text{R}})$ is the
barrier height, $\nabla^{2}V_{\text{R}}$ and $\nabla^{2}V^{\ddagger}$ are the
Hessians at the reactant and TS configurations, and $\lambda_{\text{b}}$ is
the sole negative eigenvalue of $\mathbf{\mu}\nabla^{2}V^{\ddagger}$
associated with the motion across the barrier. In Ref. 12, it has been
explicitly shown that instanton theory is equivalent to KLT in one-dimensional
systems at equilibrium. Given the equivalence of the Fokker–Planck
representation of the rate, from which KLT is commonly derived, and the path-
integral representation underlying instanton theory, it is clear that our
instanton theory correctly reduces to multi-dimensional KLT in the limit of
thermodynamic equilibrium (purely conservative forces). We illustrate this
point in Fig. SI.2, where we compare KLT and instanton results for the
potential

$\displaystyle V(\mathsf{x})$
$\displaystyle=\frac{A}{B^{4}}\left(x_{+}^{2}-B^{2}\right)^{\\!2}+x_{-}^{2}-bx_{+},$
(SI.15)

using the same parameters as in the main text [below Eq. (10)]. The rate
constants are shown for a range of different biases $b$ and diffusion
anisotropies. As expected, the two rate estimates agree to within numerical
precision.

Beyond the value of the rate constant, Fig. SI.3 further elucidates the
connection between the KLT and instanton pictures of the reaction mechanism.
It can be seen that the instanton velocity (pointing along the path) aligns
with the eigenvector $v_{+}$ associated with the sole positive eigenvalue of
$\nabla\mathbf{\mu}\mathsf{F}(\mathsf{x}^{\ddagger})$ at the TS, which
indicates the direction of the reactive flux. Instead, the canonical momentum
$\bar{\mathsf{p}}_{\text{f}}$ aligns with the eigenvector $u_{+}$ associated
with the only positive eigenvalue of
$\nabla\mathsf{F}(\mathsf{x}^{\ddagger})\mathbf{\mu}$. This eigenvector points
along the unstable diffusive normal mode at the TS, which is orthogonal to the
stochastic separatrix and thus defines the reaction coordinate [24]. Instanton
theory thus recovers these concepts from KLT at equilibrium and extends them
to far-from-equilibrium regimes.

##  VI Active field theory

Figure SI.4: (Left) Field configuration at the stationary points of the
potential [Eq. (SI.17)] associated with the stochastic field theory defined in
the main text ($\nu=0$) with $h=0.1$. (Right) Instanton of the same system,
where the color map indicates the field configuration along the path as a
function of space and time. The black dashed line highlights the TS
configuration or critical nucleus.

At equilibrium, the instanton rate equation [Eq. (13)] for active field
theories reduces to the corresponding KLT rate [25, 26, 27]

$\displaystyle\frac{k_{\text{KLT}}}{\mathcal{V}}$
$\displaystyle=\left(\frac{B_{x}}{2\pi\epsilon\kappa}\right)^{\\!\\!f/2}\frac{\lvert\lambda_{\text{b}}\rvert}{2\pi}\,\frac{Z^{\ddagger}}{Z_{\text{R}}}\,\mathrm{e}^{-V^{\ddagger}/\varepsilon}.$
(SI.16)

For a $\phi^{4}$ theory with a potential functional of the form

$\displaystyle V[\phi(x)]$
$\displaystyle=\int\left[\frac{\kappa}{2}\,\lvert\nabla\phi\rvert^{2}+W(\phi)\right]\mathrm{d}\mathsf{x},$
(SI.17)

the fluctuation factors at the reactant configuration,
$\phi(\mathsf{x})=\phi_{\text{R}}(\mathsf{x})$, and the TS configuration,
$\phi(\mathsf{x})=\phi^{\ddagger}(\mathsf{x})$, are given by

$\displaystyle Z^{\ddagger}$
$\displaystyle=\left\lvert\text{det}^{\prime}\left(-\kappa\Delta+\frac{\partial^{2}W}{\partial\phi^{2}}\right)_{\\!\\!\phi(\mathsf{x})=\phi^{\ddagger}(\mathsf{x})}\right\rvert^{-1/2},$
(SI.18a) $\displaystyle Z_{\text{R}}$
$\displaystyle=\left[\det\left(-\kappa\Delta+\frac{\partial^{2}W}{\partial\phi^{2}}\right)_{\\!\\!\phi(\mathsf{x})=\phi_{\text{R}}(\mathsf{x})}\right]^{\\!-1/2},$
(SI.18b)

where $\lambda_{\text{b}}$ is the single negative eigenvalue of
$\mu[-\kappa\Delta+\partial^{2}W/\partial\phi^{2}]$ at the TS. The translation
symmetry of the interface formed at the TS leads to zero eigenvalues in the
Hessian. The prime in the definition of $Z^{\ddagger}$ indicates that these
zero eigenvalues have been removed from the determinant.

From the stationarity condition at the TS, $\delta V/\delta\phi=0$, it follows
that $E_{x}=\kappa\lvert\nabla\phi\rvert^{2}/2-W(\phi)$ is constant
(independent of $\mathsf{x}$) along $\phi^{\ddagger}(\mathsf{x})$. We can
therefore add a zero under the integral in Eq. (SI.17), leading to

$\displaystyle V[\phi^{\ddagger}(x)]$
$\displaystyle=\int\left[\frac{\kappa}{2}\,\lvert\nabla\phi^{\ddagger}\rvert^{2}+W(\phi^{\ddagger})+E_{x}(\phi^{\ddagger})-E_{x}\right]\mathrm{d}\mathsf{x}.$
$\displaystyle=B_{x}+V[\phi_{>}(\mathsf{x})],$ (SI.19)

Here, we used that, in the absence of finite-size effects,
$\phi^{\ddagger}(\mathsf{x})$ is dominated by the less stable (uniform) state
$\phi_{>}$ with only a small nucleus (or bubble) extending towards the lower-
energy state (see Fig. SI.4), so that $V[\phi_{>}(\mathsf{x})]=-\int
E_{x}\,\mathrm{d}\mathsf{x}$. In the common case of an exothermic nucleation
process, one obtains the simple relationship
$B_{x}=V^{\ddagger}=V[\phi^{\ddagger}(\mathsf{x})]-V[\phi_{\text{R}}(\mathsf{x})]$,
in agreement with previous work on nucleation rates at equilibrium [26, 27].
This, however, does not hold out of equilibrium, where the TS is not a
stationary point of a potential.

In practice, we represent the field on a grid. Using Eq. (SI.17) in one
dimension as an example, the discretized potential on a grid of $N_{x}$ points
spaced by $\xi$ is given by

$V(\varphi_{0},...,\varphi_{N_{x}-1})=\\\
\xi\sum_{i=0}^{N_{x}-1}\frac{\kappa}{2}\left(\frac{\varphi_{i+1}-\varphi_{i}}{\xi}\right)^{\\!\\!2}+W(\varphi_{i}),$
(SI.20)

where $\varphi_{i}\equiv\phi(x_{i})$, and $\varphi_{0}=\varphi_{N}$ because we
employ periodic boundary conditions. The optimization of stationary points and
evaluation of the fluctuation terms [Eqs. (SI.18)] can then be carried out by
taking derivatives with respect to the $\varphi_{i}$ and correctly accounting
for $\xi$. This procedure is analogous to the discretization of the instanton
in time described in the main text. The corresponding discretized instanton
action in field space [Eq. (12)] is given by

$\displaystyle
S_{N}(\phi_{0},...,\phi_{N};t_{\text{f}})=\frac{\tau\xi}{4}\sum_{n=0}^{N-1}\left(\frac{\phi_{n+1}-\phi_{n}}{\tau}-\mu
F(\phi_{n})\right)^{\\!\\!2},$ (SI.21)

where each bead $\phi_{n}=(\varphi_{0},...,\varphi_{N_{x}-1})$ corresponds to
a field configuration, and $F(\phi_{n})=-\delta
V(\phi_{n})/\delta\phi+F_{\text{nc}}(\phi_{n})$ is the sum of all conservative
and non-conservative forces. The instanton connects
$\phi_{0}\equiv\phi_{\text{R}}$ to $\phi_{N}\equiv\phi^{\ddagger}$. It is
obtained by minimizing Eq. (SI.21) with respect to all intermediate beads
$(\phi_{1},...,\phi_{N-1})$.

The path-integral representation of the rate is given by

$k\,Z_{\text{R}}=\lim_{N,N_{x}\rightarrow\infty}\left(\frac{\xi}{4\pi\epsilon\tau}\right)^{\\!\\!NN_{x}/2}\\\
\times\int\dot{\phi}_{\sigma}(t_{\text{f}})\,\mathrm{e}^{-S_{N}(\phi_{0},...,\phi_{N};t_{\text{f}})/\epsilon}\,\mathrm{d}\phi_{1}...\mathrm{d}\phi_{N-1}\mathrm{d}\mathsf{s},$
(SI.22)

where $\dot{\phi}_{\sigma}(t_{\text{f}})$ is the velocity orthogonal to the
dividing surface and the generalization to $f$ dimensions is straightforward.
Carrying out a Laplace approximation in analogy to the main text then leads to
the instanton rate for active field theories in Eq. (13). The fluctuations
around the instantons are contained within the determinant of
$\nabla^{2}_{\phi}S_{N}$, where
$\nabla_{\phi}=(\partial/\partial\phi_{1},...,\partial/\partial\phi_{N-1},\partial/\partial\mathsf{s})$
is the derivative vector with respect to all intermediate beads and the
parameters of the field-space dividing surface $\mathsf{s}$. However, in
addition to the time-translation mode, also the modes corresponding to
translation of the interface must be projected out via the zero eigenvectors
of $\partial F(\phi^{\ddagger})/\partial\phi$. As discussed before, the
reaction coordinate at the TS can be projected out via
$\bar{p}_{\text{f}}=\frac{1}{\xi}\frac{\partial S_{N}}{\partial\phi_{N}}$.

Figure SI.5: Size of the critical nuclei in the phase transition of the
stochastic field theory defined in the main text over a range of applied
fields $h$ and nonequilibrium driving strengths $\nu$. The size of the
critical nucleus is measured by
${\int[\phi^{\ddagger}(x)-\phi_{\text{R}}(x)]\,\mathrm{d}x/[\text{max}(\phi^{\ddagger})-\text{min}(\phi^{\ddagger})]}$.

In Fig. SI.4, we illustrate the KLT and instanton pictures of a transition in
field space. As before, the instanton smoothly connects the stationary points
through a trajectory in time. It can be seen that, at the TS, part of the
field extends from the reactant to the product configuration, forming a
“critical nucleus” or an interface between the states. The nucleus is smaller
than the box in which the field is defined. Given the periodic boundary
conditions, the nucleus can form anywhere in the box without a change in the
activation barrier, giving rise to the previously mentioned continuous
translation symmetry at the TS.

Fig. SI.5 explores how the size of the critical nuclei, which need to be
formed to facilitate a transition, varies with the strength of the applied
field $h$ and the nonequilibrium driving $\nu$. It can be seen that with
increasing driving strength, the required size of the critical nucleus
decreases. This directly correlates with the rate enhancement due to
nonequilibrium driving observed in Fig. 3(b) of the main text.

It is worth pointing out that, for the active field theory defined in the main
text, the backward transitions proceed via a uniform TS when $\nu\neq 0$
because of the finite size of the box [4]. For our purposes, this is not a
problem since a meaningful action can still be extracted.

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