# A mean field thermodynamic framework for time dependent self-assembly and
pattern formation

Michael Nguyen1,2, Suriyanarayanan Vaikuntanathan1,2 1The James Franck
Institute, The University of Chicago, Chicago, IL,  2 Department of Chemistry,
The University of Chicago, Chicago, IL.

###### Abstract

In this work, we use a minimal model to introduce a framework for controlling
self-assembly under the influence of time-dependent driving forces. We develop
a mean-field thermodynamic framework that predicts the conditions required to
reliably self-assemble a desired spatial pattern under time-varying external
fields. We also calculate the entropy production associated with the time-
dependent self-assembly process and examine how it can be used to predict
conditions under which the external time-varying signal is reliably encoded as
a spatial pattern in the self-assembling material. While the results in this
work are developed in the context of a minimal one-dimensional model, we
anticipate that the framework can be used to establish guidelines for
controlling self-assembly in more complex scenarios.

##  I Introduction

Self-assembly is a process in which components in a system organize themselves
into structures and patterns without human intervention. Though design
principles for self-assembly for systems at global or local equilibrium have
seen many advances recently, there has not been much progress for self-
assembly far from equilibrium. Indeed, many essential processes in biophysics
and chemistry occur far from equilibrium [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12]. In our previous works [13, 14, 15], using the ideas of stochastic
thermodynamics we have developed a general framework to predict the structure
of an assembly at steady state. Specifically, we can write bounds that relate
the non-equilibrium driving forces to structures and basic kinetics of the
assembly. Our framework, however, only applies to steady-state assemblies and
does not extend to other important types of assemblies such as ones under
time-periodic force. Indeed, time-periodic forcing has been used to create
non-equilibrium states with enhanced order in many contexts [16, 17, 18, 4,
19, 20, 21, 22]. It was showed that dynamic phase transitions can occur by
tuning the frequency of inputting the energy into the system [23, 24]. In this
paper, we fill this gap by developing a general framework that can be used to
explore how the pattern of the assembly forms under periodic drive and its
connection to dissipation.

When the assembly is at a steady state with little to no internal relaxation,
one can write out mean-field equations that relate the growth rates of
particles with the components inside the assembly [13, 14, 25]. Under the
periodic drive, the self-assembled system is no longer in a time-independent
steady-state, and the aforementioned mean-field conditions are no longer
valid. The mathematical and thermodynamic treatment of such steady states is
not trivial [26]. As our first result, we show that many of our mean-field
results [13, 14] can still be adapted and used to probe the periodic assembly
effectively under certain conditions. Further, inspired by works in [27, 28,
29, 30, 31, 32, 33], we also develop a more general treatment that works for a
broader set of conditions using bi-partite Markov Networks. Besides connecting
the growth rates of the assembly with its inner structure, these models also
allow us to write out the entropy production for the self-assembly process.
The field of stochastic thermodynamics has shown how a tabulation of the
entropy production rate can be used to efficiently constrain the non-
equilibrium steady-state properties of a system. For example, the entropy
production has been highlighted as measure of a system’s distance from
equilibrium [34, 35, 36, 37, 38, 39] and has been used to characterize
critical behavior in non-equilibrium phase transitions [40, 41, 42, 43]. In
our previous works [13, 14, 15], we had demonstrated that the entropy
production can be used to predict the structure of non-equilibrium assembly.
Following these ideas, in this paper, we show that the entropy production
informs how time correlation between particles affect the patterns being
deposited into the assembly.

The paper is organized in the following way. In Section II, we introduce the
model and simulations used in this paper and explore their properties. In
Section III, we review the mean-field approach that was developed in [5, 14,
25] for steady-state setting and demonstrate how it is still applicable for a
periodic system under certain regimes. We then develop the bi-partite Markov
model in Section IV which is applicable in a broader range of conditions and
can be used to predict the properties of time-dependent non-equilibrium steady
states with surprising accuracy. Finally, we look at the entropy production of
the two models in Section V and show how they can be used to investigate how
patterns are driven into the assembly.

##  II Model And Simulations

Figure 1: Schematic of the one dimension growth assembly with rates affected
by a time-varying external field $h(t)$. A polymer is grown from a bath
containing two distinct monomers labelled by $R$ and $B$. The rates of
addition and removal are influenced by the energy of interaction between the
various monomers, an oscillating external field, and the chemical potential of
the monomers in the bath. In this work, we show that a minimal extra chemical
potential from equilibrium, $\delta\mu_{c}$, is required to consistently
encode the signal of the external field into the assembly. We also outline a
mean field theory to estimate the probability of obtaining a monomer of type
$n$ ($R/B$ in this case) a fixed distance $l\gg l_{m}$, where $l_{m}$ is the
length scale associated with a single monomer, away from the tip of the
assembly.

We consider a one-dimensional growth of an assembly made up of two kinds of
particles (denoted by $R$ and $B$) as shown in Fig. 1. In order to compactly
specify the energetics of this system, we will assign numeric values of $1$
and $-1$ to the two monomer types $R$ and $B$, respectively. As the assembly
grows, we only allow particles to attach or detach at the outermost block of
the assembly. We choose dynamics wherein particles are added into the assembly
at a constant rate:

$\begin{split}k^{add}_{t}({n_{i},n_{j}\xrightarrow{}n_{j},n_{k}})&=\exp(\mu)P_{gen}(n_{k}|n_{j})\\\
\end{split}$ (1)

Here n represents the configurations/properties of the particles that make up
the assembly (in this case $1$ or $-1$). $P_{gen}(n_{k}|n_{j})$ is the
probability of getting a particular particle $n_{k}$ from the bath given the
interface particle is $n_{j}$. This probability depends on the relative
concentrations of the particles in the bath. For this paper, we will let the
two particles have equal concentrations setting $P_{gen}(n_{k}|n_{j})=0.5$.
$\mu$ is the chemical potential of the particles on the bath. The particles,
however, are removed with a time-dependent rate:

$k^{rem}_{t}(n_{k},n_{i}\xleftarrow{}n_{i},n_{j})=\exp(-Jn_{i}n_{j}-h\cos{\left(\frac{2\pi
t}{T}\right)}n_{j})$ (2)

Here $J$ specifies the monomer-monomer interaction strength, $h$ is the
amplitude of the external field, $t$ is time, and $T$ is the period. When
there is no external field, $h=0$, the assembly will reach a steady-state, and
its configuration will be set by the chemical potential $\mu$. Specifically,
when $\mu=\mu_{\rm{eq}}$, the assembly is at equilibrium and does not grow on
average. When $\mu>\mu_{eq}$, the assembly grows, while for $\mu<\mu_{eq}$, it
will shrink. Throughout this paper, we will write
$\mu=\delta\mu+\mu_{\rm{eq}}$ with $\delta\mu>0$ to denote the extra chemical
driving force on the growing assembly. We then run simulations with the above
rates using the First Reaction Method (FRM) [44].

To analyze how well the external field $h(t)$ affects the pattern inside the
assembly, we take the whole assembly generated by the simulation through a
Fourier Transform with the convention:

$m(n)=\frac{1}{N}\sum_{q}^{N}m(q)\exp(\frac{2\pi iqn}{N})$ (3)

Here $m(n)$ denotes the state of the particle at a certain position $n$, $N$
is the total number of particles in the assembly, and $q$ is the spatial
frequency of the assembly’s pattern. We then convert the spatial frequency
into the time period, $\omega$, using the growth rate:
$\dot{N}_{\rm{ave}}=\frac{1}{T}\int_{0}^{T}\dot{N}(t)dt=\frac{1}{q\omega}$.
Fig. 2 shows the result of the Fourier Transform for an assembly consisting of
$10^{7}$ particles. The location of the peak of the Fourier Transform
corresponds to the period, $T$, of the external field. We repeat this analysis
to assemblies obtained with $\delta\mu$ from the range: $\delta\mu:4.0-9.0$
and plot the magnitude square of the peak varying with the number of particles
in the assemblies in Fig. 3.

Figure 2: Fourier Transform of the assembly obtained using simulations at
$\delta\mu=6.5$, $J=4$, $h=0.2$ and $T=1000$. The period of the peak is the
same as the period of the external field. Figure 3: Peak of the Fourier
Transform as a function of the number of particles in the assembly with
parameters: $J=4$, $h=0.2$ and $T=1000$. Top: in this range, $\delta\mu=4.0$
to $6.0$ , the peak keeps decaying as the number of particles increases. Down:
in this range, $\delta\mu=6.5$ to $9.0$, the peak stays constant with the
number of particles.

From Fig. 3, we see that the peak of the assembly first increases with
increasing $\delta\mu$. It then caps out and decreases with increasing
$\delta\mu$. Closer inspection also reveals that the peaks decay with an
increasing number of particles. To quantify the decay, we fit the peak to the
number of particles using the formula: $\ln{\max(|m|^{2})}=a\ln{N}+b$. The
parameter $a$ is the decay coefficient of the assembly peak. Fig. 4 shows how
this decay coefficient changes with $\delta\mu$: With $\delta\mu<7$, the decay
coefficient is positive, and the pattern in the assembly is not deposited
consistently throughout the assembly. Only with $\delta\mu>7$, the pattern is
deposited consistently into the assembly in every cycle, which results in a
$0$ decay coefficient. This suggests that to effectively drive a pattern from
a varying field into the assembly, a minimum amount of driving force,
$\delta\mu$, is required.

Figure 4: Decay coefficient of the assembly as a function of $\delta\mu$ at
$J=4$, $h=0.2$ and $T=1000$.. The decay coefficients, a, are extracted using
$\ln{\max(|m|^{2})}=a\ln{N}+b$.  Figure 5: Up: Probability profiles of
obtaining a red particle as a function of time. The profile of the external
field, $h(t)$, is also plotted in dash line for reference. Down: Amplitudes of
the probability profiles of obtaining a red particle. These results are
obtained from simulations with $J=4$, $h=0.2$ and $T=1000$.

In addition to the correlations obtained from the Fourier Transform, another
figure of merit to ascertain how the correlations inside the assembly are
guided by the time-dependent external field can be constructed by measuring
the probability to find a specific monomer type in the bulk of the assembly, a
fixed distance away from the tip of the assembly 1, $P^{t}_{inside}(R/B)$.
Mirroring the time dependence of the external field, this probability function
will also be a function of time. In practice, we extract this probability by
recording the moment in which a particle at the interface is covered by a new
particle coming into the assembly. For example, after growing at least
$5\crossproduct 10^{7}$ particles, the probability of obtaining a $R$ particle
at a certain time is computed using:

$P^{t}_{inside}(R)=\frac{N_{R}(t\%T)}{N_{R}(t\%T)+N_{B}(t\%T)}$ (4)

Here $t$ is the time in which the particle is added into the assembly, and $T$
is the period of the external field. Note that $P^{t}_{inside}(R/B)$ is
distinct from $P^{t}(R/B)$, the probability associated with finding a monomer
of a specific sort at the tip of the assembly. Fig 5A shows the probabilities
inside the assembly collected at different $\delta\mu$. Similar to the peaks
of the Fourier Transforms, the amplitude of the probability profiles also
maximizes at certain $\delta\mu$ as shown in Fig 5B. Note, however, the
location of this peak is at $\delta\mu=5$ while the one obtained from the
Fourier Transform is at $\delta\mu=7$. This offset is simply due to the fact
that the probabilities $P^{t}_{inside}(R/B)$ are single-particle quantities
while the Fourier transform accounts for many-particle correlations. In the
next section, we develop a mean-field theory for predicting the probability
profiles.

##  III Mean-Field Approach

We first review the mean-field approach that we develop in our previous works
[13, 14] for steady-state systems in the absence of time-varying fields.
Briefly, instead of looking at the full configurations of the assembly, we
focus on the interface of the assembly. Specifically, the system of interest
will consist of a particle that is in contact with the bath and the one right
next to it. We then write out master equations for the interface probability:
$P^{ss}(n_{i},n_{j})$ as shown in the appendix Eq. S1. The superscript $ss$ is
there to indicate that the system is now at a steady state because the
external field is no longer changing with time. For this particular 1D system,
the rates for the system are:

$k^{add}({n_{i},n_{j}\xrightarrow{}n_{j},n_{k}})=\exp(\mu_{eq}+\delta\mu)P_{gen}(n_{k}|n_{j})$
(5)
$k^{rem}(n_{k},n_{i}\xleftarrow{}n_{i},n_{j})=\exp(-Jn_{i}n_{j}-hn_{j})P_{sys}(n_{k}|n_{i})$
(6)

Notice that the external field in the removal rate is no longer time-
dependent. In addition, a new term, $P_{sys}(n_{k}|n_{i})$ has appeared in the
removal rate. This is the conditional probability of obtaining a particle
$n_{k}$ in the assembly given its neighbor $n_{i}$. This probability arises
from the uncertainty of not looking at the assembly’s full configuration. We
now make the mean-field assumption that the conditional probability,
$P_{sys}(n_{k}|n_{i})$, can be described using the bulk structure of the
assembly:

$P_{sys}(n_{k}|n_{i})=\frac{P^{ss}_{inside}(n_{k},n_{i})}{\sum_{n_{k}}P^{ss}_{inside}(n_{k},n_{i})}$
(7)

Here $P^{ss}_{inside}(n_{k},n_{i})$ is the probability of observing a
configuration $(n_{k},n_{i})$ inside the assembly at steady state. To solve
the master equations at steady state, we still need a relation between the
probabilities inside the assembly, $P^{ss}_{inside}(n_{k},n_{i})$, and the
probabilities at the interface $P^{ss}(n_{k},n_{i})$. This is established by
noticing that the rate in which a particle is deposited into the assembly is
proportional to its amount inside the assembly:

$P^{ss}_{inside}(n_{i})=\frac{\sum_{n_{k},n_{j}}J^{ss}(n_{k},n_{i}\rightarrow
n_{i},n_{j})}{\sum_{n_{k},n_{i},n_{j}}J^{ss}(n_{k},n_{i}\rightarrow
n_{i},n_{j})}$ (8)

Here $J^{ss}(n_{k},n_{i}\rightarrow n_{i},n_{j})$, the current as the assembly
grows from from $(n_{i},n_{j})$ into $(n_{j},n_{k})$, is defined as:

$\begin{split}&J({n_{i},n_{j}\xrightarrow{}n_{j},n_{k}})=\\\
&P^{ss}(n_{i},n_{j})P_{gen}(n_{k}|n_{j})k^{add}({n_{i},n_{j}\xrightarrow{}n_{j},n_{k}})\\\
&-P^{ss}(n_{j},n_{k})P_{sys}(n_{i}|n_{j})k^{rem}({n_{i},n_{j}\xleftarrow{}n_{j},n_{k}})\end{split}$
(9)

We can now solve the master equations Eq. S1 at steady state by setting its
left-hand side to zero and obtain the probability profiles inside the
assembly.

As we vary the external field with time, the rates $k^{add}$ and $k^{rem}$ are
in general time-dependent, and the system is no longer in a steady state. One
way to extend the above mean-field approach is assuming the system is in a
quasi-steady state. In other words, in solving the master equation, the left-
hand side of Eq. S1 is still set to zero, while the rates on the right-hand
side will correspond to a different point in time. However, this naive
approach only works in the limit where the oscillation is extremely slow (high
$T$) or extremely high (small $T$). In general, we cannot assume the system is
at a quasi-steady state. We can, however, solve the master equations Eq. S1
numerically with time-dependent rates and use Eq. 8 to predict the
probabilities inside the assembly.

Figure 6: Probability profile for obtaining a red particle at a certain time
in the assembly at A: This profile is at $\delta\mu=3,J=4,h=0.2,T=10$. B: This
profile is at $\delta\mu=3,J=4,h=0.2,T=1000$. C: This profile is at
$\delta\mu=6,J=4,h=0.2,T=10$. The blue dots are obtained from FRM simulations.
The orange dots are obtained from simulations of the bi-partite Markov process
with Eq. S3 and S4. The red curve is obtained by solving the mean-field master
equations, Eq. S1, numerically with time-dependent rates. The green dots are
obtained from solving the mean-field master equations, Eq. S1, by assuming the
assembly is at quasi steady-state at every point in time.

In Fig 6, we compare the probability profiles obtained from the FRM
simulations (blue dots) and those from solving the mean-field master equation,
Eq. S1, numerically (red curve), which indicates that in the regime of high
$\delta\mu$ and high $T$, the mean-field master equations can be used to
predict the structures inside a periodic assembly. This is because, in these
regimes, the rates at which particles going into the assembly are relatively
high compared to the rate that the field is changing. However, this does not
mean the assembly is in a quasi-steady state as in the naive mean-field
approach. In Fig. 6B, we also plot the probability profile by solving the
master equations using the naive mean-field approach (green dot) to contrast
the one obtained from solving the time-dependent master equations numerically
(red curve).

In this section, we have shown that the mean-field framework developed in [13,
14, 25] for a steady-state assembly can be extended to a periodic assembly.
With this approach, instead of focusing on the full configuration of the
assembly, we can look at the kinetics in the interface then deduce the
assembly’s configuration. The predictions from the mean-field are particularly
accurate in the regime of high $\delta\mu$ and high $T$ where the kinetics of
the assembly is relatively fast to the oscillating field.

##  IV Bi-Partite Markov Network

When we develop our mean-field model, one assumption we take is that the
conditional probability, $P_{sys}(n_{k}|n_{i})$, is related to the probability
of the bulk $P^{ss}_{inside}(n_{k},n_{i})$ through Eq. 7. This only applies
when the kinetics of the assembly is fast relative to the field. When the
kinetics of assembly is comparable to the oscillation of the field, however,
different parts of the assembly will experience different parts of the field,
and we cannot use Eq. 7 to predict the particle $n_{k}$ given just its
neighbor $n_{i}$. To take in account this effect, we have to include a time-
dependent term in our interface probabilities: $P^{t}(n_{i}t_{x},n_{j}t_{y})$.
Here in the notation $n_{i}t_{x}$, the $t_{x}$ represents the time that the
particle $n_{i}$ is added into the assembly. In order words, a particle will
remember the time it is deposited into the assembly. The $t_{y}$ on the
interface particle, $n_{j}$, represents the current time of the system.

Figure 7: Schematic of the bi-partite Markov model. In addition to addition
and removal of particles (horizontal transitions), the system now has moves
that advance time, $t_{x}$, at the interface (vertical transitions).

To develop an improved mean field theory that accounts for such effects,
inspired by works in [27, 28, 29, 30, 31, 32, 33], we set up a bi-partite
Markov network between time $t$ and configuration of the assembly $n$. As
shown in Fig. 7, besides the addition and removal of particles, the system now
has a move that advance time $t_{y}$ into $t_{y+1}$. The new move is
constructed as following: First, the period $T$ is partitioned into $M$ parts.
The time $t_{y}$ then moves across these partitions with rate
$\lambda=\frac{M}{T}$ as demonstrated in Ref. [32]. When the system advances
in $t_{y}$, the external field also changes according to $h\cos{(\frac{2\pi
t_{y}}{T})}$ which in turn affects the removal rate $k^{rem}$. With this
setting, all the possible transitions in this bi-partite Markov process can be
summarized as follow:

$\begin{split}&{\bf(1)}\quad
n_{i}t_{x},n_{j}t_{y}\genfrac{}{}{0.0pt}{2}{\xrightarrow{k^{add}}}{\xleftarrow[k^{rem}]{}}n_{j}t_{y},n_{k}t_{y}\\\
&\\\ &{\bf(2)}\quad
n_{i}t_{x},n_{j}t_{y}{\xleftarrow[k^{rem}]{}}n_{j}t_{z},n_{k}t_{y}\\\
&(\text{with}\quad t_{z}\neq t_{y})\\\ &\\\ &{\bf(3)}\quad
n_{i}t_{x},n_{j}t_{y}\quad{\xrightarrow{\lambda}}\quad
n_{i}t_{x},n_{j}t_{y+1}\end{split}$ (10)

We can now write the master equations for the bi-partite Markov process as
shown in the appendix (Eq. S2 and Eq. S3). After solving the bi-partite Markov
master equations consistently at a steady-state, we can connect the
probabilities at the interface and the probabilities inside the assembly using
the currents as we did in Eq. 8:

$\begin{split}&P^{t}_{inside}(n_{i}t_{x})\\\
&\propto\sum_{n_{j},n_{k},t_{y},t_{z}}\Big{[}J^{t}_{add}({n_{k}t_{y},n_{i}t_{x}\xrightarrow{}n_{i}t_{x},n_{j}t_{x}})\\\
&-J^{t}_{rem}({n_{k}t_{y},n_{i}t_{z}\xleftarrow{}n_{i}t_{x},n_{j}t_{z}})\Big{]}\end{split}$
(11)

With

$\begin{split}&J^{t}_{add}({n_{k}t_{y},n_{i}t_{x}\xrightarrow{}n_{i}t_{x},n_{j}t_{x}})=\\\
&P_{gen}(n_{j}|n_{i})k_{t}^{add}({n_{k}t_{y},n_{i}t_{x}\xrightarrow{}n_{i}t_{x},n_{j}t_{x}})P^{t}(n_{k}t_{y},n_{i}t_{x})\end{split}$
(12)

and

$\begin{split}&J^{t}_{rem}(n_{k}t_{y},n_{i}t_{z}\xleftarrow{}n_{i}t_{x},n_{j}t_{z})=\\\
&P^{t}(n_{i}t_{x},n_{j}t_{z})P^{t}_{sys}(n_{k}t_{y}|n_{i}t_{z})k_{t}^{rem}(n_{k}t_{y},n_{i}t_{z}\xleftarrow{}n_{i}t_{x},n_{j}t_{z})\
\end{split}$ (13)

Figs. 6 show that the probability profiles obtained using Eq. 11 (orange dots)
match very well with ones from FRM simulations (blue dots). We have also
tested Eq. 11 using different kinetics and harmonics; Eq. 11 (orange dots)
still captures the probability profiles very well when the mean field method
(red curve) does not (Fig. 8).

Figure 8: Probability profile for obtaining a red particle at a certain time
in the assembly with two external fields at different periods $T_{1}=2$ and
$T_{2}=3$ with $J=4,h=2$. A: This profile is for $\delta\mu=5$. B: This
profile is for $\delta\mu=3$. The results from bi-partite Markov processes
agree very well with ones from FRM simulations. The mean-field method,
however, only works well at high $\delta\mu$.

##  V Entropy Production

In the previous sections, we have shown that two approaches, mean-field and
bi-partite Markov process, do an excellent job in reproducing the probability
profiles of the periodic assembly in their proper regimes. These results also
clarified conditions under which the temporal oscillations in the external
signal could be encoded in the assembly. We now explore whether an accounting
for the entropy production rate can also provide intuition for this problem.

###  V.1 Entropy Production From the Time Dependent Mean Field Approach

In our previous works [13, 14], we have written down a general form for the
entropy production of a self-assembly process with the mean-field approach:

$\dot{S}_{\rm{mf}}=\dot{N}(\delta\mu-\epsilon_{diss})$ (14)

where $\epsilon_{diss}=\langle E^{\rm{eq}}\rangle-G^{\rm{eq}}-\langle
E^{\rm{eff}}\rangle+G^{\rm{eff}}$. Here $E^{\rm{eq}}$ is the equilibrium
energy for our system. $E^{\rm{eff}}$ is the effective energy that corresponds
to structure of the system when it is out equilibrium. The bracket
$\langle..\rangle$ is the ensemble average of the system’s non-equilibrium
configuration. $G^{\rm{eq}}$ is the free energy corresponding to the
equilibrium energy while $G^{\rm{eq}}$ is the free energy corresponding to the
effective energy. Eq. 14 shows that the entropy production rate doesn’t depend
on the detailed kinetics of the process but is simply captured in terms of the
difference between the non-equilibrium and equilibrium steady states and an
account of the non-equilibrium driving forces.

In Section III, we have extended the mean-field approach to an assembly under
period drive. Along those lines with the formalism of time-dependent entropy
production [45], we can also readily adopt the form of the entropy production
for the steady-state assembly, Eq. 14, to the periodic assembly:

$\dot{S}_{\rm{mf}}=\frac{1}{T}\int_{0}^{T}\dot{N}(t)(\delta\mu(t)-\epsilon_{diss}(t))dt$
(15)

where $\epsilon_{diss}(t)=\langle E^{\rm{eq}}\rangle(t)-G^{\rm{eq}}(t)-\langle
E^{\rm{eff}}\rangle(t)+G^{\rm{eff}}(t)$.

For our system, the $E^{\rm{eq}}(t)$ is the energy of the 1D polymer model
with coupling constant $J$ and a constant external field $h$:

$E^{\rm{eq}}(t)=\sum_{i}-Jn_{i}n_{i+1}-h\cos{\left(\frac{2\pi
t}{T}\right)}n_{i}$ (16)

For the $E^{\rm{eff}}(t)$, we use an effective coupling constant
$J_{\rm{eff}}$ and effective external field $h_{\rm{eff}}$ to describe the
configurations in the assembly at a point in time:

$E^{\rm{eff}}(t)=\sum_{i}-J_{\rm{eff}}n_{i}n_{i+1}-h_{\rm{eff}}\cos{\left(\frac{2\pi
t}{T}\right)}n_{i}$ (17)

The bracket $\langle..\rangle$ now is the ensemble average of the system at
equilibrium with the energy $E^{\rm{eff}}(t)$. The effective coupling constant
$J_{\rm{eff}}$ and effective external field $h_{\rm{eff}}$ can be extracted
using observables. Specifically, for our system, we can use results of the
1D-Ising model to relate $\langle n_{i}\rangle(t)$, $\langle
n_{i}n_{i+1}\rangle(t)$ with $J_{\rm{eff}}$ and $h_{\rm{eff}}$ as demonstrated
in [13]. The $\delta\mu(t)$ is computed using
$\delta\mu(t)=\mu-G^{\rm{eq}}(t)$. We plot the entropy production from Eq. 15
in Fig. 9.

As the system approaches the $\delta\mu$ in which pattern is added into the
assembly consistently (around $\delta\mu\approx 7$), this entropy production
rate approaches an inflection point. The inflection point is due to the
growing contributions from the $\epsilon_{\rm diss}$ term. Indeed, the
quantity $\epsilon_{\rm diss}$ can also be written as a relative entropy
between the observed distribution of patterns in the assembly and the
corresponding equilibrium steady-state distribution. As periodic spatial
correlations begin to get reliably encoded in the assembly, the increase in
the relative entropy captured by $\epsilon_{\rm diss}$ can be sufficient to
counteract $\delta\mu$. This can lead to saturation or slow down in the rate
of increase of entropy production.

These results suggest that an analysis of the entropy production rate may be
used to predict the minimum non-equilibrium forces required to reliably
sustain correlations in the non-equilibrium assembly.

###  V.2 Entropy Production From Bi-Partite Markov Process

Similar inferences can be drawn by studying the entropy production rate of the
bi-partite Markov state caricature. Because of the setup of the bi-partite
Markov process, some of the transitions are unidirectional, so the normal
definition of entropy production does not apply to each transition. However,
according to [46], the total entropy production of the system does not depend
on these unidirectional transitions but only on the bidirectional transitions
[46]. The total entropy production, thus, is:

$\begin{split}&\dot{S}=\sum_{t_{x},t_{y},n_{k},n_{i},n_{j}}\big{[}J^{t}_{add}({n_{k}t_{y},n_{i}t_{x}\xrightarrow{}n_{i}t_{x},n_{j}t_{x}})\\\
&-J^{t}_{rem}(n_{k}t_{y},n_{i}t_{x}\xleftarrow{}n_{i}t_{x},n_{j}t_{x})]\\\
&\ln{\frac{J^{t}_{add}({n_{k}t_{y},n_{i}t_{x}\xrightarrow{}n_{i}t_{x},n_{j}t_{x}})}{J^{t}_{rem}(n_{k}t_{y},n_{i}t_{x}\xleftarrow{}n_{i}t_{x},n_{j}t_{x})}}\end{split}$
(18)

Here the currents $J^{t}_{add}$ and $J^{t}_{rem}$ are defined according to Eq.
12 and Eq. 13. Thus, after solving the master equations of the bi-partite
Markov process (Eq. S2 and Eq. S3) at steady state, Eq. 18 can be used to
calculate the entropy production of the process. In Fig. 9, we plot the
entropy production of the bi-partite Markov process (Eq. 18 -green triangle)
with the entropy production of the mean-field equations (Eq. 15-blue square).
This shows that the entropy production of the bi-partite Markov process is
always higher than the one of the mean-field equations. This is because we
have artificially labeled timestamps into the particles. Reflecting the
increased information content, the entropy production also increases. As shown
in Sec. III, in the limit of high $\delta\mu$ and $T$, the bipartite master
equation approaches the mean-field master equations derived previously, and
the two entropy production rates converge in this limit. Reflecting these
limits, the entropy production rate of the bi-partite Markov process has a
peak around the non-equilibrium chemical potential $\delta\mu$ at which
temporal correlations in the external signal are reliably encoded inside the
assembly. Together, the results of this section suggest that an analysis of
the entropy production rate in such time-dependent non-equilibrium self-
assembly processes can potentially be illuminating and reveal the requirements
for encoding external temporal signals as spatial correlations in the
assembly.

Figure 9: Entropy production of the one dimension growth assembly with rates
affected by a time-varying external field h(t) with $J=4,h=0.2,T=1000$.
$\dot{N}_{\rm{ave}}$ is the average growth rate of assemblies over the period
T: $\dot{N}_{\rm{ave}}=\frac{1}{T}\int_{0}^{T}\dot{N}(t)dt$. The blue curve is
computed using the mean-field entropy production, Eq. 15. The green curve is
computed using the full bi-partite Markov Process entropy production, Eq. 18.

##  VI Conclusion

In this paper, we have developed two frameworks to study the growing of an
assembly under periodic drive. In the regime of high $\delta\mu$ and $T$, the
mean-field equations for steady-state in our previous works [13, 14] can be
extended to predict the probability profile inside the assembly. A bi-partite
Markov network has also been constructed that works in a wider regime. The
results from these two approaches agree very well with those from FRM
simulations.

We also demonstrate that a minimal driving force is required to reliably
encode the external temporal signal as a spatial correlation in the assembly.
Our work also suggests that accounting for the entropy production rate might
help predict these minimum driving force conditions. Future work will explore
the application of these ideas to higher dimensional materials with more
complex interactions.

The authors acknowledge support from the National Science Foundation under
Grant No. DMR-1848306.

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## Appendix

##  S1 The Mean Field Equations

The master equations correspond to the probabilities $P^{t}(n_{i},n_{j})$ that
are used for our mean-field approach are:

$\begin{split}&\frac{dP^{t}(n_{i},n_{j})}{dt}=-P^{t}(n_{i},n_{j})\Big{[}\sum_{n_{k}}P_{gen}(n_{k}|n_{j})k_{t}^{add}({n_{i},n_{j}\xrightarrow{}n_{j},n_{k}})+\\\
&\sum_{n_{k}}P^{t}_{sys}(n_{k}|n_{i})k_{t}^{rem}(n_{k},n_{i}\xleftarrow{}n_{i},n_{j})\Big{]}+\\\
&\Big{[}P_{sys}^{t}(n_{i}|n_{j})\sum_{n_{k}}P(n_{j},n_{k})k_{t}^{rem}({n_{i},n_{j}\xleftarrow{}n_{j},n_{k}})\Big{]}+\\\
&\Big{[}P_{gen}(n_{j}|n_{i})\sum_{n_{k}}P(n_{k},n_{i})k_{t}^{add}({n_{k},n_{i}\xrightarrow{}n_{i},n_{j}})\Big{]}\end{split}$
(S1)

##  S2 The Bi-Partite Master Equations

According to Eq. 10 in the main text, the addition of a new particle will only
result in states in which the interface particle and its neighbor have the
same timestamp. Thus, there are two kinds of master equations for the bi-
partite Markov Network. The first one is for states in which the interface
particle and its neighbor have the same timestamp:

$\begin{split}&\frac{dP^{t}(n_{i}t_{x},n_{j}t_{x})}{dt}=-P^{t}(n_{i}t_{x},n_{j}t_{x})\Big{[}\sum_{n_{k}}P_{gen}(n_{k}|n_{j})k_{t}^{add}({n_{i}t_{x},n_{j}t_{x}\xrightarrow{}n_{j}t_{x},n_{k}t_{x}})\\\
&+\sum_{n_{k},t_{y}}P^{t}_{sys}(n_{k}t_{y}|n_{i}t_{x})k_{t}^{rem}(n_{k}t_{y},n_{i}t_{x}\xleftarrow{}n_{i}t_{x},n_{j}t_{x})\Big{]}\\\
&+\Big{[}\sum_{n_{k},t_{y}}P^{t}_{sys}(n_{i}t_{x}|n_{j}t_{y})P^{t}(n_{j}t_{y},n_{k}t_{x})k_{t}^{rem}({n_{i}t_{x},n_{j}t_{x}\xleftarrow{}n_{j}t_{y},n_{k}t_{x}})\Big{]}\\\
&+\Big{[}\sum_{n_{k},t_{y}}P_{gen}(n_{j}|n_{i})P^{t}(n_{k}t_{y},n_{i}t_{x})k_{t}^{add}({n_{k}t_{y},n_{i}t_{x}\xrightarrow{}n_{i}t_{x},n_{j}t_{x}}))\Big{]}\\\
&+\lambda P^{t}(n_{i}t_{x},n_{j}t_{x-1})-\lambda
P^{t}(n_{i}t_{x},n_{j}t_{x})\end{split}$ (S2)

The second one is for state in which the interface particle and its neighbor
have different timestamp:

$\begin{split}&\frac{dP^{t}(n_{i}t_{x},n_{j}t_{z})}{dt}=-P^{t}(n_{i}t_{x},n_{j}t_{z})\Big{[}\sum_{n_{k}}P_{gen}(n_{k}|n_{j})k_{t}^{add}({n_{i}t_{x},n_{j}t_{z}\xrightarrow{}n_{j}t_{z},n_{k}t_{z}})+\\\
&\sum_{n_{k},t_{y}}P^{t}_{sys}(n_{k}t_{y}|n_{i}t_{x})k_{t}^{rem}(n_{k}t_{y},n_{i}t_{z}\xleftarrow{}n_{i}t_{x},n_{j}t_{z})\Big{]}\\\
&+\Big{[}\sum_{n_{k},t_{y}}P^{t}_{sys}(n_{i}t_{x}|n_{j}t_{y})P^{t}(n_{j}t_{y},n_{k}t_{z})k_{t}^{rem}({n_{i}t_{x},n_{j}t_{z}\xleftarrow{}n_{j}t_{y},n_{k}t_{z}})\Big{]}\\\
&+\lambda P^{t}(n_{i}t_{x},n_{j}t_{z-1})-\lambda
P^{t}(n_{i}t_{x},n_{j}t_{z})\end{split}$ (S3)

In Eq. S3, $t_{x}\neq t_{z}$. To obtain, Eq. 11 from the main-text, we
consider the probability:
$P(n_{i}t_{x},n_{j})=\sum_{t_{z}}P^{t}(n_{i}t_{x},n_{j}t_{z})$. The master
equation of this probability constructed from Eq. S2 and Eq. S3 is:

$\begin{split}&\frac{dP^{t}(n_{i}t_{x},n_{j})}{dt}=-\Big{[}\sum_{n_{k},t_{z}}P^{t}(n_{i}t_{x},n_{j}t_{z})P_{gen}(n_{k}|n_{j})k_{t}^{add}({n_{i}t_{x},n_{j}t_{z}\xrightarrow{}n_{j}t_{z},n_{k}t_{z}})+\\\
&\sum_{n_{k},t_{y},t_{z}}P^{t}(n_{i}t_{x},n_{j}t_{z})P^{t}_{sys}(n_{k}t_{y}|n_{i}t_{x})k_{t}^{rem}(n_{k}t_{y},n_{i}t_{z}\xleftarrow{}n_{i}t_{x},n_{j}t_{z})\Big{]}\\\
&+\Big{[}\sum_{n_{k},t_{y},t_{z}}P^{t}_{sys}(n_{i}t_{x}|n_{j}t_{y})P^{t}(n_{j}t_{y},n_{k}t_{z})k_{t}^{rem}({n_{i}t_{x},n_{j}t_{z}\xleftarrow{}n_{j}t_{y},n_{k}t_{z}})\Big{]}\\\
&+\Big{[}\sum_{n_{k},t_{y}}P_{gen}(n_{j}|n_{i})P^{t}(n_{k}t_{y},n_{i}t_{x})k_{t}^{add}({n_{k}t_{y},n_{i}t_{x}\xrightarrow{}n_{i}t_{x},n_{j}t_{x}})\Big{]}\end{split}$
(S4)

When the system at steady state, we can set the left side of Eq. S4 to 0 and
rewrite it into:

$\begin{split}&\sum_{n_{k},t_{y},t_{z}}\Big{[}P^{ss}(n_{i}t_{x},n_{j}t_{z})P_{gen}(n_{k}|n_{j})k_{t}^{add}({n_{i}t_{x},n_{j}t_{z}\xrightarrow{}n_{j}t_{z},n_{k}t_{z}})-\\\
&P^{ss}_{sys}(n_{i}t_{x}|n_{j}t_{y})P^{t}(n_{j}t_{y},n_{k}t_{z})k_{t}^{rem}({n_{i}t_{x},n_{j}t_{z}\xleftarrow{}n_{j}t_{y},n_{k}t_{z}})\Big{]}=\\\
&\sum_{n_{k},t_{y},t_{z}}\Big{[}P_{gen}(n_{j}|n_{i})P^{ss}(n_{k}t_{y},n_{i}t_{x})k_{t}^{add}({n_{k}t_{y},n_{i}t_{x}\xrightarrow{}n_{i}t_{x},n_{j}t_{x}})-\\\
&P^{ss}(n_{i}t_{x},n_{j}t_{z})P^{ss}_{sys}(n_{k}t_{y}|n_{i}t_{x})k_{t}^{rem}(n_{k}t_{y},n_{i}t_{z}\xleftarrow{}n_{i}t_{x},n_{j}t_{z})\Big{]}\end{split}$
(S5)

Next, we make an assumption that the rates of the process only depends on the
two further most layers of the assembly. Specifically, the rate
$k_{t}^{add}({n_{i}t_{x},n_{j}t_{y}\xrightarrow{}n_{j}t_{y},n_{k}t_{y}})$ only
depends on $(n_{j}t_{y},n_{k}t_{y})$; and the rate
$k_{t}^{rem}({n_{i}t_{x},n_{j}t_{z}\xleftarrow{}n_{j}t_{y},n_{k}t_{z}})$
depends on $(n_{j}t_{y},n_{k}t_{z})$. If this is not the case for our rates,
we can expand the element $n_{i}$ to include more layers until the rates only
depend on the two furthest elements. We then rewrite the left side of Eq. S5
into:

$\begin{split}&\sum_{t_{y}}P^{ss}_{sys}(n_{i}t_{x}|n_{j}t_{y})\Big{[}\sum_{n_{k},t_{z}}\Big{[}P^{ss}(n_{j}t_{y})P_{gen}(n_{k}|n_{j})k_{t}^{add}({n_{i}t_{x},n_{j}t_{y}\xrightarrow{}n_{j}t_{y},n_{k}t_{y}})-\\\
&P^{t}(n_{j}t_{y},n_{k}t_{z})k_{t}^{rem}({n_{i}t_{x},n_{j}t_{z}\xleftarrow{}n_{j}t_{y},n_{k}t_{z}})\Big{]}\Big{]}=\\\
&\sum_{t_{y}}P^{ss}_{sys}(n_{i}t_{x}|n_{j}t_{y})\Big{[}\sum_{n_{k},n_{i}^{{}^{\prime}},t_{z},t_{x}^{{}^{\prime}}}\Big{[}P^{ss}(n_{i}^{{}^{\prime}}t_{x}^{{}^{\prime}},n_{j}t_{y})P_{gen}(n_{k}|n_{j})k_{t}^{add}({n_{i}^{{}^{\prime}}t_{x}^{{}^{\prime}},n_{j}t_{y}\xrightarrow{}n_{j}t_{y},n_{k}t_{y}})-\\\
&P^{t}(n_{j}t_{y},n_{k}t_{z})P(n_{i}^{{}^{\prime}}t_{x}^{{}^{\prime}}|n_{j}t_{z})k_{t}^{rem}({n_{i}^{{}^{\prime}}t_{x}^{{}^{\prime}},n_{j}t_{z}\xleftarrow{}n_{j}t_{y},n_{k}t_{z}})\Big{]}\Big{]}\end{split}$
(S6)

For the sake of notation, let’s define:

$\begin{split}V(n_{j}t_{y})=&\sum_{n_{k},n_{i}^{{}^{\prime}},t_{z},t_{x}^{{}^{\prime}}}\Big{[}P^{ss}(n_{i}^{{}^{\prime}}t_{x}^{{}^{\prime}},n_{j}t_{y})P_{gen}(n_{k}|n_{j})k_{t}^{add}({n_{i}^{{}^{\prime}}t_{x}^{{}^{\prime}},n_{j}t_{y}\xrightarrow{}n_{j}t_{y},n_{k}t_{y}})-\\\
&P^{t}(n_{j}t_{y},n_{k}t_{z})P(n_{i}^{{}^{\prime}}t_{x}^{{}^{\prime}}|n_{j}t_{z})k_{t}^{rem}({n_{i}^{{}^{\prime}}t_{x}^{{}^{\prime}},n_{j}t_{z}\xleftarrow{}n_{j}t_{y},n_{k}t_{z}})\Big{]}\end{split}$
(S7)

$\begin{split}V(n_{i}t_{x},n_{j})=&\sum_{n_{k},t_{y},t_{z}}\Big{[}P^{ss}(n_{k}t_{y},n_{i}t_{x})P_{gen}(n_{j}|n_{i})k_{t}^{add}({n_{k}t_{y},n_{i}t_{x}\xrightarrow{}n_{i}t_{x},n_{j}t_{x}})-\\\
&P^{ss}(n_{i}t_{x},n_{j}t_{z})P^{ss}_{sys}(n_{k}t_{y}|n_{i}t_{x})k_{t}^{rem}(n_{k}t_{y},n_{i}t_{z}\xleftarrow{}n_{i}t_{x},n_{j}t_{z})\Big{]}\end{split}$
(S8)

Plug Eqs. S8, S7, S6 into Eq. S5 and we have:

$V(n_{i}t_{x},n_{j})=\sum_{t_{y}}P^{ss}_{sys}(n_{i}t_{x}|n_{j}t_{y})V(n_{j}t_{y})$
(S9)

Summing all over $n_{j}$ and we have:

$V(n_{i}t_{x})=\sum_{n_{j},t_{y}}P^{ss}_{sys}(n_{i}t_{x}|n_{j}t_{y})V(n_{j}t_{y})$
(S10)

Eq. S10 suggests that: $\bf{V}$ is an eigenvector of the conditional
probability matrix $\bf{P^{ss}_{sys}}$. Another eigenvector of the conditional
probability matrix is $\bf{P^{ss}_{inside}}$ since:

$P^{ss}_{inside}(n_{i}t_{x})=\sum_{n_{j},t_{y}}P^{ss}_{sys}(n_{i}t_{x}|n_{j}t_{y})P^{ss}_{inside}(n_{j}t_{y})$
(S11)

This indicates that $P^{ss}_{inside}(n_{i}t_{x})\propto V(n_{i}t_{x})$. Using
the normalization of probability, we then have:

$P^{ss}_{inside}(n_{i}t_{x})=\frac{V(n_{i}t_{x})}{\sum_{n_{i},t_{x}}V(n_{i}t_{x})}$
(S12)

##  S3 The Coarse Grain Master Equations and Its Entropy Production

As demonstrated in the main-text, the entropy production of the bi-partite
Markov process is always higher than the entropy production of the mean-field
model. To convene these two approaches and their entropy productions, we have
to look at a coarse grain probability from the bi-partite Markov model:

$\begin{split}P^{t}(n_{i},n_{j}t_{y})=\sum_{t_{x}}P^{t}(n_{i}t_{x},n_{j}t_{y})\end{split}$
(S13)

In effect, we have merge the time partitions in the bi-partite Markov process
in the coarse grain probabilities. The master equations for the coarse grain
probability, $P^{t}(n_{i},n_{j}t_{y})$ is:

$\begin{split}&\frac{dP^{t}(n_{i},n_{j}t_{y})}{dt}=\\\
&-\sum_{t_{x},t_{z},n_{k}}P^{t}(n_{i}t_{x},n_{j}t_{y})\Big{[}P_{gen}(n_{k}t_{y}|n_{j}t_{y})k^{add}({n_{i}t_{x},n_{j}t_{y}\xrightarrow{}n_{j}t_{y},n_{k}t_{y}})+\\\
&P^{t}_{sys}(n_{k}t_{z}|n_{i}t_{x})k^{rem}(n_{k}t_{z},n_{i}t_{y}\xleftarrow{}n_{i}t_{x},n_{j}t_{y})\Big{]}+\\\
&\Big{[}\sum_{n_{k},t_{x},t_{z}}P^{t}_{sys}(n_{i}t_{x}|n_{j}t_{z})P^{t}(n_{j}t_{z},n_{k}t_{y})k^{rem}({n_{i}t_{x},n_{j}t_{y}\xleftarrow{}n_{j}t_{z},n_{k}t_{y}})\Big{]}+\\\
&\Big{[}\sum_{n_{k},t_{x}}P_{gen}(n_{j}t_{y}|n_{i}t_{y})P^{t}(n_{k}t_{x},n_{i}t_{y})k^{add}({n_{k}t_{x},n_{i}t_{y}\xrightarrow{}n_{i}t_{y},n_{j}t_{y}}))\Big{]}\\\
&+\lambda P^{t}(n_{i},n_{j}t_{y-1})-\lambda P^{t}(n_{i},n_{j}t_{y})\\\
&=-P^{t}(n_{i},n_{j}t_{y})\Bigg{(}\sum_{t_{x},,t_{z},n_{k}}P^{t}_{sys}(t_{x}|n_{i},n_{j}t_{y})\\\
&\Big{[}P_{gen}(n_{k}t_{y}|n_{j}t_{y})k^{add}({n_{i}t_{x},n_{j}t_{y}\xrightarrow{}n_{j}t_{y},n_{k}t_{y}})+\\\
&P^{t}_{sys}(n_{k}t_{z}|n_{i}t_{x})k^{rem}(n_{k}t_{z},n_{i}t_{y}\xleftarrow{}n_{i}t_{x},n_{j}t_{y})\Big{]}\Bigg{)}+\\\
&\sum_{n_{k}}\Bigg{(}P^{t}(n_{j},n_{k}t_{y})\Big{[}\sum_{t_{x},t_{z}}P^{t}_{sys}(n_{i}t_{x}|n_{j}t_{z})P^{t}_{sys}(t_{z}|n_{j},n_{k}t_{y})k^{rem}({n_{i}t_{x},n_{j}t_{y}\xleftarrow{}n_{j}t_{z},n_{k}t_{y}})\Big{]}\\\
&+P^{t}(n_{k},n_{i}t_{y})\Big{[}\sum_{t_{x}}P_{gen}(n_{j}t_{y}|n_{i}t_{y})P^{t}_{sys}(t_{x}|n_{k},n_{i}t_{y})k^{add}({n_{k}t_{x},n_{i}t_{y}\xrightarrow{}n_{i}t_{y},n_{j}t_{y}})\Big{]}\Bigg{)}\\\
&+\lambda P^{t}(n_{i},n_{j}t_{y-1})-\lambda
P^{t}(n_{i},n_{j}t_{y})\end{split}$ (S14)

Unlike the master equations for the full probabilities bi-partite Markov
process, the master equations for the coarse grain probabilities (Eq. S14)
cannot be self-consistently solved at a steady state because it still depends
on the probability $P^{t}(n_{i}t_{x},n_{j}t_{y})$. So to get the coarse grain
probabilities, we have to solve the master equations of the full probabilities
first. The entropy production corresponds to the coarse grain probabilities
is:

$\begin{split}&\dot{S}_{reduced}=\sum_{t_{y},n_{k},n_{i},n_{j}}\\\
&\Big{(}P^{t}(n_{k},n_{i}t_{y})\big{[}\sum_{t_{x}}P_{gen}(n_{j}t_{y}|n_{i}t_{y})P^{t}_{sys}(t_{x}|n_{k},n_{i}t_{y})k^{add}({n_{k}t_{x},n_{i}t_{y}\xrightarrow{}n_{i}t_{y},n_{j}t_{y}})\big{]}\\\
&-P^{t}(n_{i},n_{j}t_{y})\big{[}\sum_{t_{x},t_{z}}P^{t}_{sys}(n_{k}t_{x}|n_{i}t_{z})P^{t}_{sys}(t_{z}|n_{i},n_{j}t_{y})k^{rem}({n_{k}t_{x},n_{i}t_{y}\xleftarrow{}n_{i}t_{z},n_{j}t_{y}})\big{]}\Big{)}\\\
&\ln{\frac{P^{t}(n_{k},n_{i}t_{y})\big{[}\sum_{t_{x}}P_{gen}(n_{j}t_{y}|n_{i}t_{y})P^{t}_{sys}(t_{x}|n_{k},n_{i}t_{y})k^{add}({n_{k}t_{x},n_{i}t_{y}\xrightarrow{}n_{i}t_{y},n_{j}t_{y}})\big{]}}{P^{t}(n_{i},n_{j}t_{y})\big{[}\sum_{t_{x},t_{z}}P^{t}_{sys}(n_{k}t_{x}|n_{i}t_{z})P^{t}_{sys}(t_{z}|n_{i},n_{j}t_{y})k^{rem}({n_{k}t_{x},n_{i}t_{y}\xleftarrow{}n_{i}t_{z},n_{j}t_{y}})\big{]}}}\end{split}$
(S15)

Figure S1: Entropy production of the one dimension growth assembly with rates
affected by a time-varying external field h(t) with $J=4,h=0.2,T=1000$.
$\dot{N}_{\rm{ave}}$ is the average growth rate of assemblies over the period
T: $\dot{N}_{\rm{ave}}=\frac{1}{T}\int_{0}^{T}\dot{N}(t)dt$. The blue curve is
computed using the mean-field entropy production, Eq. 15. The green curve is
computed using the full bi-partite Markov Process entropy production, Eq. 18.
The orange curve is computed using the entropy production of the coarse grain
Markov Process, Eq. S15.

As shown in Fig. S1, the entropy production of the coarse grain model, Eq.
S15, is identical to one from the mean field model, Eq. 15.