Complex and even non-monotonic responses to external control can be found in many thermodynamic systems. In such systems, nonequilibrium shortcuts can rapidly drive the system from an initial state to a desired final state. One example is the Mpemba effect, where preheating a system allows it to cool faster. We present nonequilibrium hasty shortcuts—externally controlled temporal protocols that rapidly steer a system from an initial steady state to a desired final steady state. The term “hasty” indicates that the shortcut only involves fast dynamics without relying on slow relaxations. We provide a geometric analysis of such shortcuts in the space of probability distributions by using timescale separation and eigenmode decomposition. We further identify the necessary and sufficient condition for the existence of nonequilibrium hasty shortcuts in an arbitrary system. The geometric analysis within the probability space sheds light on the possible features of a system that can lead to hasty shortcuts, which can be classified into different categories based on their temporal pattern. We also find that the Mpemba-effect-like shortcuts only constitute a small fraction of the diverse categories of hasty shortcuts. This theory is validated and illustrated numerically in the self-assembly model inspired by viral capsid assembly processes.

## I. INTRODUCTION

Systems driven out of equilibrium are prevalent across physics, biology, and material science. The Mpemba effect,^{1–17} the strong Mpemba effect,^{3,4} and certain hysteresis effects^{18} are general examples of systems demonstrating non-monotonic responses toward external control. Furthermore, such complex nonequilibrium responses are also ubiquitous in natural systems from the cellular level, such as chemical reaction networks^{19–22} and circadian clocks,^{23–25} to the molecular level, such as in DNA mechanics^{26–30} or protein folding.^{31,32} Timescale separation is common in such nonequilibrium processes, where dynamics of interest, such as collective behavior or self-assembly, occur on a much slower timescale than other components of system dynamics, such as vibrations of atoms due to thermal fluctuation.

Traditional studies focus on the slow dynamics in systems, which are best suited for equilibrium in a static environment or slow relaxations in a slowly changing environment.^{33} In both cases, the fast dynamics can be integrated out and replaced by local Boltzmann distributions. Reconstructing the slow dynamics of complex systems has been the focus of recent efforts, such as using trajectory data mined from full atomistic simulations to generate few-state interpretable Markov state models.^{34–36} However, in the case of rapidly changing stimuli prevalent in realistic systems,^{37} both fast and slow dynamics play important roles in determining how systems evolve away from equilibrium.^{38} Rapid environmental driving harnesses interesting fast dynamics and can result in a system being able to exist in a target far-from-equilibrium steady state^{39,40} or access nonequilibrium shortcuts as with the Mpemba effect.^{1–17} Such far-from-equilibrium dynamics find a clean theoretical description in stochastic thermodynamics.^{41}

Understanding both the long-timescale macroscopic behavior and the fast relaxations at different environmentally controlled conditions is essential to elucidate how systems respond to rapidly changing external control protocols. In this paper, we present nonequilibrium hasty shortcuts—temporal protocols comprising rapidly switching sequences of control parameters *u*—that can steer a system from an initial equilibrium state (corresponding to the initial parameter value *u*_{ini}) to a final equilibrium state (corresponding to the final parameter value *u*_{fin}). The rapid parameter switches only allow the system to utilize its fast relaxations, but they do not leave excess time for slow relaxations. However, because the relaxation modes are not mutually perpendicular to each other (the evolution operator is not Hermitian or symmetric), both the fast and slow relaxation modes play important roles in understanding the fast relaxation dynamics of the hasty shortcuts. We developed a geometric approach to determine if a system can allow for a hasty shortcut by rapidly tuning a one-dimensional control parameter. By writing down a family of master equations of a system at various values of the control parameter *u*’s and by performing a timescale separation analysis for each *u*, we obtain a family of fast projection operators to describe the fast evolution of the system at each transient control *u*, and a family of slow manifolds $S(u)$ spanned by the slow relaxation eigenmodes. In the probability simplex comprising all possible probability distributions of the system’s state, a fast projection operator for a given *u* rapidly maps an arbitrary probability distribution onto the corresponding slow manifold. Based on the analysis of the family of fast projections and slow manifolds in the probability simplex, we developed a formalism toward identifying nonequilibrium hasty shortcuts through geometric analysis. This theory allows us to write down the necessary and sufficient condition for the existence of such nonequilibrium shortcuts in an arbitrary system that can be described by a master equation without degeneracy.

To distinguish the hasty shortcut discussed in this paper from the many other interesting shortcuts defined in nonequilibrium thermal systems or nonequilibrium quantum systems,^{42} we provide a very brief review of several types of shortcuts. In quantum dynamics, the shortcuts to adiabaticity describe the problem where one can utilize an extra term of Hamiltonian to assist an evolving wave function in obeying adiabatic quantum evolution beyond the adiabatic limit (e.g., maintaining on the *n*th eigenstate of a rapidly changing time-dependent Hamiltonian).^{43–45} In biological physics, similar controls with counteradiabatic terms have also been developed for diverse systems.^{46,47} Similarly, in thermodynamics, there have been studies of shortcuts to isothermality,^{48–51} where an extra term is engineered to keep a system in a Boltzmann distribution when environmental parameters rapidly change. Linear response theory has also been used to develop slowly varying optimal processes or rapidly varying weak optimal processes.^{52}

There are several other types of shortcuts that can be described as problems in the optimal control theory,^{53} where one can engineer non-monotonic control of multiple environmental parameters to steer a system from one initial state to a final state with a minimum energy dissipation^{54,55} or waiting time.^{5,56} Summarized in Ref. 42, there are three broad categories of methods to determine shortcuts: (1) inverse engineering, where a protocol is inferred using a chosen evolution of the system,^{57–59} (2) counterdiabatic driving, where rapid driving must preserve the trajectory of a probability distribution function,^{60,61} and (3) fast-forward driving, where an external potential is used to accelerate, decelerate, or reverse a reference process.^{62–64}

In our work, we have defined a general class of hasty shortcuts—by controlling an arbitrary control parameter (not limited to temperature) that rapidly switches in time, the system only relaxes the fast degrees of freedom, and it does not have enough time to relax its slow degrees of freedom due to the rapid control parameter switches.

While other types of shortcuts have looked into minimizing a cost function, such as time duration or energy dissipation, the hasty shortcuts presented herein instead involve the classification of a system’s dynamics into fast and slow, which are exponentially different in their rates. Rather than focusing on searching for a specific protocol or designing a system that minimizes the target function (e.g., time duration), the hasty shortcuts emphasize the avoidance of any slow dynamics altogether as its name suggests. Thus, when hasty shortcuts exist, it is likely to be exponentially faster than other processes that involve slow dynamics.

The hasty shortcut exists if the system, by only utilizing the series of fast relaxations corresponding to the sequence of control parameter values, can achieve the desired final equilibrium distribution. Some hasty shortcuts presented in this paper partially resemble the cooling/heating shortcuts proposed by the studies of the Mpemba effect, which claims that it may be faster to cool a system by first heating it up.^{5} It has been shown that heating and cooling are inherently asymmetric,^{65} and thus interesting shortcuts may emerge by non-monotonically controlling temperature. Some of the hasty shortcuts we found in this work resemble the strong Mpemba effect: To drive a system from *u*_{ini} to *u*_{fin} > *u*_{ini}, the hasty shortcut may start by an initial downward switch of *u*_{1} < *u*_{ini}, which is followed by a final quench of *u*_{2} = *u*_{fin}.

To illustrate the geometric theory of hasty shortcuts, we demonstrate a simple assisted assembly model inspired by a viral capsid assembly problem studied by Phillip Geissler’s group.^{66} We provide a simple 24-state assembly model that demonstrates timescale separation. In this rather kinetically simple model with a single control parameter (the subunit concentration *c*_{s}), we demonstrate numerous hasty shortcuts connecting different equilibria.

## II. THEORY

### A. Timescale separation at constant control

*n*configurations and $R\u0302(u)$ is the transition rate matrix controlled by a parameter

*u*. Here, the control parameter is allowed to vary in time according to a temporal protocol

*u*(

*t*).

*u*, then regardless of the initial probability distribution $p\u20d7(0)$, the system eventually relaxes toward its corresponding stationary state $p\u20d7uss$, where $R\u0302(u)p\u20d7uss=0$. Such relaxation within a stationary environment (i.e., constant control parameter

*u*) can be separated into different relaxation eigenmodes, where each mode is an eigenvector of $R\u0302(u)$ such that

*λ*

_{i}(

*u*) dictates the relaxation rate. The time evolution of a system’s probability distribution while the control parameter is maintained at a stationary value

*u*is

*b*

_{i}’s. Here, we assume that the rate matrix $R\u0302(u)$ is diagonalizable for each

*u*involved (e.g., this assumption is guaranteed to be true when detailed balance condition is satisfied). In another description, one can define an evolution operator for a constant environmental condition

*u*over time period

*t*, which maps an initial probability vector $p\u20d7(0)$ into $p\u20d7(t)$, the probability vector at time

*t*. This evolution map can be written as

*b*

_{i}that decomposes the initial probability vector into a superposition of eigenvectors. Here, we have defined a propagation operator

^{67}In this paper, we are interested in systems that have separated relaxation timescales. In this system, for a transiently stationary control parameter value

*u*, the relaxation dynamics can be separated into fast and slow eigenmodes. Reflected in the spectrum of eigenvalues starting from the first nonzero eigenvalue

*λ*

_{2}, if

*λ*

_{c}and 1/

*λ*

_{c+1}for any

*c*≥ 2, then the separation of timescale occurs. In our example simulation, we choose the threshold of timescale separation as $\lambda c+1\lambda c>10$. The timescale separation allows us to define the persistence time

*τ*, which is effectively infinitely short for the slow relaxations and infinitely long for the fast relaxation,

*τ*, $e\lambda i\tau \u22480$ if

*i*>

*c*and $e\lambda i\tau \u22481$ if

*i*≤

*c*. In other words, the constant

*u*-dynamics of time duration

*τ*at fixed control value

*u*, denoted by $W\u0302\tau $, can be approximated by a rapid projection operator $M\u0302(u)$, which maps the full probability space to the slow manifold $S(u)$ (see Fig. 1),

*c*entries equal 1 (i.e., $D\u0302ii(01)=1$ for

*i*≤

*c*) as

*τ*is too short to allow for any slow relaxation, and $D\u0302ii(01)=0$ for

*i*>

*c*as within the time

*τ*, all the fast eigenmodes fully relax and vanish. This map $M\u0302(u)$, defined for a control parameter

*u*, is a projection operator that characterizes the rapid relaxation of any initial probability to a slow manifold $S(u)$. The slow manifold is a vector space spanned by the eigenvectors corresponding to the slow modes. At the timescale

*τ*, the slow relaxations do not have enough time to occur.

If one further waits for a time much longer than *τ*, the slow dynamics defined within the slow manifold can eventually bring the system to the ultimate stationary state, $p\u20d7uss$. As illustrated in Fig. 1, at a given stationary condition *u*, the fast projections onto the slow manifold $M\u0302(u)$ are represented by parallel bundles of dashed lines (fibers) onto the slow manifold $S(u)$ (a yellow hyperplane); given long enough time *t* ≫ *τ*, the slow relaxations toward the ultimate stationary state $p\u20d7uss$ are shown by solid arrows confined on the slow manifold $S(u)$.

### B. Nonequilibrium hasty shortcuts

*u*

_{ini},

*u*

_{1},

*u*

_{2}, …,

*u*

_{fin}that can steer the system from the initial stationary state $p\u20d7uiniss$ to the final stationary state $p\u20d7ufinss$. The term “hasty” indicates that the dwell time

*τ*at each control value

*u*

_{i}is infinitely short to allow for the slow relaxations to occur [see Eq. (8)]. Since the controlled protocol avoids waiting for slow relaxations that are exponentially slower than fast relaxations, the protocol serves as a shortcut.

^{68}In summary, the whole control protocol constructs an evolution operator comprising a sequence of fast relaxation projections $M\u0302(u)$’s, which rapidly maps the initial state to the final state,

Let us illustrate the nonequilibrium hasty shortcut by comparing it with a straightforward rudimentary control protocol that directly sets the control value from *u*_{ini} to *u*_{fin}. In Fig. 2, we sketch three slow manifolds $S(uini)$, $S(u1)$, and $S(ufin)$ within the probability simplex. The black locus consists of all stationary probability distributions $p\u20d7uss$ for arbitrary values of *u*. The control task is to steer the system from the initial steady state $p\u20d7uiniss$ to the final steady state $p\u20d7ufinss$. A hasty shortcut is illustrated by a control sequence (*u*_{ini}, *u*_{1}, *u*_{fin}). During the first step of the shortcut, the system rapidly evolves to the slow manifold $S(u1)$ by the rapid projection $M\u0302(u1)$. Then, without waiting for any slow relaxation, the second step control (of value *u*_{fin}) directly evolves the system into the desired final steady state by the rapid projection $M\u0302(ufin)$. In comparison, the rudimentary one-step quenching control, where *u*_{ini} is directly set to *u*_{fin}, involves both an initial rapid projection $M\u0302(ufin)$ that maps the system onto the slow manifold $S(ufin)$ and a long-time slow relaxation as the system slowly traverses the slow manifold toward the final state $p\u20d7ufinss$. In summary, the hasty shortcut can save time by avoiding slow relaxations, which take exponentially longer time than rapid relaxations.

The nonequilibrium hasty shortcut can be analogous to a hasty driver who is averse to traffic slowdowns. By constantly resetting the destination on the car’s GPS, the driver aims to always travel at high speed until ultimately colliding with the final destination at full speed. In this analogy, traveling at high speed corresponds to utilizing fast relaxations, traffic slowdowns represent slow dynamics on the slow manifold, and transiently resetting the destination of GPS corresponds to the rapid switches of *u*. Notice that some hasty shortcuts, e.g., the one shown in Fig. 2, may be counterintuitive and resemble the Mpemba effect, as it initially steers *u* away from the direction of the target *u*_{fin} to reach an intermediate state, which can then be rapidly relaxed to the desired final state by $M\u0302(ufin)$. In Sec. III C, we classify hasty shortcuts into different categories and show that the Mpemba-effect-like shortcuts only comprise a small fraction of possible hasty shortcuts.

### C. Condition for the existence of nonequilibrium hasty shortcuts

We examine the condition of a system to allow for the nonequilibrium hasty shortcut. The hasty shortcut presented in this paper is likely to be found in kinetically nontrivial systems, including antiferromagnetic spin systems where one can easily identify the strong Mpemba effect.^{3,5} However, the hasty shortcuts are more general than the Mpemba effect and thus could be found in systems where the Mpemba effect does not exist. Here, we use a geometric approach to identify the general kinetic features that allow for a general thermal system to have nonequilibrium hasty shortcuts.

*u*. We further assume that the rate matrix $R\u0302(u)$ is diagonalizable for each

*u*involved in our control. Both the rate matrix $R\u0302(u)$ of ergodic systems satisfying detailed balance conditions and the rate matrix $R\u0302(u)$ of non-detail-balanced systems without degenerate eigenvectors guarantee diagonalizability. For these systems, the steady states $p\u20d7uss$ for arbitrary given values of

*u*form a one-dimensional locus

*L*given by

*L*by a sequence of rapid projection operators, $M\u0302(u)$’s, then the system allows for hasty shortcuts.

The necessary and sufficient condition for hasty shortcuts can be geometrically represented by the nontrivial intersections between two sets constructed below, where both sets can be illustrated within a system’s probability simplex (or the $p\u20d7$-space).

**Reachable set**,

*R*

_{s}(

*n*), is defined as the set of probability distributions $p\u20d7$ that are reachable through steering any initial stationary distribution $p\u20d7uiniss$ by arbitrary

*n*-step sequences of fast relaxation projections $M\u0302(ui)$’s,

*n*-step reachable set

*R*

_{s}(

*n*) can be constructed iteratively by

*R*

_{s}(

*i*) ⊂

*R*

_{s}(

*i*+ 1) is true for all positive integer

*i*. Additionally, since $p\u20d7$ is bounded, we can denote the bounded infinite-step reachable set as

*R*

_{s}= lim

_{n→∞}

*R*

_{s}(

*n*). For illustration, in Fig. 3(a), we sketch the construction of

*R*

_{s}(1) in the $p\u20d7$-space. In this case, the set

*R*

_{s}(1) consists of all possible distributions $p\u20d7$’s that can be reached by any single-step rapid projection $M\u0302(u1)$ applied to any initial steady-state distribution $p\u20d7uiniss$, for any arbitrary

*u*

_{1}and

*u*

_{ini}. This reachable set is illustrated by a two-dim ribbon, which is swept by a moving one-dim curve (

*T*(

*u*)) in the $p\u20d7$-space: Given a chosen value of

*u*

_{1}, the rapid projection $M\u0302(u1)$ maps all possible steady states in

*L*to a transient set

*T*(

*u*

_{1}) is the red curve obtained by the projection of the whole steady-state manifold

*L*(as black curve) onto the slow manifold $S(u1)$ (as a yellow plane) via the single-step projection $M\u0302(u1)$ (as dashed arrows). Then, one can construct the set

*R*

_{s}(1) as the union of all transient sets

*T*(

*u*

_{1}) for all possible values for

*u*

_{1},

*R*

_{s}(1) is the pink ribbon sweep by the red curves for different choices of

*u*

_{1}. It is straightforward to see that

*L*, the set of steady states (i.e., the black curve), is a backbone of

*R*

_{s}(

*n*) for all positive integer

*n*.

**Penultimate set**, given by

*F*(

*u*) is not necessarily a one-dim object, and the number of fast eigenmodes for $R\u0302(u)$ determines

*F*(

*u*)’s dimension. In Fig. 3(b), each dashed arrow represents one fiber

*F*(

*u*) for a specific value of

*u*, and the penultimate set can be considered as a multidimensional blue ribbon swept by a family of fibers

*F*(

*u*) for all values of

*u*. One can show that the steady-state manifold

*L*is a subset of the penultimate set. In other words, the set of steady states

*L*(as black curve) is the backbone of the penultimate set.

**Existence condition of hasty shortcuts**: The existence condition of (up to

*n*-step) hasty shortcuts can be written as the nontrivial intersection between the

*n*-step reachable set

*R*

_{s}(

*n*) and the penultimate set

*P*

_{s},

*R*

_{s}(

*n*) and

*P*

_{s}. Here, trivial intersection between

*R*

_{s}(

*n*) and

*P*

_{s}is their shared backbone

*L*(i.e., the set of steady-state distributions). Intuitively, the trivial intersections between the

*R*

_{s}(

*n*) and

*P*

_{s}correspond to time-independent protocol

*u*

_{fin}=

*u*

_{n}=⋯=

*u*

_{1}=

*u*

_{ini}, where the initial and final steady-state distributions are trivially identical. In the following, we discuss the geometrical features that allow for the nontrivial intersection.

### D. Geometric analysis

The geometric representation of the existence condition for hasty shortcuts [as Eqs. (11), (12), (16), and (18)] allows us to investigate the underlying mechanisms of hasty shortcuts.

First, we show that the hasty shortcuts are inherently far from equilibrium and steady states, which allows for the necessary geometrical features for the nontrivial intersection between *R*_{s}(*n*) and *P*_{s}. This can be visualized by Fig. 3(c): Recall that *R*_{s}(1) = ∪_{u}*T*(*u*) is the union of a family of transient sets *T*(*u*) for all *u*’s, and *P*_{s} = ∪_{u}*F*(*u*) is the union of a family of “fibers” *F*(*u*) for all *u*’s. Moreover, recall that the existence condition for hasty shortcuts is that *R*_{s} and *P*_{s} intersect beyond the shared steady-state backbone *L* [Eq. (18)]. However, as shown in Fig. 3(c), at a given value *u*, *T*(*u*) and *F*(*u*) can only intersect at the corresponding steady state, $p\u20d7uss$. This is because *F*(*u*) belongs to the subspace spanned by fast eigenvectors, and $T(u)\u2282S(u)$ is confined to the slow manifold spanned by the slow eigenvectors. Thus, we argue that for systems $R\u0302(u)$’s with continuous dependence on control parameter *u*, in the vicinity of *u*, it is very unlikely to construct nontrivial intersections between the reachable set and the penultimate set. In other words, hasty shortcuts can only be achieved by far-from-steady-state driving protocols.

Furthermore, we illustrate potential geometric features that could allow for the existence of hasty shortcuts. We propose a few geometric features that allow for the nontrivial intersection between *R*_{s} and *P*_{s}. They shed light on the kinetic features that could allow a system to have hasty shortcuts. Even though *T*(*u*) and *F*(*u*) cannot intersect beyond the trivial point $p\u20d7uss$ [see Fig. 3(c)], the nontrivial intersection between *R*_{s}(*n*) and *P*_{s} may still occur if any one or more of the following geometrical features are satisfied:

(

*#*1) the steady-state locus is a highly bent curve that almost intersects itself;(

*#*2) the reachable set and/or the penultimate set are highly bent;(

*#*3) the reachable set and/or the penultimate set occupies a large space in the probability simplex.

**(#1)** states that the curve *L* is so highly curved that it almost approaches self-intersection, causing some point $p\u20d7uass$ along the curve to be in close proximity to another point $p\u20d7ubss$ at a very different *u*. In this situation, the shared backbone of the two “ribbons” *R*_{s} and *P*_{s} is highly bent and almost approaches self-intersection, increasing the chance for nontrivial intersections between *R*_{s} and *P*_{s}. To the next level, if the curve *L* is so bent that its point $puass$ intersects the fiber *F*(*u*_{b}) corresponding to $M\u0302(ub)$, it indicates that there exists a single-step quench shortcut from $puass$ to $pubss$, which can be realized by directly quenching *u*_{a} to *u*_{b}. In the extreme case, if the curve *L* intersects itself at $puass=pubss$, the system may demonstrate reentrant behaviors that resemble those studied in reentrant transitions.^{69–74} For example, the highly curved equilibrium locus *L* and its associated relaxation trajectories have been illustrated graphically in the work of Klich *et al.* focusing on the mean field antiferromagnetic Ising model, which demonstrates the strong Mpemba effect.^{3} For other general thermodynamic systems controlled by a simple control parameter *u* (such as temperature, pressure, or concentration), the feature (#1) may not be easily satisfied. For general systems without a strongly non-monotonic parameter dependence required in geometric feature (#1), the hasty shortcut may still exist under geometric features (#2) and (#3).

**(#2)** states that the geometric shape of either *R*_{s}(*n*) or *P*_{s} is highly bent. It increases the chance for the two sets to intersect beyond their shared backbone *L*, which is the condition of the existence of the hasty shortcut. Notice that here we do not assume for geometric feature (#1), and thus the contortion of *R*_{s} or *P*_{s} is not due to a bent backbone. Rather, the contortion of *R*_{s} and/or *P*_{s} can result from the change of fast and slow eigenvectors $v\u20d7i(u)$ of the rate matrix $R\u0302(u)$ as *u* varies. In other words, even for systems whose steady states $p\u20d7uss$ have a plain (not-highly curved) dependence on *u*, the hasty shortcut may still exist if the relaxation modes change dramatically as one varies the control parameter *u*.

**(#3)** states that *R*_{s}(*n*) and/or *P*_{s} occupies a large space in the $p\u20d7$-space. It increases the chance for the nontrivial intersection between *R*_{s} and *P*_{s}, which allows for the existence of hasty shortcuts. First, to allow *P*_{s} to occupy a large space (i.e., greater dimensionality), it can be realized for systems whose eigenvectors consist of a large number of fast eigenvectors and only a few slow eigenvectors [i.e., *c* is small in Eq. (7)].^{75} Second, to increase the reachable set, one may consider increasing the number of steps in the control protocol. In other words, the larger the *n*, the larger the reachable set *R*_{s}(*n*), as one can show *R*_{s}(*n*) ⊂ *R*_{s}(*n* + 1).

The increase of *R*_{s}(*n*) as the number of steps *n* increases can be analyzed geometrically. In Fig. 4, we illustrate two cases where increasing step number achieves new reachable states. For simplicity, consider a periodic control protocol oscillating between *u*_{1} and *u*_{2}. Depending on the directions of the projection operators $M\u0302(u1)$ and $M\u0302(u2)$ relative to the locations of the slow manifolds $S(u1)$ and $S(u2)$, the system may be driven to a sequence of states that either converges or diverges. In Fig. 4(a), as one increases the number of cycles between *u*_{1} and *u*_{2}, the resulting states converge. In comparison, in Fig. 4(b), the resulting system reaches a sequence of states that diverge as one increases the number of periods. In summary, geometric relations between the fast and slow eigenmodes for various values of *u* could be engineered to increase the size of the reachable set *R*_{s}(*n*) and thus increase the chance of obtaining hasty shortcuts.

## III. NUMERICAL SIMULATION

### A. Assisted assembly model

*c*

_{s}. When a site

*k*is occupied by a subunit, it is denoted by state

*s*

_{k}= 1, and when empty, it is

*s*

_{k}= 0. Thus, the configuration of each octahedron in the solution is represented by an eight-bit string, and its energy is defined in an Ising model-like form. The energy of each configuration

**s**= (

*s*

_{1}, …,

*s*

_{8}) is given by the summation of the binding energy for each subunit and the interaction energy between neighboring subunits. We have

*h*is the binding energy between each subunit and the binding site and

*J*is the interaction energy between any neighboring pair of subunits. If each binding site were distinguishable, the model would consist of 2

^{8}unique configurations. In reality, since each binding site is assumed to be identical with the same binding affinity with subunits,

*h*, the 2

^{8}configuration can be simplified to 24 distinct configurations indexed by

*i*, and their degeneracy is denoted by

*g*

_{i}. The degeneracy

*g*

_{i}was calculated using six symmetry moves, and one can verify that $\u2211i=124gi=256$. Given this consideration, we can define the free energy of a distinguishable configuration

*i*as

*F*

_{i}=

*E*

_{i}−

*β*

^{−1}ln

*g*

_{i}, where

*β*= 1/

*k*

_{B}

*T*is assumed to be unity in this work. We consider a dilute solution of octahedrons, where each octahedron is considered independent and identically distributed. Then, the state of all octahedrons can be represented by the configuration probabilities of one octahedron. The probability $p\u20d7(t)$ evolves according to the concentration of subunits in the solution

*c*

_{s}, temperature

*T*, and the interaction energies. Specifically, the probability evolves according to the master equation given by

*R*

_{ij}represent probability transition rates from configuration

*j*to

*i*. Two types of transitions considered: binding and unbinding of a single subunit. The rate of any binding event is given as

*F*

_{i}denotes the free energy of configuration

*i*. The term

*ξη*(

*i*,

*j*) captures the steric hindrance barrier of the binding or unbinding transition due to occupied neighboring subunits. Here,

*ξ*is a positive constant and

*η*(

*i*,

*j*) =

*η*(

*j*,

*i*) is the occupation number of neighboring sites around the binding/unbinding site for the transition between configurations

*i*and

*j*. In summary, the free energy barrier for the transition between configurations

*i*and

*j*is assumed to be their average free energy (

*F*

_{i}+

*F*

_{j})/2 plus the steric hindrance

*ξη*(

*i*,

*j*). One can verify that the transition rates satisfy the detailed balance condition. Some possible transitions are sketched in Fig. 5. The diagonal elements of the rate matrix are chosen such that each column of $R\u0302$ adds up to 0. Thus, the dynamics of assisted assembly can be described by a family of rate matrices $R\u0302(cs)$ parameterized by control parameter

*c*

_{s}. The theory of timescale separation and Eqs. (1), (3), and (4) apply.

### B. Discrete reachable set and penultimate set

The system’s dynamics at a given subunit concentration *c*_{s} is captured by the corresponding rate matrix $R\u0302(cs)$. The control parameter *u* for this assembly problem is chosen to be the subunit concentration *c*_{s}. To illustrate the systems’ equilibrium distributions at different control values, we discretized *c*_{s} and obtained the equilibrium distribution for each *c*_{s}, shown in Fig. 6.

Numerical eigenanalysis of *R*(*c*_{s}) reveals that the system shows timescale separation over an arbitrary selection of values of *c*_{s} (see Fig. S1 in the supplementary material). Therefore, we can use the eigenmode separation approach described in Sec. II A to perform an efficient simulation of the system’s dynamics in a reduced space. It demonstrates good agreement between the full dynamics and the reduced dynamics based on timescale separation (see Fig. S2 in the supplementary material).

With a numerically solvable model, we verify our geometrical theory for the existence of hasty shortcuts. We simplify the control protocol by requiring the values of *c*_{s} to be chosen from a discrete set *c*_{s} = 0.1, 0.6, …, 7.1. In the rest of the paper, both *u* and *c*_{s} denote the control parameter and will be used interchangeably. Then, by further discretizing the probability distribution space, we numerically obtain the discrete reachable set [as dots in Fig. 7(a)] and its nontrivial intersection with the penultimate set [as blue dots in Fig. 7(b)]. The reachable set is constructed as follows: Consider the system starts at a steady-state probability distribution $p\u20d7(t0)=p\u20d7uiniss$ for the initial control parameter to be the subunit concentration chosen from the discrete list *c*_{s} = 0.1, 0.6, …, 7.1. We perform two rapid steps of controls (*u*_{1}, *u*_{2}) to drive the system. After the two rapid projections, the system reaches $p\u20d7(t2)=M\u0302(u1)M\u0302(u2)p\u20d7uiniss$. By choosing all possible protocols (*u*_{ini}, *u*_{1}, *u*_{2}), we can steer the system’s probability distributions into a two-step reachable set *R*_{s}(2). Notice that each element of the set is a 24-dim vector in the probability simplex, and for illustrative purposes, they are shown in a two-dim plot by principal component analysis (PCA) in Fig. 7(a). Notice that if *u*_{ini} = *u*_{1} = *u*_{2}, the control protocol does not change the parameter at all, and the system remains at the initial equilibrium distribution. The reachable set contains the 15 steady-state distributions corresponding to each discrete value of *c*_{s}, shown by 15 golden stars in Fig. 7(a).

To determine the hasty shortcuts, we numerically find the intersection between the reachable set and the penultimate set. This can be done by taking any probability distribution from the reachable set, e.g., $p\u20d7(t2)$, applying an ultimate-step control of *u*_{fin}. If the rapid relaxation $M\u0302(ufin)$ projects $p\u20d7(t2)$ to the steady state $p\u20d7ufinss$, then the distribution $p\u20d7(t2)$ (from the reachable set) also belongs to the penultimate set. Numerically, the intersection between a point $p\u20d7(t2)$ in the reachable set and a point in the penultimate set (corresponding to *u*_{fin}) is judged by the following: If $maxM\u0302(ufin)p\u20d7(t2)\u2212p\u20d7ufinss<\epsilon =10\u22127$, then $p\u20d7(t2)$ can be mapped by a final control *u*_{fin} to a state that is extremely close to the ultimate steady state, $p\u20d7ufinss$. Moreover, the corresponding control sequence, (*u*_{ini}, *u*_{1}, *u*_{2}, *u*_{fin}), constructs a hasty shortcut connecting the initial steady state $p\u20d7uiniss$ to the final steady state $p\u20d7ufinss$, which only utilizes the fast dynamics and does not require waiting for any slow dynamics. Notice that rather than showing the complete overlap between the reachable set and the penultimate set, in Fig. 7(b) and Fig. S3 in the supplementary material, we have shown the intersection set corresponding to different choices of final control values *u*_{fin}. In Fig. 7(b), left, the intersection set contains four different distributions for *u*_{fin} → *c*_{s} = 5.1. It indicates that the steady-state distribution corresponding to *c*_{s} = 5.1 can be accessed through different members of the penultimate set. In contrast, for the final steady state $p\u20d7ss(cs=7.1)$, only a single nontrivial intersection exists [see blue dots shown in Fig. 7(b), right].

As demonstrated above, even in a simple self-assembly system, we can identify multistep and non-monotonic hasty shortcuts capable of driving an initial equilibrium distribution to a desired final equilibrium. One such shortcut is illustrated in Fig. 7(c) where a system is driven from $p\u20d7ss(cs=6.6)$ to $p\u20d7ss(cs=5.1)$. This non-monotonic control involves starting at the steady state of concentration *u*_{ini} = 6.6, applying a sudden decrease of subunit concentration to *u*_{1} = 4.1, immediately followed by a dramatic subunit concentration increase, *u*_{2} = 7.1, and ultimately switching the subunit concentration to the desired final value *c*_{s} = 5.1. This shortcut is one of many hasty shortcuts obtained by the reachable set-penultimate set intersection.

### C. Classification of nonequilibrium hasty shortcuts

Even for a low number of control steps *n* = 2, the system demonstrates numerous hasty shortcuts. In Fig. 8(a), we list all (*n* = 2)-step hasty shortcuts starting from initial steady state of *u*_{ini} = 3.1, 3.6, …, 7.1 and ending at a final steady state of *u*_{fin}, i.e., *u*_{ini}, *u*_{1}, *u*_{2}, *u*_{fin}. For illustrative purposes, the shortcuts are represented by two parts: (1) The initial three control values *u*_{ini}, *u*_{1}, *u*_{2} are shown as the three columns to the left of the vertical bold line, and (2) then each ultimate control *u*_{fin} that can lead the system, prepared by an initial control to reach $p\u20d7ufinss$, is shown as a filled (colored) block in the eight columns to the right of the vertical bold line. The color of each block corresponds to the value of *u*. Notice that among all possible rapid control sequences of *n* = 2, many cannot achieve any ultimate equilibrium and thus do not qualify as shortcuts. The complete list of all possible control sequences is presented in Fig. S4 in the supplementary material. Moreover, we notice that there does not exist any (*n* = 2)-step hasty shortcut starting from or ending at low initial concentrations (*u*_{ini} < 3.1 or *u*_{fin} < 3.1), which indicates that a large number of steps *n* may be required to find shortcuts in the low concentration region.

The numerical results revealed numerous hasty shortcuts of diverse temporal patterns. Here, we classify the shortcut into a few categories of interest. Notice that the Mpemba-effect-like shortcuts only constitute a small fraction of all possible hasty shortcuts. We classify hasty shortcuts as (i) monotonic, (ii) overshooting only, (iii) countershooting only, (iv) reckless control, (v) quenching, and (vi) other defined as follows:

(

*i*) “Monotonic”: The shortcut involves*u*_{1}and*u*_{2}that are within the closed range of [*u*_{ini},*u*_{fin}] (or [*u*_{fin},*u*_{ini}]) and the control is monotonic,*u*_{1}≤*u*_{2}(or*u*_{1}≥*u*_{2}).(

*ii*) “Overshooting only”: One or more intermediate controls*u*_{1},*u*_{2}go beyond the range between*u*_{ini}and*u*_{fin}along the forward direction (from*u*_{ini}to*u*_{fin}), thus creating an overshoot.(

*iii*) “Countershooting only”: One or both intermediate controls*u*_{1},*u*_{2}go beyond the range between*u*_{ini}and*u*_{fin}along the counter-forward direction (from*u*_{fin}to*u*_{ini}). By definition, the Mpemba-effect-like shortcuts must belong to this category.(

*iv*) “Reckless control”: One of the intermediate control (either*u*_{1}or*u*_{2}) overshoots, and the other intermediate control counter-shoots. This is reckless as it involves large-amplitude controls beyond*u*_{ini}and*u*_{fin}in both the forward and backward directions.(

*v*) “Quenching”: The control is a direct one-step quench from*u*_{ini}to*u*_{fin}that can immediately drive the system from initial equilibrium to the final equilibrium.(

*vi*) “Other”: This involves all kinds of controls that do not fall in the classifications above.

In Figs. 8(b) and 8(c), we illustrate the classification of shortcuts by presenting the hasty shortcuts that can reach two specific final equilibria (i.e., for *u*_{fin} = 5.1 and *u*_{fin} = 6.1). It is interesting to note that no monotonic or countershoot shortcuts can arrive at the steady state of *u*_{fin} = 5.1. Still, all categories of shortcuts are found to drive some initial equilibrium to the final equilibrium of *u*_{fin} = 6.1. Moreover, for certain pairs of initial *u*_{ini} and final *u*_{fin}, there may be no direct quenching or monotonic shortcut. In Fig. 7(c), only two shortcuts are found to start from *u*_{ini} = 4.6 and end at *u*_{fin} = 6.1, and both are overshooting shortcuts. For a complete summary of all shortcuts for all possible final equilibria, please refer to Fig. S5 in the supplementary material.

We further demonstrate that by increasing the number of steps *n*, one can reveal more hasty shortcuts by achieving a larger reachable set *R*_{s}(*n*) as argued in the geometric feature (#3) in Sec. II D. The low concentration region *c*_{s} < 3.1, which is inaccessible by (*n* = 2)-step shortcuts (see Fig. 8), can be accessed by (*n* = 4)-step shortcuts (see Fig. 9). For the purpose of demonstration, we numerically construct the reachable set *R*_{s}(4) by using $M\u0302(u4)M\u0302(u3)M\u0302(u2)M\u0302(u1)puiniss$, where the initial equilibria are restricted to the low concentration region *u*_{ini} = *c*_{s} = 0.1, …, 3.6. A list of new hasty shortcuts is found to connect from *u*_{ini} ≤ 3.6 to any ultimate value *u*_{fin} as shown in Figs. S6 and S7 in the supplementary material. In Fig. 9(a), we list all hasty shortcuts toward $p\u20d7ufinss$ for *u*_{fin} = 3.1.

In this low concentration range, where (*n* = 2)-step shortcuts do not exist, (*n* = 4)-step shortcuts are also harder to achieve: As shown in Fig. 9(a), control protocols with simple patterns, such as quench, undershoot, and monotonic, cannot serve as hasty shortcuts. Only three categories of shortcuts are observed: overshoot, reckless control, and others. This observation agrees with the geometric analysis following feature (#3) in Sec. II D, where erratic or oscillatory multistep control protocols with a number of direction turns can lead to shortcuts where simpler patterns fail to reach. Here, a direction turn is defined by one change of control direction of *u*_{i−1}, *u*_{i}, *u*_{i+1}, e.g., *u*_{1} > *u*_{2} < *u*_{3} or *u*_{1}⟨*u*_{2}⟩*u*_{3}. By quantifying the number of direction turns in each (*n* = 4)-step hasty shortcut, we find that many overshoot-type shortcuts are strongly “oscillatory,” involving three direction turns. We also observe a number of reckless control or other shortcuts that involve two, three, or four direction turns. The histograms of the number of direction turns in each category of shortcuts are shown in Fig. 9(b).

We further investigate the effect of control protocol with strictly periodic oscillations between *u*_{1} and *u*_{2} (i.e., periodic direction turns). We choose an initial equilibrium of *u*_{ini} = 0.1 and then oscillate the control parameter between *u*_{1} = 7.1 and *u*_{2} = 1.1. In Fig. 10(a), we demonstrate the evolution of the initial distribution after a single period $(M\u0302(u1)M\u0302(u2))$, double periods $([M\u0302(u1)M\u0302(u2)]2)$, and triple periods $([M\u0302(u1)M\u0302(u2)]3)$. We verify that the system evolves into a different distribution at the end of each period in a converging manner as described by Fig. 4(a) in Sec. II C. In Fig. 10(b), we also demonstrate the noncommutative property of (*u*_{1}, *u*_{2}) pairs in Fig. 10(b), where $M\u0302(ui)M\u0302(u2)M\u0302(u1)p\u20d7uiss$ (forward cycle) and $M\u0302(ui)M\u0302(u1)M\u0302(u2)p\u20d7uiss$ (backward cycle) lead to different ultimate states. The periodic oscillatory and noncommutative control analysis further underscores the importance of oscillatory, non-monotonic, and reckless patterns in achieving hasty shortcuts.

## IV. CONCLUSIONS

This paper proposes nonequilibrium hasty shortcuts, a new class of rapidly steered control protocols that can manipulate a system from an initial equilibrium (or steady state) to a different final equilibrium (or steady state). The name “hasty” indicates that throughout the process, the system’s dynamics only involve fast dynamics but not slow relaxations (due to the rapid change of the control parameter). We have described a general geometric approach to understanding the nonequilibrium hasty shortcuts in arbitrary systems that can be described by a master equation. At any transiently fixed control parameter *u*, the fast dynamics $(M\u0302(u))$ rapidly projects any initial probability distribution onto a low-dimensional slow manifold $S(u)$ that is spanned by the slow relaxation modes. By allowing the control parameter *u* to assume a range of values, we obtain a family of slow manifolds and a family of fast projection operators. This allows us to construct a reachable set and a penultimate set in the probability space. Moreover, the existence of hasty shortcuts can be determined through the nontrivial intersection between the reachable set and the penultimate set. The geometric perspective of this paper allows us to illustrate the possible mechanisms that could allow a system to have hasty shortcuts.

As an illustration, we demonstrate hasty shortcuts in a simple Ising-model-like assisted assembly process where each assembly is an eight-sided lattice. Each lattice site can bind up to one subunit. The control parameter *u* is the concentration of the subunit in the solution *c*_{s}. We numerically generate the reachable and penultimate sets and demonstrate many nonequilibrium hasty shortcuts of various patterns and step lengths. Tabulating the complete set of shortcut protocols revealed a diverse set of shortcuts, including monotonic, overshooting, countershooting, reckless control, quenching, and many others. Among these shortcuts, the countershooting shortcuts may resemble the strong Mpemba effect that was previously predicted.^{3,5}

This geometric approach lays the foundation for future research into the processes that make hasty shortcuts possible, thus enriching the field of nonequilibrium thermodynamic control strategies. In the future, such a geometric approach may be extended to controlling nonadiabatic quantum dynamics and strongly dissipative systems.

## SUPPLEMENTARY MATERIAL

See the supplementary material for appendices and additional figures.

## ACKNOWLEDGMENTS

The authors appreciate the inspiring discussions with Professor Oren Raz, Dr. Zhongmin Zhang, and Professor Christopher Jarzynski. S.S.C. acknowledges the support provided by the National Science Foundation Graduate Research Fellowship (Grant No. DGE-2040435). We also appreciate the financial support received from the startup fund at UNC-Chapel Hill and the fund from the National Science Foundation Grant No. DMR-2145256.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Supraja S. Chittari**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Zhiyue Lu**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

## REFERENCES

*Control Theory for Physicists*

In cases where timescale separation does not occur at certain values or ranges of *u*, then $M\u0302(u)=1$ for these *u*’s. However, the geometric analysis still applies as long as the system has one or more ranges of *u* where timescale separation occurs. Obviously, if timescale separation does not occur at the whole range of available *u*, then the hasty shortcut analysis does not apply. In such cases, one may still find shortcuts in controlling thermodynamic systems by using other approaches to find a continuous-time protocol that requires the shortest time.

One may conclude that smaller number of fast modes compared to slow modes (a small *c* value) may result in a smaller reachable set *R*_{s}, which is partially dependent on the dimensionality of the slow manifold $S(u)$’s. In this case, one may have to construct hasty shortcuts by increasing the number of steps *n*, which can increase the size of *R*_{s}(*n*).