Geometric approach to nonequilibrium hasty shortcuts

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 effects¹⁸ 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 nth 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 u₁ < u_ini, which is followed by a final quench of u₂ = 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.

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, $p⃗uss$⁠. As illustrated in Fig. 1, at a given stationary condition u, the fast projections onto the slow manifold $M̂(u)$ 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 $p⃗uss$ are shown by solid arrows confined on the slow manifold $S(u)$⁠.

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 $p⃗uss$ for arbitrary values of u. The control task is to steer the system from the initial steady state $p⃗uiniss$ to the final steady state $p⃗ufinss$⁠. A hasty shortcut is illustrated by a control sequence (u_ini, u₁, u_fin). During the first step of the shortcut, the system rapidly evolves to the slow manifold $S(u1)$ by the rapid projection $M̂(u1)$⁠. 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 $M̂(ufin)$⁠. In comparison, the rudimentary one-step quenching control, where u_ini is directly set to u_fin, involves both an initial rapid projection $M̂(ufin)$ 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 $p⃗ufinss$⁠. 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 $M̂(ufin)$⁠. 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.

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.

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 $p⃗$-space).

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) = ∪_uT(u) is the union of a family of transient sets T(u) for all u’s, and P_s = ∪_uF(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, $p⃗uss$⁠. This is because F(u) belongs to the subspace spanned by fast eigenvectors, and $T(u)⊂S(u)$ is confined to the slow manifold spanned by the slow eigenvectors. Thus, we argue that for systems $R̂(u)$’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 $p⃗uss$ [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 $p⃗uass$ along the curve to be in close proximity to another point $p⃗ubss$ 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 $M̂(ub)$⁠, 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 $v⃗i(u)$ of the rate matrix $R̂(u)$ as u varies. In other words, even for systems whose steady states $p⃗uss$ 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 $p⃗$-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 u₁ and u₂. Depending on the directions of the projection operators $M̂(u1)$ and $M̂(u2)$ 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 u₁ and u₂, 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.

The system’s dynamics at a given subunit concentration c_s is captured by the corresponding rate matrix $R̂(cs)$⁠. 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 $p⃗(t0)=p⃗uiniss$ 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 (u₁, u₂) to drive the system. After the two rapid projections, the system reaches $p⃗(t2)=M̂(u1)M̂(u2)p⃗uiniss$⁠. By choosing all possible protocols (u_ini, u₁, u₂), 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 = u₁ = u₂, 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., $p⃗(t2)$⁠, applying an ultimate-step control of u_fin. If the rapid relaxation $M̂(ufin)$ projects $p⃗(t2)$ to the steady state $p⃗ufinss$⁠, then the distribution $p⃗(t2)$ (from the reachable set) also belongs to the penultimate set. Numerically, the intersection between a point $p⃗(t2)$ in the reachable set and a point in the penultimate set (corresponding to u_fin) is judged by the following: If $maxM̂(ufin)p⃗(t2)−p⃗ufinss<ε=10−7$⁠, then $p⃗(t2)$ can be mapped by a final control u_fin to a state that is extremely close to the ultimate steady state, $p⃗ufinss$⁠. Moreover, the corresponding control sequence, (u_ini, u₁, u₂, u_fin), constructs a hasty shortcut connecting the initial steady state $p⃗uiniss$ to the final steady state $p⃗ufinss$⁠, 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 $p⃗ss(cs=7.1)$⁠, 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 $p⃗ss(cs=6.6)$ to $p⃗ss(cs=5.1)$⁠. This non-monotonic control involves starting at the steady state of concentration u_ini = 6.6, applying a sudden decrease of subunit concentration to u₁ = 4.1, immediately followed by a dramatic subunit concentration increase, u₂ = 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.

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, u₁, u₂, u_fin. For illustrative purposes, the shortcuts are represented by two parts: (1) The initial three control values u_ini, u₁, u₂ 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 $p⃗ufinss$⁠, 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 u₁ and u₂ that are within the closed range of [u_ini, u_fin] (or [u_fin, u_ini]) and the control is monotonic, u₁ ≤ u₂ (or u₁ ≥ u₂).

• (ii) “Overshooting only”: One or more intermediate controls u₁, u₂ 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 u₁, u₂ 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 u₁ or u₂) 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 $M̂(u4)M̂(u3)M̂(u2)M̂(u1)puiniss$⁠, 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 $p⃗ufinss$ 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., u₁ > u₂ < u₃ or u₁⟨u₂⟩u₃. 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 u₁ and u₂ (i.e., periodic direction turns). We choose an initial equilibrium of u_ini = 0.1 and then oscillate the control parameter between u₁ = 7.1 and u₂ = 1.1. In Fig. 10(a), we demonstrate the evolution of the initial distribution after a single period $(M̂(u1)M̂(u2))$⁠, double periods $([M̂(u1)M̂(u2)]2)$⁠, and triple periods $([M̂(u1)M̂(u2)]3)$⁠. 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 (u₁, u₂) pairs in Fig. 10(b), where $M̂(ui)M̂(u2)M̂(u1)p⃗uiss$ (forward cycle) and $M̂(ui)M̂(u1)M̂(u2)p⃗uiss$ (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.

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 $(M̂(u))$ 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.

See the supplementary material for appendices and additional figures.

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.

The authors have no conflicts to disclose.

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).

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