Optimal Nonequilibrium Processes

• Optimal nonequilibrium processes are finite-time transformations and control protocols that minimize quantities like irreversible work and entropy production in systems far from equilibrium.
• They leverage geometric structures, such as Riemannian metrics and thermodynamic length, to define optimal paths (geodesics) for system control and state transitions.
• Applications span discrete protocols, stochastic resetting, quantum systems, and experimental atomic cooling, illustrating practical strategies for energy-efficient control.

Searching arXiv for papers on optimal nonequilibrium processes and related frameworks. {"query":"all:(\"optimal nonequilibrium processes\" OR \"optimal control\" nonequilibrium thermodynamics)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} arXivSearch(query="all:(\"optimal nonequilibrium processes\" OR \"finite-time thermodynamics\" OR \"nonequilibrium steady states\" OR \"stochastic thermodynamics\")", max_results=10, sort_by="submittedDate", sort_order="descending") Optimal nonequilibrium processes are finite-time transformations, control protocols, or statistical procedures defined on systems that are away from equilibrium and optimized with respect to a specified objective such as irreversible work, entropy production, excess dissipation, work extraction, first-passage time, or detection power. In the literature, the term covers several distinct regimes: slowly driven isothermal processes close to equilibrium, transitions between nonequilibrium steady states, arbitrarily far-from-equilibrium control with restricted actuation, discrete jump protocols, collisionless few-particle thermalization, and even optimal statistical certification of broken detailed balance from trajectory data (Bonança et al., 2014, Zulkowski et al., 2013, Zhong et al., 2022, Mayer et al., 2019, Strang, 2024).

1. Scope of the subject and senses of “optimality”

The optimization target depends on the physical setting. For isothermal near-equilibrium driving, the central quantity is the irreversible work

$W_{\mathrm{irr}} \equiv W-\Delta F \ge 0,$

and the problem is to choose the protocol $\lambda(t)$ that minimizes dissipation over a fixed duration (Bonança et al., 2014). For overdamped systems and many related classical models, the same slow-driving structure appears as minimization of excess work or excess power (Zulkowski et al., 2015).

For transitions between nonequilibrium steady states, the objective is no longer equilibrium free-energy dissipation. One important choice is the Hatano–Sasa functional

$$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$

whose average measures irreversibility associated with changing the steady state itself rather than the housekeeping cost of maintaining it (Zulkowski et al., 2013). In open quantum systems, the analogous objective is the average information entropy production

$$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$

optimized over smooth finite-time control schedules (Zulkowski et al., 2015).

Far from equilibrium, optimality can mean exact finite-time work minimization under restricted controls, rather than asymptotic metric minimization. In that setting, the state variable is the probability density itself, evolved by the Fokker–Planck equation, and the control problem becomes a Hamiltonian boundary-value problem (Zhong et al., 2022). In discrete-state far-from-equilibrium Markov systems, finite-time optimal control can be formulated through an entropy-production action, with the optimal protocol containing discontinuous endpoint jumps in geodesic space (Mohite et al., 2 Nov 2025).

The same phrase is also used in narrower but technically important senses. For diffusion with stochastic resetting, optimality refers to minimizing the mean first-passage time to a target, and the optimal process is a genuinely nonequilibrium intermittent one rather than an equilibrium Langevin comparator with the same stationary distribution (Evans et al., 2012). For steady-state Ornstein–Uhlenbeck processes, “optimal” can refer to the best asymptotic linear test statistic for rejecting detailed balance from trajectory data, not to controlling the dynamics themselves (Strang, 2024). This suggests that the field is unified more by its variational structure than by any single dynamical model.

2. Thermodynamic objectives, entropy production, and distance-like quantities

A recurring feature is that the cost functional can be written in terms of entropy production or a quadratic form in protocol velocity. In the close-to-equilibrium isothermal regime,

$$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$

with

$$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$

so dissipation is governed by equilibrium fluctuations and a correlation time (Bonança et al., 2014). In overdamped diffusions this becomes

$$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$

with $\boldsymbol\zeta$ the generalized inverse diffusion tensor (Zulkowski et al., 2015).

For nonequilibrium steady-state transitions, the Hatano–Sasa average is approximated in the slow-driving regime by

$$\langle Y \rangle_{\boldsymbol \Lambda} \approx \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \boldsymbol \zeta(\boldsymbol \lambda(t)) \cdot \bigg[ \frac{ d \boldsymbol \lambda }{dt} \bigg],$$

where $\boldsymbol\zeta(\boldsymbol\lambda)$ is a generalized inverse diffusion tensor built from steady-state correlations of $\lambda(t)$0 (Zulkowski et al., 2013). In the quantum Lindblad setting the corresponding structure is

$\lambda(t)$1

with $\lambda(t)$2 positive semi-definite, allowing null directions in parameter space (Zulkowski et al., 2015).

Relative entropy and related divergences are equally central. In the few-atom collisionless cooling experiment, the target thermal state is

$\lambda(t)$3

and the entropic distance to this target is measured by

$\lambda(t)$4

The stepwise entropy production toward the target is

$\lambda(t)$5

and the associated statistical length is

$\lambda(t)$6

These satisfy the refined second-law bound

$\lambda(t)$7

a far-from-equilibrium version of the horse-carrot theorem (Mayer et al., 2019).

For discrete control, the one-step cost has an exact information-theoretic form: $\lambda(t)$8 so multi-step dissipation is the sum of KL divergences between consecutive equilibrium ensembles (Large et al., 2018). For Markov systems driven between NESSs, the total entropy production rate is

$\lambda(t)$9

and slow driving generates an effective Lagrangian

$$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$0

combining housekeeping dissipation with a NESS-corrected quadratic control term (Kamp et al., 17 Jun 2025).

3. Geometric structure and thermodynamic metrics

A large part of the subject is geometric. In one-parameter near-equilibrium problems, the variational functional

$$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$1

implies constant excess power along the optimum, and the minimum equals the square of a thermodynamic length divided by duration (Zulkowski et al., 2015). In multidimensional control spaces, the same object defines a Riemannian metric

$$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$2

so optimal protocols are geodesics of $$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$3 (Zulkowski et al., 2015).

For isothermal processes close to equilibrium, linear response yields the same logic in terms of the effective friction

$$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$4

Where $$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$5 is large, the optimum slows down; where it is small, it speeds up (Bonança et al., 2014). For NESS transitions of a dragged colloidal particle with controls $$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$6, the metric

$$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$7

is flat after the coordinate transformation

$$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$8

so optimal protocols are straight lines in $$Y \equiv \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \frac{\partial \phi}{\partial \boldsymbol \lambda}\big(x(t);\boldsymbol \lambda(t)\big),$$9 traversed at constant speed (Zulkowski et al., 2013).

The quantum version preserves the same pattern but changes the geometry. The metric

$$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$0

is only positive semi-definite, and the two-level bosonic-bath example has a null $$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$1-direction while dissipation depends only on $$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$2 (Zulkowski et al., 2015). In that example the minimum entropy production has the thermodynamic-length form

$$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$3

A stronger geometric claim appears in the thesis literature: the friction-tensor geometry on equilibrium control manifolds is presented as the pullback of Wasserstein geometry, and optimal protocols are decomposed as

$$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$4

with $$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$5 a geodesic part and $$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$6 a counterdiabatic part (Zhong, 26 Sep 2025). This suggests that near-equilibrium thermodynamic length and exact finite-time probability transport are not competing metaphors but different projections of the same control problem.

4. Exact finite-time control and far-from-equilibrium optimization

Beyond linear response, exact control theory works directly on density evolution. For an overdamped particle with controlled potential $$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$7,

$$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$8

the finite-time work-minimization problem under limited control can be written as a Pontryagin problem on the pair $$\langle \Sigma_I\rangle_{\boldsymbol{\Lambda}} = \int_0^\tau dt\, \mathrm{Tr}\left\{ \partial_t\rho\, \big[-\ln \rho+\ln \rho_t^{\mathrm{eq}}\big] \right\},$$9 with Hamiltonian functional

$$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$0

The canonical equations are

$$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$1

with mixed boundary conditions

$$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$2

For affine-control potentials this gives an explicit control update

$$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$3

The framework reproduces the exact harmonic result, yields monotone optimal protocols with endpoint jumps for a quartic trap, and produces a non-monotonic optimal protocol for a linearly biased double-well when the barrier is high and the duration is intermediate (Zhong et al., 2022).

A related discrete-state far-from-equilibrium theory derives an exact finite-time optimal-control law in geodesic coordinates. If $$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$4 denotes the geodesic map, the optimal finite-time protocol is

$$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$5

and the minimum entropy production is

$$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$6

The total cost splits into bulk and boundary terms,

$$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$7

which is the basis for the claim that discontinuous endpoint jumps are generic rather than pathological (Mohite et al., 2 Nov 2025).

Exact solvability also persists in some driven harmonic systems with nonequilibrium forcing. For the overdamped dynamics

$$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$8

the optimal protocol minimizing work for fixed $$W_{\mathrm{irr}} = \beta\int_{0}^{\tau}d t\,\left(\frac{d \lambda}{d t}\right)^{2} \tau^{c}[\lambda(t)]\,\mathcal{X}[\lambda(t)],$$9 and $$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$0 has the exact decomposition

$$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$1

with

$$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$2

and

$$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$3

The quasistatic bound separates an initial-state information term, a transport term from the time-averaged force, and a fluctuation-harvesting term from the force variance, so the optimal protocol automatically acts as an information engine (Olsen et al., 9 Apr 2025).

When exact PDE control is impractical, automatic differentiation through stochastic simulations provides a computational alternative. That approach reproduces the exact harmonic-trap optimum including endpoint jumps, outperforms near-equilibrium protocols for far-from-equilibrium Ising magnetization reversal, and for a $$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$4 barrier crossing finds a protocol that hastens the approach to, and slows the departure from, the barrier region compared to the near-equilibrium theoretical protocol (Engel et al., 2022).

5. Discrete protocols, search, inference, and surrogate modeling

Optimal nonequilibrium processes need not be smooth in time. In discretely driven systems, the control is a sequence

$$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$5

with instantaneous jumps $$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$6 and dwell times $$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$7. The exact infinite-time excess work is

$$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$8

and finite-time corrections depend on force autocovariances and incomplete relaxation. The resulting lower bound

$$\mathcal{X}[\lambda(t)] = \left\langle \left( \frac{\partial H}{\partial\lambda}\right)^{2} \right\rangle_{\lambda(t)} - \left\langle \frac{\partial H}{\partial\lambda}\right\rangle_{\lambda(t)}^{2},$$9

is saturated when all $$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$0 are equal, so optimal discrete protocols equalize excess work per step. In multistable landscapes, this permits genuinely discrete strategies such as avoiding barrier regions entirely (Large et al., 2018).

Search with stochastic resetting is an even simpler example of nonequilibrium advantage. A diffusive particle that resets to $$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$1 at rate $$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$2 has mean first-passage time

$$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$3

with optimum at nonzero $$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$4. An equilibrium Langevin process constructed to have the same stationary distribution has a larger optimum,

$$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$5

so the nonequilibrium intermittent strategy is more efficient than the equilibrium comparator even under matched stationary statistics (Evans et al., 2012).

Optimality also appears on the inference side. For a stationary Ornstein–Uhlenbeck process

$$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$6

nonequilibrium is encoded by

$$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$7

Among all linear stochastic line integrals, the asymptotically optimal detector of broken detailed balance has vector field

$$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$8

and the optimal scalar statistic is

$$\langle \beta W_{\rm ex}\rangle \approx \int_0^\tau dt\; \dot{\boldsymbol\lambda}^T \boldsymbol\zeta(\boldsymbol\lambda)\dot{\boldsymbol\lambda},$$9

which equals the entropy production rate (Strang, 2024). Here optimal nonequilibrium processing means optimal statistical certification, not optimal control.

Data-driven surrogate models supply yet another layer. From nonstationary molecular-dynamics trajectories, a Markovian Langevin model can be inferred in terms of a biased landscape

$\boldsymbol\zeta$0

together with local drift, friction, and noise fields (Lickert et al., 2021). That framework does not itself optimize protocols, but it provides reduced models on which optimization can plausibly be built.

6. Experimental realization, multi-objective extensions, and limitations

The most explicit experimental realization in this literature is the dilute-gas cooling experiment on a few noninteracting Cs atoms in a strongly nonharmonic optical dipole trap driven by degenerate Raman sideband cooling pulses. The target thermal state for the axial dynamics is

$\boldsymbol\zeta$1

and because each Raman pulse resets the momentum but leaves the position distribution essentially frozen, the system is driven into nonthermal states that do not self-thermalize in the absence of collisions. The optimization problem is therefore to choose the pulse spacing $\boldsymbol\zeta$2 that minimizes the entropic distance to the target thermal state (Mayer et al., 2019).

The experiment contains on average about seven Cs atoms, with peak collision rate $\boldsymbol\zeta$3, initial temperature

$\boldsymbol\zeta$4

and Raman temperature

$\boldsymbol\zeta$5

For three equally spaced Raman pulses of duration $\boldsymbol\zeta$6, the harmonic simulation gives the optimum

$\boldsymbol\zeta$7

equal to one quarter of the trap period, while the real nonharmonic trap gives

$\boldsymbol\zeta$8

At that spacing the entropic distance to the target state after the third pulse is reduced by almost a factor of two relative to nonoptimal spacings, and the overlap between measured and target axial distributions reaches about $\boldsymbol\zeta$9, roughly twice the value for $$\langle Y \rangle_{\boldsymbol \Lambda} \approx \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \boldsymbol \zeta(\boldsymbol \lambda(t)) \cdot \bigg[ \frac{ d \boldsymbol \lambda }{dt} \bigg],$$0 or $$\langle Y \rangle_{\boldsymbol \Lambda} \approx \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \boldsymbol \zeta(\boldsymbol \lambda(t)) \cdot \bigg[ \frac{ d \boldsymbol \lambda }{dt} \bigg],$$1. The first two statistical lengths are nearly equal at the optimum, while the third is much smaller, and the generalized horse-carrot theorem is verified experimentally through the $$\langle Y \rangle_{\boldsymbol \Lambda} \approx \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \boldsymbol \zeta(\boldsymbol \lambda(t)) \cdot \bigg[ \frac{ d \boldsymbol \lambda }{dt} \bigg],$$2-based surrogate divergence (Mayer et al., 2019).

Recent work also broadens the topic from single-objective optimality to trade-off geometry. In intrinsically nonequilibrium systems, Pareto-optimal control is formulated through

$$\langle Y \rangle_{\boldsymbol \Lambda} \approx \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \boldsymbol \zeta(\boldsymbol \lambda(t)) \cdot \bigg[ \frac{ d \boldsymbol \lambda }{dt} \bigg],$$3

and the stationarity condition

$$\langle Y \rangle_{\boldsymbol \Lambda} \approx \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \boldsymbol \zeta(\boldsymbol \lambda(t)) \cdot \bigg[ \frac{ d \boldsymbol \lambda }{dt} \bigg],$$4

For the active-particle example, the entire Pareto front is organized by the intrinsic scale

$$\langle Y \rangle_{\boldsymbol \Lambda} \approx \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \boldsymbol \zeta(\boldsymbol \lambda(t)) \cdot \bigg[ \frac{ d \boldsymbol \lambda }{dt} \bigg],$$5

while for the cyclic quantum-dot engine the corresponding scales are

$$\langle Y \rangle_{\boldsymbol \Lambda} \approx \int_{0}^{\tau} dt \ \bigg[ \frac{ d \boldsymbol \lambda^{T} }{dt} \bigg] \cdot \boldsymbol \zeta(\boldsymbol \lambda(t)) \cdot \bigg[ \frac{ d \boldsymbol \lambda }{dt} \bigg],$$6

In both cases Pareto-optimal protocols consist of smooth branches connected by boundary jumps, and systems with the same intrinsic scale belong to the same control equivalence class (Forão et al., 22 Jun 2026). A plausible implication is that boundary jumps are not an artifact of particular objective choices but a persistent structural feature of finite-time nonequilibrium optimization.

The limitations are correspondingly diverse. Slow-driving metric theories rely on linear response and remain most reliable when control changes are gradual compared with relaxation times (Bonança et al., 2014, Zulkowski et al., 2013, Zulkowski et al., 2015, Zulkowski et al., 2015, Kamp et al., 17 Jun 2025). Exact density-space control suffers from the curse of dimensionality (Zhong et al., 2022). Discrete-control formulas are derived in a near-equilibrium regime with truncated memory (Large et al., 2018). OU detection theory is asymptotic and model-specific (Strang, 2024). Data-driven Langevin surrogates depend on approximate Markovian closure and coordinate choice (Lickert et al., 2021). Even so, taken together these results establish a coherent picture: optimal nonequilibrium processes are controlled by a competition among dissipation, dynamical realizability, information, and control architecture, and the most successful formalisms make that competition explicit rather than hiding it inside equilibrium analogies.