Control of active field theories at minimal dissipation

Artur Soriani; Elsen Tjhung; \'Etienne Fodor; Tomer Markovich

REVIEW 2 major objections 4 minor 2 cited by

Control of active field theories at minimal dissipation

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Active field theories dissipate least total heat at a finite control duration, and dynamical response theory predicts the optimum.

desk verdict The total-heat objective is a real step forward for active-matter control, and the finite-duration minimum is robust; the quantitative scalings, however, are derived with response theory outside its controlled regime and should be treated as conditional. read the letter →

arxiv 2504.19285 v2 pith:QFDRRWI5 submitted 2025-04-27 cond-mat.stat-mech cond-mat.soft

Artur Soriani , Elsen Tjhung , Étienne Fodor , Tomer Markovich This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.soft

keywords activematterstochasticthermodynamicsfinite-timecontroldynamicalresponsetheoryfieldheatdissipationphaseseparationoptimalprotocols

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Active systems constantly dissipate energy even when unperturbed, so the standard passive rule—optimal control is quasistatic—does not apply. The paper argues that the right quantity to minimize is the total heat $Q_t$, which includes both the heat released during the manipulation and the heat released while the system relaxes afterwards, and that for thermodynamically consistent active field theories $Q_t$ always has a global minimum at a finite protocol duration. It then shows how to find that optimum with dynamical response theory, and predicts for an active phase-separation model (Chemical Model B) that the optimal duration scales as the square root of system size divided by activity, with the optimal protocol switching from monotonic to non-monotonic shape as the duration grows. If correct, this gives experimenters a systematic rule for controlling active materials at minimal dissipation in finite time.

What carries the argument

The load-bearing object is the dynamical response theory (DRT) expansion of the heat in powers of the protocol rate, which turns the total heat into a one-dimensional effective Lagrangian $L(a,\dot a) = m(a)\dot a^2 + \langle P\rangle_s(a)$, where $m(a)$ is an effective mass built from integrated response functions and $\langle P\rangle_s(a)$ is the steady-state active power. The optimization then reduces to minimizing this Lagrangian over the protocol shape, with the optimal durations given by explicit formulas such as $\tau_t^2 = (\int \dot a^2 m\,ds)/(P_0+\int \langle P\rangle_s\,ds)$, so that the crossover between monotonic and non-monotonic optimal protocols is governed by the competition between the kinetic term (dominant at short $\tau$) and the steady-state power term (dominant at long $\tau$).

What would settle it

Simulate Chemical Model B at finite noise for a series of protocol durations and compare the numerically measured $Q_t(\tau)$ with the DRT prediction; if at the $\Delta\mu/T$ and $V/\ell$ values where $\Omega\tau_t$ is not much larger than one the numerical minimum lies outside the predicted $\tau_t$, the quantitative claim fails. Alternatively, measure the total heat in an experimental active phase-separating system and test whether the minimum and its scaling with system size and activity match $\tau_t \sim (V/\ell)^{1/2}(\Delta\mu/T)^{-1}$.

Watch

Extended reading notes

Core claim

The paper's central claim is that for active field theories kept away from equilibrium by a chemical fuel, the total heat dissipated in a full control cycle—manipulation plus post-protocol relaxation—is a non-monotonic function of protocol duration, with a global minimum at an intermediate duration that is absent from the protocol heat alone. The existence of the minimum follows from two opposing contributions: at short durations the Lagrangian term $m(a)\dot a^2$ makes heat diverge as $\sim 1/\tau$, while at long durations the background active power $P_0+\langle P\rangle_s(a)$ makes it grow linearly with $\tau$. Applying this to Chemical Model B—a conserved scalar field theory with a linear Onsager coupling between density and fuel—the paper derives the scaling $\tau_t \sim (V/\ell)^{1/2}(\Delta\mu/T)^{-1}$ for homogeneous states, and shows that the optimal protocol crosses over from a monotonic master curve to a non-monotonic one that lingers near the phase boundary, because the steady-state dissipation landscape shapes the optimal strategy.

Load-bearing premise

The load-bearing premise is that the approximate method used to compute heat from slow-response theory remains accurate at the optimal protocol duration; if that approximation breaks down, the predicted optimum could move, though a finite-time minimum still exists.

Editorial extensions

If this is right

Total dissipation in active systems is minimized at a finite protocol duration, so optimal control does not require quasistatic protocols; the optimum can be located from response functions alone.
For conserved active fields, the optimal duration grows as the square root of system size and shrinks linearly with activity, $\tau_t \sim (V/\ell)^{1/2}(\Delta\mu/T)^{-1}$, so larger or more active systems favor different operating speeds.
The shape of the optimal protocol crosses over from a monotonic master curve at short durations to a non-monotonic one that keeps the control parameter near the phase boundary at long durations, reflecting the landscape of steady-state dissipation.
The protocol heat $Q_p$ alone can be monotonic (no minimum), especially at high activity and large volume, whereas the total heat $Q_t$ always has a global minimum; hence optimizing total heat, not protocol heat, is the generally valid target.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

This suggests the same trade-off should appear in experimental active systems whose steady-state dissipation grows with the control-parameter dwell time, such as light-activated colloids or ATP-driven emulsions, so the predicted finite-duration optimum could be checked with time-resolved measurements of heat or fuel consumption.
The effective-mass form of the Lagrangian hints at a geometric picture in which finite-time optimal protocols are geodesics in a parameter-space metric determined by response functions; extending the analogy to multi-parameter protocols could yield thermodynamic cycles with characteristic speeds.
The crossover protocol that hovers at the phase boundary suggests a general strategy: when control time is long, park the system in a low-dissipation state rather than interpolating monotonically; this may apply to guided self-assembly or membrane remodeling in active materials.
Extending the computation to protocols that cross phase transitions—flagged as future work—would require the Lagrangian to include interfacial contributions, and the present homogeneous-state predictions provide a baseline against which such extensions can be tested.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Key thing to know: this paper gives the field the right objective for thermodynamic control of active field theories—total heat including post-relaxation, not just protocol heat—and shows that this objective has a finite-duration minimum. That part is physically sound and well supported. The trouble is the quantitative layer: the scalings for the optimal duration and the crossover between protocol shapes are derived with dynamical response theory at points where the theory is not controlled.

What is actually new: the distinction between protocol heat Qp and total heat Qt, the general argument that Qt must have a global minimum because active work grows linearly with duration, and the Chemical Model B case study with explicit scalings and a claimed crossover from monotonic to non-monotonic optimal protocols. This goes beyond the earlier Davis-Proesmans-Fodor framework by targeting total heat and by computing the post-relaxation contribution explicitly. The numerics are public, the analytics are checked against simulation in several regimes, and there are no fitted parameters. The demonstration that the spurious drift does not contribute to the active work is a careful piece of work.

The soft spot is exactly what the stress-test note says. DRT requires Omega*tau >> 1, where Omega is the slowest relaxation rate. For conserved CMB, Omega ~ V^{-2} while the predicted optimum has tau_t ~ V^{1/2}/Delta_mu, so Omega*tau_t ~ 1/(Delta_mu V^{3/2}). Increasing V pushes the predicted optimum further outside the controlled regime, and Figures 4(c-d) show the relative errors growing there. The V^{1/2} scaling and the crossover are not rigorously established by the expansion used to derive them. They may well be correct—the exact small-noise numerics support them over the plotted range—but the derivation is not controlled. A second symptom, disclosed by the authors themselves in SM IV.E, is that for long durations the Euler-Lagrange optimal protocol would cross the spinodal and they patch it a posteriori to remain in the homogeneous phase. The long-tau master curve and the crossover are therefore not fully derived. None of this breaks the central qualitative claim: the finite-duration minimum of total heat is robust and general.

Who this is for: people working on active matter control, stochastic thermodynamics of field theories, and phase-separation kinetics. It deserves a serious referee. The right outcome is probably major revision, with a clearer statement of the DRT validity regime and an honest separation of what is asymptotically matched versus universally derived.

Send it to review.

Referee Report

2 major / 4 minor

Summary. The paper develops a thermodynamically consistent framework for optimal control of active field theories, centered on the distinction between the protocol heat Q_p, dissipated only during manipulation, and the total heat Q_t, which also includes post-protocol relaxation. The central claim is that for active systems Q_t generically has a global minimum at a finite protocol duration, because Q_t is finite at τ = 0 (relaxation after an instantaneous quench) and grows linearly at large τ through the active fuel-consumption term. The authors use dynamical response theory (DRT) around the quasistatic regime to derive explicit expressions for Q_p and Q_t, Eq. (12), and for the optimal durations, Eq. (14). They apply the framework to Chemical Model B (CMB), a conserved scalar active field theory with a φ⁴ free energy and a linear coupling between density and fuel concentration, working in the homogeneous phase and in a small-noise expansion. They report scalings τ_t ~ (V/ℓ)^{1/2}(Δμ/T)^{-1}, a crossover between monotonic and non-monotonic optimal protocols, and support the predictions with small-noise analytical calculations, DRT, and numerical simulations. The simulation code is publicly available.

Significance. If the quantitative claims hold, this is a valuable contribution to stochastic thermodynamics and active matter: it identifies the total heat as the correct control objective, demonstrates a generic finite-time minimum for active systems, and provides parameter-free predictions for the optimal duration and protocol shape in a thermodynamically consistent field theory. The paper has notable strengths: no fitted parameters enter the predictions; the response functions, steady-state power, and boundary terms are computed from the model; the small-noise calculations and simulations are cross-checked; and the code is public. The qualitative argument for a finite-duration minimum of Q_t is robust, as it relies only on the finite value at τ = 0 and the linear large-τ growth of active work. However, the quantitative scalings and the protocol-shape crossover rest on DRT in a regime where its control parameter Ωτ is not asymptotically large, and the long-duration optimal protocol is altered a posteriori to avoid crossing the phase transition. These issues are load-bearing for the advertised predictions, though not for the existence of the finite-τ minimum.

major comments (2)

[Eq. (14), Eq. (23), SM IV.B, Figs. 4(c-d)] The quantitative scalings of the optimal duration are derived within DRT, whose validity requires Ωτ ≫ 1 where Ω is the slowest relaxation frequency. For conserved CMB, SM IV.B gives Ω ≡ ω₁(a_min, k_min) ∼ V⁻² because k_min = 2π/V, while Eq. (14) predicts τ_t ∼ V^{1/2}/Δμ, so Ωτ_t ∼ 1/(Δμ V^{3/2}) at fixed Δμ. Thus increasing V pushes the predicted optimum outside the controlled regime, and Figs. 4(c-d) indeed show relative errors growing with V and Δμ. The paper itself states that DRT captures the global minimum of Q_t only when its validity regime contains τ_t. Consequently, the advertised V^{1/2} and Δμ⁻¹ scalings and the quantitative location of the crossover are not established by the expansion used to derive them. The existence of a finite-duration minimum is not affected, since it follows from general energetics, but the quantitative control predictions need either an explicit error bound, a higher-order estimate, or a reformulation as qualitative tendencies.
[SM IV.E and Fig. 5] SM IV.E shows that for τ > τ̂ the assumption of a monotonic optimal protocol is false, and the Euler-Lagrange solution would cross the phase-transition line; the authors then 'forcefully alter a posteriori all optimal protocols to never cross the phase transition.' This means the non-monotonic master curve γ_∞ and the associated crossover are not derived from the stated optimization problem. Because the crossover is a headline claim, the authors should either solve the constrained optimization within the homogeneous phase or provide numerical evidence that the a posteriori restriction does not change the minimal Q_t or its minimizing duration. The main text notes that crossing phase transitions is left for future work, but Eq. (23), which estimates the optimal duration using γ_∞, is precisely built from this constrained object, so the caveat is load-bearing for the quantitative predictions.

minor comments (4)

[Throughout (typeset version)] The version I received has many symbol substitutions (e.g., 'k 1' for '≫ 1', 'Ä' for τ, '¼φ' for λ_φ), which makes verification unnecessarily difficult; please ensure the final typeset version uses correct mathematical symbols.
[Eq. (23)] Equation (23) is presented as an approximation obtained by matching asymptotics, and the text says it reproduces the numerically estimated value for the specific parameters, but no accuracy estimate is given; a brief discussion of how the accuracy degrades as Ωτ_t approaches the boundary of the DRT regime would strengthen the reproducibility of this central quantity.
[Captions of Figs. 4(c-d)] The definition of the relative error ε_x = τ_x^DRT/τ_x^anal − 1 appears only in the caption; it would help to define it explicitly in the main text when the scalings are first discussed.
[Code availability] The public repository [68] is a strength; consider adding a version identifier or DOI so that the exact simulation parameters and post-processing scripts used for Figs. 3–5 can be cited unambiguously.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the predictions are computed from model-derived response functions and checked against independent small-noise analytics and public simulations, with no fitted input renamed as a prediction.

full rationale

The central finite-duration minimum of Qt and the scaling claims are not equivalent to any fitted input. The paper rederives the dynamical-response expansion in Methods (Eqs. (31)-(35) and SM Sec. I): the effective mass m(a), the boundary terms B_t/B_p, and the steady-state power <P>_s are explicit integrals of first- and second-order response functions (SM Secs. II-III), evaluated for CMB from the model parameters (SM Sec. IV.C), not adjusted to reproduce the target minimum. The existence of a finite-duration minimum follows from the derived Lagrangian form (Eqs. (12)-(13)): at large tau the total heat grows linearly because P0 > 0 and <P>_s is bounded in steady state, while at small tau the m(a) a-dot^2 term makes Qt ~ 1/tau; this is a consequence of the energy balance, not a normalization choice. The scaling tau_t ~ (V/ell)^{1/2} (Delta-mu/T)^{-1} comes from the explicit V and Delta-mu dependence of the integrated response functions (SM Sec. IV.D), and the paper compares DRT with exact small-noise solutions and public simulations (Figs. 3-5; github link in Ref. [68]); no fitted parameter is renamed as a prediction. The acknowledged DRT validity restriction (main text: 'DRT predictions quantitatively capture the global minimum of Qt whenever their regime of validity contains tau_t') and the a-posteriori constraint that optimal protocols avoid the spinodal (SM Sec. IV.E: 'we forcefully alter a posteriori all optimal protocols to never cross the phase transition') are stated limitations of the quantitative layer, not evidence that the derivation is circular. Self-citations to [4,38] provide the modeling and response-theory starting point, but the load-bearing response relations are rederived in this paper, so the self-citations are not load-bearing reductions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

All quantitative predictions are parameter-free once the model is specified: the model constants (lambda_phi, lambda_n, b, kappa, mu, T, Delta-mu, V, a0, a_tau, phi_0) are inputs, and the response functions m(a), the steady-state power <P>_s, and boundary terms are derived rather than fitted. The main caveats are the modeling assumptions listed above, especially the a posteriori constraint on optimal protocols in the supplement.

assumptions (7)

standard math Standard stochastic thermodynamics: heat, work, and energy are defined through path weights and Stratonovich integrals, with the first law Qt = Wext + Wact + E(0) - E(tau+tau_r).
Invoked in Methods to derive Qp and Qt; a well-established framework, not introduced for this paper.
domain assumption The active system is described by a thermodynamically consistent field theory with a constant chemical potential difference Delta-mu and a symmetric Onsager matrix L (Eqs. 3-6); the active work is Wact = integral Delta-mu n-dot.
This restricts the class of active theories considered and is the basis for the explicit forms of Qp and Qt.
domain assumption Protocols are smooth and slow enough that DRT applies, Omega tau >> 1, with the total change a(tau)-a(0) allowed to be large.
Stated before Eq. (12); used to expand non-stationary averages in powers of a-dot; breaks down at small Omega tau and at large Delta-mu and V.
domain assumption The post-protocol relaxation time tau_r is taken far larger than the slowest relaxation mode, so exponentially decaying terms in the relaxation contribution are dropped.
Used in SM Sec. III.D to cancel the boundary term and obtain the compact Qt formula; for conserved fields Omega^{-1} grows with volume, making this a nontrivial choice.
domain assumption Weak-noise, small-T expansion around a homogeneous state with Gaussian Fourier modes; Isserlis theorem is used for higher correlation functions.
Used for all analytic CMB results in SM Secs. III-IV; limits applicability to near-homogeneous configurations.
ad hoc to paper For Chemical Model B the cross-coupling is taken as C = gamma partial_x phi (Eq. 16).
Chosen as the simplest gradient coupling; it makes phi statistically equilibrium-like, which is an illustrative assumption.
ad hoc to paper Optimal protocols are altered a posteriori so they never cross the phase transition line, because the small-noise homogeneous expansion is invalid there.
SM Sec. IV.E; the reported optimal protocol is not the unconstrained minimizer of the DRT Lagrangian in that region.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Control of active field theories at minimal dissipation." pith.science (2026). https://pith.science/paper/QFDRRWI5

@misc{pith2026250419285,
  author       = {Pith},
  title        = {Pith review of: Control of active field theories at minimal dissipation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QFDRRWI5}},
  note         = {Machine review of arXiv:2504.19285}
}

read the original abstract

Advances in experimental techniques enable the precise manipulation of a large variety of active systems, which constantly dissipate energy to sustain nonequilibrium phenomena without any equilibrium equivalent. To design novel materials out of active systems, an outstanding challenge is to rationalize how material properties can be optimally controlled by applying external perturbations. However, equilibrium thermodynamics is inadequate to guide the control of such nonequilibrium systems. Therefore, there is a dire need for a novel framework to provide a systematic toolbox for the thermodynamic control of active matter. Here, we build an optimization procedure for generic active field theories within a thermodynamically consistent formulation. Central to our approach is the distinction between the protocol heat, which is dissipated only during manipulation, and the total heat, which also accounts for the post-manipulation dissipation. We demonstrate that the latter generically features a global minimum with respect to the protocol duration. We deploy our versatile approach to an active theory of phase separation, and examine the scalings of the optimal protocol duration with respect to activity and system size. Remarkably, we reveal that the landscape of steady-state dissipation regulates the crossover between optimal control strategies for a finite duration.

Figures

Figures reproduced from arXiv: 2504.19285 by the authors.

**Figure 2.** FIG. 2. (a) Phase diagram of Chemical Model B (CMB) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Protocol heat [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Extremal protocol durations ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. (a) Optimal protocol [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 1.** Figure 1: FIG. 1. Plots of the external work [PITH_FULL_IMAGE:figures/full_fig_p029_1.png]

**Figure 2.** Figure 2: FIG. 2. Plots comparing the volume and activity scalings of th [PITH_FULL_IMAGE:figures/full_fig_p030_2.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Topology of pulsating active matter: Defect asymmetry controls emergent motility
cond-mat.soft 2026-05 unverdicted novelty 7.0 of 10

Defect motility in pulsating active matter arises from a ratchet effect of asymmetric rotating cores, regulating the spiral-to-fiber wave crossover.
Optimal active engines obey the thermodynamic Lorentz force law
cond-mat.stat-mech 2025-12 conditional novelty 6.0 of 10

Finite-time work extraction from nonconservative active engines equals a Lorentz-force Lagrangian on thermodynamic state space, so optimal protocols are counterdiabatic paths satisfying a thermodynamic Lorentz force law.

Reference graph

Works this paper leans on

85 extracted references · 59 canonical work pages · cited by 2 Pith papers

[1]

in terms of steady-state averages [ 38]. For a given protocol a(t), the standard chain rule, valid within the Stratonovitch convention [ 59], yields ∫ V ¶F ¶ϕ ˙ϕ dr = ˙F − ˙a∂aF , (27) from which we deduce Qt = Wext + Wact + E(t = 0) − E(t = Ä + Är) , (28) where we have used E = ïF ð, along with the deﬁnitions of the external work Wext [Eq. ( 8)] and the ...
[2]

Jarzynski, Equalities and inequalities: Irreversibilit y and the second law of thermodynamics at the nanoscale, Annu

C. Jarzynski, Equalities and inequalities: Irreversibilit y and the second law of thermodynamics at the nanoscale, Annu. Rev. Condens. Matter Phys. 2, 329 (2011)

2011
[3]

M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrody- namics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013)

2013
[4]

Bechinger, R

C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016)

2016
[5]

L. K. Davis, K. Proesmans, and ´E. Fodor, Active Matter under Control: Insights from Response Theory, Phys. Rev. X 14, 11012 (2024)

2024
[6]

Guillamat, J

P. Guillamat, J. Ign´ es-Mullol, and F. Sagu´ es, Control of active liquid crystals with a magnetic ﬁeld, Proc. Natl. Acad. Sci. U.S.A. 113, 5498 (2016)

work page 2016
[7]

Matsunaga, J

D. Matsunaga, J. K. Hamilton, F. Meng, N. Bukin, E. L. Martin, F. Y. Ogrin, J. M. Yeomans, and R. Golestanian, Controlling collective rotational patterns of magnetic ro- tors, Nat. Commun. 10, 1 (2019)

work page 2019
[8]

Palacci, S

J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, Living crystals of light-activated colloidal surfers, Science 339, 936 (2013)

2013

Show all 85 references

[9]

Y. Shin, J. Berry, N. Pannucci, M. P. Haataja, J. E. Toettcher, and C. P. Brangwynne, Spatiotemporal Con- trol of Intracellular Phase Transitions Using Light- Activated optoDroplets, Cell 168, 159 (2017)

2017
[10]

Frangipane, D

G. Frangipane, D. Dell’Arciprete, S. Petracchini, C. Maggi, F. Saglimbeni, S. Bianchi, G. Vizsnyiczai, M. L. Bernardini, and R. Di Leonardo, Dynamic density shaping of photokinetic E. coli , eLife 7, e36608 (2018)

2018
[11]

Zhang, S

R. Zhang, S. A. Redford, P. V. Ruijgrok, N. Kumar, A. Mozaﬀari, S. Zemsky, A. R. Dinner, V. Vitelli, Z. Bryant, M. L. Gardel, and J. J. de Pablo, Spatiotem- poral control of liquid crystal structure and dynamics through activity patterning, Nat. Mater. 20, 875 (2021)

2021
[12]

Nishiyama, J

K. Nishiyama, J. Berezney, M. M. Norton, A. Aggar- wal, S. Ghosh, M. F. Hagan, Z. Dogic, and S. Fraden, Closed-loop control of active nematic ﬂows (2024), arXiv:2408.14414 [cond-mat.soft]

2024
[13]

V´ elez-Ceron, R

I. V´ elez-Ceron, R. C. V. Coelho, P. Guillamat, M. T. da Gama, F. Sagu´ es, and J. Ign´ es-Mullol,Active nematic pumps (2024), arXiv:2407.09960 [cond-mat.soft]

2024 arXiv
[14]

Seifert, Stochastic thermodynamics, ﬂuctuation the- orems and molecular machines, Rep

U. Seifert, Stochastic thermodynamics, ﬂuctuation the- orems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012)

2012
[15]

Schmiedl and U

T. Schmiedl and U. Seifert, Optimal ﬁnite-time pro- cesses in stochastic thermodynamics, Phys. Rev. Lett. 98, 108301 (2007)

2007
[16]

Blaber and D

S. Blaber and D. A. Sivak, Optimal control in stochastic thermodynamics, J. Phys. Commun. 7, 033001 (2023)

2023
[17]

S. A. Loos, S. Monter, F. Ginot, and C. Bechinger, Uni- versal Symmetry of Optimal Control at the Microscale, Phys. Rev. X 14, 21032 (2024)

2024
[18]

G. M. Rotskoﬀ and G. E. Crooks, Optimal control in nonequilibrium systems: Dynamic riemannian geometry of the ising model, Phys. Rev. E 92, 060102 (2015)

2015
[19]

T. R. Gingrich, G. M. Rotskoﬀ, G. E. Crooks, and P. L. Geissler, Near-optimal protocols in complex nonequilib- rium transformations, Proc. Natl. Acad. Sci. USA 113, 10263 (2016)

2016
[20]

G. M. Rotskoﬀ, G. E. Crooks, and E. Vanden-Eijnden, Geometric approach to optimal nonequilibrium control: Minimizing dissipation in nanomagnetic spin systems, Phys. Rev. E 95, 012148 (2017)

2017
[21]

Alvarado, E

J. Alvarado, E. Teich, D. Sivak, and J. Bechhoe- fer, Optimal control in soft and active matter (2025), arXiv:2504.08676 [cond-mat.soft]

2025 arXiv
[22]

Schneider and H

E. Schneider and H. Stark, Optimal steering of a smart active particle, EPL 127, 64003 (2019)

2019
[23]

Liebchen and H

B. Liebchen and H. L¨ owen, Optimal navigation strategies for active particles, EPL 127, 34003 (2019)

2019
[24]

L. Piro, E. Tang, and R. Golestanian, Optimal navigation strategies for microswimmers on curved manifolds, Phys. Rev. Res. 3, 023125 (2021)

2021
[25]

Sinha, V

S. Sinha, V. Krishnan, and L. Mahadevan, Optimal con- trol of interacting active particles on complex landscapes 11 (2023), arXiv:2311.17039 [cond-mat.soft]

2023 arXiv
[26]

Baldovin, D

M. Baldovin, D. Gu´ ery-Odelin, and E. Trizac, Control of active brownian particles: An exact solution, Phys. Rev. Lett. 131, 118302 (2023)

2023
[27]

Garcia-Millan, J

R. Garcia-Millan, J. Sch¨ uttler, M. E. Cates, and S. A. M. Loos, Optimal closed-loop control of active particles and a minimal information engine (2024), arXiv:2407.18542 [cond-mat.stat-mech]

2024 arXiv
[28]

Y. Wang, E. Lei, Y.-H. Ma, Z. C. Tu, and G. Li, Ther- modynamic geometric control of active matter (2024), arXiv:2409.09994 [cond-mat.stat-mech]

2024 arXiv
[29]

Sch¨ uttler, R

J. Sch¨ uttler, R. Garcia-Millan, M. E. Cates, and S. A. M. Loos, Active particles in moving traps: minimum work protocols and information eﬃciency of work extraction (2025), arXiv:2501.18613 [cond-mat.stat-mech]

2025 arXiv
[30]

M. J. Falk, V. Alizadehyazdi, H. Jaeger, and A. Murugan, Learning to control active matter, Phys. Rev. Res. 3, 033291 (2021)

2021
[31]

Casert and S

C. Casert and S. Whitelam, Learning protocols for the fast and eﬃcient control of active matter, Nat. Commun. 15, 9128 (2024)

2024
[32]

M. M. Norton, P. Grover, M. F. Hagan, and S. Fraden, Optimal Control of Active Nematics, Phys. Rev. Lett. 125, 178005 (2020)

2020
[33]

Shankar, V

S. Shankar, V. Raju, and L. Mahadevan, Optimal trans- port and control of active drops, Proc. Natl. Acad. Sci. U.S.A. 119, 1 (2022)

2022
[34]

Ghosh, C

S. Ghosh, C. Joshi, A. Baskaran, and M. F. Hagan, Spa- tiotemporal control of structure and dynamics in a polar active ﬂuid, Soft Matter 20, 7059 (2024)

2024
[35]

Ghosh, A

S. Ghosh, A. Baskaran, and M. F. Hagan, Achieving de- signed texture and ﬂows in bulk active nematics using optimal control theory (2024), arXiv:2408.14596 [cond- mat.soft]

2024 arXiv
[36]

Krishnan, S

V. Krishnan, S. Sinha, and L. Mahadevan, Hamiltonian bridge: A physics-driven generative framework for tar- geted pattern control (2024), arXiv:2410.12665 [cond- mat.soft]

2024 arXiv
[37]

P. M. Chaikin and T. C. Lubensky, Principles of Con- densed Matter Physics (Cambridge University Press, 1995)

1995
[38]

Fodor and M

E. Fodor and M. Cristina Marchetti, The statistical physics of active matter: From self-catalytic colloids to living cells, Physica A 504, 106 (2018)

2018
[39]

Markovich, ´E

T. Markovich, ´E. Fodor, E. Tjhung, and M. E. Cates, Thermodynamics of Active Field Theories: Energetic Cost of Coupling to Reservoirs, Phys. Rev. X 11, 21057 (2021)

2021
[40]

M. E. Cates and J. Tailleur, Motility-induced phase sepa - ration, Annu. Rev. Condens. Matter Phys. 6, 219 (2015)

2015
[41]

Chat´ e, Dry aligning dilute active matter, Annu

H. Chat´ e, Dry aligning dilute active matter, Annu. Rev. Condens. Matter Phys. 11, 189 (2020)

2020
[42]

Markovich and T

T. Markovich and T. C. Lubensky, Odd viscosity in active matter: Microscopic origin and 3d eﬀects, Phys. Rev. Lett. 127, 048001 (2021)

2021
[43]

Fruchart, C

M. Fruchart, C. Scheibner, and V. Vitelli, Odd viscosity and odd elasticity, Annu. Rev. Condens. Matter Phys. 14, 471 (2023)

2023
[44]

Markovich and T

T. Markovich and T. C. Lubensky, Nonreciprocity and odd viscosity in chiral active ﬂuids, Proc. Natl. Acad. Sci. U.S.A 121, e2219385121 (2024)

2024
[45]

Pietzonka and U

P. Pietzonka and U. Seifert, Entropy production of active particles and for particles in active baths, J. Phys. A Math. Theor. 51, 01LT01 (2018)

2018
[46]

Gaspard and R

P. Gaspard and R. Kapral, Thermodynamics and statis- tical mechanics of chemically powered synthetic nanomo- tors, Adv. Phys. X 4, 1602480 (2019)

2019
[47]

Datta, P

A. Datta, P. Pietzonka, and A. C. Barato, Second Law for Active Heat Engines, Phys. Rev. X 12, 31034 (2022)

2022
[48]

Chatzittoﬁ, J

M. Chatzittoﬁ, J. Agudo-Canalejo, and R. Golestanian, Entropy production and thermodynamic inference for stochastic microswimmers, Phys. Rev. Res. 6, 1 (2024)

2024
[49]

Bebon, J

R. Bebon, J. F. Robinson, and T. Speck, Thermodynam- ics of active matter: Tracking dissipation across scales (2024), arXiv:2401.02252

2024 arXiv
[50]

Agranov, R

T. Agranov, R. L. Jack, M. E. Cates, and ´E. Fodor, Thermodynamically consistent ﬂocking: from discontin- uous to continuous transitions, New J. Phys. 26, 063006 (2024)

2024
[51]

Sorkin, H

B. Sorkin, H. Diamant, G. Ariel, and T. Markovich, Sec- ond law of thermodynamics without einstein relation, Phys. Rev. Lett. 133, 267101 (2024)

2024
[52]

Fodor, R

E. Fodor, R. L. Jack, and M. E. Cates, Irreversibility and Biased Ensembles in Active Matter: Insights from Stochastic Thermodynamics, Annu. Rev. Condens. Mat- ter Phys. 13, 215 (2022)

2022
[53]

Aslyamov, F

T. Aslyamov, F. Avanzini, E. Fodor, and M. Esposito, Nonideal reaction-diﬀusion systems: Multiple routes to instability, Phys. Rev. Lett. 131, 138301 (2023)

2023
[54]

Falasco and M

G. Falasco and M. Esposito, Macroscopic stochastic ther- modynamics, Rev. Mod. Phys. 97, 015002 (2025)

2025
[55]

D. A. Sivak and G. E. Crooks, Thermodynamic metrics and optimal paths, Phys. Rev. Lett. 108, 190602 (2012)

2012
[56]

Hatano and S.-i

T. Hatano and S.-i. Sasa, Steady-state thermodynamics of langevin systems, Phys. Rev. Lett. 86, 3463 (2001)

2001
[57]

Gupta, S

D. Gupta, S. H. L. Klapp, and D. A. Sivak, Eﬃcient con- trol protocols for an active ornstein-uhlenbeck particle, Phys. Rev. E 108, 024117 (2023)

2023
[58]

S. R. D. Groot and P. Mazur, Non-Equilibrium Thermo- dynamics (North-Holland Publishing Company, 1962)

1962
[59]

For instance, a ﬂux of photons has a dif- ferent time-reversal signature than chemical driving [ 38]

What is important is the time-reversal signature of the active driving. For instance, a ﬂux of photons has a dif- ferent time-reversal signature than chemical driving [ 38]
[60]

C. W. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences (Springer, 2009)

2009
[61]

M. E. Cates, ´E. Fodor, T. Markovich, C. Nardini, and E. Tjhung, Stochastic Hydrodynamics of Complex Flu- ids: Discretisation and Entropy Production, Entropy 24, 1 (2022)

2022
[62]

Supplemental material
[63]

A. Y. Grosberg and J. F. Joanny, Nonequilibrium statis- tical mechanics of mixtures of particles in contact with diﬀerent thermostats, Phys. Rev. E 92, 1 (2015)

2015
[64]

Y. I. Li, R. Garcia-Millan, M. E. Cates, and ´Etienne Fodor, Towards a liquid-state theory for active matter, EPL 142, 57004 (2023)

2023
[65]

O. Azzaroni, Polymer brushes here, there, and every- where: Recent advances in their practical applications and emerging opportunities in multiple research ﬁelds, Journal of Polymer Science Part A: Polymer Chemistry 50, 3225 (2012)

2012
[66]

Nardini, ´E

C. Nardini, ´E. Fodor, E. Tjhung, F. van Wijland, J. Tailleur, and M. E. Cates, Entropy production in ﬁeld theories without time-reversal symmetry: Quantify- ing the non-equilibrium character of active matter, Phys. Rev. X 7, 1 (2017) . 12

2017
[67]

A. A. Hyman, C. A. Weber, and F. J¨ ulicher, Liquid-liquid phase separation in biology, Annu. Rev. Cell Dev. Biol. 30, 39 (2014)

2014
[68]

C. A. Weber, D. Zwicker, F. J¨ ulicher, and C. F. Lee, Physics of active emulsions, Rep. Prog. Phys. 82, 064601 (2019)

2019
[69]

https://github.com/artursoriani/control-of-active-ﬁeld- theories-at-minimal-dissipation
[70]

Benamou and Y

J.-D. Benamou and Y. Brenier, A computational ﬂuid mechanics solution to the monge-kantorovich mass trans- fer problem, Numer. Math. 84, 375 (2000)

2000
[71]

Aurell, C

E. Aurell, C. Mej ´ ıa-Monasterio, and P. Muratore- Ginanneschi, Optimal protocols and optimal transport in stochastic thermodynamics, Phys. Rev. Lett. 106, 250601 (2011)

2011
[72]

Ito and A

S. Ito and A. Dechant, Stochastic time evolution, infor- mation geometry, and the cram´ er-rao bound, Phys. Rev. X 10, 021056 (2020)

2020
[73]

Van Vu and K

T. Van Vu and K. Saito, Thermodynamic uniﬁcation of optimal transport: Thermodynamic uncertainty relation, minimum dissipation, and thermodynamic speed limits, Phys. Rev. X 13, 011013 (2023)

2023
[74]

Chennakesavalu and G

S. Chennakesavalu and G. M. Rotskoﬀ, Uniﬁed, geomet- ric framework for nonequilibrium protocol optimization, Phys. Rev. Lett. 130, 107101 (2023)

2023
[75]

Pietzonka and U

P. Pietzonka and U. Seifert, Universal trade-oﬀ between power, eﬃciency, and constancy in steady-state heat en- gines, Phys. Rev. Lett. 120, 190602 (2018)

2018
[76]

A. G. Frim and M. R. Deweese, Optimal ﬁnite-time Brownian Carnot engine, Phys. Rev. E 105, 1 (2022)

2022
[77]

A. G. Frim and M. R. Deweese, Geometric Bound on the Eﬃciency of Irreversible Thermodynamic Cycles, Phys. Rev. Lett. 128, 230601 (2022)

2022
[78]

A. W. C. Lau and T. C. Lubensky, State-dependent dif- fusion: Thermodynamic consistency and its path integral formulation, Phys. Rev. E 76, 011123 (2007)

2007
[79]

Sorkin, G

B. Sorkin, G. Ariel, and T. Markovich, Consistent ex- pansion of the langevin propagator with application to entropy production, J. Stat. Mech. 2025, 013208 (2025)

2025
[80]

M. V. S. Bonan¸ ca and S. Deﬀner, Optimal driving of isothermal processes close to equilibrium, J. Chem. Phys. 140, 244119 (2014)

2014
[81]

Control of active ﬁeld theories at minimal dissipation

R. Kubo, M. Toda, and N. Hashitsume, Statisti- cal Physics II: Nonequilibrium Statistical Mechanics (Springer Berlin, 1991). Supplemental Material for “Control of active ﬁeld theories at minimal dissipation” Artur Soriani, 1, 2 Elsen Tjhung, 3 ´Etienne Fodor, 4 and Tomer Marko...

1991
[82]

L. K. Davis, K. Proesmans, and ´E. Fodor, Active Matter under Control: Insights from Response Th eory, Phys. Rev. X 14, 11012 (2024)

2024
[83]

https://github.com/artursoriani/control-of-active-ﬁeld-theories-at-minimal-dissipation
[84]

Markovich, ´E

T. Markovich, ´E. Fodor, E. Tjhung, and M. E. Cates, Thermodynamics of Active Fi eld Theories: Energetic Cost of Coupling to Reservoirs, Phys. Rev. X 11, 21057 (2021)

2021
[85]

M. E. Cates, ´E. Fodor, T. Markovich, C. Nardini, and E. Tjhung, Stochastic Hyd rodynamics of Complex Fluids: Discreti- sation and Entropy Production, Entropy 24, 1 (2022)

2022

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

in terms of steady-state averages [ 38]. For a given protocol a(t), the standard chain rule, valid within the Stratonovitch convention [ 59], yields ∫ V ¶F ¶ϕ ˙ϕ dr = ˙F − ˙a∂aF , (27) from which we deduce Qt = Wext + Wact + E(t = 0) − E(t = Ä + Är) , (28) where we have used E = ïF ð, along with the deﬁnitions of the external work Wext [Eq. ( 8)] and the ...

[2] [2]

Jarzynski, Equalities and inequalities: Irreversibilit y and the second law of thermodynamics at the nanoscale, Annu

C. Jarzynski, Equalities and inequalities: Irreversibilit y and the second law of thermodynamics at the nanoscale, Annu. Rev. Condens. Matter Phys. 2, 329 (2011)

2011

[3] [3]

M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrody- namics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013)

2013

[4] [4]

Bechinger, R

C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016)

2016

[5] [5]

L. K. Davis, K. Proesmans, and ´E. Fodor, Active Matter under Control: Insights from Response Theory, Phys. Rev. X 14, 11012 (2024)

2024

[6] [6]

Guillamat, J

P. Guillamat, J. Ign´ es-Mullol, and F. Sagu´ es, Control of active liquid crystals with a magnetic ﬁeld, Proc. Natl. Acad. Sci. U.S.A. 113, 5498 (2016)

work page 2016

[7] [7]

Matsunaga, J

D. Matsunaga, J. K. Hamilton, F. Meng, N. Bukin, E. L. Martin, F. Y. Ogrin, J. M. Yeomans, and R. Golestanian, Controlling collective rotational patterns of magnetic ro- tors, Nat. Commun. 10, 1 (2019)

work page 2019

[8] [8]

Palacci, S

J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, Living crystals of light-activated colloidal surfers, Science 339, 936 (2013)

2013