Pith. sign in

REVIEW 3 major objections 3 minor 49 references

Optimal Finite-Time Control of Nonreciprocal Brownian Dimers: Thermodynamic Anomaly and Multiple Transitions

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read Nonreciprocally interacting Brownian dimers have a critical transport duration: before it, an oscillatory protocol minimizes mean work; after it, the unregularized optimal work is unbounded below, with extractable work and power becoming in

desk verdict The exact nonreciprocal dimer control solution is a real, clean new result, but the unbounded-work anomaly is a model artifact unless the cost of maintaining nonreciprocity is accounted for — and that accounting is deferred to the missing SM. read the letter →

arxiv 2607.20420 v1 pith:DIXPT4ZK submitted 2026-07-22 cond-mat.stat-mech cond-mat.softphysics.bio-ph

Ruicheng Bao This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.softphysics.bio-ph
keywords stochasticthermodynamicsoptimalcontrolnonreciprocalinteractionsBrowniandimersworkextractionfinite-timeprotocolsthermodynamicanomalyopticaltweezers
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

This paper solves exactly the finite-time optimal-control problem for two overdamped Brownian particles held in independently movable harmonic traps and coupled by a nonreciprocal interaction. It shows that the optimal protocol is oscillatory and deliberately uses the extension channel even when only the center is to be moved, because the nonreciprocal interaction contributes an area term that can do work. The central claim is a thermodynamic anomaly: past a finite critical duration Tc, the mean-work functional is unbounded below, so both extractable work and output power are infinite unless the model is regularized. Adding a physical constraint—finite trap range or saturating active force—restores a finite optimum and turns the anomaly into well-defined transitions: a first-order-like jump for a hard range and a second-order-like continuous onset for smooth saturation. With a finite range, the optimal protocol can switch repeatedly as duration grows, producing multiple work-duration kinks.

What carries the argument

The central object is the mean-work functional W[y,s]=∫₀ᵀ(2ẏ²−2αsẏ+½ṡ²)dt plus boundary terms, with the nonreciprocal contribution −2α∫s ẏ dt coupling center and extension channels. This area term is what allows extension motion to help or oppose center transport. The Euler–Lagrange equations reduce to the oscillator v̈+α²v=0, and the stability of the quadratic form is governed by the determinant Δα(T,ρ)=2α²cos(αT)+α(ρ+1)sin(αT)+ρ[1−cos(αT)]; Tc is its first positive zero, the first conjugate time where the second variation loses positive definiteness.

What would settle it

Numerically minimize the work functional W[y,s]=∫₀ᵀ(2ẏ²−2αsẏ+½ṡ²)dt with no bound on s and a purely linear force: for T just above the first positive zero of Δα(T,ρ), the minimum should go to −∞ as the endpoint jumps A and B are scaled up along the negative eigenmode, and finite-difference solutions should show no finite minimizer. If a finite minimum or bounded jumps are observed, the claimed anomaly would be contradicted.

Watch

Extended reading notes

Core claim

The paper derives a closed-form mean-work functional for the two-trap dimer and shows that for nonreciprocal coupling α≠0 the center velocity obeys a harmonic oscillator v̈ + α²v = 0, with optimal mean paths given by sinusoids. The minimal work is an explicit quadratic form in the prescribed center displacement L and excess-separation change M. The central discovery is that this quadratic form ceases to be positive definite at the first conjugate time Tc: for T<Tc the explicit sinusoid protocol is the unique global minimizer; at Tc the endpoint jumps diverge as (Tc−T)^{-1}; for T>Tc there is a negative direction, so the infimum of the external work is −∞. The paper then shows that imposing a

Load-bearing premise

The anomaly rests on modeling the nonreciprocal interaction as an unbounded, cost-free linear force αs; if that force saturates or its maintenance energy is charged, the unbounded-work infimum disappears, as the paper's own regularized versions demonstrate.

Editorial extensions

If this is right

  • For durations shorter than Tc, the explicit sinusoid protocol is the unique global work minimizer, so any alternative two-trap protocol costs more mean work.
  • For durations longer than Tc, the unregularized mean-work infimum is −∞, making the raw problem ill-posed and forcing any physically meaningful treatment to include a cutoff or saturation.
  • A hard finite trap range converts the anomaly into a first-order-like jump from the zero protocol to the maximum-range protocol in the zero-target case, while smooth force saturation gives a continuous, second-order-like onset with the order parameter growing as (T−Tc)^{1/2}.
  • Under finite-range constraints, successive optimal strategies (e.g., one-wall versus two-wall) can alternate as duration increases, producing multiple kinks in the work-duration curve that have no single-particle analog.
  • The long-time output power of the regularized optimal protocol approaches the plateau α²Rmax²/(2ρ²), with possible finite-time local shoulders or peaks.
  • Following the paper's claims, nonreciprocal control can extract finite negative work even before Tc, and the reciprocal limit recovers the familiar one-particle and free-particle results.

Reading between the lines

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

  • The divergence of the endpoint jumps as Tc is approached from below is best read as a diagnostic that the linear nonreciprocal force is a coarse-grained model; any real system will saturate or pay an energy cost, so the observable content of the anomaly is the regularized transition rather than the unregularized infinity.
  • The jump vector in the (Y, R/2) plane rotates by angle αT during the protocol, suggesting a geometric-phase interpretation of nonreciprocal optimal control and a possible link to area-pumping or topological effects in active-matter control.
  • The multiple-kink phenomenon should be generic whenever two competing strategies—area harvesting versus drag dissipation—are in balance under a control bound; analogous finite-time transitions may appear in discrete-state, quantum, or multi-particle optimal-control problems.
  • A dual-trap experiment with a nonreciprocally coupled pair, such as two optical traps around a motor-driven filament or optically coupled nanoparticles, could test for the predicted oscillatory extension excursions during pure center translation and for the sharp growth of control amplitudes near Tc.
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.

Referee Report

3 major / 3 minor

Summary. The paper solves exactly a finite-time mean-work optimal control problem for two overdamped Brownian particles held by two independently movable harmonic traps, with either reciprocal or nonreciprocal pairwise interactions. The controls are eliminated in favor of the mean center and separation dynamics, yielding Euler-Lagrange equations, closed-form oscillatory optimal protocols, and an explicit quadratic work formula. For reciprocal interactions the solution reduces to known single-particle transport plus a finite-time separation cost. For nonreciprocal interactions, the central claim is that beyond a critical duration Tc the work functional acquires a negative mode, so the unregularized infimum of the external work is -infinity and extractable work/output power are unbounded. Finite trap range or force saturation regularize this divergence and, in the zero-target case, convert it into first-order-like or second-order-like optimal-protocol transitions, including multiple duration kinks. The paper also discusses experimental relevance to dual-trap actomyosin and levitated nanoparticle platforms.

Significance. If the central claims hold, this is a rare exactly solvable finite-time optimal-control problem for an interacting, genuinely nonequilibrium system. The closed-form solution, the clean reduction of the reciprocal case to prior single-particle results, and the explicit regularized bounds are strengths. The paper appears internally consistent in its variational derivation: the Euler-Lagrange equations, boundary conditions, and the determinant condition leading to Tc are mutually consistent, and there is no evident circularity or fitting. The main significance is therefore twofold: it provides a benchmark for interacting stochastic-thermodynamic control, and it identifies a sharp divergence and subsequent transitions that could, in principle, be probed in dual-trap or levitated-nanoparticle experiments. However, the headline 'thermodynamic anomaly' depends on treating the nonreciprocal force as cost-free; the physical interpretation is not yet established because the energy-input regularization is only mentioned and deferred.

major comments (3)
  1. [Thermodynamic anomaly and its restoration, Eqs. (19)-(20), Fig. 3] The central unboundedness conclusion is asserted rather than proven in the reviewed text: the positive-definiteness of the second variation for all admissible perturbations, the divergence of A,B as (Tc-T)^-1, and the resulting global infimum -infinity for T>Tc are all assigned to SM Sec. S6, which is not present. Since claiming unbounded external work and output power is the paper's headline result, please provide these derivations in the main text or a complete appendix. A negative direction establishes local unboundedness along one mode, but the global variational statement, including endpoint-jump perturbations, requires the missing argument.
  2. [Thermodynamic anomaly and its restoration; Discussion] The anomaly is generated by Eq. (7)'s term -2 alpha integral s ydot dt, in which the nonreciprocal force alpha s is cost-free. The manuscript itself notes that incorporating the energy input that maintains nonreciprocity is a possible regularization, but defers this to SM Sec. S7. This is load-bearing for the title's 'Thermodynamic Anomaly': if that energy cost is nonzero, the negative mode may be stabilized and the unbounded-work divergence may disappear. The finite-range bound (20) and the saturating-force prescription are external constraints or model changes, not thermodynamic bookkeeping. Please either include an energy-accounting calculation or revise the claims to state explicitly that the anomaly is a property of the cost-free linear active-force model, not an established physical prediction.
  3. [First- and second-order-like protocol transitions and multiple kinks, Fig. 4] The regularized-transition results -- hard-cutoff first-order-like jump, smooth-saturation scaling ||s*||_inf/s0 proportional to (T-Tc)^{1/2}, and the one-wall/two-wall switch producing a second kink -- are presented graphically or by assertion, with derivations deferred to SM Sec. S9, which is not included. Since these transitions are advertised as principal findings, the equations governing the regularized optima and the phase-boundary calculations should be shown, at least in an appendix, rather than only cited as SM.
minor comments (3)
  1. [Model, Eqs. (1)-(5)] The notation a,b and alpha,rho is mostly clear, but the stable regime rho=1+a+b>0 and the restriction rho>=1 used throughout should be stated more prominently; the reader must infer that the nonreciprocity parameters are dimensionless and that the extension-restoring condition is assumed from the start.
  2. [References, Ref. [31]] The Supplemental Material reference contains unresolved placeholders ('Refs. [? ? ?]'). This should be fixed, and the SM sections cited in the main text should be listed explicitly.
  3. [End Matter, Eq. (24)] The statement that the inequality W_nr*(M=0) <= 2L^2/(T+2) strengthens monotonically with |alpha| is plausible but only sketched via the argument W0 - 2|alpha||integral s ydot dt|. A short derivation or a reference to the SM would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: exact variational derivation from stated model; the thermodynamic anomaly is a mathematical consequence of the quadratic work functional, not an imposed prediction.

full rationale

The paper's derivation chain is self-contained. The mean-work functional (Eq. 7) is obtained by eliminating the controls in favor of mean paths using the stated dynamics (Eq. 5); the Euler-Lagrange equations (Eq. 14) and the oscillator equation (Eq. 15) follow from minimizing that functional with fixed endpoints and natural terminal conditions. The optimal solution (Eq. 16), the linear system for the jump constants (Eq. 17), the determinant (Eq. 19), and the critical time Tc are all derived from the same variational problem. The unbounded infimum for T>Tc is identified by analysis of the second variation and the determinant, not assumed. The subsequent regularizations (finite range, force saturation) are additional constraints applied to the same model, and the paper explicitly presents the anomaly as a diagnostic of the unregularized model. No fitted parameters are relabeled as predictions, no self-citation is load-bearing, and no existing result is merely renamed. The only notable limitation is that the cost of maintaining nonreciprocity is deferred to SM S7 and not computed, which bears on physical interpretation rather than circularity of the derivation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivation uses no fitted parameters: α, ρ, L, M, T are inputs. The only hand-chosen quantities are the regularizer scales (Rmax, s0, smax) used to restore boundedness; the exact unregularized anomaly follows from the stated linear model. No new particles, fields, or conserved quantities are introduced.

free parameters (3)
  • Rmax (finite trap-range cutoff)
    Hard constraint |R(t)|≤Rmax imposes the reachable state bound |s|≤Rmax/ρ and makes the regularized optimum finite; the value is chosen by hand and sets the scale of the finite work and the location of protocol transitions.
  • s0 (force-saturation scale)
    Saturating force g_sat(s)=α s0 tanh(s/s0) is introduced ad hoc to restore a finite optimum; the transition order parameter is normalized by s0.
  • smax (hard state cutoff)
    Hard cutoff on the mean extension used in the zero-target regularized calculation; an ad hoc regularizer analogous to Rmax.
assumptions (5)
  • domain assumption The mean particle dynamics are exactly the linear equations (5): ẏ=-(y-Y)+αs, ṡ=-ρs+R.
    Averaging Eq. (1) is exact because the noise and forces are linear, but the model itself (linear nonreciprocal forces, unit friction/stiffness/temperature) is the physical postulate.
  • domain assumption The trap controls Y(t) and R(t) may jump at t=0 and t=T, while the mean positions y,s remain continuous.
    This is the standard admissible set in this literature (cf. Schmiedl-Seifert); it creates the boundary terms in Eq. (7).
  • domain assumption The objective is the mean external trap work, Eq. (6), and the nonreciprocal interaction's maintenance cost is excluded.
    This choice is what makes W*_nr able to be negative and unbounded; the paper notes charging that cost as an alternative regularization in SM Sec. S7.
  • domain assumption The system is restricted to the extension-restoring regime ρ≥1.
    This excludes unstable fixed-trap configurations and ensures the boundary term (ρ-1)s_T²/4 is nonnegative.
  • standard math Piecewise-smooth protocols and finite-action mean paths are sufficient for the variational calculus.
    The Euler-Lagrange equations and integration by parts assume sufficient regularity; standard for this problem.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimal Finite-Time Control of Nonreciprocal Brownian Dimers: Thermodynamic Anomaly and Multiple Transitions." pith.science (2026). https://pith.science/paper/DIXPT4ZK

@misc{pith2026260720420,
  author       = {Pith},
  title        = {Pith review of: Optimal Finite-Time Control of Nonreciprocal Brownian Dimers: Thermodynamic Anomaly and Multiple Transitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DIXPT4ZK}},
  note         = {Machine review of arXiv:2607.20420}
}
abstract

We solve exactly a finite-time thermodynamic optimal control problem for two nonreciprocally interacting Brownian particles translated by two harmonic traps. The controller manipulates both the center and separation of the pair. Nonreciprocal interactions generate an internal active force that couples these two channels. The optimal protocol is oscillatory, deliberately opens the dimer even when the target separation is unchanged, and can extract work during transport. A central finding is a finite critical time beyond which the external-work infimum is $-\infty$: at any prescribed duration beyond this threshold, both extractable work and output power are unbounded. Physical regularizations such as finite trap range and force saturation restore a finite optimum and convert the anomaly into optimal-protocol transitions: in the zero-target case, a hard finite range produces a first-order-like jump from the zero protocol to a maximum-range protocol, whereas smooth force saturation gives a continuous, second-order-like onset. Under finite-range constraints, the optimal protocol can further undergo multiple finite-time transitions, producing multiple work-duration kinks with no qualitative analog in prior studies.

Figures

Figures reproduced from arXiv: 2607.20420 by the authors.

Figure 1
Figure 1. FIG. 1. Two overdamped Brownian particles are held by in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Rigid target translation with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Minimum trap-work landscape for [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Physical regularizations of the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 3 linked inside Pith

  1. [1]

    Seifert, Stochastic thermodynamics, fluctuation the- orems and molecular machines, Reports on Progress in Physics75, 126001 (2012)

    U. Seifert, Stochastic thermodynamics, fluctuation the- orems and molecular machines, Reports on Progress in Physics75, 126001 (2012)

    2012

  2. [2]

    Peliti and S

    L. Peliti and S. Pigolotti,Stochastic Thermodynamics: An Introduction(Princeton University Press, 2021)

    2021

  3. [3]

    Cao and S

    Y. Cao and S. Liang, Stochastic thermodynamics for bi- ological functions, Quantitative Biology13, e75 (2025)

    2025

  4. [4]

    Schmiedl and U

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

    2007

  5. [5]

    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

  6. [6]

    Blaber and D

    S. Blaber and D. A. Sivak, Optimal control in stochastic thermodynamics, Journal of Physics Communications7, 033001 (2023)

    2023

  7. [7]

    Bustamante, J

    C. Bustamante, J. Liphardt, and F. Ritort, The nonequi- librium thermodynamics of small systems, Physics Today 58, 43 (2005)

    2005

  8. [8]

    Ciliberto, Experiments in stochastic thermodynamics: Short history and perspectives, Physical Review X7, 021051 (2017)

    S. Ciliberto, Experiments in stochastic thermodynamics: Short history and perspectives, Physical Review X7, 021051 (2017)

    2017

Show all 49 references
  1. [9]

    Gomez-Marin, T

    A. Gomez-Marin, T. Schmiedl, and U. Seifert, Optimal protocols for minimal work processes in underdamped stochastic thermodynamics, The Journal of chemical physics129(2008)

    2008

  2. [10]

    Schmiedl and U

    T. Schmiedl and U. Seifert, Efficiency at maximum power: An analytically solvable model for stochastic heat engines, EPL (Europhysics Letters)81, 20003 (2008)

    2008

  3. [11]

    K. S. Olsen, R. Goerlich, Y. Roichman, and H. L¨ owen, Harnessing non-equilibrium forces to optimize work ex- traction, Nature Communications16, 11031 (2025)

    2025

  4. [12]

    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, Phys. Rev. Lett.135, 088301 (2025)

    2025

  5. [13]

    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, Phys. Rev. E112, 024119 (2025)

    2025

  6. [14]

    Meibohm, S

    J. Meibohm, S. Monter, S. A. Loos, and C. Bechinger, Finite-time transitions in optimal control and non- equilibrium relaxation, arXiv preprint arXiv:2604.25798 (2026)

    2026 arXiv

  7. [15]

    S. A. M. Loos, S. Monter, F. Ginot, and C. Bechinger, Universal symmetry of optimal control at the microscale, Phys. Rev. X14, 021032 (2024)

    2024

  8. [16]

    Monter, S

    S. Monter, S. A. Loos, and C. Bechinger, Optimal transi- tions between nonequilibrium steady states, Proceedings of the National Academy of Sciences122, e2510654122 (2025)

    2025

  9. [17]

    Welker and P

    T. Welker and P. Pietzonka, Accuracy comes at a cost: Optimal localisation against a flow, arXiv preprint arXiv:2602.13173 (2026)

    2026

  10. [18]

    L. K. Davis, K. Proesmans, and E. Fodor, Active matter under control: Insights from response theory, Phys. Rev. 6 X14, 011012 (2024)

    2024

  11. [19]

    Monter, L

    S. Monter, L. T. Stutzer, S. A. Loos, and C. Bechinger, Energy-efficient control of interacting microscopic sys- tems: When longer paths save energy, arXiv preprint arXiv:2603.16205 (2026)

    2026

  12. [20]

    A. V. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V. Nosenko, and H. L¨ owen, Statistical mechanics where newton’s third law is broken, Physical Review X5, 011035 (2015)

    2015

  13. [21]

    S. A. M. Loos and S. H. L. Klapp, Irreversibility, heat and information flows induced by non-reciprocal interactions, New Journal of Physics22, 123051 (2020)

    2020

  14. [22]

    Fruchart, R

    M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature592, 363 (2021)

    2021

  15. [23]

    Cocconi, H

    L. Cocconi, H. Alston, and T. Bertrand, Active bound states arising from transiently nonreciprocal pair inter- actions, Phys. Rev. Res.5, 043032 (2023)

    2023

  16. [24]

    Dechant and E

    A. Dechant and E. Lutz, Fundamental limits on nonequi- librium sensing, Nature Communications16, 10227 (2025)

    2025

  17. [25]

    J. T. Finer, R. M. Simmons, and J. A. Spudich, Sin- gle myosin molecule mechanics: piconewton forces and nanometre steps, nature368, 113 (1994)

    1994

  18. [26]

    J. W. Shaevitz, E. A. Abbondanzieri, R. Landick, and S. M. Block, Backtracking by single rna polymerase molecules observed at near-base-pair resolution, Nature 426, 684 (2003)

    2003

  19. [27]

    J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Busta- mante, Recent advances in optical tweezers, Annu. Rev. Biochem.77, 205 (2008)

    2008

  20. [28]

    Ribezzi-Crivellari and F

    M. Ribezzi-Crivellari and F. Ritort, Free-energy infer- ence from partial work measurements in small systems, Proceedings of the National Academy of Sciences111, E3386 (2014)

    2014

  21. [29]

    C. J. Bustamante, Y. R. Chemla, S. Liu, and M. D. Wang, Optical tweezers in single-molecule biophysics, Nature Reviews Methods Primers1, 25 (2021)

    2021

  22. [30]

    Marang, B

    C. Marang, B. Scott, J. Chambers, L. K. Gunther, C. M. Yengo, and E. P. Debold, A mutation in switch i alters the load-dependent kinetics of myosin va, Nature com- munications14, 3137 (2023)

    2023

  23. [31]

    See Supplemental Material at [url] for complementary derivations, which includes Refs. [? ? ?]

  24. [32]

    Define the initial and final jumps as ∆ 0Y=Y(0 +)−Y(0),∆ 0R=R(0 +)−R(0) and ∆T Y=Y(T)−Y(T −),∆ T R=R(T)−R(T −), re- spectively

    It is useful to view the jumps in the two collective control channels as a vector. Define the initial and final jumps as ∆ 0Y=Y(0 +)−Y(0),∆ 0R=R(0 +)−R(0) and ∆T Y=Y(T)−Y(T −),∆ T R=R(T)−R(T −), re- spectively. If j0 = ∆0Y ∆0R/2 = A B ,j T = ∆T Y ∆T R/2 , then the optimality e...

  25. [33]

    In optimal-control terminology, it is the first conjugate time

    Throughout, we refer toT c as the critical duration to emphasize that the total protocol duration is prescribed. In optimal-control terminology, it is the first conjugate time

  26. [34]

    Then the jumps need not diverge exactly at T=T c; the quadratic form is only semidefinite and the minimizer is degenerate

    For nongeneric endpoints, for example the zero-target caseL=M= 0, the prescribed target vector (L, M) can have zero projection onto the singular direction at T=T c. Then the jumps need not diverge exactly at T=T c; the quadratic form is only semidefinite and the minimizer is d...

  27. [35]

    The contribution of the boundary term is non-negative forρ≥1

  28. [36]

    Gardini, S

    L. Gardini, S. M. Heissler, C. Arbore, Y. Yang, J. R. Sell- ers, F. S. Pavone, and M. Capitanio, Dissecting myosin- 5b mechanosensitivity and calcium regulation at the single molecule level, Nature communications9, 2844 (2018)

    2018

  29. [37]

    Rieser, M

    J. Rieser, M. A. Ciampini, H. Rudolph, N. Kiesel, K. Hornberger, B. A. Stickler, M. Aspelmeyer, and U. Deli´ c, Tunable light-induced dipole-dipole interaction between optically levitated nanoparticles, Science377, 987 (2022)

    2022

  30. [38]

    Liˇ ska, T

    V. Liˇ ska, T. Zem´ ankov´ a, P. J´ akl, M.ˇSiler, S. H. Simpson, P. Zem´ anek, and O. Brzobohat` y, Pt-like phase transition and limit cycle oscillations in non-reciprocally coupled optomechanical oscillators levitated in vacuum, Nature Physics20, 1622 (2024)

    2024

  31. [39]

    Reisenbauer, H

    M. Reisenbauer, H. Rudolph, L. Egyed, K. Hornberger, A. V. Zasedatelev, M. Abuzarli, B. A. Stickler, and U. Deli´ c, Non-hermitian dynamics and non-reciprocity of optically coupled nanoparticles, Nature Physics20, 1629 (2024)

    2024

  32. [40]

    A. C. Barato and U. Seifert, Thermodynamic cost of ex- ternal control, New Journal of Physics19, 073021 (2017)

    2017

  33. [41]

    Zhong and M

    A. Zhong and M. R. DeWeese, Limited-control optimal protocols arbitrarily far from equilibrium, Phys. Rev. E 106, 044135 (2022)

    2022

  34. [42]

    A. Dechant, Minimum entropy production, detailed balance and wasserstein distance for continuous-time markov processes, Journal of Physics A: Mathematical and Theoretical55, 094001 (2022)

    2022

  35. [43]

    van der Meer and A

    J. van der Meer and A. Dechant, Near-optimality of conservative driving in discrete systems, arXiv preprint arXiv:2602.18321 (2026)

    2026 arXiv

  36. [44]

    Dechant and J

    A. Dechant and J. van der Meer, Renormalized en- tropy production for optimal transport in jump pro- cesses: Make conservative forces optimal again, arXiv preprint arXiv:2605.00656 (2026)

    2026 arXiv

  37. [45]

    Cavina, A

    V. Cavina, A. Mari, A. Carlini, and V. Giovannetti, Op- timal thermodynamic control in open quantum systems, Phys. Rev. A98, 012139 (2018)

    2018

  38. [46]

    Rolandi, M

    A. Rolandi, M. Perarnau-Llobet, and H. J. D. Miller, Optimal control of dissipation and work fluctuations for rapidly driven systems, New Journal of Physics25, 073005 (2023)

    2023

  39. [47]

    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. X13, 011013 (2023)

    2023

  40. [48]

    Zhong and M

    A. Zhong and M. R. DeWeese, Beyond linear response: Equivalence between thermodynamic geometry and op- timal transport, Phys. Rev. Lett.133, 057102 (2024). 7

    2024

  41. [49]

    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). End Matter Explicit constants and work decomposition—ForT < Tc, ∆α >0 and Eq. (17) can be inverted explicitly. With Q= 1−C, one obtains A...

    2023

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.