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Active charge and discharge of a capacitor: scaling solution and energy optimization

T0 review · 0 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For a capacitor charged to a fixed final state in a fixed time, the minimum Joule heat is achieved by a linear charge ramp, and the heat–time trade-off has a sharp speed limit.

desk verdict A correct, well-executed pedagogical re-derivation of known fast-forward and optimal-control results for an RC circuit, with clean experimental support; worth publishing as a teaching resource. read the letter →

arxiv 2501.12028 v1 pith:SCJA7GZN submitted 2025-01-21 physics.ed-ph physics.class-ph

S. Faure , C. A. Plata , A. Prados , D. Guéry-Odelin This is my paper · ORCID
classification physics.ed-phphysics.class-ph
keywords RCcircuitcapacitorchargingJouleheatenergyoptimizationvariationalcalculusspeedlimitfast-forwardprotocolquasistatic
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 asks a simple control question: if you want to bring a capacitor to a target charge in a finite time, what driving voltage wastes the least energy as Joule heat, and how much heat is unavoidable? For an ideal RC circuit, the answer is a linear charge ramp: charge grows at constant current, the source voltage rises linearly on top of an offset, and the minimum dissipated heat is exactly $2\tau/t_f$ times the stored energy $\Delta U^+$. The paper also builds the same result from time-rescaled exponential protocols, showing that speeding a standard RC charge by a factor $\alpha$ multiplies the dissipated heat by $\alpha$, and it confirms both predictions in a classroom circuit for acceleration factors from 0.2 to 5. The practical payoff is a sharp speed limit: no finite-time charging protocol can beat the trade-off $t_f|Q| \ge 2\tau\Delta U^+$, so the linear ramp is the benchmark for low-loss capacitor charging.

What carries the argument

Two objects carry the argument. First, the time-rescaled reference solution $q(t)=q_r(\Lambda(t))$ with $\Lambda(t)=\alpha t$ maps the standard exponential charge $q_r(t)=C\varepsilon_f(1-e^{-t/\tau})$ onto accelerated or decelerated trajectories; substituting it into the circuit law gives the required voltage drive and, via the first law of thermodynamics, the heat scaling $Q(\alpha)=\alpha Q_r$. Second, the variational minimization of the heat functional $-Q^+ = R\int \dot q^2 \, dt$ has an Euler-Lagrange equation whose solution is $\ddot q=0$, producing the linear ramp and the companion speed limit $t_f|Q| \ge 2\tau\Delta U^+$; the optimal voltage is linear with an offset, with step and linear shapes emerging as limiting cases.

What would settle it

Measure the dissipated heat while charging a capacitor from zero to $q_f$ in a time $t_f$ much shorter than $\tau=RC$, using the prescribed linear-ramp voltage $\varepsilon_{\rm opt}(t)$ and a source with known slew rate; if the measured $-Q$ exceeds $2\tau\Delta U^+/t_f$, the ideal-circuit equality is violated in practice. More decisively, numerically minimize $-Q = R\int_0^{t_f} \dot q^2 \, dt$ over all smooth charge trajectories with $q(0)=0$ and $q(t_f)=q_f$; the linear ramp is the unique minimizer, so any trajectory yielding lower heat would refute the central claim.

Watch

Extended reading notes

Core claim

The central discovery is a closed-form minimum for Joule dissipation in a resistively charged capacitor driven by an arbitrary voltage waveform. For a pure charge from $q(0)=0$ to $q(t_f)=q_f$, the heat functional is $-Q^+[q] = \int_0^{t_f} R \dot q^2 \, dt$; minimizing it under the fixed endpoints gives $\ddot q = 0$, i.e. $q_{\rm opt}(t) = q_f t/t_f$. The voltage that realizes this charge is $\varepsilon_{\rm opt}(t)=\varepsilon_f(\tau/t_f + t/t_f)$ over $0<t<t_f$ and $\varepsilon_f$ afterwards, and the resulting heat is $-Q_{\rm opt} = 2\tau \Delta U^+/t_f$. Since every other protocol has more heat, the inequality $t_f|Q| \ge 2\tau\Delta U^+$ is a speed limit. The same inverse-engineering framework yields the heat scaling $Q(\alpha)=\alpha Q_r$ for the time-rescaled exponential protocol, verified experimentally for $\alpha$ between 0.2 and 5.

Load-bearing premise

Everything rests on treating the RC loop as an ideal first-order linear circuit with constant resistance and capacitance and a generator that can instantly follow any requested voltage, since the paper itself finds that the generator's finite slew rate already perturbs the response for large acceleration factors.

Editorial extensions

If this is right

  • The linear ramp, or constant-current charging, is the minimum-dissipation charging strategy for any RC circuit with fixed resistance and capacitance and a prescribed duration $t_f$; no other waveform can beat it.
  • The speed limit $t_f|Q| \ge 2\tau\Delta U^+$ quantifies the quasistatic trade-off: dissipation vanishes only as $t_f\to\infty$, and accelerating a reference charge by a factor $\alpha$ raises the dissipated heat proportionally to $\alpha$.
  • The same Euler-Lagrange minimization applies to the overdamped mechanical analog (a damped harmonic oscillator driven by an external force), so the linear-protocol result carries over directly to that classroom problem.
  • For capacitor-based energy storage, the result sets a floor: any finite-time charge of an ideal RC buffer must dissipate at least $2\tau\Delta U/t_f$ in the series resistance.

Reading between the lines

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

  • A direct extension of the paper's logic is that partial charging between arbitrary voltages $\varepsilon_i$ and $\varepsilon_f$ should obey the same speed limit with heat scaled by $C(\varepsilon_f-\varepsilon_i)^2/2$, so the bound is more general than the pure-charge case emphasized in the main text.
  • For very short target times, the optimal voltage approaches a delta-function-like spike, so any real generator's finite voltage slew rate will push the measured heat above the ideal bound; this suggests a practical test of how closely a physical source can approach the speed limit.
  • Applying the same variational principle to an RLC circuit would require constant current with finite jumps at the endpoints, implying that in underdamped systems the practical speed limit is set not by Joule heat alone but by the realizability of those current discontinuities.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies active charge and discharge of an RC circuit driven by a programmable voltage source, with a pedagogical thermodynamics/control-theory framing. It first constructs time-rescaled reference protocols q(t)=C εf(1−e^{-αt/τ}), derives the required driving voltage ε+(t)=εf+εf(α−1)e^{-αt/τ} (Eq. 7), and shows that the dissipated heat scales as Q(α)=αQ_r (Eq. 16). These predictions are tested experimentally for α between 0.2 and 5, including the effective time-constant ratio τ/τ_exp=α. The paper then solves the optimal-control problem of minimizing Joule heat for a prescribed final charge qf and finite time tf: the optimal trajectory is the linear ramp q_opt(t)=qf t/tf, the optimal voltage is ε_opt(t)=εf(τ/tf + t/tf), the minimum heat is −Q_opt = 2τ/tf ΔU+, and the resulting speed limit is tf|Q| ≥ 2τΔU+ (Eqs. 20–26). The optimal, step, and linear voltage protocols are compared experimentally through the total work, with no adjustable parameters.

Significance. If the results stand, this is a valuable and unusual contribution to physics education: it connects elementary circuit theory to finite-time thermodynamics, variational calculus, control, and speed limits, and it provides a complete experimental project using a waveform generator, an oscilloscope, and data processing. The theoretical derivations are transparent and checkable by hand, and the experimental validation is parameter-free: the scaling law Q(α)=αQ_r and the optimal-protocol predictions are derived from first principles and then compared directly with measured voltages and currents. The paper also explicitly identifies the ideal-source assumption (constant R, C, negligible parasitic inductance/leakage, and effectively infinite source bandwidth) and documents the finite-slew-rate deviations that appear for large α. The central physical claims are the linear-ramp optimality and the speed limit, both of which are sound and are supported by the reported data.

minor comments (6)
  1. [III, Fig. 2 and Fig. 4] The text states that discrepancies are “typically below 5%” and later that the measured error is “always within 2–3%”, but no error bars or a precise definition of the discrepancy metric are provided; please specify how this percentage is computed and whether it reflects systematic or random deviations across repeated trials.
  2. [Eqs. (17) and (29)] The experimental voltage used in the heat and work integrals is denoted εexp(t) in one place and εgen in Eq. (29); please clarify whether the programmed waveform or the measured generator voltage is used, as this affects the interpretation of the quoted agreement.
  3. [V, Eq. (23) and Fig. 5] The optimal voltage has discontinuities at t=0 and t=tf, so its experimental implementation depends on the source slew rate; the paper acknowledges this qualitatively, but a short quantitative statement of the range of tf/τ over which the optimal protocol was actually implemented would strengthen the comparison in Fig. 5a.
  4. [Fig. 5 caption] The caption does not label the horizontal axis explicitly; please state that it is tf/τ and define the symbols (disk, square, triangle) directly in the caption rather than only in the body text.
  5. [III and Acknowledgments] There are several typos: “discretize the the theoretical curve” in Section III, and “Rhode et Schwartz” in the experimental description should be “Rohde & Schwarz”.
  6. [Eqs. (16), (A3), and footnote 17] Because Q is defined as a negative quantity, expressions such as Q±(α)=αQ_r± and Qopt=2τ/tf Q_r can be misread as positive heats; a one-sentence reminder immediately after Eq. (16) that Q_r is negative and that the dissipated heat is −Q would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central derivations are self-contained and experimentally verified without adjustable parameters.

full rationale

The paper's derivation chain is self-contained. Eq. (2) is the RC ODE; Eqs. (6)-(7) construct the voltage that realizes the rescaled trajectory by direct substitution, and Eq. (16) follows from Q = -R∫i²dt with i(t)=α(Cεf/τ)e^{-αt/τ}, an integral requiring no fitted input. The optimal protocol is obtained by minimizing the convex functional R∫qdot²dt subject to fixed endpoints, giving q_opt=qf t/tf and Eq. (22), and the voltage (23) is inferred from the same ODE. Experimental checks (Figs. 2-5) use measured voltages to compute work and heat and compare with predictions without adjustable parameters; the exponential fit of τ_exp is a diagnostic, not an input. Citations to the authors' prior fast-forward work (Refs. 5, 12, 14, 16) provide motivation and context for the time-rescaling ansatz, but the ansatz is stated explicitly and the subsequent mathematics is independent of any external theorem. Therefore no circular step reduces a prediction to its input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; R, C, epsilon_f, tau, alpha, and tf are inputs or control parameters. The model relies on the ideal RC assumption and the thermodynamic identification of Joule heating with dissipated heat. No new entities are introduced.

assumptions (4)
  • domain assumption The RC circuit is an ideal first-order linear system with constant R and C, no parasitic inductance or leakage.
    Invoked in Eq. (2); the entire derivation rests on this linear ODE.
  • domain assumption The voltage source can generate arbitrary time-dependent waveforms with sufficient bandwidth.
    Needed for the inverse-engineered driving in Eq. (7); the paper notes finite slew rate causes deviations for large alpha.
  • domain assumption Energy bookkeeping identifies W = integral epsilon i dt as work and -Q = R integral i^2 dt as dissipated heat, with first law Delta U = Q + W.
    Used in Section IV to split generator energy into stored energy and Joule heat; a standard modeling choice.
  • standard math The Euler-Lagrange equation gives the global minimum for the quadratic functional R integral q_dot^2 dt.
    Used in Section V to derive the optimal ramp; the functional is convex, so the stationary point is the global minimum.

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Cite this review

Pith. "Pith review of Active charge and discharge of a capacitor: scaling solution and energy optimization." pith.science (2026). https://pith.science/paper/SCJA7GZN

@misc{pith2026250112028,
  author       = {Pith},
  title        = {Pith review of: Active charge and discharge of a capacitor: scaling solution and energy optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SCJA7GZN}},
  note         = {Machine review of arXiv:2501.12028}
}
read the original abstract

Capacitors are ubiquitous in electronic and electrical devices. In this article, we study -- both theoretically and experimentally -- the charging and discharging of capacitors using active control of a voltage source. The energy of these processes is analyzed in terms of work and heat. We show how to approach the quasistatic regime by slowing down the charging or discharging processes. Conversely, we study the price to be paid in terms of Joule heat when we speed up these processes. Finally, we develop optimal processes that minimize energy consumption for a finite charging time. Our work combines fundamental concepts from thermodynamics, classical mechanics and electrical circuits, thus blurring the artificial frontiers at the undergraduate level between these disciplines. Also, it provides a simple example of a prominent problem in current science, the optimization of energy resources. Moreover, our study lends itself well to an experimental project in the classroom, involving computer control of a voltage source, data acquisition, and processing.

Figures

Figures reproduced from arXiv: 2501.12028 by the authors.

Figure 1
Figure 1. FIG. 1. RC circuit under a properly engineered voltage [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Charge of a capacitor. (a) Measured voltage [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Dissipated heat for the charging (left) and discharg [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Charge of a capacitor [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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Reference graph

Works this paper leans on

41 extracted references · 26 canonical work pages · cited by 1 Pith paper

  1. [1]

    Callen ,\ @noop title Thermodynamics and an Introduction to Thermostatistics \ ( publisher Wiley ,\ year 1985 ) NoStop

    author author H. Callen ,\ @noop title Thermodynamics and an Introduction to Thermostatistics \ ( publisher Wiley ,\ year 1985 ) NoStop

    work page 1985

  2. [2]

    Goldstein ,\ @noop title Classical mechanics ,\ edition 2nd \ ed.,\ Addison-Wesley series in physics\ ( publisher Addison-Wesley ,\ year 1980 ) NoStop

    author author H. Goldstein ,\ @noop title Classical mechanics ,\ edition 2nd \ ed.,\ Addison-Wesley series in physics\ ( publisher Addison-Wesley ,\ year 1980 ) NoStop

    work page 1980

  3. [3]

    author author L. D. \ Landau \ and\ author E. M. \ Lifshitz ,\ @noop title Course of Theoretical Physics: Vol. 1, Mechanics ,\ edition 3rd \ ed.,\ Course of Theoretical Physics\ ( publisher Butterworth-Heinemann ,\ year 1976 ) NoStop

    work page 1976

  4. [4]

    Sekimoto ,\ @noop title Stochastic Energetics \ ( publisher Springer ,\ year 2010 ) NoStop

    author author K. Sekimoto ,\ @noop title Stochastic Energetics \ ( publisher Springer ,\ year 2010 ) NoStop

    work page 2010

  5. [5]

    Guéry-Odelin , author C

    author author D. Guéry-Odelin , author C. Jarzynski , author C. A. \ Plata , author A. Prados ,\ and\ author E. Trizac ,\ title title Driving rapidly while remaining in control: classical shortcuts from Hamiltonian to stochastic dynamics , \ https://doi.org/10.1088/1361-6633/acacad journal journal Reports on Progress in Physics \ volume 86 ,\ pages 035902...

    doi:10.1088/1361-6633/acacad

  6. [6]

    Blaber \ and\ author D

    author author S. Blaber \ and\ author D. A. \ Sivak ,\ title title Optimal control in stochastic thermodynamics , \ https://doi.org/10.1088/2399-6528/acbf04 journal journal Journal of Physics Communications \ volume 7 ,\ pages 033001 ( year 2023 ) NoStop

    work page doi:10.1088/2399-6528/acbf04

  7. [7]

    Liberzon ,\ @noop title Calculus of Variations and Optimal Control Theory : A Concise Introduction \ ( publisher Princeton University Press ,\ year 2012 ) NoStop

    author author D. Liberzon ,\ @noop title Calculus of Variations and Optimal Control Theory : A Concise Introduction \ ( publisher Princeton University Press ,\ year 2012 ) NoStop

    work page 2012

  8. [8]

    Schmiedl \ and\ author U

    author author T. Schmiedl \ and\ author U. Seifert ,\ title title Optimal Finite - Time Processes In Stochastic Thermodynamics , \ https://doi.org/10.1103/PhysRevLett.98.108301 journal journal Physical Review Letters \ volume 98 ,\ pages 108301 ( year 2007 ) NoStop

    doi:10.1103/physrevlett.98.108301

Show all 41 references
  1. [9]

    Aurell , author C

    author author E. Aurell , author C. Mejía-Monasterio ,\ and\ author P. Muratore-Ginanneschi ,\ title title Optimal Protocols and Optimal Transport in Stochastic Thermodynamics , \ https://doi.org/10.1103/PhysRevLett.106.250601 journal journal Physical Review Letters \ volume 1...

    doi

  2. [10]

    author author D. A. \ Sivak \ and\ author G. E. \ Crooks ,\ title title Thermodynamic Metrics and Optimal Paths , \ https://doi.org/10.1103/PhysRevLett.108.190602 journal journal Physical Review Letters \ volume 108 ,\ pages 190602 ( year 2012 ) NoStop

    doi

  3. [11]

    Masuda \ and\ author K

    author author S. Masuda \ and\ author K. Nakamura ,\ title title Fast-forward of adiabatic dynamics in quantum mechanics , \ https://doi.org/10.1098/rspa.2009.0446 journal journal Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences \ volume 466 ...

    2009

  4. [12]

    Guéry-Odelin , author A

    author author D. Guéry-Odelin , author A. Ruschhaupt , author A. Kiely , author E. Torrontegui , author S. Martínez-Garaot ,\ and\ author J. G. \ Muga ,\ title title Shortcuts to adiabaticity: Concepts , methods, and applications , \ https://doi.org/10.1103/RevModPhys.91.04500...

    doi

  5. [13]

    Nakamura , author J

    author author K. Nakamura , author J. Matrasulov ,\ and\ author Y. Izumida ,\ title title Fast-forward approach to stochastic heat engine , \ https://doi.org/10.1103/PhysRevE.102.012129 journal journal Physical Review E \ volume 102 ,\ pages 012129 ( year 2020 ) NoStop

    doi

  6. [14]

    author author C. A. \ Plata , author A. Prados , author E. Trizac ,\ and\ author D. Gu \'e ry-Odelin ,\ title title Taming the Time Evolution in Overdamped Systems : Shortcuts Elaborated from Fast - Forward and Time - Reversed Protocols , \ https://doi.org/10.1103/PhysRevLett....

    doi

  7. [15]

    Impens , author F

    author author F. Impens , author F. M. \ D'Angelis , author F. A. \ Pinheiro ,\ and\ author D. Guéry-Odelin ,\ title title Time scaling and quantum speed limit in non- Hermitian Hamiltonians , \ https://doi.org/10.1103/PhysRevA.104.052620 journal journal Physical Review A \ vo...

    doi

  8. [16]

    Faure , author S

    author author S. Faure , author S. Ciliberto , author E. Trizac ,\ and\ author D. Guéry-Odelin ,\ title title Shortcut to stationary regimes: A simple experimental demonstration , \ https://doi.org/10.1119/1.5082933 journal journal American Journal of Physics \ volume 87 ,\ pa...

    doi

  9. [17]

    we follow the usual sign convention in which energy exchanges, in the form of either heat or work, increasing (decreasing) the internal energy of the system are positive (negative)

    note Note that we have written the first law as U = Q+W , i.e. we follow the usual sign convention in which energy exchanges, in the form of either heat or work, increasing (decreasing) the internal energy of the system are positive (negative). Hence, dissipation involves a ne...

  10. [18]

    Deffner \ and\ author S

    author author S. Deffner \ and\ author S. Campbell ,\ title title Quantum speed limits: from Heisenberg ’s uncertainty principle to optimal quantum control , \ https://doi.org/10.1088/1751-8121/aa86c6 journal journal Journal of Physics A: Mathematical and Theoretical \ volume ...

    doi

  11. [19]

    Okuyama \ and\ author M

    author author M. Okuyama \ and\ author M. Ohzeki ,\ title title Quantum Speed Limit is Not Quantum , \ @noop journal journal Physical Review Letters \ volume 120 ,\ pages 070402 ( year 2018 ) NoStop

    2018

  12. [20]

    Shiraishi , author K

    author author N. Shiraishi , author K. Funo ,\ and\ author K. Saito ,\ title title Speed Limit for Classical Stochastic Processes , \ @noop journal journal Physical Review Letters \ volume 121 ,\ pages 070601 ( year 2018 ) NoStop

    2018

  13. [21]

    Shanahan , author A

    author author B. Shanahan , author A. Chenu , author N. Margolus ,\ and\ author A. del Campo ,\ title title Quantum Speed Limits across the Quantum -to- Classical Transition , \ https://doi.org/10.1103/PhysRevLett.120.070401 journal journal Physical Review Letters \ volume 120...

    doi

  14. [22]

    Funo , author N

    author author K. Funo , author N. Shiraishi ,\ and\ author K. Saito ,\ title title Speed limit for open quantum systems , \ https://doi.org/10.1088/1367-2630/aaf9f5 journal journal New Journal of Physics \ volume 21 ,\ pages 013006 ( year 2019 ) NoStop

    doi

  15. [23]

    Shiraishi \ and\ author K

    author author N. Shiraishi \ and\ author K. Saito ,\ title title Speed limit for open systems coupled to general environments , \ https://doi.org/10.1103/PhysRevResearch.3.023074 journal journal Phys. Rev. Research \ volume 3 ,\ pages 023074 ( year 2021 ) NoStop

    doi

  16. [24]

    author author C. A. \ Plata , author D. Guéry-Odelin , author E. Trizac ,\ and\ author A. Prados ,\ title title Finite-time adiabatic processes: Derivation and speed limit , \ https://doi.org/10.1103/PhysRevE.101.032129 journal journal Physical Review E \ volume 101 ,\ pages 0...

    doi

  17. [25]

    author author S. Deffner ,\ title title Quantum speed limits and the maximal rate of information production , \ https://doi.org/10.1103/PhysRevResearch.2.013161 journal journal Physical Review Research \ volume 2 ,\ pages 013161 ( year 2020 ) NoStop

    doi

  18. [26]

    Ito \ and\ author A

    author author S. Ito \ and\ author A. Dechant ,\ title title Stochastic time-evolution, information geometry and the Cramer - Rao Bound , \ https://doi.org/10.1103/PhysRevX.10.021056 journal journal Physical Review X \ volume 10 ,\ pages 021056 ( year 2020 ) NoStop

    doi

  19. [27]

    Van Vu \ and\ author Y

    author author T. Van Vu \ and\ author Y. Hasegawa ,\ title title Geometrical Bounds of the Irreversibility in Markovian Systems , \ https://doi.org/10.1103/PhysRevLett.126.010601 journal journal Physical Review Letters \ volume 126 ,\ pages 010601 ( year 2021 ) NoStop

    doi

  20. [28]

    author author A. Prados ,\ title title Optimizing the relaxation route with optimal control , \ https://doi.org/10.1103/PhysRevResearch.3.023128 journal journal Physical Review Research \ volume 3 ,\ pages 023128 ( year 2021 ) NoStop

    2021 doi

  21. [29]

    author author J. S. \ Lee , author S. Lee , author H. Kwon ,\ and\ author H. Park ,\ title title Speed Limit for a Highly Irreversible Process and Tight Finite - Time Landauer ’s Bound , \ https://doi.org/10.1103/PhysRevLett.129.120603 journal journal Physical Review Letters \...

    doi

  22. [30]

    Patrón , author A

    author author A. Patrón , author A. Prados ,\ and\ author C. A. \ Plata ,\ title title Thermal brachistochrone for harmonically confined Brownian particles , \ https://doi.org/10.1140/epjp/s13360-022-03150-3 journal journal The European Physical Journal Plus \ volume 137 ,\ pa...

    doi

  23. [31]

    author author A. Dechant ,\ title title Minimum entropy production, detailed balance and Wasserstein distance for continuous-time Markov processes , \ https://doi.org/10.1088/1751-8121/ac4ac0 journal journal Journal of Physics A: Mathematical and Theoretical \ volume 55 ,\ pag...

    doi

  24. [32]

    This behavior is analogous to that found in the minimization of the irreversible work of an underdamped particle under harmonic confinement in a thermal bath at temperature T

    note These finite jump discontinuities do not contribute to the dissipated heat but make the source voltage have two delta peaks at the initial and final times, diverging to + and - , respectively. This behavior is analogous to that found in the minimization of the irreversibl...

  25. [33]

    Hou , author Y

    author author J. Hou , author Y. Yang , author H. He ,\ and\ author T. Gao ,\ title title Adaptive Dual Extended Kalman Filter Based on Variational Bayesian Approximation for Joint Estimation of Lithium - Ion Battery State of Charge and Model Parameters , \ https://doi.org/10....

    doi

  26. [34]

    Dreyer , author C

    author author W. Dreyer , author C. Guhlke ,\ and\ author M. Herrmann ,\ title title Hysteresis and phase transition in many-particle storage systems , \ @noop journal journal Continuum Mechanics and Thermodynamics \ volume 23 ,\ pages 211--231 ( year 2011 ) NoStop

    2011

  27. [35]

    Dreyer , author C

    author author W. Dreyer , author C. Guhlke ,\ and\ author R. Huth ,\ title title The behavior of a many-particle electrode in a lithium-ion battery , \ @noop journal journal Physica D: Nonlinear Phenomena \ volume 240 ,\ pages 1008--1019 ( year 2011 ) NoStop

    2011

  28. [36]

    Dreyer , author J

    author author W. Dreyer , author J. Jamnik , author C. Guhlke , author R. Huth , author J. Mo s kon ,\ and\ author M. Gaber s c ek ,\ title title The thermodynamic origin of hysteresis in insertion batteries , \ @noop journal journal Nature materials \ volume 9 ,\ pages 448--4...

    2010

  29. [37]

    Li , author J

    author author M. Li , author J. Lu , author Z. Chen ,\ and\ author K. Amine ,\ title title 30 Years of Lithium ‐ Ion Batteries , \ https://doi.org/10.1002/adma.201800561 journal journal Advanced Materials \ volume 30 ,\ pages 1800561 ( year 2018 ) NoStop

    doi

  30. [38]

    Wang , author J

    author author Y. Wang , author J. Tian , author Z. Sun , author L. Wang , author R. Xu , author M. Li ,\ and\ author Z. Chen ,\ title title A comprehensive review of battery modeling and state estimation approaches for advanced battery management systems , \ https://doi.org/10...

  31. [39]

    author author A. Manthiram ,\ title title A reflection on lithium-ion battery cathode chemistry , \ https://doi.org/10.1038/s41467-020-15355-0 journal journal Nature Communications \ volume 11 ,\ pages 1550 ( year 2020 ) NoStop

    2020 doi

  32. [40]

    note Note that the notation employed here is slightly different from that of the main text: here, _ i and _ f always stand for the initial and final voltage of the corresponding process, whereas in the main text we only introduced _ f as the final voltage of the charging proce...

  33. [41]

    Gomez-Marin , author T

    author author A. Gomez-Marin , author T. Schmiedl ,\ and\ author U. Seifert ,\ title title Optimal protocols for minimal work processes in underdamped stochastic thermodynamics , \ https://doi.org/10.1063/1.2948948 journal journal The Journal of Chemical Physics \ volume 129 ,...

    doi

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