Classical Ergotropy Extraction Driving

• The paper introduces a quench–adiabat protocol that instantaneously quenches the system and then adiabatically returns to achieve optimal work extraction through passive-state rearrangement.
• It leverages Gardner’s construction and ergodicity to determine the minimal final energy, defining classical ergotropy as the energy gap relative to the initial distribution.
• The work further distinguishes coherent and incoherent ergotropy contributions, providing a classical analogue to quantum passive-state engineering in Hamiltonian systems.

Searching arXiv for recent and foundational papers on classical ergotropy extraction and related formulations. Classical ergotropy extraction driving is the problem of identifying a time-dependent perturbation that extracts the maximal amount of work from a thermally isolated classical Hamiltonian system prepared in a nonequilibrium distribution. In the formulation developed for classical systems, the maximal extractable work is the classical ergotropy, defined as the gap between the initial mean energy and the minimal final energy reachable by volume-preserving dynamics. Under an ergodic assumption, the optimal driving is given by a two-step quench–adiabat protocol: an instantaneous quench to a Hamiltonian for which the initial distribution is already passive, followed by an adiabatic return to the original Hamiltonian. This construction provides a classical analogue of the quantum passive-state rearrangement problem and supports a coherent–incoherent decomposition of ergotropy closely paralleling the quantum case (Campisi, 18 Aug 2025).

1. Problem statement and definition of classical ergotropy

A classical system is specified by canonical coordinates $z=(q,p)$ and an unperturbed Hamiltonian $H_0(z)$. The initial condition is a nonequilibrium phase-space distribution $\rho_0(z)\ge 0$ with $\int dz\,\rho_0(z)=1$, and the system is then isolated from heat baths. One applies a time-dependent perturbation $V(z,t)$ over a finite interval $t\in[0,\tau]$, so that the full Hamiltonian is

$H(z,t)=H_0(z)+V(z,t).$

Because the dynamics is Hamiltonian, Liouville’s theorem enforces phase-space volume preservation. The extracted work is the decrease in the average energy with respect to the original Hamiltonian:

$$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$

The central optimization problem is therefore to determine which driving $V(z,t)$ maximizes $W$ (Campisi, 18 Aug 2025).

Under very general arguments identified with Gardner’s theorem, the minimal reachable final energy $H_0(z)$0 over all volume-preserving maps coincides with the Gardner free energy. The classical ergotropy is then

$H_0(z)$1

This definition places classical ergotropy on the same conceptual footing as quantum ergotropy, where the maximum cyclic-unitary work is determined by rearranging a fixed spectrum into a passive ordering (Campisi, 18 Aug 2025).

A complementary line of work formulates classical ergotropy in terms of classical relative entropies. In that approach, for an initial distribution on an energy shell $H_0(z)$2, a final distribution on shell $H_0(z)$3, and a Gibbs reference distribution $H_0(z)$4, one defines

$H_0(z)$5

This makes explicit that classical ergotropy quantifies the additional nonequilibrium structure produced by Liouville evolution relative to thermal equilibrium on the target shell (Sone et al., 2021). A plausible implication is that the Gardner-style rearrangement picture and the relative-entropy picture are complementary descriptions of the same optimization problem.

2. Passive rearrangement and the Gardner construction

The optimal classical rearrangement is expressed through two cumulative phase-space functions. The first is

$H_0(z)$6

which is strictly decreasing in $H_0(z)$7. The second is

$H_0(z)$8

which is strictly increasing in $H_0(z)$9. Gardner’s construction then identifies the minimal-energy reachable distribution as a function of the unperturbed energy alone,

$\rho_0(z)\ge 0$0

with energy

$\rho_0(z)\ge 0$1

Accordingly,

$\rho_0(z)\ge 0$2

Equivalently,

$\rho_0(z)\ge 0$3

where $\rho_0(z)\ge 0$4 is the one-dimensional rearrangement of $\rho_0(z)\ge 0$5 that makes the final distribution a decreasing function of energy (Campisi, 18 Aug 2025).

This is the classical passive-state construction. Its significance is structural: the minimizing state is not obtained by thermalization, but by a measure-preserving sorting of phase-space weight so that larger values of $\rho_0(z)\ge 0$6 are assigned to lower-energy regions of $\rho_0(z)\ge 0$7. That is the direct classical counterpart of the quantum passive state, where the largest eigenvalues are aligned with the lowest eigenenergies (Campisi, 18 Aug 2025).

The relative-entropy formulation sharpens the same point in a different language. In that framework, any deviation from uniformity on a constant-energy surface stores active energy, and one can write

$\rho_0(z)\ge 0$8

where $\rho_0(z)\ge 0$9 is the inhomogeneity that develops on the final shell. This suggests that passive rearrangement and energy-shell homogenization are dual descriptions of the same resource structure (Sone et al., 2021).

3. The quench–adiabat driving protocol

The central result for classical ergotropy extraction is the quench–adiabat, or QA, protocol. It consists of two steps (Campisi, 18 Aug 2025).

First, one performs an instantaneous quench at $\int dz\,\rho_0(z)=1$0,

$\int dz\,\rho_0(z)=1$1

Since the quench is instantaneous, the phase-space distribution remains $\int dz\,\rho_0(z)=1$2. However, by construction, $\int dz\,\rho_0(z)=1$3 becomes a decreasing function of the new Hamiltonian $\int dz\,\rho_0(z)=1$4, so it is passive relative to $\int dz\,\rho_0(z)=1$5.

Second, one performs an adiabatic return from $\int dz\,\rho_0(z)=1$6 to $\int dz\,\rho_0(z)=1$7 over $\int dz\,\rho_0(z)=1$8. Under the ergodic hypothesis for each frozen Hamiltonian $\int dz\,\rho_0(z)=1$9, the enclosed phase-space volume

$V(z,t)$0

is an adiabatic invariant. Each phase-space point is therefore transported from a level set of $V(z,t)$1 to a level set of $V(z,t)$2 with the same volume label. The result is exactly the Gardner passive rearrangement, so the final energy is $V(z,t)$3 and the extracted work is maximal (Campisi, 18 Aug 2025).

The protocol has a direct operational interpretation. The quench “locks in” the initial distribution as a passive state of an auxiliary Hamiltonian, and the adiabatic return uses Liouville volume preservation together with adiabatic invariance to map that passive ordering back onto the original Hamiltonian. The paper explicitly states that this is the classical analogue of the quantum “reorder-and-adiabatic-ramp” proof (Campisi, 18 Aug 2025).

This protocol should be distinguished from work on classically driven quantum systems, where a classical field controls the ergotropy of a quantum emitter or battery. In those settings, “classical drive” denotes an externally applied field acting on a quantum system, whereas in classical ergotropy extraction driving the system itself is classical and the objective is the optimal Hamiltonian deformation implementing the passive rearrangement (George et al., 2024).

4. Why the QA protocol is optimal

The optimality proof proceeds by introducing the phase-space volume function of the quenched Hamiltonian,

$V(z,t)$4

and the shell probability density

$V(z,t)$5

Because $V(z,t)$6, one obtains

$V(z,t)$7

Adiabatic invariance of $V(z,t)$8 implies that this same $V(z,t)$9 becomes the probability density over shells of the original Hamiltonian after the return. The final mean energy is therefore

$t\in[0,\tau]$0

Choosing the special monotone $t\in[0,\tau]$1 makes $t\in[0,\tau]$2 and $t\in[0,\tau]$3, which yields

$t\in[0,\tau]$4

The QA protocol thus exactly attains the Gardner minimum (Campisi, 18 Aug 2025).

The same result can be re-expressed in action–angle variables when the unperturbed dynamics admits coordinates $t\in[0,\tau]$5 with $t\in[0,\tau]$6. Adiabatic invariance then states that the action is approximately conserved during the slow stage, so the final map permutes action “rings” in the way required to minimize the average energy. This is again described as the classical counterpart of the quantum reorder-and-ramp argument (Campisi, 18 Aug 2025).

A plausible implication is that the proof isolates the genuinely dynamical content of classical ergotropy extraction: the problem is not merely combinatorial sorting of probabilities, but the physical implementation of that sorting by Hamiltonian flows constrained by Liouville’s theorem and adiabatic invariance.

5. Coherent and incoherent parts of classical ergotropy

The 2025 treatment shows that classical ergotropy splits into coherent and incoherent contributions, just as in the quantum case (Campisi, 18 Aug 2025). The relevant classical dephasing, or homogenization, operator acts on the energy shells of $t\in[0,\tau]$7:

$t\in[0,\tau]$8

This map preserves the mean energy while erasing the fine structure of $t\in[0,\tau]$9 along each iso-$H(z,t)=H_0(z)+V(z,t).$0 shell. One then defines

$H(z,t)=H_0(z)+V(z,t).$1

and

$H(z,t)=H_0(z)+V(z,t).$2

The incoherent part is therefore the ergotropy stored in radial shell populations, while the coherent part is the additional ergotropy contained in angular inhomogeneities within each shell (Campisi, 18 Aug 2025).

The paper also gives an information-theoretic identity:

$H(z,t)=H_0(z)+V(z,t).$3

where $H(z,t)=H_0(z)+V(z,t).$4 is the passive rearrangement of $H(z,t)=H_0(z)+V(z,t).$5, $H(z,t)=H_0(z)+V(z,t).$6 is the Kullback–Leibler divergence, and

$H(z,t)=H_0(z)+V(z,t).$7

measures deviation from the dephased distribution, identified as “classical coherence” (Campisi, 18 Aug 2025).

This decomposition is closely related to the earlier relative-entropy formulation of classical ergotropy, in which energy-shell inhomogeneity is the active resource and the classical limit emerges from the geometric formulation of quantum mechanics when probability densities are supported on energy-eigenstate poles (Sone et al., 2021). It is also consistent with the distinction between incoherent and coherent ergotropy extraction in isolated quantum settings, where incoherent extraction corresponds to level permutations and coherent extraction addresses off-diagonal structure in the energy basis (Łobejko, 2021). The classical case replaces Hilbert-space coherence by phase-space inhomogeneity on constant-energy surfaces.

6. Examples, topology, and the role of ergodicity

A canonical example is the harmonic oscillator with displaced Gaussian initial distribution

$H(z,t)=H_0(z)+V(z,t).$8

Under the unperturbed flow, this Gaussian rotates and does not become passive. The QA protocol first quenches to

$H(z,t)=H_0(z)+V(z,t).$9

with

$$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$0

so that the state is stationary and passive relative to $$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$1. One then adiabatically moves the potential minimum from $$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$2 back to $$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$3, producing the final passive Gibbs-form distribution

$$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$4

and the extracted work is

$$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$5

Thus the ergotropy equals the potential-energy drop of the displaced center (Campisi, 18 Aug 2025).

The analysis becomes subtler for topologically nontrivial initial distributions, such as a ring-shaped or “Mexican-hat” distribution with a central valley. In that case, the monotone $$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$6 generates an auxiliary Hamiltonian $$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$7 with level sets that are not simply connected. A purely adiabatic global return cannot permute all such structures in a single step. The proposed resolution is to choose an implementable $$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$8 sharing the same level-set topology, perform the adiabatic return, and then supplement it with a finite sequence of small non-adiabatic “ring-permutation” drivings that permute subregions and approach the true passive ordering arbitrarily closely. The paper states that this combination of QA plus controlled mixing fully recovers $$W = E_i - E_f = \int dz\,\rho_0(z)\,H_0(z)-\int dz\,\rho_\tau(z)\,H_0(z).$$9 (Campisi, 18 Aug 2025).

The role of ergodicity is central. The adiabatic invariance of enclosed phase-space volume $V(z,t)$0 relies on ergodic motion on each energy shell, in the sense discussed through Khinchin’s ergodic-theorem-based proofs and extensions by Hertz, Jarzynski, and Ott (Campisi, 18 Aug 2025). If the instantaneous Hamiltonians are not ergodic, volume invariance may fail for a global adiabatic deformation. The same work nonetheless emphasizes two extensions: one may often select an $V(z,t)$1 with the same shell topology as $V(z,t)$2, and when shells are multiply connected one can replace the purely adiabatic step by topologically targeted field pulses that permute small subregions. This suggests that ergodicity is sufficient for the clean QA construction, but not always necessary for near-optimal or fully optimal extraction in more structured phase spaces.

7. Relation to quantum ergotropy and broader significance

The classical theory is explicitly framed as an analogue of the solved quantum ergotropy problem. In finite-dimensional quantum systems, ergotropy is the maximal cyclic work

$V(z,t)$3

where the passive state $V(z,t)$4 is obtained by assigning the largest eigenvalues of $V(z,t)$5 to the lowest energy levels of $V(z,t)$6 (Francica, 2022). The classical QA protocol implements the corresponding rearrangement through Hamiltonian deformation rather than unitary diagonalization and eigenvalue sorting (Campisi, 18 Aug 2025).

This correspondence has several consequences. First, it strengthens the claim that there is a unified passive-state principle underlying both classical and quantum ergotropy. Second, it clarifies that coherent versus incoherent contributions are not uniquely quantum; the classical setting admits an analogous decomposition once “coherence” is interpreted as inhomogeneity on energy shells (Campisi, 18 Aug 2025, Sone et al., 2021). Third, it places classical ergotropy extraction alongside work on practical ergotropy estimation and control in quantum platforms, where passive states are reached through feedback or externally driven protocols, such as FQErgo for unknown quantum states (Joshi et al., 2024). The conceptual parallel is that both cases seek constructive procedures that attain the passive rearrangement rather than merely proving its existence.

A common misconception is to identify “classical ergotropy extraction driving” with the use of a classical field to control a quantum battery. The literature distinguishes these notions sharply. In one case, the controlled object is a classical Hamiltonian system and the optimal driving is the QA protocol (Campisi, 18 Aug 2025). In the other, a classical field acts on a quantum system, such as a two-level emitter or battery, and the analysis concerns the evolution of quantum ergotropy under continuous or pulsed driving (George et al., 2024, Morrone et al., 2023). The shared term “classical” refers to the controller in the latter case, but to the dynamical regime of the working medium in the former.

Within this broader landscape, classical ergotropy extraction driving establishes that the maximal-work problem for isolated classical systems can be solved constructively under the ergodic hypothesis, while also providing a framework for more general non-ergodic and topologically constrained cases. The result is a precise classical counterpart to passive-state engineering in quantum thermodynamics and a basis for practical energy-recovery protocols in classical Hamiltonian systems (Campisi, 18 Aug 2025).