Classical Ergotropy Overview

• Classical ergotropy is defined as the maximum average energy extractable by cyclic Hamiltonian driving, achieved by rearranging the phase-space density to form a passive state.
• The theory employs rearrangement techniques, including the Gardner construction, to minimize mean energy through optimal ordering of phase-space probabilities.
• Extraction protocols such as the quench-adiabat method under ergodic assumptions demonstrate how inhomogeneities on energy shells enable practical work extraction.

Classical ergotropy is the maximum average energy that can be extracted from a thermally isolated classical mechanical system by cyclic, time-dependent driving. In the modern literature, it is formulated as a minimization of the mean energy of a phase-space density under Hamiltonian, phase-space-volume-preserving evolution, with the difference between the initial energy and the minimal reachable energy defining the extractable work. This quantity is also called available energy, and in several treatments it coincides with the Gardner free energy (Campisi, 18 Aug 2025, Campisi, 30 Mar 2026).

1. Phase-space definition and passive reference state

In the classical formulation, the system is described by a phase-space point $\mathbf z=(\mathbf q,\mathbf p)$, an unperturbed Hamiltonian $H_0(\mathbf z)$, and a normalized phase-space density $\rho(\mathbf z)$ or $\rho_0(\mathbf z)$. The allowed dynamics are Hamiltonian and therefore phase-space-volume preserving. The work-extraction problem is to find, among all states reachable by cyclic driving, the one with the lowest mean value of $H_0$ (Campisi, 18 Aug 2025, Campisi, 30 Mar 2026).

One convenient definition is the ergotropy functional

$$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$

where $\varphi$ ranges over Hamiltonian flows generated by cyclic driving. Equivalently, if $\breve\rho$ denotes the evolved state of minimum possible energy expectation among all such Hamiltonian rearrangements, then

$$\mathcal E[\rho] = \int d\mathbf z\, H_0(\mathbf z)\,[\rho(\mathbf z)-\breve\rho(\mathbf z)].$$

The state $\breve\rho$ is the passive companion of $H_0(\mathbf z)$0: it has the same measure distribution as the initial density, but arranged so that the energy expectation is minimal. A density is passive when $H_0(\mathbf z)$1, meaning that no cyclic Hamiltonian operation can lower its mean energy any further (Campisi, 30 Mar 2026).

A complementary expression, emphasized in the extraction problem for a prescribed initial state $H_0(\mathbf z)$2, writes the initial mean energy as

$H_0(\mathbf z)$3

and the minimal reachable mean energy as the passive or Gardner ground-state energy

$H_0(\mathbf z)$4

where

$H_0(\mathbf z)$5

is the phase-space volume occupied by points with density larger than $H_0(\mathbf z)$6, and

$H_0(\mathbf z)$7

is the phase-space volume enclosed by the energy shell $H_0(\mathbf z)$8, with inverse $H_0(\mathbf z)$9. The classical ergotropy is then

$\rho(\mathbf z)$0

This is identified as the Gardner free energy (Campisi, 18 Aug 2025).

2. Rearrangement theory and the Gardner construction

A central development in the recent classical literature is the recasting of the ergotropy problem as a function rearrangement problem. Earlier explicit formulas required strong regularity assumptions, notably continuity of $\rho(\mathbf z)$1 and the absence of flat plateaus. Under those assumptions one introduces

$\rho(\mathbf z)$2

and the phase volume

$\rho(\mathbf z)$3

If the passive state has the form $\rho(\mathbf z)$4 with $\rho(\mathbf z)$5 strictly decreasing, then one obtains

$\rho(\mathbf z)$6

and therefore

$\rho(\mathbf z)$7

This formula is explicit but not fully general, because it presupposes that $\rho(\mathbf z)$8 is strictly decreasing and continuous (Campisi, 30 Mar 2026).

The generalization removes those limitations by defining an ergotropic rearrangement of sets and densities. For a measurable set $\rho(\mathbf z)$9, the ergotropic rearrangement is

$\rho_0(\mathbf z)$0

where $\rho_0(\mathbf z)$1 is the Lebesgue measure of $\rho_0(\mathbf z)$2. For a generic nonnegative measurable density,

$\rho_0(\mathbf z)$3

Using the cumulative superlevel-set measure $\rho_0(\mathbf z)$4, this becomes

$\rho_0(\mathbf z)$5

This is the general passive-state formula, valid even when $\rho_0(\mathbf z)$6 has jumps or flat regions (Campisi, 30 Mar 2026).

The construction generalizes the symmetric decreasing rearrangement of measure theory. When the Hamiltonian is spherically symmetric, for example

$\rho_0(\mathbf z)$7

rearrangement by energy reduces to the usual symmetric decreasing rearrangement by radius. In this sense, ergotropic rearrangement replaces radial ordering by ordering with respect to the Hamiltonian (Campisi, 30 Mar 2026).

3. Optimal extraction protocols

Beyond the static variational characterization, recent work gives an explicit driving protocol that realizes the classical optimum under an ergodic assumption. The protocol is a quench-adiabat protocol consisting of an instantaneous quench followed by an adiabatic return (Campisi, 18 Aug 2025).

The first step is an instantaneous quench at $\rho_0(\mathbf z)$8 to an auxiliary Hamiltonian

$\rho_0(\mathbf z)$9

where $H_0$0 is monotonically decreasing. This makes the initial distribution passive relative to $H_0$1: regions of higher probability density are assigned lower auxiliary energy. The second step is an adiabatic return from $H_0$2 back to $H_0$3 (Campisi, 18 Aug 2025).

The mechanism is the adiabatic invariance of phase-space volume. If the dynamics generated by each frozen Hamiltonian is ergodic on its energy shells, then the enclosed phase volume $H_0$4 is an adiabatic invariant. Iso-$H_0$5 hypersurfaces are then mapped into iso-$H_0$6 hypersurfaces enclosing the same phase volume. Because $H_0$7, the final state becomes a function of $H_0$8 with the same ordering structure, and the paper states that the final state is exactly the passive or Gardner ground state (Campisi, 18 Aug 2025).

This construction is made explicit through the probability density over phase-volume shells for the quench Hamiltonian,

$H_0$9

Using $$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$0 and the phase-volume parametrization gives

$$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$1

and, under ergodicity and adiabatic invariance, the final mean energy is

$$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$2

The appendix of the paper shows that this equals the Gardner expression for the passive energy (Campisi, 18 Aug 2025).

The same work emphasizes that the ergodic assumption makes the protocol clean and exact, but the construction also serves as a design principle more broadly. In non-ergodic or topologically nontrivial cases, it can be supplemented by additional rearrangement or permutation steps, in analogy with the classical restacking ideas of plasma physics. This suggests that the quench-adiabat protocol is both an exact solution in the ergodic case and a constructive template in more general settings (Campisi, 18 Aug 2025).

4. Relative-entropic formulation and coherent/incoherent split

A distinct line of work reformulates classical ergotropy in information-theoretic terms. In this approach, extractable work is identified with the part of a distribution’s distance from equilibrium that cannot be removed by the allowed dynamics. For a classical phase-space distribution that is inhomogeneous on energy surfaces, the classical ergotropy is defined by the relative-entropy difference

$$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$3

where $$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$4 is the thermal distribution on the final energy surface $$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$5 (Sone et al., 2021).

In this formulation, classical ergotropy quantifies work stored in inhomogeneities on energy surfaces, described as the classical analogue of quantum coherence. The same paper rewrites it as

$$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$6

with

$$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$7

The physical interpretation is that the extractable work resides in the mismatch between the actual post-dynamics distribution and the reference distribution associated with the initial energy-surface population (Sone et al., 2021).

A second decomposition, developed directly in the phase-space framework, introduces a classical dephasing operator relative to $$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$8,

$$\mathcal E[\rho] = \max_{\varphi} \int d\mathbf z\, H_0(\mathbf z)\,\Big[\rho(\mathbf z)-\rho(\varphi^{-1}(\mathbf z))\Big],$$9

which homogenizes the distribution along each energy shell. The classical ergotropy then splits as

$\varphi$0

The corresponding relation to Kullback–Leibler divergences is

$\varphi$1

where $\varphi$2 is the passive state associated with $\varphi$3, $\varphi$4, and

$\varphi$5

quantifies the amount of inhomogeneity or coherence with respect to $\varphi$6 (Campisi, 18 Aug 2025).

These two information-theoretic perspectives are conceptually aligned. Both treat classical ergotropy as a measure of energy that is not exhausted by a coarse energy-shell description. In one language this appears as inhomogeneity on energy surfaces; in the other it appears as a coherent contribution relative to shell dephasing. A plausible implication is that the classical theory supports an internal distinction between shell-averaged population work and finer phase-space structure, even though no quantum superposition is involved.

5. Relation to quantum ergotropy and terminological usage

The quantum definition of ergotropy is

$\varphi$7

or, equivalently, the difference between the energy of $\varphi$8 and that of its passive rearrangement. In the absence of coherence in the energy basis, this reduces to work obtainable from rearranging populations only (Francica et al., 2020). Several quantum papers therefore use “classical” or “classical-like” language for the population-only sector of the quantum problem, even when they do not define a distinct classical phase-space theory (Hovhannisyan et al., 2024, Malavazi et al., 2024).

Taken together, these works suggest that “classical ergotropy” is used in two related but nonidentical ways.

Usage	Object	Representative source
Classical phase-space ergotropy	Phase-space density under Hamiltonian, volume-preserving dynamics	(Campisi, 18 Aug 2025, Campisi, 30 Mar 2026)
Classical-like or incoherent ergotropy	Work from populations after dephasing in the energy basis	(Francica et al., 2020, Malavazi et al., 2024)
Population-reordering analogue inside quantum theory	Probability distribution over energy levels rearranged without changing entropy	(Hovhannisyan et al., 2024)

In the dedicated classical literature, the object is a phase-space density and the passive state is a rearranged density in phase space. In much of the quantum-battery literature, by contrast, the closest analogue to a classical contribution is the incoherent ergotropy of the dephased state. One paper states explicitly that if the state has no coherence in the energy basis, then

$\varphi$9

so ergotropy reduces to the work obtainable from rearranging populations only (Francica et al., 2020). Another defines the incoherent component as

$\breve\rho$0

and treats it as the closest analogue of a classical extractable-work measure (Malavazi et al., 2024).

This terminological bifurcation is important. Classical ergotropy in the strict mechanical sense is not merely the diagonal restriction of a density matrix; it is a theory of phase-space rearrangement under Hamiltonian flow. Conversely, the population-only sector of the quantum problem is not a separate classical mechanics, but rather the diagonal limit of the quantum passive-state construction. Several papers emphasize the closeness of the two problems, and one argues that there is “very little, if not nothing at all,” genuinely quantum in the ergotropy problem beyond discreteness of the spectrum (Campisi, 18 Aug 2025).

6. Thermodynamic-limit behavior and conceptual status

Recent classical work also studies the fate of ergotropy in the thermodynamic limit. For densities of the form

$\breve\rho$1

one finds

$\breve\rho$2

These states are therefore asymptotically passive (Campisi, 30 Mar 2026).

The mechanism is illustrated by an ideal gas of $\breve\rho$3 particles in three dimensions, for which

$\breve\rho$4

and the phase volume scales as

$\breve\rho$5

with inverse

$\breve\rho$6

For a uniform density on an energy shell,

$\breve\rho$7

the passive rearrangement can be computed explicitly, and the ergotropy vanishes as $\breve\rho$8. The paper attributes this to concentration of measure: in high dimension, almost all the measure of a shell is concentrated near its outer boundary, leaving essentially no room to move probability inward to lower energy (Campisi, 30 Mar 2026).

This asymptotic passivity clarifies a frequent misconception. Classical ergotropy is not simply “energy above equilibrium,” nor is it equivalent to mean energy itself. It is the mechanically available part of the energy relative to the optimal passive rearrangement. A state may have substantial average energy and still have negligible ergotropy if its phase-space distribution is already ordered in the relevant sense. Conversely, inhomogeneity on energy shells can store extractable work even when no quantum coherence is present (Sone et al., 2021, Campisi, 30 Mar 2026).

The present state of the subject therefore has two complementary conclusions. First, classical ergotropy is a fully developed analogue of quantum ergotropy, with passive states, optimal rearrangements, explicit extraction protocols, and information-theoretic reformulations. Second, the term remains context-dependent across the literature: in dedicated classical mechanics it denotes a phase-space resource, whereas in quantum thermodynamics it often denotes the population-only or incoherent sector of a broader quantum quantity (Campisi, 18 Aug 2025, Francica et al., 2020).