A geometric measure of extractable energy

Energy is not the same as available work

A physical system can contain energy without being able to release it as useful work. The distinction depends on its state and on which operations are allowed.

In the setting studied by Michele Campisi, a system is thermally isolated: it does not exchange heat with its surroundings. An external drive changes a parameter and eventually returns it to its original value. The maximum energy that can be extracted through such cyclic driving is called ergotropy, also known as available energy.

After this maximum extraction, the system reaches its associated passive state. Under the same class of operations, no more work can be extracted from that state. Passive does not mean empty of energy, nor does it necessarily mean ordinary thermal equilibrium. Thermal states are examples, but the family of passive states is broader.

Campisi’s theoretical preprint, submitted on 8 November 2026, asks whether ergotropy can be expressed as a precise mathematical distance from this passive counterpart.

Comparing two energy profiles

The paper’s answer involves an energy profile: a probability distribution describing how likely different energies are. For a quantum system, it gives the probabilities of the possible outcomes of an energy measurement.

Two profiles enter the comparison. One belongs to the initial state. The other belongs to its associated passive state, constructed by rearranging that state according to the rules of the problem.

The distance between them is the 1-Wasserstein distance, a concept from optimal transport. Imagine redistributing a pile of material into a target shape. Moving more material, or moving it farther, costs more. The Wasserstein distance is the smallest total cost among all redistributions that produce the target.

Here, the material is probability and the distance travelled is a difference in energy. The resulting transport cost therefore has units of energy. Campisi reports that it equals the ergotropy, for both the classical and quantum formulations considered.

This is not a distance between arbitrary physical states. It compares a specific pair of energy distributions: the original profile and the profile of that state’s passive companion.

Why the equality emerges

The proof uses cumulative probabilities. Instead of asking how likely one particular energy is, a cumulative distribution asks how likely the energy is to fall at or below a chosen threshold.

The crucial ordering is that the passive companion has at least as much cumulative probability below every energy threshold as the initial state. Its energy distribution is shifted toward lower energies in this precise sense.

For distributions along a single axis, the 1-Wasserstein distance is the area between their cumulative curves. Because these curves have the ordering established in the paper, that area also equals the difference in their mean energies. That difference is the ergotropy.

The classical argument uses rearrangements of probability in position-and-momentum space. The quantum argument orders the state’s eigenvalues from largest to smallest and assigns them to energy levels from lowest to highest. Both lead to the same energy-profile interpretation.

Two oscillator examples

Campisi illustrates the result with classical and quantum harmonic oscillators.

In the classical example, a Gaussian distribution in position and momentum is displaced from the oscillator’s centre. Its passive companion is the centred Gaussian. In the paper’s chosen units, a position displacement of magnitude a gives an ergotropy of a²/2. The width of the Gaussian does not affect that result: the extractable part comes from the displacement, rather than its statistical spread.

For the quantum example, the oscillator starts in a coherent state. Its passive companion is the quantum ground state. With the energy-level spacing set to one, the ergotropy is |α|², where α describes the coherent state. The ground-state energy remains after extraction; it is not counted as available work.

A distance, not an extraction recipe

The equality does not mean that an actual work-extraction process follows an optimal-transport plan. Campisi explicitly distinguishes the two. Physical evolution obeys additional constraints, whereas the Wasserstein calculation optimizes over probability transport plans with the required starting and ending distributions.

The work is a mathematical preprint, not an experimental demonstration or a practical engine design. Its contribution is a precise interpretation of available energy relative to an associated passive state—not a universal measure of departure from every meaning of equilibrium.