Quantum thermodynamics

Push a large machine and the work it costs is a single, reliable number. Push a few quanta — drive a small system between two configurations — and the work is a distribution: repeat the protocol and each run costs something different, and some runs cost less than the free-energy difference, as if they cheated the second law. They did not. The Jarzynski equality shows that the right average of that fluctuating work recovers the equilibrium free energy exactly, and that what is wasted is never negative once you average properly. Drive one below.

Work fluctuates; the average does not lie

For a protocol driven at finite speed, the work $W$ has a distribution. The Jarzynski equality ties its exponential average to the equilibrium free-energy change: $\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}$. The ordinary average obeys the second law, $\langle W\rangle\ge\Delta F$, and the gap is the dissipated work — the energy irreversibly lost to the bath. Drive slowly and the distribution narrows toward $\Delta F$ with nothing dissipated; drive hard and it broadens, wasting more. For a Gaussian work distribution the whole story is closed-form.

$$\langle e^{-\beta W}\rangle=e^{-\beta\Delta F},\qquad \Delta F=\langle W\rangle-\tfrac{\beta\sigma^2}{2},\qquad W_{\rm diss}=\langle W\rangle-\Delta F=\tfrac{\beta\sigma^2}{2}\ \ge 0$$

Live — the work distribution, the Jarzynski free energy, and the dissipation (exact, Gaussian work)

P(W) ΔF (Jarzynski) W < ΔF tail

mean work ⟨W⟩ 1.00 drive strength σ 0.50 temperature T=1/β 1.00

ΔF: — W_diss = ⟨W⟩−ΔF: — runs with W<ΔF: —

The left panel is the work distribution; the green line is the free energy the exponential average recovers, always to the left of the mean $\langle W\rangle$, and the orange tail is the fraction of runs that cost less than $\Delta F$ — the apparent second-law violations the equality accounts for. The right panel plots the dissipated work against the drive strength: it is a parabola through the origin, so slowing the protocol toward equilibrium ($\sigma\to0$) drives the waste to zero, the reversible ideal. It never goes negative — that is the second law, recovered from fluctuations.

Deeper: fluctuation theorems and what is claimed

The Jarzynski equality is one of a family of fluctuation theorems that make the second law a statement about a distribution rather than a single trajectory; a stronger cousin, the Crooks theorem, relates the forward and reverse work distributions directly. The library's quantum-thermodynamics surface estimates the free energy and the dissipated work from calibrated work samples and tracks the entropy-production rate, but it carries an explicit claim boundary: it is a readiness and calibrated-protocol estimate only, with no thermodynamic-peak claim and no hardware submission. This panel takes the exactly-solvable Gaussian work distribution, where $\Delta F=\langle W\rangle-\beta\sigma^2/2$ in closed form, and its free energy and dissipation match a numerical evaluation of the Jarzynski average to better than $10^{-12}$.

Where it fits

Every gate and every pulse in the rest of the portal has a thermodynamic cost, and at the few-qubit scale that cost fluctuates. Fluctuation theorems are how the platform reasons about that cost honestly — not as a single efficiency number but as a distribution with a floor set by the free energy — and how it separates the work a protocol must spend from the work it merely wastes. It is the same coherence budget seen from the energy side.

Quantity	Meaning
$\langle W\rangle$	average work to run the protocol
$\Delta F$	equilibrium free-energy change — the unavoidable minimum
$W_{\rm diss}=\langle W\rangle-\Delta F$	dissipated work, $\ge 0$, zero only if reversible
Jarzynski $\langle e^{-\beta W}\rangle$	recovers $e^{-\beta\Delta F}$ exactly, from fluctuations

Quantum sensing Error correction Algorithms Back to overview

Evidence boundary: the panel computes the exactly-solvable Gaussian case of the Jarzynski equality live in your browser — the closed-form free energy and dissipated work checked against a numerical evaluation of the work average to better than $10^{-12}$. It illustrates the fluctuation theorem and the second law; it is not a hardware measurement, and the library's thermodynamics surface is explicit that it is a readiness estimate only.