Work Distributions

A work distribution describes run-to-run fluctuations of energy transferred by a driving protocol. In macroscopic thermodynamics one often tracks only average work. In a small quantum system, the whole distribution matters: individual realizations fluctuate, rare events control exponential averages, and the definition of a work random variable depends on the measurement protocol.

This page explains how work distributions are organized and interpreted. The standard closed-system construction is the Two-Point Measurement Scheme; here the focus is the distribution as a statistical object, with examples and diagnostics. Quantum Quenches owns the corresponding sudden many-body initial-value problem, Driven Many-Body Systems owns extensive absorption and heating, and Loschmidt Echo and Dynamical Phase Transitions Preview owns the pure-eigenstate characteristic-function relation, Fisher zeros, and return-rate singularities.

What a Work Distribution Is

For a finite isolated system with discrete energy measurements, a work distribution often has the form

$$P(W) = \sum_j p_j\, \delta(W-W_j), \qquad \sum_j p_j=1.$$

The labels $j$ may represent pairs of measured energy outcomes, monitored trajectories, or another operational record. The work values $W_j$ and probabilities $p_j$ are meaningful only after the protocol is specified.

The average work is

$$\langle W\rangle = \int dW\,W P(W),$$

and higher moments are

$$\langle W^k\rangle = \int dW\,W^k P(W).$$

For a discrete distribution, the integrals are shorthand for sums over the delta peaks.

Characteristic Functions and Cumulants

The characteristic function is

$$G(u) = \left\langle e^{iuW}\right\rangle = \int dW\,e^{iuW}P(W).$$

Moments follow from derivatives at $u=0$:

$$\langle W^k\rangle = \frac{1}{i^k} \left. \frac{d^kG}{du^k} \right|_{u=0}.$$

The cumulant generating function is

$$K(u) = \ln G(u).$$

The first two cumulants are the mean and variance:

$$\kappa_1 = \langle W\rangle, \qquad \kappa_2 = \langle W^2\rangle-\langle W\rangle^2.$$

Characteristic functions are often easier to access than $P(W)$ itself. Interferometric and full-counting-statistics protocols typically measure a generating function and reconstruct distributional information from it.

TPM Distribution

For a closed driven protocol with initial Hamiltonian $H_0$, final Hamiltonian $H_\tau$, and unitary $U_{\tau,0}$, the two-point measurement scheme gives

$$P_{\mathrm{TPM}}(W) = \sum_{m,n} p(m,n) \delta\!\left( W-E_m^\tau+E_n^0 \right),$$

where

$$p(m,n) = \operatorname{Tr} \left[ \Pi_m^\tau U_{\tau,0} \Pi_n^0\rho_0\Pi_n^0 U_{\tau,0}^\dagger \Pi_m^\tau \right].$$

This distribution is positive and normalized. If the initial state is thermal, it is the work distribution used in the standard quantum Jarzynski equality and Crooks relation. If the initial state has energy coherence, it is the distribution for the experiment in which the first projective energy measurement was actually performed.

Driven Two-Level System

A useful finite example has ground energy $0$ and excited energy $\epsilon_0$ initially, and ground energy $0$ and excited energy $\epsilon_\tau$ finally. Let the initial excited-state population be $p_e^0$, and set $p_g^0=1-p_e^0$.

Assume the drive produces transition probabilities

$$P_{g\to e} = P_{e\to g} = r, \qquad P_{g\to g} = P_{e\to e} = 1-r.$$

The four TPM work values are:

Initial Branch	Final Branch	Work
$g$	$g$	$0$
$g$	$e$	$\epsilon_\tau$
$e$	$g$	$-\epsilon_0$
$e$	$e$	$\epsilon_\tau-\epsilon_0$

The distribution is

$$\begin{aligned} P(W) = {}& p_g^0(1-r)\delta(W) + p_g^0 r\,\delta(W-\epsilon_\tau) \\ &+ p_e^0 r\,\delta(W+\epsilon_0) + p_e^0(1-r) \delta\!\left( W-\epsilon_\tau+\epsilon_0 \right). \end{aligned}$$

The mean work is

$$\langle W\rangle = p_g^0 r\,\epsilon_\tau - p_e^0 r\,\epsilon_0 + p_e^0(1-r)(\epsilon_\tau-\epsilon_0).$$

This example shows three generic features:

• work can be negative in individual runs even when the average is positive;
• nonadiabatic transitions broaden the distribution;
• changing gaps and changing populations contribute differently to the work values.

When $r=0$, each initial energy branch stays on its corresponding final branch. When $r$ is nonzero, the drive can promote the system from ground to excited or de-excite it while the Hamiltonian changes.

Sudden Harmonic Oscillator Quench

The frozen-state criterion and final-basis overlap probabilities are derived in Sudden Approximation. Here those probabilities are used to construct a work distribution.

A harmonic oscillator with frequency $\omega$ has Hamiltonian

$$H(\omega) = \frac{p^2}{2m} + \frac12m\omega^2x^2,$$

with energy levels

$$E_n(\omega) = \hbar\omega \left( n+\frac12 \right).$$

In a sudden frequency quench, the frequency changes from $\omega_0$ to $\omega_\tau$ so rapidly that the state does not have time to evolve during the change. The initial and final energy eigenstates are different oscillator bases. The TPM distribution is

$$P(W) = \sum_{m,n} p_n^0 \left| \langle m;\omega_\tau|n;\omega_0\rangle \right|^2 \delta\!\left( W - E_m(\omega_\tau) + E_n(\omega_0) \right).$$

The overlaps encode the mismatch between the two oscillator bases. For a sudden quench they are not generally diagonal in $m,n$, so the distribution has many peaks. Parity is conserved by the instantaneous frequency change, so only same-parity overlaps contribute.

The mean work can be found without summing all overlaps. For an initial thermal state of $H(\omega_0)$,

$$\langle H(\omega_0)\rangle_0 = \frac{\hbar\omega_0}{2} \coth \left( \frac{\beta\hbar\omega_0}{2} \right).$$

The sudden final average energy is

$$\langle H(\omega_\tau)\rangle_0 = \frac{\hbar}{4} \left( \omega_0 + \frac{\omega_\tau^2}{\omega_0} \right) \coth \left( \frac{\beta\hbar\omega_0}{2} \right).$$

Therefore

$$\langle W\rangle = \frac{\hbar}{4} \left( \frac{\omega_\tau^2}{\omega_0} - \omega_0 \right) \coth \left( \frac{\beta\hbar\omega_0}{2} \right).$$

This mean is useful, but it hides the distributional structure: the quench creates a squeezed-state energy distribution in the final oscillator basis.

Exponential Averages and Rare Events

Fluctuation relations often involve

$$\left\langle e^{-\beta W}\right\rangle.$$

This average weights low-work and negative-work events strongly. In simulations or experiments, it can converge slowly even when the ordinary mean $\langle W\rangle$ appears stable. A small number of rare events can dominate the estimate.

For a Gaussian work distribution with mean $\mu$ and variance $v$,

$$\left\langle e^{-\beta W}\right\rangle = \exp \left( -\beta\mu + \frac{\beta^2v}{2} \right).$$

This formula is not generally exact for quantum work distributions; it is a useful warning. Fluctuations are not a small correction inside exponential work identities.

Coherence and Quasiprobabilities

When the initial state has energy coherence, a projective initial energy measurement changes the state. There are then several inequivalent questions:

• What work distribution is obtained if the initial energy is strongly measured?
• What average energy change occurs if the initial measurement is not performed?
• Can one construct a weak-measurement or interferometric quasiprobability that keeps some coherence information?
• Which distribution appears in a particular fluctuation theorem?

Quasiprobability approaches can retain phase-sensitive information, but they may become negative or complex. That is not a defect if the object is not being claimed as an ordinary probability distribution. It is a sign that the protocol is answering a different operational question from TPM.

The practical rule is simple: do not compare work distributions unless their measurement protocols match.

Open and Monitored Systems

For open systems, the work distribution is not determined by the reduced system energy difference alone. A stochastic first-law statement along a monitored trajectory often has the schematic form

$$\Delta e[\gamma] = w[\gamma] + \sum_\alpha q_\alpha[\gamma],$$

where $w[\gamma]$ is work assigned to external driving and $q_\alpha[\gamma]$ is heat exchanged with reservoir $\alpha$. This expression requires a trajectory model, a sign convention, and jump or measurement records that correspond to physical energy exchanges.

Examples include:

• quantum-jump trajectories with thermally consistent jump operators;
• full counting statistics for reservoir energy or particle number;
• continuously monitored systems with feedback;
• inclusive system-plus-bath energy measurements.

A reduced master equation may predict $\rho(t)$ accurately while still being insufficient to define a trustworthy work distribution. The record-level model matters.

Diagnostics for a Claimed Distribution

Before using a work distribution, check:

• Is the protocol closed, open, monitored, or inclusive system-plus-bath?
• Which energy measurements or counting fields define $W$?
• Does the initial state commute with the initial Hamiltonian?
• Are degeneracies handled by energy projectors or by refined measurements?
• Is the distribution positive, or is it a quasiprobability?
• Does the mean agree with the relevant energy balance?
• Does the distribution satisfy the intended normalization and fluctuation relation?
• Are rare events sampled well enough for exponential averages?

These checks prevent a common failure mode: computing a plausible histogram and then interpreting it with the wrong theorem.

Common Mistakes

1. Calling any histogram of final-minus-initial system energy “the work distribution” without specifying the measurement protocol.

2. Inferring Jarzynski or Crooks relations from a distribution whose initial state, dynamics, or reverse protocol does not satisfy the theorem’s assumptions.

3. Ignoring negative-work events because they are rare.

4. Treating quasiprobability negativity as a numerical error without checking the protocol.

5. Using average work alone to validate a distribution.

6. Applying closed-system TPM formulas to dissipative dynamics without tracking heat.

Exercises

Two-Level Mean Work

Starting from the two-level distribution above, derive the mean work.

Solution

The distribution has four branches:

$$\begin{array}{c|c|c} \text{branch} & \text{probability} & \text{work} \\ \hline g\to g & p_g^0(1-r) & 0 \\ g\to e & p_g^0 r & \epsilon_\tau \\ e\to g & p_e^0 r & -\epsilon_0 \\ e\to e & p_e^0(1-r) & \epsilon_\tau-\epsilon_0 \end{array}$$

Multiplying probability by work and summing gives

$$\langle W\rangle = p_g^0 r\,\epsilon_\tau - p_e^0 r\,\epsilon_0 + p_e^0(1-r)(\epsilon_\tau-\epsilon_0).$$

The $g\to g$ branch contributes zero.

Moments from the Characteristic Function

Show that $G(u)=\langle e^{iuW}\rangle$ satisfies

$$\langle W\rangle = \frac{1}{i}G'(0), \qquad \langle W^2\rangle = \frac{1}{i^2}G''(0).$$

Solution

Differentiate under the average:

$$G'(u) = \left\langle iW e^{iuW} \right\rangle.$$

At $u=0$,

$$G'(0)=i\langle W\rangle,$$

so

$$\langle W\rangle=\frac{1}{i}G'(0).$$

Differentiating twice gives

$$G''(u) = \left\langle (iW)^2e^{iuW} \right\rangle.$$

Thus

$$G''(0)=i^2\langle W^2\rangle,$$

and

$$\langle W^2\rangle = \frac{1}{i^2}G''(0).$$

Sudden Oscillator Quench Mean

For an initial thermal oscillator at frequency $\omega_0$, use

$$\left\langle \frac{p^2}{2m}\right\rangle_0 = \left\langle \frac12m\omega_0^2x^2\right\rangle_0 = \frac{\hbar\omega_0}{4} \coth \left( \frac{\beta\hbar\omega_0}{2} \right)$$

to derive the mean work for a sudden quench to $\omega_\tau$.

Solution

The initial energy is

$$\langle H(\omega_0)\rangle_0 = \frac{\hbar\omega_0}{2} \coth \left( \frac{\beta\hbar\omega_0}{2} \right).$$

The kinetic part immediately after the quench is unchanged:

$$\left\langle \frac{p^2}{2m}\right\rangle_0 = \frac{\hbar\omega_0}{4} \coth \left( \frac{\beta\hbar\omega_0}{2} \right).$$

The position variance follows from the initial potential energy:

$$\frac12m\omega_0^2\langle x^2\rangle_0 = \frac{\hbar\omega_0}{4} \coth \left( \frac{\beta\hbar\omega_0}{2} \right),$$

so

$$\frac12m\omega_\tau^2\langle x^2\rangle_0 = \frac{\hbar\omega_\tau^2}{4\omega_0} \coth \left( \frac{\beta\hbar\omega_0}{2} \right).$$

Therefore

$$\langle H(\omega_\tau)\rangle_0 = \frac{\hbar}{4} \left( \omega_0 + \frac{\omega_\tau^2}{\omega_0} \right) \coth \left( \frac{\beta\hbar\omega_0}{2} \right).$$

Subtracting the initial energy gives

$$\langle W\rangle = \frac{\hbar}{4} \left( \frac{\omega_\tau^2}{\omega_0} - \omega_0 \right) \coth \left( \frac{\beta\hbar\omega_0}{2} \right).$$

References

• P. Talkner, E. Lutz, and P. Hänggi, “Fluctuation theorems: Work is not an observable,” Physical Review E 75, 050102, 2007.
• M. Campisi, P. Hänggi, and P. Talkner, “Colloquium: Quantum fluctuation relations: Foundations and applications,” Reviews of Modern Physics 83, 771, 2011.
• M. Esposito, U. Harbola, and S. Mukamel, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,” Reviews of Modern Physics 81, 1665, 2009.
• S. Deffner and S. Campbell, Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information, Morgan & Claypool, 2019.
• H. T. Quan and W. H. Zurek, “Testing quantum adiabaticity with quench echoes,” New Journal of Physics 12, 093025, 2010.