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. For extended systems, see Quantum Quenches and Driven Many-Body Systems. For an initial energy eigenstate, the characteristic function also connects to the Loschmidt amplitude.

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

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

Choose the logarithm continuously near $u=0$, with $K(0)=0$. Zeros of $G(u)$ can obstruct a single logarithm over the whole real axis. When the required moments exist, 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, the two-point measurement scheme supplies joint probabilities $p(m,n)$ for initial and final energies. Group branches with the same difference to obtain

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

This positive distribution describes the experiment with the initial energy measurement included. For coherent initial states its mean can differ from the energy change in an experiment that omits that measurement.

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 $p_g^0=1-p_e^0$.

A unitary drive on a two-dimensional Hilbert space has equal transition probabilities in the two directions. Write

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

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

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

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 add or redistribute work branches; the variance need not grow monotonically with transition probability;
• 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 initial Gibbs state is a squeezed thermal state relative to the final oscillator modes, with a structured energy distribution. It is pure only for an initial ground state.

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 $v$,

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

The formula is exact for a Gaussian distribution. A generic quantum work distribution is not Gaussian, so its higher cumulants cannot simply be dropped.

For a concrete sampling example, take a thermal qubit with $\beta\epsilon=5$ and a cyclic drive that swaps its two energy levels. Almost every run has $W=+\epsilon$, while $W=-\epsilon$ occurs with probability $p_e=1/(1+e^5)\simeq0.00669$. Nevertheless, the rare branch contributes $p_e e^5\simeq0.9933$ to the exact exponential average of one. In 100 independent runs, the chance of seeing no such branch is $(1-p_e)^{100}\simeq0.51$. A stable-looking sample mean therefore does not guarantee an accurate Jarzynski estimate.

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.

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

• M. Campisi, P. Hänggi, and P. Talkner, “Colloquium: Quantum fluctuation relations: Foundations and applications,” Reviews of Modern Physics 83, 771–791 (2011). arXiv:1012.2268.
• S. Deffner and S. Campbell, Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information, Morgan & Claypool, 2019.
• M. Esposito, U. Harbola, and S. Mukamel, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,” Reviews of Modern Physics 81, 1665–1702 (2009).
• H. T. Quan and W. H. Zurek, “Testing quantum adiabaticity with quench echoes,” New Journal of Physics 12, 093025 (2010).
• P. Talkner, E. Lutz, and P. Hänggi, “Fluctuation theorems: Work is not an observable,” Physical Review E 75, 050102(R) (2007).