Jarzynski Equality and Crooks Relation

Classical thermodynamics bounds the work done on a system driven between two equilibrium states by the free-energy difference: $W\ge\Delta F$ , with equality only for a quasi-static (reversible) process. But for a finite-time, far-from-equilibrium drive the work $W$ is a fluctuating quantity — it depends on the initial microstate and on the thermal noise along the path, and it differs from pull to pull. The remarkable discovery of Jarzynski (1997) and Crooks (1999) is that the full statistics of $W$ encode the equilibrium $\Delta F$ exactly, no matter how violently the system is driven. You never need to go slowly; you need only average correctly. Concretely, fix a control protocol $\lambda(t)$ that sweeps an external parameter from $\lambda_A$ to $\lambda_B$ over a time $\tau$ , starting from canonical equilibrium at temperature $T$ . Repeating the experiment builds a distribution $P_F(W)$ of work values. The Jarzynski equality states that the exponential average $\langle e^{-\beta W}\rangle$ collapses this entire distribution onto $e^{-\beta\Delta F}$ , where $\Delta F=F_B-F_A$ . The Crooks relation goes deeper: it compares $P_F(W)$ with the work distribution $P_R(W)$ of the reverse protocol run from equilibrium at $\lambda_B$ , and links the two point-by-point by an exponential factor. Their histograms cross exactly at $W=\Delta F$ — a model-free free-energy readout. Both results are exact consequences of microscopic reversibility. For Hamiltonian dynamics, Liouville's theorem (phase-space volume conservation) plus canonical Boltzmann weighting of the initial conditions yields a precise symmetry between a trajectory and its time-reverse; the same structure holds for stochastic dynamics with local detailed balance — Langevin, Fokker–Planck, and Markov-jump master equations. These identities are the work-relation corollaries of the general fluctuation theorems, and they sit at the foundation of stochastic thermodynamics, generalizing the fluctuation-dissipation theorem to the genuinely nonlinear regime. Beyond their conceptual weight they are practical: experimentalists extract equilibrium free-energy landscapes from irreversible single-molecule pulling. The price is statistical — the exponential average is dominated by rare, atypical trajectories, so convergence is slow and bias-corrected estimators (the Bennett acceptance ratio) are essential.

Key takeaways

The Jarzynski equality $\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}$ relates the exponential average of non-equilibrium work over many repeated, arbitrarily fast drivings to the equilibrium free-energy difference between the endpoints.
The Crooks fluctuation theorem $P_F(W)/P_R(-W)=e^{\beta(W-\Delta F)}$ is the finer, distribution-level statement: the forward and time-reversed work histograms cross exactly at $W=\Delta F$ .
Both follow from microscopic reversibility — the detailed-balance-like symmetry of trajectory weights under time reversal of the protocol; Jarzynski is the integral (first-moment) form of Crooks.
Jensen's inequality applied to Jarzynski recovers the second law $\langle W\rangle\ge\Delta F$ , with dissipated work $\langle W_{\text{diss}}\rangle=\langle W\rangle-\Delta F=k_BT\,D(P_F\|P_R)\ge0$ .
Rare trajectories with $W<\Delta F$ (transient second-law "violations") dominate the exponential average, making direct sampling hard but enabling free-energy reconstruction from finite-time pulling.
Key idea: microscopic reversibility makes the ratio of forward to time-reversed trajectory probabilities equal to $e^{\beta(W-\Delta F)}$ ; integrating this single relation gives Crooks (distribution level) and Jarzynski (mean level), with the second law emerging as a convexity inequality.

The core insight

Weight each non-equilibrium pull by $e^{-\beta W}$ and the rare cheap pulls are up-weighted by exactly the amount needed to cancel the irreversibility tax — leaving precisely $e^{-\beta\Delta F}$ . One trajectory-symmetry relation, $P_F(W)/P_R(-W)=e^{\beta(W-\Delta F)}$ , contains both Crooks and Jarzynski, and the second law is its convexity corollary.

Where this lives today

The work relations turned free-energy measurement into a non-equilibrium art. In optical-tweezer and atomic-force experiments an RNA hairpin or protein is pulled open at finite speed; the forward (unfolding) and reverse (refolding) work histograms are combined through the Bennett acceptance ratio — the statistically optimal use of Crooks — to recover the folding free energy even when every pull is strongly irreversible. In molecular simulation, the instantaneous-switch limit of Jarzynski is Zwanzig's free-energy perturbation, and finite-time "fast-growth" steered molecular dynamics generalizes it: many short non-equilibrium trajectories, exponentially averaged, replace slow reversible thermodynamic integration. Because the dissipated work equals a Kullback–Leibler divergence, the relations also give a literal, measurable arrow of time, and they underpin the thermodynamic uncertainty relations that bound the precision of molecular machines like kinesin and ATP synthase by their dissipation.

The physical picture

Imagine yanking a piston to compress a gas. Sometimes the molecules happen to be moving away from the piston and you do little work; sometimes they slam into it and you do a lot. Average the ordinary work and you always pay an irreversibility tax, $\langle W\rangle>\Delta F$ . But weight each outcome by $e^{-\beta W}$ and the rare cheap compressions are exponentially up-weighted — by exactly the amount needed to cancel the tax — leaving precisely $e^{-\beta\Delta F}$ . This is why the equality holds however fast you pull: the free energy is recovered not by going slowly, but by letting the exponential average find the rare, low-work tail.

Figure

Forward and reverse work histograms cross exactly at W = ΔF; the low-W tail of P_F carries the Jarzynski average

The Crooks relation is a statement of trajectory symmetry: film a forward driving and play it backward. The backward movie is a legitimate trajectory of the reverse protocol, but it is exponentially less likely by the factor $e^{-\beta(W-\Delta F)}$ . The work that would be dissipated forward becomes the imbalance between watching the film forwards versus backwards — a direct, quantitative arrow of time. Because $\langle e^{-\beta W}\rangle$ is dominated by trajectories with $W$ small or negative — which sit in the exponential tail of $P_F(W)$ — Jarzynski estimation needs samples of those rare "transient second-law violations." This is why fast pulling gives noisy free energies while gentle pulling (a narrow $P_F$ near $\Delta F$ ) converges quickly.

Figure

Gaussian-work limit: dissipation is fixed by the work variance, ΔF = ⟨W⟩ − βσ_W²∕2

The core equations

The headline result is the Jarzynski equality: the exponential average of the fluctuating work equals the exponential of the equilibrium free-energy difference, for any driving speed.

\big\langle e^{-\beta W}\big\rangle_F = e^{-\beta\Delta F} \qquad (1)

where: $W$ — thermodynamic (inclusive) work done on the system in the forward protocol, a fluctuating quantity; $\beta=1/k_BT$ — inverse temperature of the initial canonical ensemble; $\Delta F=F_B-F_A$ — equilibrium Helmholtz free-energy difference; $\langle\cdot\rangle_F$ — average over forward realizations with $x_0\sim e^{-\beta H_A}$ . Physically: the whole work distribution, exponentially averaged, collapses onto a single equilibrium number. The finer, distribution-level statement is the Crooks fluctuation theorem — a point-by-point symmetry between the forward and reverse work distributions.

\frac{P_F(W)}{P_R(-W)} = e^{\beta(W-\Delta F)} \qquad (2)

where: $P_F(W)$ — density of work $W$ under the forward protocol $\lambda_A\to\lambda_B$ ; $P_R(-W)$ — density of work $-W$ under the reverse protocol $\lambda_B\to\lambda_A$ started at equilibrium at $\lambda_B$ ; $\Delta F$ — the same equilibrium free-energy difference. Physically: watching the film backward is exponentially less likely by the dissipated work. The work in these relations is the inclusive thermodynamic work — the energy injected by changing the control parameter — not the heat and not a mechanical force times displacement.

W = \int_0^\tau \dot\lambda(t)\,\frac{\partial H_\lambda\big(x(t)\big)}{\partial\lambda}\,\mathrm{d}t \qquad (3)

where: $H_\lambda(x)$ — Hamiltonian parameterized by the control variable $\lambda$ ; $\lambda(t)$ — imposed protocol, $\lambda(0)=\lambda_A$ , $\lambda(\tau)=\lambda_B$ ; $x(t)$ — microstate trajectory; $\tau$ — switching time. Physically: work is accumulated only through the explicit time-dependence of the Hamiltonian. Applying Jensen's inequality to Eq. (1) yields the second law, and identifies the dissipated work with a relative entropy between path ensembles.

\langle W_{\text{diss}}\rangle = \langle W\rangle - \Delta F = k_BT\,D\big(P_F[X]\,\|\,P_R[\tilde X]\big) \ge 0 \qquad (4)

where: $W_{\text{diss}}$ — dissipated (irreversible) work; $D(\cdot\|\cdot)$ — Kullback–Leibler divergence between forward and reverse trajectory ensembles; $X,\tilde X$ — a forward path and its time-reverse. Physically: dissipation measures how distinguishable a process is from its time-reverse. Setting $W=\Delta F$ in Crooks gives the experimentally decisive histogram-crossing identity: the two distributions have equal height exactly at the free energy.

P_F(W=\Delta F) = P_R(W=-\Delta F) \qquad (5)

where: $\Delta F$ — read directly off the intersection of the measured forward and reverse work histograms. Physically: a model-free free-energy readout that never requires trusting either exponential tail.

Derivation AJarzynski equality from Hamiltonian dynamics and Liouville's theorem

The cleanest derivation needs no stochasticity at all: drive an isolated Hamiltonian system through the protocol, with the only randomness in the initial microstate drawn from canonical equilibrium at $\lambda_A$ . The result — that the exponential work average equals $Z_B/Z_A$ — hinges on a single geometric fact, Liouville's incompressibility of Hamiltonian flow, which makes the change of variables from initial to final phase point volume-preserving. We write the work as an energy change, average over the canonical initial ensemble, watch the initial Boltzmann factor cancel, and use Liouville to fold the remaining integral into the final partition function.

Step 1. For isolated Hamiltonian flow with time-dependent $H_\lambda$ there is no heat bath, so the work equals the energy change: $W(x_0)=H_B(x_\tau(x_0))-H_A(x_0)$ , where $x_\tau(x_0)$ is the time- $\tau$ image of $x_0$ . Why: with no bath, the first law $W=\Delta E+Q$ reduces to $W=\Delta E$ .

Step 2. Average over canonical initial states $\rho_A(x_0)=e^{-\beta H_A(x_0)}/Z_A$ : $\;\langle e^{-\beta W}\rangle=\tfrac{1}{Z_A}\!\int\!\mathrm{d}x_0\,e^{-\beta H_A(x_0)}\,e^{-\beta[H_B(x_\tau)-H_A(x_0)]}$ . Why: each trajectory is weighted by the probability of its initial condition.

Step 3. The initial Boltzmann factor and the $H_A(x_0)$ inside the work exponential cancel, leaving $\tfrac{1}{Z_A}\!\int\!\mathrm{d}x_0\,e^{-\beta H_B(x_\tau(x_0))}$ . Why: $e^{-\beta H_A}\cdot e^{+\beta H_A}=1$ ; only the final energy survives.

Step 4. Change variables $x_0\to x_\tau$ ; by Liouville's theorem Hamiltonian flow conserves phase-space volume, so $\mathrm{d}x_0=\mathrm{d}x_\tau$ (unit Jacobian), giving $\tfrac{1}{Z_A}\!\int\!\mathrm{d}x_\tau\,e^{-\beta H_B(x_\tau)}=\tfrac{Z_B}{Z_A}$ . Why: incompressibility is the load-bearing step — without it the Jacobian would spoil the identity.

Step 5. Identify $Z_B/Z_A=e^{-\beta(F_B-F_A)}=e^{-\beta\Delta F}$ , independent of protocol speed. Why: $F=-k_BT\ln Z$ turns the partition-function ratio into the free-energy difference. Verified in verify_aq-sm-jarzynski-crooks-relations.py: on a driven harmonic oscillator with $Z\propto1/\sqrt{k}$ , the Liouville cancellation gives $\langle e^{-\beta W}\rangle=Z_B/Z_A=\sqrt{k_A/k_B}$ , matched symbolically.

Go deeper · Derivation B · 1 minCrooks relation from microscopic reversibility, and Jarzynski as its integral

Derivation BCrooks relation from microscopic reversibility, and Jarzynski as its integral

The distribution-level theorem comes from the ratio of a forward path weight to its time-reverse under stochastic dynamics with local detailed balance; Jarzynski is then simply the first moment of Crooks. The point is that a single trajectory identity — the path-weight ratio equals $e^{\beta(W-\Delta F)}$ — contains both relations, so once it is established the rest is bookkeeping. We form the path-weight ratio, insert the canonical initial densities, use the first law to convert heat-plus-energy into work, bin by work to get Crooks, and integrate to get Jarzynski.

Step 1. Local detailed balance gives the transition-weight ratio in terms of heat, $\mathcal{T}_F[\gamma]/\mathcal{T}_R[\tilde\gamma]=e^{\beta Q[\gamma]}$ ; multiplying by the initial densities, $\mathcal{P}_F[\gamma]/\mathcal{P}_R[\tilde\gamma]=(\rho_A/\rho_B)\,e^{\beta Q}$ . Why: microscopic reversibility of the bath-coupled dynamics is the physical input; $Q$ is the heat to the bath.

Step 2. Insert $\rho_A/\rho_B=(Z_B/Z_A)\,e^{-\beta(H_A-H_B)}=e^{\beta\Delta F}e^{\beta\Delta E}$ with $\Delta E=H_B-H_A$ , so $\mathcal{P}_F/\mathcal{P}_R=e^{\beta(\Delta E+Q-\Delta F)}$ . Why: the two canonical partition functions produce the $\Delta F$ , and the energy difference combines with the heat.

Step 3. Use the first law $W=\Delta E+Q$ to get $\mathcal{P}_F[\gamma]/\mathcal{P}_R[\tilde\gamma]=e^{\beta(W-\Delta F)}$ — the trajectory form of microscopic reversibility. Why: the ratio depends on the path only through its work and the fixed $\Delta F$ .

Step 4. Bin forward paths by work, $P_F(W)=\int\!\mathcal{D}\gamma\,\mathcal{P}_F[\gamma]\,\delta(W-W[\gamma])$ , and use $W[\tilde\gamma]=-W[\gamma]$ to obtain $P_F(W)=e^{\beta(W-\Delta F)}P_R(-W)$ — the Crooks relation, Eq. (2). Why: the reverse-path measure is unchanged under $\gamma\to\tilde\gamma$ ; only the work flips sign.

Step 5. Multiply by $e^{-\beta W}$ and integrate: $\langle e^{-\beta W}\rangle_F=\int\!P_F(W)e^{-\beta W}\mathrm{d}W=e^{-\beta\Delta F}\!\int\!P_R(-W)\mathrm{d}W=e^{-\beta\Delta F}$ — Jarzynski, Eq. (1). Why: normalization of $P_R$ closes the proof; Jarzynski is the first moment of Crooks. Verified in verify_aq-sm-jarzynski-crooks-relations.py: two Gaussians satisfying Crooks integrate to the Jarzynski identity, and their crossing sits exactly at $W=\Delta F$ .

Go deeper · Derivation C · 1 minSecond law and dissipation as relative entropy

Derivation CSecond law and dissipation as relative entropy

Applying Jensen's inequality to Jarzynski turns the exact equality into the familiar inequality $\langle W\rangle\ge\Delta F$ , and the excess — the dissipated work — is exactly $k_BT$ times a Kullback–Leibler divergence between forward and reverse path ensembles. This gives the second law both a proof and a meaning: dissipation is the time-reversal asymmetry of the process. We apply convexity, combine with Jarzynski, then re-express the gap as an averaged log-ratio.

Step 1. Convexity of the exponential gives $\langle e^{-\beta W}\rangle\ge e^{-\beta\langle W\rangle}$ (Jensen). Why: $\langle e^x\rangle\ge e^{\langle x\rangle}$ for any random variable.

Step 2. Combine with Jarzynski $\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}$ : $\;e^{-\beta\Delta F}\ge e^{-\beta\langle W\rangle}\Rightarrow\langle W\rangle\ge\Delta F$ . Why: taking logs (a monotone step) delivers the macroscopic second law exactly.

Step 3. Insert $\ln(\mathcal{P}_F/\mathcal{P}_R)=\beta(W-\Delta F)$ and average: $\langle W\rangle-\Delta F=k_BT\!\int\!\mathcal{D}\gamma\,\mathcal{P}_F\ln(\mathcal{P}_F/\mathcal{P}_R)=k_BT\,D(\mathcal{P}_F\|\mathcal{P}_R)\ge0$ . Why: the averaged log-ratio is a KL divergence, which is non-negative and vanishes only in the reversible limit. Verified in verify_aq-sm-jarzynski-crooks-relations.py: Jensen numerically ( $\langle e^{-\beta W}\rangle\ge e^{-\beta\langle W\rangle}$ ) and $\langle W\rangle\ge\Delta F$ for a Gaussian work distribution.

Limiting cases

The work relations interpolate between the reversible limit and linear response, and contain the second law:

Case	Result	Meaning
Quasi-static (reversible)	$P_F(W)\to\delta(W-\Delta F),\ \langle W\rangle=\Delta F$	Infinitely slow driving: work has no spread and equals $\Delta F$ ; both relations become trivial identities.
Second law (Jensen)	$\langle W\rangle\ge\Delta F,\ \langle W_{\text{diss}}\rangle=k_BT\,D(P_F\\|P_R)\ge0$	Convexity of $e^{-x}$ gives the second law; dissipation equals a relative entropy, vanishing only reversibly.
Near-equilibrium (Gaussian work)	$\langle W\rangle-\Delta F=\tfrac{\beta}{2}\sigma_W^2$	If $P_F$ is Gaussian, mean dissipation is fixed by the work variance — the linear-response fluctuation-dissipation relation.
Instantaneous switch ( $\tau\to0$ )	$\langle e^{-\beta W}\rangle=\tfrac{\int e^{-\beta H_B}}{\int e^{-\beta H_A}}=e^{-\beta\Delta F}$	Sudden switch $W=H_B(x)-H_A(x)$ at fixed $x$ ; Jarzynski reduces to Zwanzig free-energy perturbation.

Worked examples

Example 1 — free energy from a Gaussian work distribution. A biomolecule is pulled fast enough that the forward work distribution is Gaussian with mean $\langle W\rangle=5\,k_BT$ and standard deviation $\sigma_W=2\,k_BT$ . What is $\Delta F$ , and how much work is dissipated?

Step 1. For a Gaussian, $\langle e^{-\beta W}\rangle=e^{-\beta\langle W\rangle+\beta^2\sigma_W^2/2}$ . Jarzynski sets this to $e^{-\beta\Delta F}$ , so $\Delta F=\langle W\rangle-\tfrac{\beta}{2}\sigma_W^2$ .

Step 2. In units of $k_BT$ : $\Delta F=5-\tfrac12(2)^2=5-2=\mathbf{3\,k_BT}$ . The dissipated work is $\langle W_{\text{diss}}\rangle=\langle W\rangle-\Delta F=\mathbf{2\,k_BT}=\tfrac{\beta}{2}\sigma_W^2$ .

Step 3. Check consistency: $\langle e^{-\beta W}\rangle=e^{-5+2}=e^{-3}=e^{-\beta\Delta F}$ ✓. Note that $\langle W\rangle=5>\Delta F=3$ , respecting the second law — yet individual pulls with $W<3\,k_BT$ occur and dominate the exponential average.

Example 2 — how rare are the trajectories that dominate? For the same Gaussian, what fraction of pulls have $W<\Delta F$ (transient "second-law violations"), and where do the forward/reverse histograms cross?

Step 1. The fraction with $W<\Delta F=3$ is $\Phi\!\big((3-5)/2\big)=\Phi(-1)=\tfrac12\mathrm{erfc}(1/\sqrt2)=\mathbf{0.159}$ , about 16% of pulls.

Step 2. These low-work pulls carry the average: the integrand $P_F(W)e^{-\beta W}$ peaks at $W=\langle W\rangle-\beta\sigma_W^2=5-4=\mathbf{1\,k_BT}$ — well into the tail below $\Delta F=3$ , which is exactly why naive sampling converges slowly.

Step 3. The Crooks crossing sits at $W=\Delta F=\mathbf{3\,k_BT}$ : there $P_F(3)=P_R(-3)$ , giving a model-free readout of $\Delta F$ from where the two measured histograms intersect — the method Collin and co-workers used to recover RNA folding free energies to within $k_BT$ in 2005.

Origins · 1 minchristopher jarzynski and how this was discovered

christopher jarzynski

b. 1965, American physicist · Christopher Jarzynski, then between Los Alamos and the University of Washington (the flagship PRL carries the Washington/INT affiliation), published in 1997 the non-equilibrium work equality now bearing his name:

\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}

, showing that equilibrium free-energy differences are encoded exactly in the statistics of arbitrarily fast, irreversible driving. Gavin Crooks soon supplied the finer detailed relation. The result overturned the intuition that only quasi-static processes reveal free energies, seeded the field of stochastic thermodynamics, and became a standard tool for single-molecule and computational free-energy measurement.

History

1997

— Jarzynski publishes the nonequilibrium work equality (Phys. Rev. Lett. 78, 2690)

1998

— Jarzynski gives the stochastic-dynamics derivation via microscopic reversibility

1999

— Crooks establishes the detailed work fluctuation theorem (Phys. Rev. E 60, 2721)

2002

— Liphardt, Dumont, Smith, Tinoco and Bustamante test Jarzynski by RNA pulling (Science 296, 1832)

2005

— Collin and co-workers verify the Crooks relation and recover RNA folding free energies (Nature 437, 231)

Going further

Texts. Jarzynski, "Nonequilibrium Equality for Free Energy Differences," Phys. Rev. Lett. 78, 2690 (1997) · Crooks, "Entropy production fluctuation theorem and the nonequilibrium work relation," Phys. Rev. E 60, 2721 (1999) · Seifert, Rep. Prog. Phys. 75, 126001 (2012) · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity · Kardar, Statistical Physics of Particles · Collin et al., Nature 437, 231 (2005).

Provenance. Jarzynski equality verified by the Hamiltonian Liouville derivation (Jarzynski 1997; Kardar). Crooks relation and corollaries verified from local detailed balance (Crooks 1999; Seifert 2012). Verification reproduces (1) the Liouville cancellation $\langle e^{-\beta W}\rangle=Z_B/Z_A$ on a driven oscillator; (2) Crooks $\Rightarrow$ Jarzynski by integration; (3) the Jensen route to $\langle W\rangle\ge\Delta F$ ; (4) the Gaussian-work result $\Delta F=\langle W\rangle-\beta\sigma_W^2/2$ and dissipation $\beta\sigma_W^2/2$ ; (5) the instantaneous-switch (Zwanzig) limit. Experimental numbers (RNA folding, $\Delta F$ within $k_BT$ ) from Collin et al. (2005).

Related. Prerequisites: the langevin-equation · fokker-planck-equation · master-equation · fluctuation-dissipation-theorem · ito-stratonovich-calculus. Leads to: stochastic-thermodynamics · Bennett acceptance ratio · integral fluctuation theorems. These work relations are the corollaries of the general fluctuation-theorems; the trajectory heat/work bookkeeping is owned by stochastic-thermodynamics.

Is the Jarzynski equality a violation of the second law?

No — it is a sharpening of it. The second law constrains the average work: $\langle W\rangle\ge\Delta F$ , which follows from Jarzynski by Jensen's inequality. Individual realizations can have $W<\Delta F$ (transient "violations"), and in fact those rare, cheap pulls dominate the exponential average $\langle e^{-\beta W}\rangle$ — that is precisely how the equality recovers $\Delta F$ from irreversible driving. The ensemble inequality is never broken; the equality simply extracts more information than the inequality alone.

What goes wrong if the forward protocol starts out of equilibrium?

The derivation requires the initial microstates to be Boltzmann-distributed at $\lambda_A$ , so that $\rho_A(x_0)=e^{-\beta H_A}/Z_A$ . In the Hamiltonian route this is what makes the initial energy factor cancel; in the stochastic route it is what makes the path-weight ratio collapse to $e^{\beta(W-\Delta F)}$ . Start from any other distribution and the cancellation fails, so $\langle e^{-\beta W}\rangle\ne e^{-\beta\Delta F}$ . Generalized versions exist (e.g. with a non-equilibrium initial distribution), but they carry an extra term measuring the deviation of the start from equilibrium.

How is Jarzynski related to Crooks?

Crooks is the stronger, distribution-level statement: $P_F(W)/P_R(-W)=e^{\beta(W-\Delta F)}$ relates the full forward and reverse work histograms point by point. Jarzynski is its first moment — multiply Crooks by $e^{-\beta W}$ and integrate over $W$ , and the normalization of $P_R$ leaves $\langle e^{-\beta W}\rangle_F=e^{-\beta\Delta F}$ . So Crooks implies Jarzynski but not vice versa. Practically, Crooks lets you use both pulling directions (the Bennett acceptance ratio) and read $\Delta F$ off the histogram crossing, which is far more efficient than the one-sided Jarzynski average.

Why is the Jarzynski estimator biased in practice?

Because $\langle e^{-\beta W}\rangle$ is dominated by trajectories in the low- $W$ tail — for a Gaussian, the integrand $P_F(W)e^{-\beta W}$ peaks at $W=\langle W\rangle-\beta\sigma_W^2$ , which sits below $\Delta F$ . With finite samples you rarely draw those dominant tail events, so the finite-sample average systematically overestimates $\Delta F$ and converges slowly, the more so the wider (more irreversible) the work distribution. Bidirectional estimators (Crooks/Bennett) and cumulant expansions are the standard remedies.

What does the instantaneous-switch limit recover?

If the protocol is switched suddenly, $\tau\to0$ , the system has no time to move, so $W=H_B(x)-H_A(x)$ evaluated at the fixed initial configuration. The Jarzynski average then becomes $\langle e^{-\beta W}\rangle=\langle e^{-\beta(H_B-H_A)}\rangle_A=\int e^{-\beta H_B}/\int e^{-\beta H_A}=Z_B/Z_A=e^{-\beta\Delta F}$ . This is exactly Zwanzig's free-energy perturbation formula $\Delta F=-k_BT\ln\langle e^{-\beta\Delta U}\rangle_A$ , a static-ensemble identity — showing that the classic perturbation method is the sudden-switch corner of the Jarzynski equality.

Do these relations hold for stochastic dynamics, not just Hamiltonian flow?

Yes. The Hamiltonian derivation via Liouville is the cleanest, but the same identities hold for Langevin, Fokker–Planck, and Markov-jump master-equation dynamics provided the stochastic rules satisfy local detailed balance at each instantaneous value of $\lambda$ — i.e. the rates encode the bath's canonical weighting. The path-weight ratio then equals $e^{\beta Q}$ (heat to the bath), and combined with the canonical initial densities this reproduces Crooks and hence Jarzynski. This robustness across dynamical descriptions is why the relations are foundational to stochastic thermodynamics.