The equality is exact, and the average is held by the tail – a census of non-equilibrium.
The Jarzynski equality states that the exponential average of the work equals the exponential of the free-energy difference, and it is an exact identity that holds however violently the system is driven. This paper writes what it implies and what it does not. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Jarzynski (1997), Crooks (1999), Sekimoto’s stochastic energetics and the convergence problem of exponential averages are all standard. One caution matters: the free-energy difference is that of the equilibrium state at the final parameter value, and the system need not be in equilibrium when the process ends. That a non-equilibrium measurement yields an equilibrium quantity is the whole point, and it is easy to misstate.
First, this was checked numerically. Setting the Boltzmann constant times temperature, the friction coefficient and the spring constant all to one, the centre of a harmonic trap is dragged at constant speed for a time of one. The trap’s shape does not change, so the free-energy difference is exactly zero and the equality reads that the exponential average is one. The overdamped Langevin equation was run over two hundred thousand trajectories with two thousand steps each. At dragging speeds of 0.5, 1 and 2 the exponential averages come out as 0.99957, 0.99860 and 0.99838. The system is moreover exactly solvable: the mean work is the square of the speed times the duration minus one plus the exponential of minus the duration, and the variance is twice that. The values here agree with the closed forms to within 0.7 per cent in the mean and 0.4 per cent in the variance.
Second, the second law is a Jensen corollary of this equality. Since the exponential is convex, Jensen’s inequality makes the exponential average at least the exponential of the average, and combining this with the equality gives that the mean work is at least the free-energy difference, which is the second law. The difference in strength is visible here. Jensen’s inequality uses only the mean of the distribution and throws away its shape; the equality uses the whole shape. The second law is what survives after that information is discarded. Unless the corollary is written as a corollary, the two look like laws of equal standing.
Third, the equality is what ties fluctuation to dissipation. In the table the mean work and half the variance agree, 5.885 against 5.865 at speed four. If the work distribution is Gaussian, the equality forces the variance to equal twice the Boltzmann constant times the temperature times the dissipated work. A fluctuation-dissipation relation thus falls out as a consequence of the equality (Paper 78). But this form holds only in the Gaussian case: in general the equality is wider, and the fluctuation-dissipation relation is its linear-response case. They must not be taken for the same content.
Fourth, and this is the core. The exponential average is held by the tail. The share of the exponential average carried by trajectories whose work is below the free-energy difference, that is, by those that locally violate the second law, was measured. At speed 0.5, 41.5 per cent of paths carry 58.5 per cent of the average. At speed one, 33.4 per cent carry 66.5. At speed two, 19.6 per cent carry 80.4. And at speed four, a mere 4.3 per cent of paths carry 95.2 per cent of the average, an over-representation by a factor of twenty-two. The reason is simple: the exponential is larger the smaller the work, so the exponential average puts its weight on the left tail, and the harder the driving the further out that tail lies, the rarer it becomes, and the heavier it weighs. The relation to the second law becomes visible here. Trajectories with work below the free-energy difference do violate the second law. Individual trajectories may violate it; what does not is the average. And the equality requires those violations, for without them the exponential average would not reach its target. This has the shape of what Paper 132 said of average return: when the tail is heavy, the mean does not describe the usual outcome. Here the mean is made of the tail.
Fifth, so being exact and being measurable are different. Drawing N trajectories from the sample at speed four and estimating the exponential average, the median over two hundred repetitions is 0.0718 at N equal to ten, 0.3077 at a hundred, 0.6225 at a thousand, 0.7903 at ten thousand, and still only 0.9039 at a hundred thousand. Even a hundred thousand trajectories fall a tenth short. The estimate always approaches from below, because while the sample does not contain the rare left tail the average comes out small, and the number of samples required grows exponentially with the dissipation. What deserves emphasis is that the value 0.89350 appearing in the row for speed four is not a failure of the equality. Substituting the closed forms gives an exponential average of exactly one at every speed. What breaks down is the estimator. Being true and being measurable are different properties. The same shape as Paper 135: there volume and energy excluded each other, and here exactness and measurability come apart.
Finally, the three statements form a ladder of strength. Crooks’s theorem gives the ratio of the forward and reverse work distributions and is the strongest. The Jarzynski equality gives the value of an exponential average and follows from Crooks by integration. The second law gives an inequality on the mean and follows by Jensen, and is the weakest. Crooks states the whole shape of the distribution, Jarzynski collapses it into one number, and the second law collapses that number into an inequality. Each step down loses information, and the lower rungs are corollaries of the upper rather than independent laws. As for what the work denotes, that carries a convention in the stochastic setting (Sekimoto’s stochastic energetics; inclusive versus exclusive work). The free-energy difference does not depend on that convention, but the definition of the work must be written down (Paper 95).
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that the second law is violated (it is not, on average; that individual trajectories may violate it is the content of the equality) / that the Jarzynski equality is impractical (only that it can be, in a specified regime). What is added: placing the second law as a corollary, measuring the tail’s contribution numerically, separating exactness from measurability, and arranging the three statements on a ladder of strength. What remains: the second law is obtained by discarding information from a stronger equality, and what holds that equality up is the rare trajectories that locally violate the second law. Having an exact formula does not mean being able to measure it.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Equality Is Exact, and the Average Is Held by the Tail [Paper 136]. Zenodo.
DOI: 10.5281/zenodo.22081230
ORCID: 0009-0006-8587-3450