Entropy really does decrease sometimes ── and the odds are exact
The second law of thermodynamics ── entropy never decreases ── is treated as the most inviolable law in physics. In fact it is violated all the time. In small systems over short times, entropy decreases with a perfectly measurable probability. And this is not hand-waving: since the 1990s an exact equality has been known, \(P(+\sigma)/P(-\sigma)=e^{\sigma/k_B}\) (the fluctuation theorem), verified experimentally with colloidal beads and single RNA molecules. Look at that expression and notice what sits in the denominator of the exponent ── \(k_B\). \(k_B\) is the scale of the exception to the second law. Which is exactly why the second law is airtight in our world: \(k_B\) is unimaginably small.
Suspend a micron-sized plastic bead in water and drag it with optical tweezers. Work is done on the bead, some of which is dissipated into the water as heat. Ordinarily.
But sometimes ── the bead is struck by water molecules and gets pushed forward. In that instant, heat flows from the water into the bead, and the work you did comes out negative. Entropy has decreased. Just for a moment, but genuinely.
In 2002 Wang and collaborators put this on record for the first time with exactly such an experiment (Phys. Rev. Lett. 89, 050601). It appeared for durations of order a second, in a system a few microns across.
What is remarkable is that this is not merely "it happens sometimes." The probability of going backwards is fixed exactly.
\(\sigma\) is the entropy produced over some stretch of time. Read it plainly: "the process that produces entropy \(\sigma\) is \(e^{\sigma/k_B}\) times more likely than the one that destroys the same \(\sigma\)."
Not "the reverse is impossible" ── merely overwhelmingly less likely. And the amount of overwhelming is measured in units of \(k_B\).
Feed numbers in and you see why this never comes up in daily life.
| Entropy produced σ | σ/k_B | ratio of forward to reverse | Comment |
|---|---|---|---|
| bead, 1 second (small) | ≈ 2 | ≈ 7× | reverse in about 1 trial in 8 ── observable |
| bead, 10 seconds | ≈ 20 | ≈ 5×10⁸ | essentially never seen |
| a drop of ink diffusing | ≈ 10¹⁵ | \(e^{10^{15}}\) | would never happen in the age of the universe |
| a cup of coffee cooling | ≈ 10²² | \(e^{10^{22}}\) | a number beyond writing |
The second law is not a prohibition; it is an enormous inequality of odds. And what converts entropy into odds is \(k_B\).
The figure simulates dragging a bead through water (an overdamped Langevin equation) many times over and histogramming the work \(W\) done each time. The red region is \(W<0\) ── trials where the bead extracted energy from the water and pushed back at you.
Lengthen the pull with the slider and the red region vanishes rapidly. The lower panel plots \(\ln[P(W)/P(-W)]\) against \(W\); it lies on a straight line of slope 1 (in units of \(k_BT\)). That straight line is the fluctuation theorem.
Sitting behind the fluctuation theorem is an even more striking equality, found by Jarzynski in 1997.
\(W\) is the work in one trial, \(\Delta F\) the free-energy difference, and \(\langle\ \rangle\) an average over many trials. The astonishing part: this is an equality no matter how far from equilibrium you drive the system. Jerk it as hard as you like and it still holds exactly.
And the second law follows in one line
Jensen's inequality gives \(\langle e^{-x}\rangle\ge e^{-\langle x\rangle}\), so
$$e^{-\Delta F/k_BT}=\langle e^{-W/k_BT}\rangle\ \ge\ e^{-\langle W\rangle/k_BT}\quad\Longrightarrow\quad \boxed{\ \langle W\rangle\ \ge\ \Delta F\ }$$That last line is the second law ── "the work required is at least the free-energy difference." The second law is a corollary of an equality, obtained by taking an average. Averaging is where the inequality is born; individual trials are free to go the other way.
If the second law is only statistical, could a clever demon exploit it? That is Maxwell's demon (1867): a being that watches molecules and opens a door only for the fast ones, separating hot from cold with no work.
The resolution took over a century, and it lands on information.
The demon must record what it observes. Its memory is finite, so eventually it must erase. Erasure is an irreversible operation ── two states collapsing into one ── and it necessarily dissipates \(k_BT\ln2\) of heat. Adding that up exactly cancels the demon's winnings.
Note again the appearance of \(k_BT\). Episode 1 said temperature is the exchange rate between energy and bits; here the rate is written out with the \(\ln2\) explicit. 1 bit = \(k_BT\ln2\) of energy. Bérut and colleagues measured this experimentally in 2012 (Nature 483, 187).
Pulling this episode together in one line:
The second law is not a prohibition but a ratio of odds, \(e^{\sigma/k_B}\). Since \(k_B=1.38\times10^{-23}\) J/K is fantastically small, macroscopic entropy production makes \(\sigma/k_B\) astronomically large and the odds of the reverse become unwritable. The second law appears absolute because \(k_B\) is small.
Conversely, in a world with a large \(k_B\) the second law would be a mere tendency. \(k_B\) is not just a unit converter ── it sets how strict thermodynamics is.
Established: the fluctuation theorem (Evans–Cohen–Morriss 1993, Gallavotti–Cohen 1995, Crooks 1999) and the Jarzynski equality (1997) are proved theorems within their stated assumptions; that the second law \(\langle W\rangle\ge\Delta F\) follows from Jarzynski via Jensen's inequality; the experimental verifications (Wang et al. 2002 with colloids, Liphardt et al. 2002 with RNA, Bérut et al. 2012 for Landauer); and Landauer's principle \(k_BT\ln2\) per bit erased ── all established results.
Points to note: (1) The fluctuation theorem's precise form depends on the setup (transient versus steady-state, the definition of work, whether the initial state is equilibrium); the \(P(+\sigma)/P(-\sigma)=e^{\sigma/k_B}\) written here is the simplest representative form. (2) The Jarzynski equality is exact but evaluating the average is hard in practice ── it is dominated by rare small-\(W\) trials, so it converges slowly (the "rare-event problem"). (3) Landauer's principle is now widely accepted, but the completeness of the resolution of Maxwell's demon is still debated by some (Earman–Norton's critique, etc.). (4) The figure is a schematic simulation, not a reproduction of any particular experiment.
Entropy decreases all the time in small systems over short times, and the odds are exact: \(P(+\sigma)/P(-\sigma)=e^{\sigma/k_B}\) (the fluctuation theorem), verified with colloids and RNA. Behind it stands the Jarzynski equality \(\langle e^{-W/k_BT}\rangle=e^{-\Delta F/k_BT}\), which holds however violently you drive the system ── and applying Jensen's inequality to it produces the second law \(\langle W\rangle\ge\Delta F\) in a single line. The second law is what you get by averaging an equality.
Maxwell's demon is defeated by the same constant: erasing one bit costs \(k_BT\ln2\) (Landauer), which exactly cancels the winnings. The second law is not a prohibition but a ratio of odds \(e^{\sigma/k_B}\), and it appears absolute only because \(k_B\) is small.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen the slider changes the pull duration and you can watch the W<0 region shrink. The lower panel's slope-1 line is the fluctuation theorem itself. "See the answer" opens each solution.