Reversa: Loschmidt's Reversal Paradox
Thermodynamics & Statistical Physics arc, page 45 of 193. Back to The Pixelated Physics Universe
Who they are
Reversa flips every molecule's velocity, and the gas should un-mix. So why does time only run one way?
Reversa is a character of the Thermodynamics & Statistical Physics arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
Newton's laws run equally well backwards, yet Boltzmann's H-theorem says a gas always heads toward equilibrium. Reverse every velocity and the gas would retrace its steps, so the arrow of time cannot come from the dynamics alone. The H-theorem rests on the molecular-chaos assumption. Josef Loschmidt raised the objection in 1876.
$$ \begin{gathered} m\,\frac{d^{2}x}{dt^{2}} = F(x) \quad \text{is unchanged by } t \to -t,\ v \to -v \\ H = \int f \ln f \; d^{3}v, \qquad \frac{dH}{dt} \le 0 \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $m$ | particle mass | kg |
| $x$ | position | m |
| $F(x)$ | force depending only on position | N |
| $t$ | time | s |
| $v$ | velocity | m/s |
| $H$ | Boltzmann's H function (per volume, up to constants) | 1 |
| $f$ | velocity distribution function | s³/m⁶ |
| $d^{3}v$ | velocity-space volume element | m³/s³ |
Honest caveat: This does not break the second law. Why the past had low entropy is still debated.
Status: part of this topic is unsolved, unconfirmed or debated. The text above says which part; treat it as an open question, not settled fact.
Sources
- Brown, Myrvold & Uffink (2009) Boltzmann's H-theorem, its discontents, and the birth of statistical mechanics, Stud. Hist. Phil. Mod. Phys. 40, 174
- Stanford Encyclopedia of Philosophy (Uffink) Boltzmann's Work in Statistical Physics (reversibility objection)
- Lebowitz (1993) Boltzmann's Entropy and Time's Arrow, Physics Today 46(9), 32
Crossovers
Reversa meets Eventide Prophet (Black Hole Information Paradox). Reversa asks how time-symmetric laws give one-way behaviour. Eventide Prophet asks whether black-hole evaporation loses information, which unitary quantum mechanics forbids. Both puzzles sit where reversibility meets the second law.
Reversa meets Erasa (Forgetting Costs Heat). Reversa can flip every velocity, but Erasa reminds her that resetting a memory costs at least k_B T ln 2 of heat per bit. Undoing the record of what happened isn't free, so Reversa can't run the world backwards without paying the second law.
Reversa meets Arnold's Cat (Cat Map Chaos). Arnold's Cat snaps back to her starting picture after a finite number of steps on a finite grid. Reversa sees the same lesson as Poincaré recurrence: reversible dynamics can return, but for a real gas the wait is far longer than the age of the universe.
Wear the lore
- Reversa – Loschmidt's Reversal Paradox Physics T-Shirt
- Related infographic poster: Boltzmann's Stone: Counting W – Plate 012 Physics Poster
Concept keywords: Loschmidt's Reversal Paradox, Reversa, Thermodynamics & Statistical Physics, physics lore, Pixelated Physics