• Reversa Loschmidt's Paradox pixel-art physics T-shirt, black tee, front view

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Reversa

Loschmidt's Reversal Paradox Physics

\[m\,\frac{d^{2}x}{dt^{2}} = F(x) \quad \text{is unchanged by } t \to -t,\ v \to -v\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Reversa flips every molecule's velocity, and the gas should un-mix. So why does time only run one way?

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.

H = ∫ f ln f d³v, dH/dt ≤ 0

  • H: Boltzmann's H function (dimensionless, per volume)
  • f: velocity distribution (s³/m⁶)
  • d³v: velocity-space volume element (m³/s³)
  • t: time (s)

Josef Loschmidt raised the objection in 1876.

Honest caveat: This does not break the second law. Why the past had low entropy is still debated.

On the shirt

Front: Reversa, the pixel-art character, with the title. Back: the reversibility of Newton's laws, Boltzmann's H-theorem, the molecular-chaos note and the 1876 objection. Printed on a unisex tee.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Para hacerse una idea: la talla M mide 20 in (50.80 cm) de ancho, unas 3.143 × 1034 longitudes de Planck (lP = 1.616255 × 10−35 m). Una cinta métrica es más rápida.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

I am the one who flips every molecule's velocity at once. Newton's laws do not mind: replace t by −t and v by −v and every trajectory runs perfectly well backwards.

Boltzmann's H-theorem says a gas always heads toward equilibrium, with H never increasing. But if I reverse every velocity in a gas that has just spread out, its own laws of motion carry it back to where it started, and H must rise. Josef Loschmidt raised this objection in 1876: the mechanics alone cannot produce a one-way arrow of time.

The answer is that the H-theorem has an extra assumption, molecular chaos, and that our universe began in a very low-entropy state. Entropy can fluctuate downward in small systems, and the second law is not broken. Why the past had such low entropy is still debated.

Equations

\[m\,\frac{d^{2}x}{dt^{2}} = F(x) \quad \text{is unchanged by } t \to -t,\ v \to -v\]
\[H = \int f \ln f \; d^{3}v, \qquad \frac{dH}{dt} \le 0\]
SymbolMeaningUnit
\(m\) particle mass \(\mathrm{kg}\)
\(x\) position \(\mathrm{m}\)
\(F(x)\) force depending only on position \(\mathrm{N}\)
\(t\) time \(\mathrm{s}\)
\(v\) velocity \(\mathrm{m/s}\)
\(H\) Boltzmann's H function (per volume, up to constants) \(\mathrm{1}\)
\(f\) velocity distribution function \(\mathrm{s³/m⁶}\)
\(d^{3}v\) velocity-space volume element \(\mathrm{m³/s³}\)

Sources

  1. Brown, Myrvold & Uffink (2009) Boltzmann's H-theorem, its discontents, and the birth of statistical mechanics, Stud. Hist. Phil. Mod. Phys. 40, 174
  2. Stanford Encyclopedia of Philosophy (Uffink) Boltzmann's Work in Statistical Physics (reversibility objection)
  3. Lebowitz (1993) Boltzmann's Entropy and Time's Arrow, Physics Today 46(9), 32

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