In thermodynamics, Loschmidt’s paradox states that according to the laws of mechanics, a system of particles interacting with any force law, which has gone through a sequence of states starting from some specified initial conditions, will go through the same sequence in reverse and return to its initial state if one reverses the velocities of all the particles; and that this conflicts with the second law of thermodynamics, which asserts that for any such sequence of states the entropy must always increase, meaning that such a system cannot be reversed back to its initial state. [1]

The reversibility paradox was pointed out by Austrian physical chemist Joseph Loschmidt in 1876 to Austrian physicist Ludwig Boltzmann in commentary on his 1872 paper on his H-theorem. In short, Loschmidt said that the H-theorem singled out the direction in time in which H decreases, whereas the underlying mechanics was the same whether time flowed forwards or backwards. [2]

Thought experiment
In 1876, in sum, Loschmidt suggested a (gedanken) thought experiment to Boltzmann. The proposal went something along the lines of the following: suppose a gas is released at 12:00 noon in the corner of an evacuated room, with the molecules independently moving according to Boltzmann’s statistical assumption, but also obeying Newton’s laws of motion.

Now suppose at 12:05, the ‘hand of God’ (fact check) reverses the direction of all the molecules. It follows from reversibility that at exactly 12:10 all the molecules must return to the corner of the room. Surely its entropy must also have the same value as at the beginning. Yet Boltzmann’s H-theorem suggests that entropy has increased continuously. Loschmidt noted that there seems to be a contradiction.

Boltzmann, supposedly, responded (fact check) to this by asserting that the H-theorem is a statistical statement, which must be true in the overwhelming majority of scenarios because the number of microstates associated with equilibrium is overwhelmingly greater than the number associated with any non-equilibrium state. In effect, according to a recent summary by Robert Ayres, Boltzmann conceded that occasional exceptions might be possible. [3]

See also
Poincaré recurrence theorem

1. Boltzmann, Ludwig. (1877). “On the Relation of a General Mechanical Theorem to the Second Law of Thermodynamics” (“Uber die Beziehung eines Allgemeine Mechanischen Satzes zum zweiten Hauptsatze der Warmetheorie”), Sitzungsberichte Akad. Wiss., Vienna, Part II, 75: 67-73.
2. Flamm, Dieter. (1999). “Boltzmann: a Disordered Genius”,, 9 April.
3. Ayres, Robert U. (1994). Information, Entropy, and Progress: a New Evolutionary Paradigm (pg. 5). Springer.

External links
Loschmidt’s paradox – Wikipedia.

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