Loschmidt's paradox

Loschmidt's paradox is the challenge that reversible microscopic laws seem unable to explain the irreversible increase of entropy. Reversing every particle's velocity in an entropy-increasing trajectory would produce a trajectory in which entropy decreases.

Boltzmann's H-theorem: A kinetic-theory result showing that Boltzmann's H-function decreases under assumptions about molecular collisions. Its claimed one-way decrease prompted Loschmidt to ask why reversing molecular velocities would not reverse the result.
Josef Loschmidt: An Austrian physicist and chemist who studied molecular theory and raised the reversibility objection to Boltzmann. His 1876 critique gave the paradox its name.
Poincaré recurrence theorem: A theorem stating that finite measure-preserving systems return arbitrarily close to their initial states after sufficient time. It raises a related challenge to permanent entropy growth, but concerns long-term recurrence rather than reversed trajectories.
Statistical mechanics: The framework that derives macroscopic physical behavior from statistical descriptions of microscopic states. The paradox exposed the assumptions needed to connect its reversible mechanics with irreversible thermodynamics.
Entanglement entropy: A measure of quantum entanglement between parts of a composite quantum system. Its growth in quantum dynamics offers a modern setting for examining entropy increase under reversible evolution.
Boltzmann equation: A kinetic equation describing how a gas's particle distribution changes through free motion and collisions. Its collision term yields the H-theorem, whose assumptions make microscopic reversibility contentious.
Ludwig Boltzmann: An Austrian physicist who developed statistical mechanics and a kinetic account of gases. His H-theorem became the central target of Loschmidt's challenge.
Zermelo's recurrence objection: An objection to irreversible statistical behavior based on the recurrence of mechanical systems to near-earlier states. It challenges the H-theorem's compatibility with mechanics through recurrence, complementing Loschmidt's velocity-reversal argument.
Past hypothesis: The proposal that the universe began in an exceptionally low-entropy macrostate. It explains an observed entropy arrow through a special boundary condition rather than asymmetric microscopic laws.
Quantum recurrence: The return of a quantum system to a state arbitrarily close to an earlier state under suitable conditions. It tests whether quantum evolution postpones or qualifies the same recurrence concerns raised in classical mechanics.

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