Loschmidt's paradox

Loschmidt's paradox, or the reversibility objection, points out that the microscopic equations of motion are invariant under time reversal, so for every entropy-increasing evolution of a system there exists a perfectly legal entropy-decreasing one, obtained by reversing all the velocities. If the dynamics permit both, how can a strictly one-way macroscopic law — entropy always increases — be derived from them?

Boltzmann's resolution does not deny the reversed trajectories; it relocates the asymmetry into the initial conditions. Entropy increases toward the future because systems begin in low-entropy states and there are vastly more high-entropy microstates to evolve into. A reversed, entropy-decreasing trajectory exists but would require preparing molecular velocities with impossible precision; the slightest perturbation destroys it and entropy resumes its climb. The arrow is in the boundary condition, not the equations.

The paradox sharpened the understanding that the second law is statistical and probabilistic rather than absolute, and it foreshadowed the modern appeal to the universe's low-entropy past.

Josef Loschmidt raised the objection to his friend and colleague Ludwig Boltzmann in 1876. Boltzmann's replies, invoking initial conditions and the statistical nature of entropy, were developed over the following two decades.