Loschmidt's Paradox

Loschmidt's Paradox is the objection raised by Josef Loschmidt in 1876 against Ludwig Boltzmann's H-theorem, which purported to prove that entropy always increases. Loschmidt observed that the microscopic laws of mechanics are time-reversible: if every particle's velocity were reversed, the system would retrace its path backward, and entropy would decrease. Therefore, for every trajectory in which entropy increases, there exists a corresponding trajectory in which entropy decreases. The paradox is not that entropy decrease is possible — it is that Boltzmann's derivation of irreversibility from reversible laws appears to contain a logical gap. The standard resolution invokes the Stosszahlansatz (molecular chaos hypothesis), an assumption about initial conditions that is itself time-asymmetric. But this resolution has never been fully satisfactory, because it replaces one mystery with another: instead of asking why entropy increases, we must ask why the molecular chaos assumption holds at one time but not the other. The paradox remains a live concern at the intersection of statistical mechanics and the philosophy of time, and any purported solution to the arrow of time must either dissolve it or explain why time-asymmetric assumptions are physically privileged.