Category: physics
Type: thought experiment
Associated thinker(s): Josef Loschmidt, Ludwig Boltzmann
Original publication or approximate period: Loschmidt’s published objection (1876); Boltzmann’s immediate reply (1877); Zermelo’s distinct recurrence objection (1896)
Difficulty: Advanced
Themes: entropy, time asymmetry, statistical mechanics
Related experiments: Maxwell’s Demon; Laplace’s Demon
Imagine an idealized, mechanically isolated gas whose trajectory carries it from a low-entropy macrostate toward equilibrium. At an instant, keep every particle’s position fixed and reverse every velocity (equivalently, every momentum). If the microscopic equations are invariant under time reversal, the new state follows the exact earlier trajectory in reverse: the gas returns to its earlier macrostate. Along this specially prepared trajectory, Boltzmann’s macro-entropy decreases.
The gas spreading from a corner is a modern teaching illustration, not Loschmidt’s exact historical setup. In his 1876 discussion, he considered a gas evolving from a highly special initial state to a stationary state, then argued that reversing all atomic velocities would retrace that evolution. The result requires an exact, coordinated reversal of the complete relevant system and time-reversal-invariant dynamics; an arbitrary change of some velocities would not make the system retrace its history.
This decrease concerns entropy assigned to macrostates (often called Boltzmann entropy). It should not be confused with the fine-grained Gibbs entropy, which is preserved under ideal Hamiltonian evolution.
The objection is that mechanics alone cannot establish that entropy rises monotonically along every mechanically possible trajectory. It is directed at an unrestricted, universal derivation of the second law from reversible microscopic dynamics, not at the empirical usefulness of thermodynamics or at statistical mechanics as a whole.
If the microscopic dynamics permit both an entropy-increasing history and its time-reversed, entropy-decreasing counterpart, what additional assumptions explain the observed thermodynamic arrow of time?
Boltzmann’s immediate 1877 reply to Loschmidt acknowledged that exceptional initial states can produce the opposite of the H-theorem’s predicted trend, and treated such states as extremely improbable. In a separate 1877 paper, he developed a statistical argument in which equilibrium corresponds to overwhelmingly many compatible microstates (or, in modern language, a much larger region of phase space) than non-equilibrium macrostates. This makes entropy increase typical under specified probability assumptions; it does not make it mechanically necessary for every microstate or by itself select a direction of time. The precise interpretation and adequacy of Boltzmann’s replies have been debated.
A low-entropy condition in the distant past is a related but distinct ingredient in many modern accounts. Called the Past Hypothesis in contemporary discussion, it supplies a temporally asymmetric boundary condition used with statistical-mechanical probabilities and dynamics to explain why entropy rises toward what we call the future. It is not Loschmidt’s objection, nor simply another name for Boltzmann’s immediate reply. Boltzmann also discussed the observable universe as a fluctuation from equilibrium in a larger universe; that proposal should not be conflated with every modern Past-Hypothesis formulation.
The objection does not predict that ordinary gases will spontaneously run backward. The required state is extraordinarily fine-tuned and not a practical operation on a macroscopic gas. Nor does the objection show that statistical mechanics is false: it shows that the microscopic reversible laws do not entail entropy increase for every allowed microstate without additional assumptions. The Boltzmann H-theorem itself depends on kinetic assumptions, including an asymmetric assumption about pre-collision molecular correlations (the Stoßzahlansatz or “molecular chaos” assumption); it is not a consequence of reversible mechanics alone.
This is distinct from Maxwell’s demon. Both topics concern the second law, but the demon raises questions about information, measurement, and the physical cost of operations; Loschmidt’s objection concerns time-reversal-invariant dynamics and exceptional reversed trajectories.
These are teaching prompts, not claims attributed to Loschmidt or Boltzmann:
The Poincaré recurrence theorem and Zermelo’s recurrence objection are related but distinct. Poincaré’s result concerns recurrence in suitable bounded mechanical systems; Zermelo used recurrence in a separate 1896 challenge to the H-theorem. Loschmidt’s objection instead relies on time-reversal invariance and velocity reversal. Recurrence is not the same argument as retracing a trajectory by reversing velocities.
The Stanford Encyclopedia of Philosophy entries listed above were checked in the relevant full-text sections. Darrigol’s published article page was inspected, and relevant translated primary passages and commentary were read in the openly available early draft; the draft itself says to cite the published version. The original Loschmidt scan was consulted directly at p. 139 during the independent follow-up review, alongside Darrigol’s translation and the SEP quotation. The rest of the original paper was not read. The article’s corner-gas example and reflection questions are editorial teaching devices. This entry is an educational explanation, not a substitute for the primary texts or a claim of expert review.