The molecular chaos assumption (Stosszahlansatz) states that pre-collision velocities are statistically uncorrelated. Why does this assumption break time-reversal symmetry in the H-theorem?
ABecause the Boltzmann equation itself is time-asymmetric and does not permit time reversal under any conditions
BBecause molecular chaos applies to pre-collision pairs but not post-collision pairs — time-reversing the system creates correlated pre-collision velocities, violating the assumption
CBecause entropy is automatically larger in the time-reversed state, ensuring H always decreases in both directions
DBecause molecular chaos is an empirical regularity that, by definition, only holds in the forward time direction
The asymmetry is subtle. Molecular chaos says pre-collision velocities are uncorrelated. After a collision, the two molecules carry correlated velocities (they 'remember' the collision). If you time-reverse the system — reversing all velocities — what were post-collision pairs become pre-collision pairs with correlated velocities. This violates molecular chaos. So the time-reversed trajectory starts from a specially correlated state to which the Stosszahlansatz does not apply, and the H-theorem cannot be applied to it. The assumption treats pre-collision and post-collision correlations differently, and that asymmetry is where the time-reversal breaking enters.
Question 2 Multiple Choice
Loschmidt's paradox points out that for every gas trajectory where H decreases (entropy increases), there exists a time-reversed trajectory. What does the time-reversed trajectory look like, and why is it problematic for the H-theorem?
AThe time-reversed trajectory also shows H decreasing — so there is no paradox, just confirmation of the theorem
BThe time-reversed trajectory shows H increasing — a gas spontaneously evolving from equilibrium toward lower entropy — which appears to contradict the theorem's claim that H always decreases
CThe time-reversed trajectory keeps H constant, since the system starts at equilibrium in the reversed direction
DThe time-reversed trajectory is physically impossible because velocity reversal cannot be performed in practice
This is the core of Loschmidt's paradox. If Newton's laws are time-reversible and the H-theorem says H always decreases, then for every entropy-increasing trajectory there must exist a time-reversed entropy-decreasing one — a gas spontaneously contracting from equilibrium to a low-entropy state. Boltzmann's resolution is that the time-reversed initial state has special correlated velocities that violate molecular chaos, so the theorem doesn't apply to it. Such states exist but are extraordinarily rare; we'd never observe them because the overwhelming majority of initial conditions satisfy molecular chaos.
Question 3 True / False
The H-theorem requires the molecular chaos assumption in addition to the Boltzmann equation; time-reversible mechanics alone is insufficient to derive entropy increase.
TTrue
FFalse
Answer: True
This is precisely what Loschmidt's paradox establishes. Time-reversible mechanics permits both entropy-increasing and entropy-decreasing trajectories. The H-theorem holds only for initial states satisfying molecular chaos (uncorrelated pre-collision velocities). This assumption carries the time-asymmetric information: it is satisfied by macroscopically prepared states but violated by the special time-reversed states. Irreversibility does not follow from the equations of motion alone — it requires a statistical assumption about the kind of initial conditions we actually prepare and encounter.
Question 4 True / False
The H-theorem implies that a macroscopic gas can seldom spontaneously decrease in entropy — such a decrease is absolutely forbidden by the laws of physics.
TTrue
FFalse
Answer: False
The H-theorem is a statistical result, not an absolute prohibition. By Poincaré recurrence, any finite mechanical system will eventually return arbitrarily close to any initial state, including a low-entropy one — because the equations of motion are deterministic and the phase space is bounded. For a macroscopic gas, such a recurrence is not forbidden but takes a time astronomically longer than the age of the universe (~10^(10^23) years). The H-theorem says that for macroscopically prepared states satisfying molecular chaos, entropy decrease is overwhelmingly improbable — not that it is physically impossible.
Question 5 Short Answer
How does Boltzmann resolve Loschmidt's paradox — and what does the resolution reveal about the status of the second law of thermodynamics?
Think about your answer, then reveal below.
Model answer: Boltzmann's resolution is that the H-theorem does not apply to all initial conditions — only those satisfying molecular chaos (uncorrelated pre-collision velocities). The time-reversed entropy-decreasing trajectories Loschmidt points to start from states with special post-collision correlations, which violate this assumption. Such states exist in phase space but are extraordinarily rare: macroscopically prepared systems almost always satisfy molecular chaos, so entropy increase is overwhelmingly probable. This reveals that the second law is not an absolute consequence of mechanics but a statistical claim about the overwhelming probability of entropy increase for the kinds of initial conditions the universe actually presents.
The deeper lesson is that irreversibility is an emergent statistical phenomenon, not a fundamental mechanical law. It requires two ingredients: time-reversible equations of motion plus the statistical assumption that initial conditions are 'generic' (satisfying molecular chaos). The second law holds not because entropy-decreasing trajectories are forbidden by physics, but because they require initial conditions that are infinitely improbable compared to entropy-increasing ones — a fact that ultimately traces back to the universe beginning in an unusually low-entropy state.