Note: This post is an English adaptation of my original Chinese article (URL). Some parts have been modified for clarity, cultural relevance, or to better fit the English-speaking audience.
This question is functionally equivalent to Loschmidt’s Paradox, also known as the reversibility paradox. At its core, it addresses a fundamental conflict in physics.
The fundamental mechanical equations for a single particle in classical mechanics are symmetric under time reversal. Consequently, the evolution of an entire system of particles should also be time-reversal symmetric and, therefore, perfectly reversible. However, according to Boltzmann’s H-theorem (which is a specific instance of the second law of thermodynamics, though not as general) the entropy of a particle system is overwhelmingly likely to not decrease. While entropy reduction is possible, its probability is infinitesimally small. This implies that the evolution of the particle system is not time-reversal symmetric, so it is irreversible. This presents a direct contradiction.
Later, in his response to Loschmidt’s critique, Boltzmann acknowledged that his proof of the H-theorem introduced the “molecular chaos” assumption (which is called “Stosszahlansatz”). More precisely, this assumption was intrinsically embedded in the construction of the Boltzmann equation, from which the H-theorem is derived. This assumption, by its mathematical nature, lacks time-reversal symmetry, and as a result, the Boltzmann equation itself does not possess this property.
Does this imply that, without the molecular chaos assumption, one could construct an equation that does satisfy time-reversal symmetry to describe the state of a particle system? Indeed, this is the case. The Liouville equation is a perfect example of such an equation, and it is fully time-reversal symmetric.
However, the crucial issue is that “reversibility” in the context of the paradox has distinct meanings: microscopic reversibility and thermodynamic reversibility. These two concepts are fundamentally different. Time-reversal symmetry is equivalent to microscopic reversibility, but it does not imply thermodynamic (or macroscopic) reversibility. Time-reversal symmetry and microscopic reversibility concern the possibility of reverse evolution, whereas thermodynamic reversibility concerns the probability of a macroscopic state evolving in reverse.
Microscopic reversibility physically means that “the probability of a system’s forward evolution is non-zero if and only if the probability of its reverse evolution is non-zero.” Mathematically, this is expressed as $P_F[ A \to B] \ne 0 \Longleftrightarrow P_R[B \to A] \ne 0$. In contrast, thermodynamic reversibility physically means that “the probability of the system’s forward evolution is equal to the probability of its reverse evolution,” expressed mathematically as $P_F[A \to B] = P_R[B \to A]$. These definitions are derived from the Crooks fluctuation theorem. For those interested, this theorem can be used to derive the Jarzynski equality, which, when combined with Jensen’s inequality, yields the Clausius statement of the second law of thermodynamics. Therefore, a process can be defined as irreversible as long as “the probability of the forward evolution is greater than the probability of the reverse evolution,” although typically we require that the forward probability be substantially greater.
Leave a Reply