Summary of Key Findings
The work of Deng Yu (2026 Fields Medalist) and his team has resolved a classic paradox in physics: while microscopic particles obey the time-reversible laws of Newtonian mechanics (for example, reversing the velocity would cause them to move in the opposite direction), the macroscopic world exhibits clear irreversibility (for instance, gas diffuses and does not spontaneously return to its original state). They have rigorously demonstrated that under the conditions of a low-density system of hard spheres, the microscopic reversible particle dynamics can be used to derive the Boltzmann equation, which describes macroscopic irreversibility. This breakthrough has overcome the previous limitation that the Boltzmann equation was only valid for extremely short periods of time. As long as the solution to the Boltzmann equation remains stable within a certain timeframe, the behavior of the hard sphere system can be accurately described by this equation. This indicates that there is no contradiction between microscopic reversibility and macroscopic irreversibility; the key lies in how the interactions between particles are accounted for in the process of making macroscopic predictions from microscopic data.
1. Why are microscopic particles reversible, but the macroscopic world not?
Imagine a transparent box divided into two halves by a barrier, with colored gas on one side only. Once the barrier is removed, the gas quickly fills the entire box and hardly ever returns to its original state—this is the most straightforward example of macroscopic irreversibility.
At the microscopic level, each particle (such as a hard sphere) follows Newtonian laws, and if all particles' velocities are reversed simultaneously, the system can move back along the same path (reversibility is possible). However, at the macroscopic level, the density and temperature of an isolated system always tend from non-uniform to uniform (for example, gas diffuses and temperature differences disappear), and this process cannot be reversed.
The paradox arises from the question of how a large number of reversible interactions can lead to irreversible macroscopic phenomena. The issue is not that the microscopic laws fail, but rather that when we describe the macroscopic behavior, we do not track the details of each individual particle (such as position and velocity); instead, we only consider a few macroscopic variables (such as density and temperature). This “reduced description” loses information about the interactions between particles, resulting in irreversibility.
2. Boltzmann’s “molecular chaos”: The key to making the macroscopic equation self-consistent
To describe a macroscopic gas, Boltzmann sought an equation that only depended on the “velocity distribution of individual particles” (the Boltzmann equation). However, particles collide with each other, and the behavior of one particle is influenced by the interactions with multiple particles. For example, if particle A collides with B, and B then collides with C, there is an indirect interaction between A and C. This leads to an infinitely complex system of equations that cannot be closed (since higher-order interactions must be considered).
Boltzmann proposed the assumption of “molecular chaos,” which states that particles before a collision are independent of each other. In everyday terms, when two people are about to collide on the road, their paths are random and not predetermined. This assumption allows the distribution of two particles to be represented as the product of their individual distributions, effectively cutting off the chain of interactions, thus making the equation self-consistent (it only requires knowledge of the single-particle distributions).
3. The breakthrough by Deng Yu’s team: Extending the applicability of the Boltzmann equation
In 1975, Landau first rigorously proved that the Boltzmann equation could be derived from hard sphere dynamics, but it was only valid for very short periods of time (shorter than the average free time of the particles, meaning particles did not have enough time to collide twice). Over longer times, particles might collide multiple times, and previous interactions could affect the current behavior, potentially invalidating the assumption of molecular chaos.
Deng Yu’s team addressed this issue by representing the history of particle collisions as a “graph” (e.g., cycles formed by chains of collisions) and using mathematical methods (such as expanding cumulative quantities and analyzing graph structures) to precisely calculate the overall impact of these interactions. They found that at low densities, the effects of these complex interactions are minimal and can be controlled. As long as the solution to the Boltzmann equation remains stable within a certain timeframe, the behavior of the hard sphere system can be described by this equation, overcoming the time limitation.
4. Macroscopic irreversibility has nothing to do with quantum mechanics
Many people believe that irreversibility is caused by quantum mechanics (for example, quantum measurements are thought to destroy reversibility). However, this is not the case:
- Classical computer simulations of hard sphere collisions can also show irreversible phenomena such as gas diffusion and temperature equalization without relying on quantum mechanics.
- Even when simulation precision is improved (e.g., by reducing time steps or increasing computational accuracy), the macroscopic results remain stable, indicating that irreversibility is not due to numerical errors.
- Boltzmann’s “H-theorem” has proven that, under the assumption of molecular chaos, the distribution of particles evolves towards increased entropy (i.e., irreversibility).
This shows that macroscopic irreversibility is a classical phenomenon and is not related to quantum mechanics.
5. Other “bridges” from microscopic to macroscopic phenomena: The Langevin equation and pattern coupling theory
In addition to the Boltzmann equation, there are two other common approaches to describe macroscopic behavior from microscopic principles:
- The Langevin equation: This equation is used for large particles moving in a liquid, where the influence of surrounding small particles (the environment) can be approximated as friction and random noise. Since the environment changes faster than the large particles, we do not need to consider “memory effects” (e.g., the previous state of the environment on the current behavior), making the equation self-consistent.
- Pattern coupling theory: This theory is applied to supercooled liquids (such as glass that solidifies rapidly). It approximates higher-order density interactions as products of lower-order ones, resulting in a self-consistent equation. While it can describe the thickening of liquids upon cooling, its predictions about the glass transition are too abrupt (real glass does not solidify suddenly), indicating that further improvements are needed.
The core of these methods is to account for previously neglected degrees of freedom (such as the environment and higher-order interactions) to turn an infinitely complex system into a self-consistent one that can be solved mathematically.
Conclusion
There is no contradiction between microscopic reversibility and macroscopic irreversibility; the key lies in how we handle the interactions between particles when making predictions from microscopic data. Deng Yu’s team’s work has significantly extended the applicability of the Boltzmann equation, providing a more solid theoretical foundation for understanding macroscopic irreversibility. In the future, we still need to explore the origins of irreversibility in other systems (such as dense gases and liquids), but this breakthrough represents an important step forward in our understanding of these phenomena.