Overview
This lecture discusses a recent mathematical breakthrough addressing Hilbert's sixth problem: deriving macroscopic, irreversible physical laws from microscopic, reversible ones, especially in the context of fluid dynamics.
Hilbert’s Sixth Problem
- David Hilbert’s sixth problem calls for an axiomatic foundation of physics—deriving physical laws from well-defined assumptions.
- One major puzzle: why time is irreversible in daily life, though atomic laws are time-reversible.
- Hilbert suggested understanding how fluid dynamics equations emerge from atomic motion as a starting point.
Theories Across Scales
- At the microscopic level, particles follow Newton’s laws.
- The mesoscopic (intermediate) level uses the Boltzmann equation for statistical behavior.
- At the macroscopic level, fluid dynamics is described by the Euler and Navier-Stokes equations.
- It's expected, but not always shown, that macroscopic laws can be derived from microscopic ones.
The New Breakthrough
- Mathematicians recently provided a derivation: tracking many hard spheres obeying Newton's laws yields the Boltzmann equation and then fluid dynamics equations as you zoom out.
- Previous derivations worked only for short times; this breakthrough extends results to longer times by systematically tracking all possible collision histories.
- The result strengthens the justification for using fluid dynamic equations in engineering and science—they follow from fundamental laws.
Significance and Limitations
- This work shows how irreversible behaviors can emerge from reversible laws, explaining the arrow of time.
- The solution does not fully resolve Hilbert’s sixth problem, as it excludes quantum mechanics, relativity, turbulence, and other complexities.
Key Terms & Definitions
- Hilbert’s Sixth Problem — The challenge to derive all physics from well-defined mathematical axioms.
- Boltzmann Equation — Describes statistical behavior of a many-particle system.
- Euler and Navier-Stokes Equations — Fundamental equations for fluid flow at the macroscopic scale.
- Time-reversible Laws — Physical laws that do not prefer a direction for time.
- Irreversible Behavior — Processes that naturally progress in one time direction, like entropy increase.
Action Items / Next Steps
- Review the derivation of macroscopic equations from Newtonian mechanics.
- Read more about Hilbert's 23 problems and their current status.
- Explore further developments in statistical mechanics and the emergence of irreversibility.