Mathematicians Crack 125-Year-Old Problem, Unite Three Physics Theories
A breakthrough in Hilbert’s sixth problem is a major step in grounding
physics in math
Can one single mathematical framework describe the motion of a fluid and
the individual particles within it? This question, first asked in 1900, now
has a solution that could help us understand the complex behavior of the
atmosphere and oceans.
The Challenge: Unifying Two Perspectives
Fluids, such as air and water, exhibit motion governed by intricate
mathematical equations. Two key approaches have traditionally been used to
describe this motion:
The Eulerian Approach: This perspective examines the flow of the fluid as a
whole, describing properties like velocity, pressure, and density at every
point in space. The famous Navier-Stokes equations, which remain one of the
unsolved Millennium Prize Problems, are a cornerstone of this approach.
The Lagrangian Approach: This focuses on individual particles within the
fluid, tracking their paths and interactions. The Boltzmann equation,
developed in the late 19th century, describes how particles move and
collide within a gas or liquid.
Despite the success of these models, they have remained largely separate.
The Navier-Stokes equations are powerful for large-scale fluid behavior but
fail to capture individual particle interactions. Conversely, the Boltzmann
equation excels at modelling particles but struggles to provide insights
into large-scale fluid dynamics.
Hilbert’s sixth problem was one of the loftiest. He called for
“axiomatizing” physics, or determining the bare minimum of mathematical
assumptions behind all its theories. Broadly construed, it’s not clear that
mathematical physicists could ever know if they had resolved this
challenge. Hilbert mentioned some specific subgoals, however, and
researchers have since refined his vision into concrete steps toward its
solution.
In March mathematicians Yu Deng of the University of Chicago and Zaher Hani
and Xiao Ma of the University of Michigan posted a new paper to the
preprint server arXiv.org that claims to have cracked one of these goals.
If their work withstands scrutiny, it will mark a major stride toward
grounding physics in math and may open the door to analogous breakthroughs
in other areas of physics.
In the paper, the researchers suggest they have figured out how to unify
three physical theories that explain the motion of fluids. These theories
govern a range of engineering applications from aircraft design to weather
prediction—but until now, they rested on assumptions that hadn’t been
rigorously proven. This breakthrough won’t change the theories themselves,
but it mathematically justifies them and strengthens our confidence that
the equations work in the way we think they do.
Each theory differs in how much it zooms in on a flowing liquid or gas. At
the microscopic level, fluids are composed of particles—little billiard
balls bopping around and occasionally colliding—and Newton’s laws of motion
work well to describe their trajectories.
But when you zoom out to consider the collective behaviour of vast numbers
of particles, the so-called mesoscopic level, it’s no longer convenient to
model each one individually. In 1872 Austrian theoretical physicist Ludwig
Boltzmann addressed this when he developed what became known as the
Boltzmann equation. Instead of tracking the behaviour of every particle,
the equation considers the likely behaviour of a typical particle. This
statistical perspective smooths over the low-level details in Favor of
higher-level trends. The equation allows physicists to calculate how
quantities such as momentum and thermal conductivity in the fluid evolve
without painstakingly considering every microscopic collision.
The three levels of analysis each describe the same underlying reality—how
fluids flow. In principle, each theory should build on the theory below it
in the hierarchy: the Euler and Navier-Stokes equations at the macroscopic
level should follow logically from the Boltzmann equation at the mesoscopic
level, which in turn should follow logically from Newton’s laws of motion
at the microscopic level. This is the kind of “axiomatization” that Hilbert
called for in his sixth problem, and he explicitly referenced Boltzmann’s
work on gases in his write-up of the problem. We expect complete theories
of physics to follow mathematical rules that explain the phenomenon from
the microscopic to the macroscopic levels. If scientists fail to bridge
that gap, then it might suggest a misunderstanding in our existing theories.
Unifying the three perspectives on fluid dynamics has posed a stubborn
challenge for the field, but Deng, Hani and Ma may have just done it. Their
achievement builds on decades of incremental progress. Prior advancements
all came with some sort of asterisk, though; for example, the derivations
involved only worked on short timescales, in a vacuum or under other
simplifying conditions.
The new proof broadly consists of three steps: derive the macroscopic
theory from the mesoscopic one; derive the mesoscopic theory from the
microscopic one; and then stitch them together in a single derivation of
the macroscopic laws all the way from the microscopic ones.
The first step was previously understood, and even Hilbert himself
contributed to it. Deriving the mesoscopic from the microscopic, on the
other hand, has been much more mathematically challenging. Remember, the
mesoscopic setting is about the collective behavior of vast numbers of
particles. So, Deng, Hani and Ma looked at what happens to Newton’s
equations as the number of individual particles colliding and ricocheting
grows to infinity and their size shrinks to zero. They proved that when you
stretch Newton’s equations to these extremes, the statistical behaviour of
the system—or the likely behavior of a “typical” particle in the
fluid—converges to the solution of the Boltzmann equation. This step forms
a bridge by deriving the mesoscopic math from the extremal behaviour of the
microscopic math.
The major hurdle in this step concerned the length of time that the
equations were modelling. It was already known how to derive the Boltzmann
equation from Newton’s laws on very short timescales, but that doesn’t
suffice for Hilbert’s program, because real-world fluids can flow for any
stretch of time. With longer timescales comes more complexity: more
collisions take place, and the whole history of a particle’s interactions
might bear on its current behavior. The authors overcame this by doing
careful accounting of just how much a particle’s history affects its
present and leveraging new mathematical techniques to argue that the
cumulative effects of prior collisions remain small.
Gluing together their long-timescale breakthrough with previous work on
deriving the Euler and Navier-Stokes equations from the Boltzmann equation
unifies three theories of fluid dynamics. The finding justifies taking
different perspectives on fluids based on what’s most useful in context
because mathematically they converge on one ultimate theory describing one
reality. Assuming that the proof is correct, it breaks new ground in
Hilbert’s program. We can only hope that with just such fresh approaches,
the dam will burst on Hilbert’s challenges and more physics will flow
downstream.
For over a century, physicists and mathematicians have sought a single
mathematical framework to describe the motion of fluids and the behaviour
of individual particles within them. Now, a team led by Zaher Hani at the
University of Michigan has solved this problem, fulfilling one of David
Hilbert’s famous 1900 challenges and providing deeper insight into the
physics of the atmosphere, oceans, and plasma flows. WHAT ARE THE THREE?
The Challenge: Unifying Three Scales of Motion
• Microscopic Scale: Individual particles follow Newton’s laws, colliding
and interacting randomly.
• Mesoscopic Scale: As particle interactions grow, their behavior follows
statistical mechanics—the foundation of kinetic gas theory.
• Macroscopic Scale: When particle numbers become immense, they obey fluid
dynamics, governed by the famous Navier-Stokes equations.
The problem has been stitching these descriptions together—ensuring that
the transition from particles to gases to fluids happens in a
mathematically consistent way.
The Breakthrough
Hani and his team found a way to derive fluid equations from particle
behaviour by carefully analysing the interactions between microscopic chaos
and large-scale stability. Their framework formally links:
• Newton’s Laws → Boltzmann Equation (kinetic theory) → Fluid Dynamics
(Navier-Stokes)
• This provides mathematical rigor to a process physicists long suspected
but couldn’t prove.
Why This Matters
• Better Climate Models: This solution enhances our ability to simulate
the atmosphere and ocean currents, leading to more accurate weather
forecasts and climate predictions.
• Plasma & Fusion Research: Understanding how charged particles behave at
different scales is key to nuclear fusion and space physics.
• Bridging Physics & Math: This result unifies major physical laws,
bringing us closer to a universal mathematical framework for fluid motion.
Physicists previously believed this problem was beyond reach, making this
achievement a landmark in mathematical physics—one that could reshape our
understanding of fluids in nature and technology.
The "125-year-old problem" refers to a challenge in fluid dynamics,
specifically the unification of three different scales of motion:
microscopic, mesoscopic, and macroscopic. The breakthrough involves
deriving the macroscopic behavior of fluids from the microscopic
interactions of individual particles. This unified framework addresses one
of David Hilbert's problems, which aimed to provide a rigorous mathematical
foundation for physics, according to Scientific American.
Here's a more detailed breakdown:
1. The Three Scales of Motion:
Microscopic Scale:
This describes the motion of individual particles, governed by Newton's
laws of motion.
Mesoscopic Scale:
As particle interactions become more significant, their behavior is
described by statistical mechanics, the foundation of kinetic gas theory.
Macroscopic Scale:
When the number of particles is extremely large, fluid dynamics takes over,
and the motion is described by the Navier-Stokes equations.
2. The Challenge:
The challenge was to create a single mathematical framework that seamlessly
connects these three scales. This meant ensuring that the transition from
individual particles to a gas and then to a fluid was mathematically
consistent, according to a LinkedIn post.
3. The Breakthrough:
A team of mathematicians, including Zaher Hani, have developed a method to
derive the macroscopic fluid equations from the microscopic particle
interactions. This involves analyzing how chaotic microscopic behavior can
lead to stable, large-scale fluid motion, according to the LinkedIn post.
4. Significance:
This breakthrough is significant because it provides a more fundamental
understanding of fluid dynamics and could have implications for various
fields, including:
Atmospheric and oceanic science:
Understanding the behavior of fluids in the atmosphere and oceans relies on
a unified framework for these different scales.
Plasma physics:
Similar principles apply to the study of plasma, a state of matter found in
stars and other hot environments.
FOR MATHS WWIZARDS: WHAT IS THAT?
1 The Eulerian Perspective: Navier-Stokes Equations
2 Conservation of Mass (Continuity Equation):
∂ρ/∂t+∇⋅(ρu)=0
3 Conservation of Momentum:
ρ(∂u//∂t+(u⋅∇)u)=−∇p+μ∇2u+F
The Lagrangian Perspective: Boltzann Equation
∂f//∂t+v⋅∇xf+F⋅∇ vf=(∂f//t)collision
The Unified Framework Scaling the Boltzmann Equation:
Kn=λL≪1.
Zero-order Approximation (Local Equilibrium):
f0=ρ(2πRT)3/2exp(−|v−u|22RT)
First-order Approximation (Navier-Stokes Limit):
μ=13ρλ¯v,
SO WE WILL KNOW THE AIR TURBULENCE VERY WELL. BUT A FEW STILL DOUBT THE
THEORY.
K RAJARAM IRS 19425
--
You received this message because you are subscribed to the Google Groups
"Thatha_Patty" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to thatha_patty+unsubscr...@googlegroups.com.
To view this discussion visit
https://groups.google.com/d/msgid/thatha_patty/CAL5XZorHAi9n1GinD%3DEtxZn-vR9ZqqS6Xn0HbiD0VZBGk4UC4w%40mail.gmail.com.
