Are all macroscopic phenomena derivable

from microscopic laws? Mahendra K. Verma

mkv@iitk.ac.in http://turbulencehub.org

Physics, IIT Kanpur

https://dfcollin.files.wordpress.com/2010/05

“To prove, with a single equation, that time had a beginning.”

“With one simple elegant equation to explain everything.”

Hawking at his thesis defence!

References

• P. Anderson, More is different, Science, 1972.

• R. Feynman, Character of Physical Laws

Ancient model Aristotle

wikipedia

• Objects are made of 4 elements:

• Fire, air, water, earth

• Indian thoughts-5 elements: Above 4 + Ether

• Objects try to go its natural state

• Earth is dynamic, impure, and changing

• Heavens is pure and unchanging!

• Laws of the Earth different from that of heavens!!

Verma, Introduction to Mechanics

Renaissance 1500-1700

Credits: wikipedia

• Heavens are changing too!

• Jupiter has moons as well, and they go around Jupiter!

• So, may be laws of heavens and Earth are the same!!

• Biggest argument—Law of universal gravitation

Newton’s laws; Law of universal

gravitation

S

E

Ma = ω2 R ∝R/T2 ∝ 1/R2

Kepler’s third law: T2 ∝ R3

F=ma ∝ 1/R2

Credits: wikipedia

E

M

Now treat Earth as centre

Moon, ball

a1/a2 = (R2/R1)2

a2 = 9.8 m/s2

Newton measured it… found it to be quite close to 9.8 m/s2!

Thus, discovered the Law of universal gravitation

• Electric and Magnetic force have same origin (Maxwell’s equation)

• EM + weak nuclear (Same force-different manifestation): Glashow, Salam, Weinberg

• EM + weak nuclear + strong nuclear (Standard model)

• EM + nuclear + gravity: ???

• Race to unify gravity is ON!

Unified theory/ Single Lagrangian

Theory of everything! TOE!

Is it?

Reductionist paradigm

Critic on Reductionism!

“The main fallacy in this kind of thinking is that the reductionist hypothesis does not by any means imply a constructionist one: The ability to reduce everything to simple fundamental laws does not imply ability to start from those laws and reconstruct the universe.”

Phil Anderson

Science, 1972

Stephan Hawking“I think the next [21st] century will be the century of complexity. We have already discovered the basic laws that govern matter and understand all the normal situations. We don’t know how the laws fit together, and what happens under extreme conditions.

But I expect we will find a complete unified theory sometime this century. There is no limit to the complexity that we can build using those basic laws.”

https://en.wikiquote.org/wiki/Complexity

X

Same forces works in all these systems. But this is only one aspect of things.

But the systems are very differently organised!

e.g., knowing the physics of bricks will not help us understand the beauty in architecture of beautiful buildings!!

Human metabolism To human behaviour ????

More complexity coming from laws of structure—stresses and strains in beams, columns, etc.

Laws at different levels stand on their own merit!

Physics, biology, psychology, and economics.

Laws @ all levels

in Biology

Hierarchical structures…DNA, cells, muscles, bones, blood, neutrons

systems like digestive, skeleton,…

senses, mind (psychology)

Anderson, 1972

in Physics

Microscopic laws!

Lagrangian for standard model is known!

Complex ones!

Atmosphere, stars, dynamo, bio-physics

Macroscopic laws

Fluids and gases, … Universe

(superfluids)

Which is a bigger challenge: Equation or the solution?

Is stating the equation of a system a complete description of the problem?

Role of computer solution or data science?

Illustrate using physics examples

Phase transition

• Landau & Wilson showed universality in PT.

• Liquid-vapor transition ⇔ Para-ferro transition.

• Large-scale physics: φ4 and φ6 theory

• Independent of microphysics (except its broad nature like local interactions).

• Mob behaviour— very different from individual behaviour.

Turbulence

NS equation

∂tu+ (u.∇)u = −∇σ +ν∇2u+ f

∇.u = 0

Velocity field

PressureExt. Force

Kinematic viscosity

viscous term

Turbulent limit: Re = UL/ν ≫1

Equations known—Solution Not! (Millennium problem)

Prove or give a counter-example of the following statement:

“In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both

smooth and globally defined, that solve the Navier–Stokes equations.”

$1 Million Clay prize

Not clear if smooth solution exists!

XY

Z

Verma, Physics of buoyant flows

Vorticity surface counters

Credits: Abhishek Kumar

Kolmogorov’s theory of turbulence

Energy supplied at large scale

Energy cascades … scale by scale (Fourier analysis)

Constant flux Π(k)

E(k) = KKoΠ2/3k−5/3

!

Π"(!)

#(!)!d

k

!f

Π"(!) Π"(!+dk)

Verma, Physics of buoyant flows

Verma et al., Fluid Dyn, 2018

What do we learn from turbulence?

• Kolmogorov’s theory is macroscopic law.

• Though NS equation can be derived from kinetic theory (microscopic), concepts like flux are macroscopic quantities.

• Energy transfers are asymmetric in time (LS to Small scales). Nonequilibrium

Arrow of time

• Asymmetric energy transfer easily sets the arrow of time.

• This is possibly the solution for many driven nonlinear systems.

• We can also define dissipation as a loss of coherent kinetic energy to heat energy of microscopic particles.

Variable energy flux

(k+dk)ββ

ΠΠ(k)

𝓕(k)dk

D(k)dk

𝓕(k) < 0

𝓕(k) > 0

⇒ Π(k) decreases

⇒ Π(k) increases

ddk

Π(k) = ℱ(k) − D(k)

Connection to finance!Money supply from the centre!

Goes to states, district, … villages

Corruption — F(k) <0

Philanthropy — F(k) >0

Buoyancy-driven turbulence

XY

Z

Hot Plate

Cold Plate

Periodic

Periodic

Eu(k) ∼ k-11/5

Πu(k) ∼ k-4/5

Not Kolmogorov-like

Kumar et al., PRE 2014

Bolgiano-Obhukhov

Kolmogorov-like

Verma et al, 2017 Verma 2018

Cold

Hot

Verma, Physics of buoyant flows

t=49 t=149t=98-4 0 4

XY

Z

Y

X X X

(a) (b) (c)

(d) (e) (f)

Rotating turbulence

Sharma et al., 2017

Dynamic pressure vs. thermodynamic pressure

P +½ ρu2 = CBernoulli's principle

http://mad-vik-low.blogspot.com/2013/08/bernoullis-principle.html

Fluid — A multiscale picture

Lessons from turbulence!

• Kolmogorov’s law would be very very difficult (impossible) to derive from the first principles.

• Concepts like energy flux and dynamic pressure come from macroscopic description—not kinetic theory.

• Variable energy flux (macrophysics idea) help us understand these varied systems better.

Conclusions

Different (emergent) behaviour at different scales?

Different laws at different levels

Not all macroscopic laws are derivable from microscopic laws.

At least, conveniently!

There may be some common principles.Large-scale modes play bigger role

Energy flux ideas!

These laws stand on their own merit! (with connections)

Thank you!