# Presentation: 940702

## Emergence and asymptotics in physics: how one theory can live inside another

**Michael Berry**
**Physics Department**
**University of Bristol**

- http://www.phy.bris.ac.uk/people/berry_mv/index.html

## more general theories encompass less general ones

- in physics, theories are mathematical, relations between them involve a parameter **

- encompassing theoryless general theory as **0

- special relativityNewtonian mechanics
***=speed of particle*/*speed of light ** *0 (slow)

- statistical mechanicsthermodynamics,
*	***=1/*number of particles*0 (many)

- general relativityspecial relativity
***=*(*gravity constantxmass*)/(*speed of light*)2/size* *0* (*light)

## viscous (Navier-Stokes) flowinviscid (Euler) flow 

**=1/Reynolds number=*viscosity*/(*densityxsizexspeed*)0 (slippery)

- wave opticsray optics
**=*wavelength*/*size*0 (short waves)

- quantum physicsclassical physics
**=*Planck’s constant*/*action of system*0 (big)

- relations between theories must involve the study of limits: ***mathematical*** ***asymptotics***

## in rare cases, the limit is simple

- special relativity involves

- usually, though, the limit is ***singular***

## singular limits

**example 1: statistical mechanics********thermodynamics**

- for many particles in a box, statistical mechanics enables continuum (thermodynamic) properties to be calculated from intermolecular interactions: pressure, as a function of volume and temperature

- this fails at ***critical points**** p*c,* V*c,* T*c

## the singularity reflects the breakdown of the continuum limit: at criticality, the fluid is not smooth under low resolution, because there are density fluctuations on all scales

- this ***new state of matter*** emerges in the large-*N* limit, and is borderland phenomenon, associated with the fact that the limit is singular, obstructing the smooth reduction of statistical mechanics to thermodynamics

## zoom out

- normal state

- critical state

## example 2: viscous (Navier-Stokes) flowinviscid (Euler) flow

- flow in a pipe

- exact solution predicts mass flow rate proportional to *P*,
- whatever the viscosity

- but for low viscosity (large Reynolds number *Re*) the observed flow rate is proportional to *P*

***turbulent*** ***flow***

- smooth flow

- in turbulence, the dissipation is not smoothly distributed, but concentrates onto a ***fractal set*** as Re∞: a singular limit

## turbulence: an asymptotically emergent phenomenon

## example 3: rays and waves

- travelling wave withspeed v :

- short wave limit: wavelength **

- oscillates infinitely fast as **: pathological function!

- we perceive the time-averaged intensity (period 10-15s for light)

- two waves in opposite directions:

- abundance of singular limits

## 

- so, showing that two light rays are twice as bright as one (1+1=2) involves a singular limit and a double average

## travelling waves: cos(1/) singularity

- evanescent waves: reaching places that rays do not (outside glass with total internal reflection, dark side of a rainbow, thin layer of air near a hot road):

## singularities of ray optics: caustics - boundaries between rays and no-rays, the result of focusing

## Slide 14

## supernumerary rainbows 

- interference fringes near an angular caustic

## singularity on the singularity: a cusp

## diffraction catastrophes, decorating a hierarchy of singularities

- inhabiting the borderland between rays and waves

## example 4: quantum and classical physics

- a useful guide is the ***correspondence principle***:
- ‘quantum observables tend to their classical counterparts’

- contexts:
- 1. spectra of atoms and molecules near the classical limit involve ***statonary states***
- 2. experiments on atoms in oscillating strong fields
- 3. ***quantum chaology***

- (meaning: *ћ*<< classical quantities with dimensions of action)

## the semiclassical and long-time limits do not commute

- a bound quantum system with *D * freedoms has discrete ***energy levels***, with spacing of order *ћ**D*

- in classical physics, energy is a continuum, so the semiclassical limit is singular - to regain the continuum, must average over detector resolution, coupling to vacuum, etc...

## viewing the spectrum with a microscope of power ћ-D, the mean level spacing is O(1), and new borderland phenomena emerge   

## universal distributions of quantum energy-level spacings S for regular and chaotic classical motion

- semiclassically emergent phenomena, whose derivation from Schrödinger’s equation involves wave asymptotics, and ***universality*** in the distribution of long classical orbits

## divergent infinite series

**divergent infinite series**

- signals of singular limits

- recall Airy’s rainbow function

- short-wave limitlarge-*X *limit of Ai(*X*)

## one (decaying exponential) ray

- dark side

- two (complex exponential) rays

- bright side

- how can one exponential (dark side) become two (bright side)?

- on the dark side, short-wave series is

- divergent series

## contributions get smaller (“head”), then bigger (“tail”)

- Stokes 1847: the second exponential is born in the complex *X* plane from the divergent tail of the series for the first exponential

## conclusions

- in all non-trivial reductions in physics, the encompassing theory is a singular perturbation (parameterised by **) of the less general one

- the singularities make the quantities of the encompassing theory nonanalytic in **

- the nonanalyticities describe emergent phenomena in the borderland between the theories

- the philosophical problem of theory reduction is itself ‘reduced’ - in physics at least - to the mathematical asymptotics of singularities

- application of physics to philosophy

- to understand the philosophical problem of theory reduction, words are inadequate

## Berry, M V, 1994, ‘Asymptotics, singularities and the reduction of theories’, Proc. 9th Int. Cong. Logic, Method., and Phil. of Sci. IX, 597-607, eds. D Prawitz, B Skyrms and D Westerståhl. (paper 260 on home page)

**Batterman, Robert W, 2001, ‘The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction and Emergence (Oxford University Press)**