Physics and Complexity David Sherrington University of Oxford & - - PowerPoint PPT Presentation
Physics and Complexity David Sherrington University of Oxford & Santa Fe Institute Physics Dictionary definition: Branch of science concerned with the nature and properties of matter and energy But today I want to use it as much as a
University of Oxford & Santa Fe Institute
Dictionary definition: Branch of science concerned with the nature and properties of matter and energy But today I want to use it as much as a mind-set with valuable methodologies And to show application to many complex systems in many different arenas
as sometimes portrayed
‘Fundamental’ particles How it all began Search for the
‘Fundamental’ particles How it all began
‘Theory of everything’ TOE is by no means the whole story Many body systems often give new behaviour through co-operation
Both ‘fundamental’ and applicable
– Flux quantization
– Basis of modern high capacity data storage
– Quantized conductivity plateaux
Highest accuracy measurements of fundamental constants even in dirty systems
– non-trivial and new – not simply anticipated from microscopics – even with simple individual units – and simple interaction rules
superficially different systems
– Minimal models – Comparisons/checks: e.g. simulation – Analysis: maths & ansätze
Build
Hard Optimization Information Science Computer Science Biology Economics Glassy Materials Mathematical Physics Probability Theory
Hard Optimization Information Science Computer Science Biology Economics Glassy Materials Mathematical Physics Probability Theory
Two-dimensional cartoon of high dimensional concept Many metastable states Hierarchy
Valleys within valleys
Hard to minimise: sticks: glassiness
Coordinate to minimise Dynamics c.f. motion
Fixed Variable
(variable)
Slow Fast External influences .. .. ..
ij ij k
– Competitive magnetic interactions – No periodicity → no simple best compromise
Ferro Antiferro
non-equilibrium equilibrium
Field-cooled Zero-field cooled
Tg
( ) ij i j ij
Cost or Hamiltonian
Spin up/down
( ) ij i j ij
( ) ij i j ij
Satisfaction Dorm A/B To maximise
Allocate N students to 2 residences with maximum happiness Also
i i
s =
No freezing
Ferromagnetic freezing GFM
Temperature/noise/uncertainty/Dean’s impatience Attractive bias
Many metastable states
‘Rugged landscape Slow dynamics Easy to equilibrate Hard to equilibrate
Hard Optimization Information Science Computer Science Biology Economics Glassy Materials Mathematical Physics Probability Theory
– e.g. distribution of tasks, partitioning
– Simultaneous satisfaction of ‘clauses’
– Capacity and accuracy
– Analogue: “statics”/equilibrium
– Needs algorithms = dynamics
simultaneous satisfiability
# of clauses # of variables M N
2 3 3 4 5
Recent example of hard optimization from computer science
HARD-SAT N/M UNSAT SAT αc
α d
Simple algorithms stick Theoretically achievable limit
Physicists recognised this subtlety through comparison with K-spin glass Phase transitions
RSB1 T Td Ts RS
Dynamical transition Thermodynamical transition
Where the idea came from
RSB=Glassy
RSB2
Originally looked at as a purely intellectually interesting extension
HARD TO RETRIEVE
Redundancy
UNRETRIEVABLE RETRIEVABLE
Shannon limit
RETRIEVABLE
Normal algorithms stick And now we know why
α* αd αc αs
SAT UNSAT EASY HARD In fact, more regimes
– Algorithms given
– We can design algorithms
– Simulated annealling – Simulated tempering – Clustering……. Great advance: Survey propagation
Artificial ‘temperature’ Tanneal
Achieving it requires (algorithmic) dynamics Frustration & disorder → glassiness But we can choose the dynamics
anneal configurations
effective stat. mech./thermodynamics
A
A T
→
Steepest descent gets stuck
Add ‘temperature’: freedom
Annealing temperature
( ) ~ exp( / )
A
P move C T −∆
TA
Annealing temperature
TA
Annealing temperature
Good basic tool but now better ones
Hard Optimization Information Science Computer Science Biology Economics Glassy Materials Mathematical Physics Probability Theory
(a) (b)
Input Output
i ij j j
U J V =
Rounding ~ “temperature” T
ij i j ij i j ij
µ µ µξ ξ
Synaptic response
‘attractors’
~ memorized patterns
‘basins of attraction’ determined by {Jij}
~ many attractors require frustration
Phase space
{Jij} Different timescales fast retrieval slow learning
Retrieval ‘Spin glass’
(metastable attractors unrelated to memories)
Para Synaptic ‘temperature’ Capacity: Pattern interference noise
(c.f. ferromagnet) (No attractors)
Retrieval
c.f. ferro
– We design
– self-train/select
Hard Optimization Information Science Computer Science Biology Economics Glassy Materials Mathematical Physics Probability Theory
Price Time
Different strategies (Disorder) Common information (Mean field) Learn from Experience (Dynamics II) Not all can win (Frustration) Buy & sell (Dynamics I)
Minimalist model
Phase transition: α c
minimum in volatility α < α c non-ergodic α > α c ergodic
Random Non- ergodic Ergodic
Random strategies, random histories c.f. spin glass susc.
Effective interaction between agents
Quasi-random J and h related to agent strategies c.f. spin glass or neural network ** Strategy point-score dynamics for agents with 2 strategies
{ sgn ( )}
( 1) ( ) /
i i
i i i s p t
p t p t H s
=
+ = − ∂ ∂
ij i j i i ij i
a
Phase space
ij i j ij
ij i j
µ µ µ
Difference from Hopfield neural network
Map to macroscopic variables (multi-time) Effective ensemble of single agents with ensemble-self-consistent memory and coloured noise
1 ' '
( 1) ( ) sgn ( ') ( ) ( )
tt t t
p t p t p t t η α α
− ≤
+ + = − +
G
“Representative agent ensemble”
pi(0)=0 pi(0)=1
Open = simulations Solid = numerical iteration of analytic effective agent equations
pi(0)=0.5 Initial bias
Current price (t) buy sell
Price-line
c.f. Evaporation-deposition-annihilation Agents place or remove orders: buy, sell, market. May be executed.
Speculators gain on price changes. Manufacturers must absorb → liquidity.
But how do they choose what to do? Evolution of strategies?
Driven by individual attitudes, co-operative actions, learning? More realistic extension of minority game?
Glassy Materials Hard Optimization Information Science Economics Biology Computer Science Mathematical Physics Probability Theory
– Minimalist modelling – Sophisticated mathematical analysis – Computer simulation
– Conceptualization
+ Conclusion I
Not only of
but also of
for
And there are many more applications still to consider
through physics
ConclusionII
– Needs much fleshing – but I hope illustrative of possibilities
– Not individuals
– e.g. Not a guide for stockmarket speculation
– And there is much more to do
Tomaso Aste Jay Banavar Arnaud Buhot Andrea Cavagna Premla Chandra Tuck Choy Ton Coolen Dinah Cragg Lexie Davison Malcolm Dunlop Alex Duering Sam Edwards David Elderfield Fernando Nobre Dominic O’Kane Reinhold Oppermann Richard Penney Albrecht Rau Hans-Juergen Sommers Nicolas Sourlas Byron Southern Mike Thorpe Tim Watkin Andreas Wendemuth Werner Wiethege Michael Wong Julio Fernandez Juan Pedro Garrahan S.K.Ghatak Irene Giardina Peter Gillin Paul Goldbart Lev Ioffe Peter Kahn Scott Kirkpatrick Stephen Laughton Esteban Moro Peter Mottishaw Normand Mousseau Hidetoshi Nishimori
Teachers, colleagues, students, postdocs, friends
Tomaso Aste Jay Banavar Ludovic Berthier Stefan Boettcher Arnaud Buhot Andrea Cavagna Premla Chandra Tuck Choy Ton Coolen Dinah Cragg Lexie Davison Andrea De Martino Malcolm Dunlop Alex Duering David Elderfield Julio Fernandez Tobias Galla Juan Pedro Garrahan S.K.Ghatak Irene Giardina Peter Gillin Paul Goldbart Lev Ioffe Robert Jack Alexandre Lefevre Turab Lookman Peter Kahn Scott Kirkpatrick Helmut Katzgraber Stephen Laughton Francesco Mancini Marc Mezard Esteban Moro Peter Mottishaw Normand Mousseau Hidetoshi Nishimori Fernando Nobre Dominic O’Kane Reinhold Oppermann Giorgio Parisi Richard Penney Albrecht Rau Avadh Saxena Manuel Schmidt Hans-Juergen Sommers Nicolas Sourlas Byron Southern Mike Thorpe Tim Watkin Andreas Wendemuth Werner Wiethege Stephen Whitelam Peter Wolynes Michael Wong Phil Anderson Sam Edwards Walter Kohhn
– Partition function
– Generating functional * Transform to macrovariables: average over disorder
Multi-replica/ multi-time correlation & response fns
* Infinite-range
extremal dominance ~ solubility + subtlety)
For aficionados