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gabriele.perfetto@uni-tuebingen.de

Fenomeni critici fuori equilibrio: cosa hanno in comune infezioni, reazioni chimiche e incendi?

About myself

2006-2011

About myself

2006-2011

Bachelor

Physics Engineering

2011-2014

Master

Physics of complex systems

2014-2016

About myself

2006-2011

Bachelor

Physics Engineering

2011-2014

Master

Physics of complex systems

2014-2016

Ph.D. Statistical Physics 2016-2020

About myself

2006-2011

Bachelor

Physics Engineering

2011-2014

Master

Physics of complex systems

2014-2016

Ph.D. Statistical Physics 2016-2020

Researcher 2021-2025

Researcher 2025 - ...

About myself

About today

How to pass from microscopic to macroscopic?

Central question of Theoretical Physics

How to pass from microscopic to macroscopic?

Central question of Theoretical Physics

Disclaimer: no general and final answer

Part I: critical phenomena and scale invariance

Phase transitions

Vapor

Temperature

Phase diagram

Phase diagram

Phase diagram

Phase diagram

Latent heat

1^st order phase transition

Phase diagram

Latent heat

1^st order phase transition

No latent heat, 2^nd order phase transition

The Ising model

M

Ernst Ising (Köln 1900- USA 1998)

Idea from Wilhelm Lenz (1920); solved by Ising (1924)

Spontaneous magnetization

Phases in the Ising model

Ferromagnetic ordering: aligned spins are energetically favourable

Temperature: thermal fluctuations favour entropy and destroy order

Phases in the Ising model

Ferromagnetic ordering: aligned spins are energetically favourable

Temperature: thermal fluctuations favour entropy and destroy order

Phases in the Ising model

Ferromagnetic ordering: aligned spins are energetically favourable

Temperature: thermal fluctuations favour entropy and destroy order

Phases in the Ising model

Ferromagnetic ordering: aligned spins are energetically favourable

Temperature: thermal fluctuations favour entropy and destroy order

Phases in the Ising model

?

Critical point – correlation length

Critical point – correlation length

Scale invariance and scaling hypothesis

• The correlation length ξ is the only relevant length scale at criticality
• ξ(T) diverges at the critical point T=T_C: 2^nd order phase transition
• The system has no characteristic length: it is scale invariant

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Scale invariance and scaling hypothesis

“Great fleas have lesser fleas upon their backs to bite them. And lesser fleas have lesser still. And so ad infinitum”

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Johnathan Swift

• The correlation length ξ is the only relevant length scale at criticality
• ξ(T) diverges at the critical point T=T_C: 2^nd order phase transition
• The system has no characteristic length: it is scale invariant

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Scale invariance: if a part of the system is magnified until it is as large as the original system, one would not be able to tell the difference between the magnified part and the original system

Scale invariance and scaling hypothesis

“Great fleas have lesser fleas upon their backs to bite them. And lesser fleas have lesser still. And so ad infinitum”

​

Johnathan Swift

• The correlation length ξ is the only relevant length scale at criticality
• ξ(T) diverges at the critical point T=T_C: 2^nd order phase transition
• The system has no characteristic length: it is scale invariant

​

Scale invariance: if a part of the system is magnified until it is as large as the original system, one would not be able to tell the difference between the magnified part and the original system

Scale invariance and scaling hypothesis

“Great fleas have lesser fleas upon their backs to bite them. And lesser fleas have lesser still. And so ad infinitum”

​

Johnathan Swift

• The correlation length ξ is the only relevant length scale at criticality
• ξ(T) diverges at the critical point T=T_C: 2^nd order phase transition
• The system has no characteristic length: it is scale invariant

​

Scale invariance: if a part of the system is magnified until it is as large as the original system, one would not be able to tell the difference between the magnified part and the original system

Power law

Scale invariance in the critical Ising model

Scale invariance in the critical Ising model

Theory of universality: renormalization group

Kenneth G. Wilson

Leo Kadanoff

If the physics is the same on all length scales, then we should be able to able to rescale our problem, to cast it on a different length scale, and get back the same thing.

Theory of universality: renormalization group

Kenneth G. Wilson

Leo Kadanoff

If the physics is the same on all length scales, then we should be able to able to rescale our problem, to cast it on a different length scale, and get back the same thing.

Coarse graining: block spins together and vote

Theory of universality: renormalization group

Kenneth G. Wilson

Leo Kadanoff

If the physics is the same on all length scales, then we should be able to able to rescale our problem, to cast it on a different length scale, and get back the same thing.

Coarse graining: block spins together and vote

Theory of universality: renormalization group

Theory of universality: renormalization group

Theory of universality: renormalization group

Universality: liquid and gas

At the critical point all length scales are identical: microscopic details become irrelevant

Universality: liquid and gas

At the critical point all length scales are identical: microscopic details become irrelevant

Universality: liquid and gas

At the critical point all length scales are identical: microscopic details become irrelevant

Universality and critical exponents

FERRO

PARA

Different materials fit into a small number of universality classes that share broad properties

Universality and critical exponents

FERRO

PARA

Different materials fit into a small number of universality classes that share broad properties

Universality and critical exponents

FERRO

PARA

Critical Point

Different materials fit into a small number of universality classes that share broad properties

Critical Point

Universality and critical exponents

FERRO

PARA

Critical Point

Different materials fit into a small number of universality classes that share broad properties

Critical Point

Universality classes: liquid, gas and beyond

Ferromagnets

Liquid

Universality classes: liquid, gas and beyond

Ferromagnets

Liquid

Universality classes: liquid, gas and beyond

Ferromagnets

Liquid

Part II: Nonequilibrium critical phenomena and scale invariance

Percolation

The term percolation (from the Latin percolare = to filter) means to make a liquid to pass through fine interstices and is often used in the context of filtering.

The french word percolateur=coffee machine makes this even more explicit.

Percolation

Percolation

Percolation

Directed Percolation

Directed Percolation

Contact process

healing

infection

healthy

infected

What happens at long times? Does the disease spread or disappear?

Forest fires

Storm

About rabbits...

B

About rabbits...

D

B

About rabbits...

D

B

C

About rabbits...

Diffusion

D

B

C

About rabbits...

Diffusion

D

B

C

Discrete, stochastic, interacting: bacteria, genes, fish, chemical elements, etc...

And directed percolation

Diff.

D

B

C

And directed percolation

Diff.

D

B

C

And directed percolation

Diff.

D

B

C

And directed percolation

Diff.

D

B

C

And directed percolation

Diff.

D

B

C

Absorbing-state phase transition

Continuous phase transition

• Critical exponents

Absorbing-state phase transition

Continuous phase transition

• Critical exponents

Predator-prey models

Predator-prey models: Lotka-Volterra

Alfred James Lotka (March 2, 1880 – December 5, 1949) was a US mathematician, physical chemist, and statistician, famous for his work in population dynamics and energetics.

Vito Volterra (3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology.

Predator-prey models: Lotka-Volterra

• Umberto D'Ancona studied the fish catches in the Adriatic Sea and had noticed that the percentage of predatory fish caught had increased during the years of World War I (1914–18).
• This puzzled him, as the fishing effort had been very much reduced during the war years.
• Volterra developed his model independently from Lotka and used it to explain d'Ancona's observation.

Predator-prey models

Hudson's

Bay Company (Canada)

Predator-prey models: Lotka-Volterra

Cycles: population oscillations

Predator-prey models: Lotka-Volterra

Rich spatiotemporal structures. Need to include diffusion and spatial fluctuations

Predator-prey models: Lotka-Volterra

Rich spatiotemporal structures. Need to include diffusion and spatial fluctuations

Predator-prey models: Lotka-Volterra

Rich spatiotemporal structures. Need to include diffusion and spatial fluctuations

Predator-prey models: Lotka-Volterra

Rich spatiotemporal structures. Need to include diffusion and spatial fluctuations

Extinction threshold and directed percolation

Extinction threshold and directed percolation

Again Directed Percolation universality class!

Conclusions

• We talked about universality in physics.
• its discovery, in my opinion, is one of the landmarks in modern physics. It means you can take information from a magnet and make sensible comments about a neural network or a complex colloidal liquid.
• It means that simple models like the Ising model can make exact predictions for real materials. This is precisely why simple theoretical models are so effective in making predictions about everyday life
• At macroscopic scales, patterns emerge. These patterns are not sensible to microscopic details. Only the essential features survive.

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