Why Complexify? Why Complexify? Universality Principles of Complex Systems Symmetry CSYS/MATH 300, Spring, 2013 | #SpringPoCS2013 Breaking The Big Theory Final words Prof. Peter Dodds For your consideration @peterdodds References Department of Mathematics & Statistics | Center for Complex Systems | Vermont Advanced Computing Center | University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . 1 of 28
Why Complexify? These slides brought to you by: Universality Symmetry Breaking The Big Theory Final words For your consideration References 2 of 28
Why Complexify? Outline Universality Symmetry Universality Breaking The Big Theory Final words Symmetry Breaking For your consideration References The Big Theory Final words For your consideration References 3 of 28
Why Complexify? Limits to what’s possible: Universality Universality ( ⊞ ): Symmetry Breaking ◮ The property that the macroscopic aspects of a The Big Theory system do not depend sensitively on the system’s Final words details. For your consideration ◮ Key figure: Leo Kadanoff ( ⊞ ). References Examples: ◮ The Central Limit Theorem: 1 e − ( x − µ ) 2 / 2 σ 2 d x . P ( x ; µ, σ ) d x = √ 2 πσ ◮ Navier Stokes equation for fluids. ◮ Nature of phase transitions in statistical mechanics. 4 of 28
Why Complexify? Limits to what’s possible: Universality Universality ( ⊞ ): Symmetry Breaking ◮ The property that the macroscopic aspects of a The Big Theory system do not depend sensitively on the system’s Final words details. For your consideration ◮ Key figure: Leo Kadanoff ( ⊞ ). References Examples: ◮ The Central Limit Theorem: 1 e − ( x − µ ) 2 / 2 σ 2 d x . P ( x ; µ, σ ) d x = √ 2 πσ ◮ Navier Stokes equation for fluids. ◮ Nature of phase transitions in statistical mechanics. 4 of 28
Why Complexify? Limits to what’s possible: Universality Universality ( ⊞ ): Symmetry Breaking ◮ The property that the macroscopic aspects of a The Big Theory system do not depend sensitively on the system’s Final words details. For your consideration ◮ Key figure: Leo Kadanoff ( ⊞ ). References Examples: ◮ The Central Limit Theorem: 1 e − ( x − µ ) 2 / 2 σ 2 d x . P ( x ; µ, σ ) d x = √ 2 πσ ◮ Navier Stokes equation for fluids. ◮ Nature of phase transitions in statistical mechanics. 4 of 28
Why Complexify? Limits to what’s possible: Universality Universality ( ⊞ ): Symmetry Breaking ◮ The property that the macroscopic aspects of a The Big Theory system do not depend sensitively on the system’s Final words details. For your consideration ◮ Key figure: Leo Kadanoff ( ⊞ ). References Examples: ◮ The Central Limit Theorem: 1 e − ( x − µ ) 2 / 2 σ 2 d x . P ( x ; µ, σ ) d x = √ 2 πσ ◮ Navier Stokes equation for fluids. ◮ Nature of phase transitions in statistical mechanics. 4 of 28
Why Complexify? Limits to what’s possible: Universality Universality ( ⊞ ): Symmetry Breaking ◮ The property that the macroscopic aspects of a The Big Theory system do not depend sensitively on the system’s Final words details. For your consideration ◮ Key figure: Leo Kadanoff ( ⊞ ). References Examples: ◮ The Central Limit Theorem: 1 e − ( x − µ ) 2 / 2 σ 2 d x . P ( x ; µ, σ ) d x = √ 2 πσ ◮ Navier Stokes equation for fluids. ◮ Nature of phase transitions in statistical mechanics. 4 of 28
Why Complexify? Universality Universality Symmetry Breaking ◮ Sometimes details don’t matter too much. The Big Theory Final words ◮ Many-to-one mapping from micro to macro For your consideration ◮ Suggests not all possible behaviors are available References at higher levels of complexity. Large questions: ◮ How universal is universality? ◮ What are the possible long-time states (attractors) for a universe? 5 of 28
Why Complexify? Universality Universality Symmetry Breaking ◮ Sometimes details don’t matter too much. The Big Theory Final words ◮ Many-to-one mapping from micro to macro For your consideration ◮ Suggests not all possible behaviors are available References at higher levels of complexity. Large questions: ◮ How universal is universality? ◮ What are the possible long-time states (attractors) for a universe? 5 of 28
Why Complexify? Universality Universality Symmetry Breaking ◮ Sometimes details don’t matter too much. The Big Theory Final words ◮ Many-to-one mapping from micro to macro For your consideration ◮ Suggests not all possible behaviors are available References at higher levels of complexity. Large questions: ◮ How universal is universality? ◮ What are the possible long-time states (attractors) for a universe? 5 of 28
Why Complexify? Universality Universality Symmetry Breaking ◮ Sometimes details don’t matter too much. The Big Theory Final words ◮ Many-to-one mapping from micro to macro For your consideration ◮ Suggests not all possible behaviors are available References at higher levels of complexity. Large questions: ◮ How universal is universality? ◮ What are the possible long-time states (attractors) for a universe? 5 of 28
Why Complexify? Universality Universality Symmetry Breaking ◮ Sometimes details don’t matter too much. The Big Theory Final words ◮ Many-to-one mapping from micro to macro For your consideration ◮ Suggests not all possible behaviors are available References at higher levels of complexity. Large questions: ◮ How universal is universality? ◮ What are the possible long-time states (attractors) for a universe? 5 of 28
Why Complexify? Universality Universality Symmetry Breaking ◮ Sometimes details don’t matter too much. The Big Theory Final words ◮ Many-to-one mapping from micro to macro For your consideration ◮ Suggests not all possible behaviors are available References at higher levels of complexity. Large questions: ◮ How universal is universality? ◮ What are the possible long-time states (attractors) for a universe? 5 of 28
Why Complexify? Fluid mechanics Universality Symmetry ◮ Fluid mechanics = One of the great successes of Breaking The Big Theory understanding complex systems. Final words ◮ Navier-Stokes equations: micro-macro system For your consideration evolution. References ◮ The big three: Experiment + Theory + Simulations. ◮ Works for many very different ‘fluids’: ◮ the atmosphere, ◮ oceans, ◮ blood, ◮ galaxies, ◮ the earth’s mantle... ◮ and ball bearings on lattices...? 6 of 28
Why Complexify? Fluid mechanics Universality Symmetry ◮ Fluid mechanics = One of the great successes of Breaking The Big Theory understanding complex systems. Final words ◮ Navier-Stokes equations: micro-macro system For your consideration evolution. References ◮ The big three: Experiment + Theory + Simulations. ◮ Works for many very different ‘fluids’: ◮ the atmosphere, ◮ oceans, ◮ blood, ◮ galaxies, ◮ the earth’s mantle... ◮ and ball bearings on lattices...? 6 of 28
Why Complexify? Fluid mechanics Universality Symmetry ◮ Fluid mechanics = One of the great successes of Breaking The Big Theory understanding complex systems. Final words ◮ Navier-Stokes equations: micro-macro system For your consideration evolution. References ◮ The big three: Experiment + Theory + Simulations. ◮ Works for many very different ‘fluids’: ◮ the atmosphere, ◮ oceans, ◮ blood, ◮ galaxies, ◮ the earth’s mantle... ◮ and ball bearings on lattices...? 6 of 28
Why Complexify? Fluid mechanics Universality Symmetry ◮ Fluid mechanics = One of the great successes of Breaking The Big Theory understanding complex systems. Final words ◮ Navier-Stokes equations: micro-macro system For your consideration evolution. References ◮ The big three: Experiment + Theory + Simulations. ◮ Works for many very different ‘fluids’: ◮ the atmosphere, ◮ oceans, ◮ blood, ◮ galaxies, ◮ the earth’s mantle... ◮ and ball bearings on lattices...? 6 of 28
Why Complexify? Fluid mechanics Universality Symmetry ◮ Fluid mechanics = One of the great successes of Breaking The Big Theory understanding complex systems. Final words ◮ Navier-Stokes equations: micro-macro system For your consideration evolution. References ◮ The big three: Experiment + Theory + Simulations. ◮ Works for many very different ‘fluids’: ◮ the atmosphere, ◮ oceans, ◮ blood, ◮ galaxies, ◮ the earth’s mantle... ◮ and ball bearings on lattices...? 6 of 28
Why Complexify? Lattice gas models Collision rules in 2-d on a hexagonal lattice: Universality Symmetry Breaking The Big Theory Final words For your consideration References ◮ Lattice matters... ◮ No ‘good’ lattice in 3-d. ◮ Upshot: play with ‘particles’ of a system to obtain new or specific macro behaviours. 7 of 28
Why Complexify? Lattice gas models Collision rules in 2-d on a hexagonal lattice: Universality Symmetry Breaking The Big Theory Final words For your consideration References ◮ Lattice matters... ◮ No ‘good’ lattice in 3-d. ◮ Upshot: play with ‘particles’ of a system to obtain new or specific macro behaviours. 7 of 28
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