Complexity Explorer Santa Few Institute

Computation in Complex Systems (Summer 2020)

This course is no longer in session.

• Two Kinds of Paths
• Polynomials vs. Exponentials
• Divide and Conquer
• Big O and All that
• When the details don't matter
• Quiz 1 (self-assessment)
• Exam
• Divide and Conquer Redux
• Dynamic Programming
• Greedy Algorithms
• Reductions and Translations
• Lessons So Far
• The Best of All Possible Algorithms
• Complexity Wrap-Up
• Quiz 2 (self-assessment)
• Exam
• Finding versus Checking
• Circuits and Formulas
• More NP-complete Problems
• P versus NP Problem
• Existence and Nonexistence
• Above and Beyond
• Exam
• Real World Problems
• Phase Transitions
• Random Problems
• Solvability Threshold
• Modeling Differential Equations
• Landscapes, Clustering, Freezing, and Hardness
• Exam
• Building Blocks: Recursive Functions
• Building Blocks: Partial Recursive Functions
• λ Calculus
• Turing Machines
• The Halting Problem
• The Grand Unified Theory of Computation
• The Analytical Engine
• Cellular Automata
• Tile-Based Computation
• Dynamical Systems
• Quiz 5 (self-assessment)
• Exam
• More from Cris Moore
• Other ComplexityExplorer resources

The Ising Model: temperature dependence

In his 1924 Ph.D. thesis, physicist Ernst Ising presented a mathematical model that described the properties of iron magnetism. The one-dimensional model consists of a series of fixed vertices or sites – iron atoms – that exist in either of two states, or spins, -1 and +1. Each site can interact with its neighbors; neighbors with the same spin have lower energy. The model was extended to two dimensions by Lars Onsager, an advance that importnatly showed that the system undergoes a phase transition. (Onsager recieved the Nobel Prize for his work on thermodynamic irreversibility.)

Ferromagnetism exhibits a critical dependence on temperature. The lattice Ising Model recapitulates these temperature-dependent phase transitions. The simulation here lets you investigate how temperature affects the coupling of spin states throughout the lattice.