Ising Models, Statistical Mechanics, and Critical Phenomenon

1. Ising Models, Statistical Mechanics, and Critical Phenomenon

2. Original 1-Dimensional Ising Model

3. Ferromagnetic Or

4. Antiferromagnetic

5. Paramagnetic

6. Hamiltonian • Represents the total Energy in the System • Classically this is usually kinetic and potential energy • In Ising Models the following is generally taken to be the Hamiltonian, where J is the interaction

7. Sign Conventions • Jij > 0, the interaction is ferromagnetic • Jij = 0, the interaction is non-interacting • Jij < 0, the interaction is antiferromagnetic • Bj > 0, the site prefers to line up with external field • Bj = 0, the site has no external influence • Bj < 0, the site prefers to line up opposite to the external field

8. Entropy • Is proportional to the natural logarithm of the number of microstates which could produce the observed macrostate (configuration S) of the system.

9. Free Energy • The free energy is a measure of the useful work obtainable from a closed thermodynamic system. • The negative change in free energy of a process is equal to the maximum amount of work extractable

10. Heat Capacity • represents the ratio of the amount of energy added to a system to the resulting change in temperature • Extensive property

11. Specific Heat Capacity • Typically represents the ratio of the amount of energy needed to raise the temperature of one Kg of a substance one degree Kelvin. • However, for our purposes we are interested in the dimensionless specific heat capacity • Intensive property

12. Magnetization • Is just the average value of the spins • Is one measure of the amount of order in the system • Intensive property

13. Magnetic Susceptibility • Measures how much the magnetic field changes in the presence of a changing magnetic field.

14. “Truncated” Correlation Function • Measures how much two sites are correlated. • If the spins are independent this quantity is near zero • Another measure of the order of a system

15. Correlation Length • The correlation length, ,is a quantity that is related to the truncated correlation length as so… • is a length scale that characterizes the exponential decay as one gets farther from some point.

16. Configuration Probability • The configuration probability is the probability of the system being in a particular configuration at equilibrium. • Where is the so called inverse temperature • And Z is a normalization constant called the partition function

17. Use of the Configuration Probability • For a function f(S) of the configuration, the configuration probability allows for the calculation of the expectation value of f(S). • For example, the expectation value of the Hamiltonian, the average Energy is:

18. The Canonical Partition Function • is a Boltzmann Factor • The partition function describes the statistical properties of a system at equilibrium. • It describes a system that is allowed to exchange heat with the environment at a fixed temperature, volume, and number of particles.

19. Partition Function is Fundamental Average Energy Variance in Energy Specific Heat Capacity

20. Partition Function is Fundamental Entropy Free Energy Free Energy Per Site

21. In Practice Free Energy Per Site is Often Used Magnetic field Magnetic Susceptibility Specific Heat Capacity

22. Exact Solution of the 1 Dimensional Ising Model

23. Exact Solution of the 2 Dimensional Ising Model

24. Exact Solutions of the Critical Point or Curie Point for the Ising Model Ising 1925 Onsager 1944

25. Metropolis-Hastings and Monte Carlo Solutions to Ising Model • Is a Markov Chain Monte Carlo (MCMC) method • A 20x20 lattice would have 2400 states • Chooses states based on Boltzman Factors • Is a kind of random walk • Used to guarantee equilibrium conditions

26. Metropolis-Hastings Algorithm • Randomly select a spin site • Calculate the energy difference between the site with its spin flipped and the site as it is, which is the energy required to make it flip. • If the flipped state is selected. If accept only if 4. Repeat

27. Average Energy vs. Temperature

28. Specific Heat vs. Temperature

29. Magnetization vs. Temperature

30. Magnetic Susceptibility vs. Temperature

31. Ising Simulations • Metropolis Simulation

32. Phenomena Associated with Phase Transitions and Critical Phenomenon • Abrupt change at a critical point (1st order, 2nd order) • Order to disorder, or differently ordered, transition (1st and 2nd order) • Latent Heat (1st order) • Discontinuous in 1st derivative of free energy (1st order) • Not analytic, discontinuous in 2nd derivative of free energy(2nd order) • Correlation Length diverges (2nd order) • Micro fluctuations don’t tend to cause macro fluctuations beyond the critical point (2nd order) • Boundary conditions no longer matter past critical point (2nd order)

33. Example of Diverging Correlation Length

34. Example of Macro-fluctuations that are relatively independent of Micro-fluctuations

35. Example of Macro-fluctuations that are relatively dependent upon Micro-fluctuations

36. Some Critical Exponents of 2nd order Phase Transitions * the beta exponent here is not the inverse temperature

37. Critical Exponents and Universality Classes • Critical exponents are the result of power laws that appear to characterize the macroscopic behavior of a system, especially near the critical point of a phase transition. • Power laws frequently, but not always, characterize certain aspects of a system. For instance, in the 2D Ising model specific heat diverges as • If systems with different underlying rules and Hamiltonians have the same critical exponents and scaling laws they may be said to belong to the same universality class. • Which is to say that some macroscopic behavior of systems is relatively independent of the microscopic mechanisms at work.

38. Critical Exponents of the Ising Model * the beta exponent here is not the inverse temperature ** exact results from mean field-theory

39. Scaling Laws of the Ising Model

40. Generalizations of IsingModel • Potts (discrete states greater than 2) • XY Model (continuous states) • N-Vector Model (more than 2 components) • All kinds of networks rather than square lattice. • Generalized Hamiltonians with varying interactions parameters (e.g. Jij)

41. Network of Model Generalizions N-Vector Model (> 2 components) Standard Model Particle Physics (4 Components) Heisenberg Model Quantum Mechanics (3 components) XY-Model (2 components, continuous states) Self Avoiding Random Walks (0 components) Potts Model (2 components , > 2 discrete states) IsingModel ( 1 component, 2 discrete states) Voter Model Lattice Gas, Dilutions, or Percolation Models (lattice sites missing)

42. Types of Analysis • Analytical or exact solutions (typically assumes that a system has infinitely many parts) • Markov Chain Monte Carlo (e.g. Metropolis, used to numerically approximate a static equilibrium, or to study the dynamics of a system) • Mean field theory (replace all complex individual interactions with an average representative interaction , more exact in higher dimensions) • Renormalization group (modern investigation of the changes of a system at different length scales)

43. Social Science Applications • Discrete Choice Models • Opinion Dynamics • Price Formation • (DSGE) General Equilibrium Models in Economics • Theory Choice in Science and Paradigm Shifts • Community Detection • Information Diffusion • Advertising • Patent Law Explosion • Majority rule