The Ising Models-Simulations in Statistical Physics-Lecture Slides, Slides of Statistical Physics

Download The Ising Models-Simulations in Statistical Physics-Lecture Slides and more Slides Statistical Physics in PDF only on Docsity!

The Ising Models

Simulations in

Statistical Physics

Ising Model

Dr. Ernst Ising May 10, 1900 – May 11, 1998

Ensemble

• System of N particles characterized by macro variables:

N, V, E

• macro state refers to a set of these variables
• There are many micro states which give the same

values of {N,V,E} or macro state.

• micro states refers to points in phase space
• All these micro states constitute an ensemble

Microcanonical Ensemble

• Isolated System
  • N particles in volume V
  • Total energy is conserved
  • External influences can be ignored
• Microcanonical Ensemble
  • The set of all micro states corresponding to the macro state with value N,V,E is called the Microcanonical ensemble
• Generate Microcanonical Ensemble
  • Start with an initial micro state
  • Demon algorithm to produce the other micro states

One-Dimensional Classical Ideal Gas

• Ideal Gas
  • The energy of a configuration is independent of the positions of the particles
  • The total energy is the sum of the kinetic energies of the individual particles
• Interesting physical quantity
  • velocity
• A Program of the 1-D Classical Ideal Gas
  • Using the demon algorithm

Canonical ensemble

• Normally system is not isolated.
  • surrounded by a much bigger system
  • exchanges energy with it.
• Composite system of laboratory system and

surroundings may be consider isolated.

• Analogy:
  • lab system <=> demon
  • surroundings <=> ideal gas
• Surroundings has temperature T which also

characterizes macro state of lab system

Phase transitions

• Examples:
  • Gas - liquid, liquid - solid
  • magnets, pyroelectrics
  • superconductors, superfluids
• Below certain temperature Tc the state of the system

changes structure

• Characterized by order parameter
  • zero above Tc and non zero below Tc
  • e.g. magnetisation M in magnets, gap in superconductors

Why Ising model?

• Simplest model which exhibit a phase transition in two

or more dimensions

• Can be mapped to models of lattice gas and binary

alloy.

• Exactly solvable in one and two dimensions
• No kinetic energy to complicate things
• Theoretically and computationally tractable
  • can make dedicated ‘Ising machine’

the idea behind a Monte Carlo simulation

• Many systems cannot be described by equations
• Many equations can not be solved
• We forget about finding a solution and compile all the possible solutions and determine their probabilities
• We take the solution of the highest probability
• This works for systems with many individual components, because on average, they will all behave like the solution of the largest probability
• We are interested in the average behavior, the most common behavior, because that’s what is predictable or controllable
• Monte Carlo methods are statistical methods to find solutions of high probability

Intro to Metropolis Algorithm

• One of Monte Carlo methods to arrive at a stable solution
• Start with a random initial configuration
• Suggest a change with probability p
• Accept the change with probability q
• Generate a random number from a random number generator of uniform distribution between 0 and 1
• Let the action be carried out if the random number generated < probability of action
• Reiterate process starting again by suggesting a change

A One-D Ising model

• Sets up a 1-D lattice of n points
• Each point in the lattice is randomly assigned a value of 1 or -
• Calculates the energy of the system according to the Hamiltonian

H = - K Σ si sJ - B Σ si Where J=1 , B=

• Periodic boundary conditions - sn+1 = s 1 - the system becomes a circle
• Picks a random point and switches its magnetic moment
• Calculates the energy of the configuration.

… one-D program overview

• Compares energy of the system with and without the change
• If the energy of the perturbed system is lower, the change is accepted with probability = 1
• If the energy of the perturbed system is higher, the change is accepted with probability = exp (-Δ/ k T)
• Iterations of the routine lead to a configuration of global minimum of energy

Where this probability comes from:

1902 - Gibbs derived that the expression for the probability of an equilibrium configuration P (^) i = 1/Z exp(-E (^) i / kT) Z = Σi exp( – E (^) i / kT )

• the partition function
• the normalizing constant, sum of all probabilities for all possible configurations.
• Most times, a near impossibility to calculate
• Due to the way nature works, a system changes in small steps and does not go very far from the thermal equilibrium situation. Taking advantage of this, we will create a random change and then compare the probability of either configuration as a thermal equilibrium configuration.
• P1= 1/Z exp(-E1/ kT) and P2= 1/Z exp(-E2/ kT)
• P = P2/P1 = exp((E1-E2) / kT)

Josiah Willard Gibbs, 1839-

Markov Chain

• The current situation depends

only on the situation one time

step before it

• If the day is one time unit and

weather is a Markov process,

tomorrow's weather depends only on today’s weather. Prior days have no influence.

• The Ising model is a Markov process.

Andrei Andreyevich Markov 1856-