Final Project: Simulating Ising Spin Relaxation with Glauber Dynamics#

Background#

In real magnetic systems, spins don’t instantly jump between configurations—they evolve in time, often driven by thermal fluctuations and local interactions. The Glauber dynamics provides a simple and physically motivated way to model how Ising spins flip over time.

This model introduces stochastic dynamics where spins flip with a temperature-dependent rate, allowing you to study:

  • Relaxation to equilibrium from non-equilibrium initial states,

  • Magnetization dynamics and response functions,

  • Time-dependent correlation functions.

Unlike Metropolis dynamics (which samples from the Boltzmann distribution), Glauber dynamics describes physical time evolution, making it perfect for exploring dynamical aspects of the Ising model.

Project Objective#

Implement Glauber spin-flip dynamics for the 1D or 2D Ising model, and investigate:

  • Relaxation of magnetization from a non-equilibrium initial condition

  • Correlation functions and dynamical time scales

  • Temperature dependence of the relaxation time

Tasks#

1. Glauber Spin-Flip Dynamics#

For a spin \( s_i = \pm 1 \), the flip probability is given by:

\[ w(s_i \to -s_i) = \frac{1}{1 + \exp(\beta \Delta E)} \]

where \( \Delta E \) is the energy change if spin \( s_i \) flips:

  • \( \Delta E = 2 J s_i \sum_{\langle i,j \rangle} s_j \)

Steps:

  1. Pick a random site \( i \)

  2. Compute \( \Delta E \)

  3. Flip the spin with probability:

\[ P_{\text{flip}} = \frac{1}{1 + e^{\beta \Delta E}} \]

Repeat this \( N \) times per time unit (where \( N \) = number of spins), to define one Monte Carlo sweep.

2. Set Up Simulations#

  • Use a 1D or 2D lattice (start with 2D: \( L \times L \), e.g., \( L = 32 \))

  • Start with fully magnetized state (e.g., all spins up)

  • Set parameters: \( J = 1 \), various temperatures \( T = 1.0 \text{ to } 3.0 \)

  • Simulate Glauber dynamics for many time steps (sweeps)

3. nalyze Relaxation#

  • Track total magnetization \( M(t) \)

  • Observe how \( M(t) \) decays toward equilibrium

  • Fit \( M(t) \sim e^{-t/\tau} \) and extract relaxation time \( \tau \)

  • Plot \( \tau \) as a function of temperature

4. Compute Correlation Functions#

Define the time autocorrelation function:

\[ C(t) = \langle s_i(0) s_i(t) \rangle \]

  • Choose a set of spins (or average over all)

  • Compute how quickly individual spins forget their initial state

Optional Challenges#

  • Compare Glauber vs. Metropolis dynamics

  • Study critical slowing down near \( T_c \)

  • Add a small magnetic field and study relaxation toward field-aligned equilibrium

  • Implement spatial correlation functions and structure factors

Learning Outcomes#

  • Understand the physical meaning of spin dynamics and relaxation

  • Connect microscopic rules to macroscopic observables (e.g., magnetization decay)

  • Explore how temperature influences dynamics

  • Practice simulating stochastic processes with physically meaningful rates

Starter Parameters#

Parameter

Value

Lattice size

\( L = 32 \)

\( J \)

1.0

\( T \)

1.0 – 3.0

\( \beta \)

\( 1 / T \)

Time step

1 sweep (N flips)