Post your questions on this hwk in the blog at the end of the page. If you hand your Hwk by e-mail, PLEASE write the file in the format HW9_YOUR-NAME and send it to G. Roux. ANDM. Civellibefore the deadline Friday December 11 at noon, otherwise it will be impossible to have it corrected before your oral exam and hence it will not be considered.

Introduction

In this short exercise, we apply the simulation method of dynamic Monte Carlo algorithms to the single spin in a field.

A spin in a field

Figure 1. Energy E of one spin in a field h.

Consider a single Ising spin σ=±1 in a magnetic field h, as shown in Figure 1. We consider its Monte Carlo dynamics at inverse temperature β discussed during the lecture.

Compute analytically the mean value of the magnetization for the spin at equilibrium.

Write a 5-line program which implements the ("faster-than-the-clock") rejection-free dynamical algorithm for this model. Compare with the analytical solution for the magnetization. (Notice that if the spin is negative at time t, it will be positive again at time t + 1). Determine the magnetization at low, even at very low temperature.

Write another 5-line program which implements the usual Metropolis algorithm for this model. Compare again with the analytical expression of the magnetization. Can one say that this algorithm is slower than the rejection-free algorithm? Discuss different temperature regimes.

Determine the autocorrelation function C(t,Δt) = ⟨σ(t)σ(t+Δt)⟩ for all times t and Δt, analytically (use the transfer matrix!). Compare your result to numeric evaluations with the two Monte Carlo algorithms, and at both low and high temperatures. You have the freedom to chose the initial condition (e.g. fixed direction, or distributed with the equilibrium distribution), but be consistent: choose the same for the analytical computation and for the numerical evaluation.

Bonus: find the initial distribution(s) which ensure(s) that the autocorrelation funciton C(t,Δt) does not depend on t.

Figure 2. A possible sequence of spin configuration.

Dynamic Monte Carlo algorithmsPost your questions on this hwk in the blog at the end of the page.If you hand your Hwk by e-mail, PLEASE write the file in the format HW9_YOUR-NAME and send it to G. Roux.

ANDM. Civelliotherwise it will be impossible to have it corrected before your oral exam and hence it will not be considered.before the deadline Friday December 11 at noon,## Introduction

In this short exercise, we apply the simulation method of dynamic Monte Carlo algorithms to the single spin in a field.## A spin in a field

Consider a single Ising spin

σ=±1 in a magnetic fieldh, as shown in Figure 1. We consider its Monte Carlo dynamics at inverse temperatureβdiscussed during the lecture.t, it will be positive again at timet+ 1). Determine the magnetization at low, even at very low temperature.C(t,Δt) = ⟨σ(t)σ(t+Δt)⟩ for all timestandΔt, analytically (use the transfer matrix!). Compare your result to numeric evaluations with the two Monte Carlo algorithms, and at both low and high temperatures. You have the freedom to chose the initial condition (e.g.fixed direction, or distributed with the equilibrium distribution), but be consistent: choose the same for the analytical computation and for the numerical evaluation.Bonus: find the initial distribution(s) which ensure(s) that the autocorrelation funcitonC(t,Δt) does not depend ont.[Print this page]