Understanding the Ising Model: Probability Calculations and Spin Configurations

[oasis2007]

A simple and famous model of certain magnetic materials is the Ising model in
which each of N atoms is modeled as a simple spin which can point upwards or
downwards. The magnetization of the material is proportional to the number of up
spins minus the number of down spins (each spin has a certain magnetization associated
with it).
Let's consider N spins in a row (as shown below). Let the "magnetization" be M,
where M is the number of up spins minus the number of down spins. Each spin is
constantly changing its direction from up to down and vice versa according to some
complex dynamics. Assume the dynamics of one spin are independent of other spins.
↑↓↓↓↑↑↓↑↓↓↑↓↓↑↓↑↑↓↓↓↑↓
(i) Clearly state a postulate on the spin configurations which will help us calculate
probabilities of different spin configurations.
(ii) Using (i), calculate the probability of a certain M value P(M) exactly.
(iii) Show P(M) is a Gaussian in the large N limit and sketch it.
(iv) Estimate the probability that M is larger than 0.1% of N when N takes the value of
Avogadro’s constant, 6x1023

 

[Valkarie]

.(i) Postulate: All spin configurations are equally likely and the probability of a spin configuration is independent of other spins. (ii) The probability of a certain M value P(M) can be calculated by summing up all possible spin configurations with the same M value. For example, when N = 6, M = 2, there are three possible spin configurations: ↑↑↓↓↓↑, ↑↓↑↓↓↑ and ↓↑↑↓↓↑. Therefore, the probability of M = 2 is P(M = 2) = 3/2^6 = 0.09375.(iii) In the large N limit, the probability of a certain M value P(M) is a Gaussian given by: P(M) = 1/√(2πσ^2)exp(-(M-μ)^2/2σ^2) where μ = 0 and σ^2 = N/4. The graph below shows the Gaussian distribution for large N: Graph: P(M) | | * | * * | * * | * * | * * | * * | * * | * * | * * | * * | * * | * * |* * |---------------------------------------------------------------M (iv) The probability that M is larger than 0.1% of N when N takes the value of Avogadro’s constant, 6x1023, is given by: P(M > 0.001N) = 1/√(2πσ^2) ∫_(0.001N)^∞ exp(-(M-μ)^2/2σ^2) dM Using the fact that the integral of a Gaussian is itself a Gaussian, P(M > 0.001N) = 1/√(2πσ^2) [1 - exp(-0.

 

[piercas]

I would like to provide a response to the content on the Ising Model and its probability calculations and spin configurations.

(i) The postulate for calculating probabilities of different spin configurations is that each spin has an equal chance of being in an up or down state at any given time. This means that the probability of a spin being in an up state is 0.5 and the probability of it being in a down state is also 0.5.

(ii) Using this postulate, we can calculate the probability of a certain M value by considering the number of possible spin configurations that will result in that particular M value. For example, if we want to calculate the probability of M = 2, we need to consider all the possible configurations where there are 2 more up spins than down spins. This can be calculated using the binomial distribution formula, which gives us P(M) = (N choose (N+M)/2) * (0.5)^(N+M)/2 * (0.5)^(N-M)/2.

(iii) In the large N limit, the probability distribution P(M) becomes a Gaussian distribution due to the central limit theorem. This means that as N increases, the probability of a certain M value will be concentrated around the mean value, with a standard deviation that decreases as N increases. A sketch of this distribution would show a bell-shaped curve, with the mean at the center and the tails becoming smaller as N increases.

(iv) To estimate the probability that M is larger than 0.1% of N, we can use the Gaussian distribution formula and plug in the values of N = Avogadro's constant (6x10^23) and M = 0.1% of N (6x10^20). This gives us a very small probability, which is expected since the magnetization of the material is proportional to the number of spins and N is a very large number. In this case, the probability would be approximately 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000