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I've written in Matlab a code for a Ising model in 1 dimension with 40 spins at $k_{B}T=1$.

I record the energy of every step in a Metropolis Monte Carlo algorithm, and then I made an histogram like this.

enter image description here

I want to show the theoretical Boltzmann distribution. What is the exact formula to get this shape? $Ax \exp(-\beta x)$?

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    $\begingroup$ What exactly are the axes of your graph? $\endgroup$
    – Philip
    Jun 23, 2020 at 0:52
  • $\begingroup$ y label counts normalized x label energy per spin $\endgroup$
    – Simo7
    Jun 23, 2020 at 23:46

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I have to make a number of assumptions, as you did not state all the necessary information. So, I am going to assume that you are using periodic boundary conditions, that is, your Hamiltonian is $$ \mathcal{H}(\sigma) = -\sum_{i=1}^N \sigma_i\sigma_{i+1}, $$ where I have denoted by $N$ the number of spins (that is, $N=40$ in your case) and used the convention that $\sigma_{N+1}=\sigma_1$.

Write $D(\sigma)$ the number of $i\in \{1,\dots,N\}$ such that $\sigma_i\neq \sigma_{i+1}$, still using the convention that $\sigma_{N+1} = \sigma_1$. Note that $D(\sigma)$ is necessarily an even number (because of the periodic boundary conditions).

Then, the total energy can be rewritten as $$ \mathcal{H}(\sigma) = D(\sigma) - (N-D(\sigma)) = 2D(\sigma) - N \,. $$ The probability that $D(\sigma)=\delta N$ (with $\delta \in \{0,\frac{2}{N},\frac{4}{N},\dots,\frac{2\lfloor N/2 \rfloor}{N}\}$) is $$ \mathrm{Prob}(D = \delta N) = \binom{N}{\delta N}\frac{\exp(-2\beta\delta N)}{\frac{1}{2}\bigl(1-\exp(-2\beta)\bigr)^N + \frac{1}{2}\bigl(1+\exp(-2\beta)\bigr)^N}, $$ since there are $\binom{N}{\delta N}$ ways of choosing the $\delta N$ pairs of disagreeing neighbors.

This can be easily reformulated in terms of the energy per spin $$ \frac{1}{N}\mathcal{H}(\sigma) = \frac{2}{N}D(\sigma) - 1. $$ Note that the possible values of the latter are $-1$, $-1+\frac{4}{N}$, $-1+\frac{8}{N}$, ... , $-1+\frac{4\lfloor N/2\rfloor}{N}$.

The probability of observing an energy per spin equal to $\epsilon$ is then given by $$ \mathrm{Prob}(\mathcal{H}=\epsilon N) = \binom{N}{\frac{1+\epsilon}{2}N} \frac{\exp\bigl(-\beta (1+\epsilon) N\bigr)}{\frac{1}{2}\bigl(1-\exp(-2\beta)\bigr)^N + \frac{1}{2}\bigl(1+\exp(-2\beta)\bigr)^N} . $$

Here is a plot of the distribution for your parameters $N=40$ and $\beta=1$ (only values of $\epsilon$ smaller than $-0.2$ are indicated as higher values have too small probability at this temperature):

enter image description here

(The computation using other boundary conditions are similar.)

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  • $\begingroup$ it helps me so much thanks $\endgroup$
    – Simo7
    Jul 1, 2020 at 19:53
  • $\begingroup$ why the division by 2 in the binomial coefficient? $\endgroup$
    – Simo7
    Jul 1, 2020 at 20:52
  • $\begingroup$ Do you mean in $\binom{N}{(1+\epsilon)N/2}$? I have just rewritten the original binomial coefficient $\binom{N}{\delta N}$ using the identity $\epsilon=2\delta - 1$ (which follows from the identity $\mathcal{H} = 2D - N$ by dividing by $N$ and using $D=\delta N$ and $\mathcal{H}=\epsilon N$), which gives $\delta = (1+\epsilon)/2$. $\endgroup$ Jul 2, 2020 at 7:42
  • $\begingroup$ What is $D(\sigma)$? $\endgroup$
    – AlphaOmega
    Feb 21, 2021 at 10:19
  • $\begingroup$ @AlphaOmega The number of pairs $\sigma_i, \sigma_{i+1}$ with $\sigma_i\neq\sigma_{i+1}$. $\endgroup$ Feb 21, 2021 at 14:15

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