I am want to calculate the specific heat for 2D 100x100 square lattice ising model. I have calculated the correlation time, viz., $\tau$. Now I want to calculate the specific heat and error in specific heat using the Jackknife method. Following are the steps which I am following, I want to know that I am going in right direction.

  1. I have arrays of Energy for each temperature for a particular Monte Carlo step. (1 Monte Carlo step = 1 iteration). I will divide the set of energies into $n=\frac{t_{MAX}}{2\tau}$, where $t_{MAX}$ is total number of Monte Carlo steps.

  2. Each divided set of energies contains $2\tau$ number of energies. For each set we calculate the variance, multiplying variance with $\beta^2$ gives specific heat for that set which we call $c_i$.

  3. Now, we use the Jackknife method: discarding $n^{th}$ measurement, we calculate $\overline c_{n_{JK}}$: $$\overline{c}_{n_{JK}}=\frac{1}{n-1}\sum_{i\ne n}c_i $$

  4. We calculate specific heat as average if $\{\overline{c}_{n_{JK}}\}$, viz., $\overline c_{JK}$: $$\overline{c}_{JK}=\frac{1}{n}\sum_{i=1}^N\overline{c}_{i_{JK}}$$

  5. Now comes the error estimation, which is given by square root of the Jackknfe varience : $$\sigma^2_{JK}=\frac{n-1}{n}\sum_{i=1}^{N}(\overline{c}_{JK}-\overline{c}_{i_{JK}})^2$$ For large $n$ the error is simply: $$\sigma_{JK}=\sqrt{\sum_{i=1}^{N}(\overline{c}_{JK}-\overline{c}_{i_{JK}})^2}$$

Is this that all correct?

PS: I do not know if we are allowed to ask a question for clarification on physics stack exchange. I hope this is not against the physics stack exchange policy.


1 Answer 1


No, you are incorrect at point number 1 and following.

  1. In your point 1, After system has reached Equilibrium, viz., after $\tau$, you need to take uncorrelated energies, i.e., energies after every 2$\tau$. In short, you take energies at $\tau, 3\tau, 5\tau, \dots$ and discard others.

  2. If you calculate variance of new set of energies at $\tau, 3\tau, 5\tau, \dots$ and multiply by $\beta^2$, you will get specific heat.

  3. Calculate $c_n$ which is, variance of all Energy excluding energy at n-th index.

  4. No need of this step.

  5. This step is fine.


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