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I want to calculate the correlation time. First I calculate the auto-correlation function:

$$ \begin{align} \chi(t) = &\frac{1}{t_{max}-t}\sum\limits_{t'=0}^{t_{max}-t}m(t')m(t'+t)\\ &-\frac{1}{t_{max}-t}\sum\limits_{t'=0}^{t_{max}-t}m(t')\frac{1}{t_{max}-t}\sum\limits_{t'=0}^{t_{max}-t}m(t'+t)\\ &\sim e^{-t/\tau} \end{align} $$

The correlation-time can be calculated by fitting autocorrelation function. I.e., \$\chi(t)\$ in exponential. For this I have written the following code:

def autocorrelation(M):
    start_time = time.time()
    tau = 1
    sweeps = len(M)
    auto = np.zeros(sweeps)
    for t in range(sweeps):
        some_time = sweeps-t
        first_term = np.average(M[:some_time]*M[t:sweeps])
        S1 = np.average(M[:some_time])
        S2 = np.average(M[t:sweeps])
        auto_temp = first_term - S1*S2
        if auto_temp > 0:
            auto[t] = auto_temp
        else:#remove oscillating part
            break 
    if auto[0] != 0:
        auto = auto[auto>0]
        auto = auto/auto[0] #normalization
        len_auto = len(auto)
        if len_auto > 1: #draw a straight line if you have atleast two points
            tau = int(-1/np.polyfit(np.arange(len_auto), np.log(auto), 1, w=np.sqrt(auto))[0])
    tau = max(tau,1)
    logging.info(f"Correlation time = {tau}")
    return tau

If you put in list M = m, the function will calculate autocorrelation function and return correlation time, i.e, \$\tau\$ I also thought of using NumPy's correlate and convolve, but I could not understand them well. I tried auto_convolve = numpy.convolve(m,m,mode = "same"), after normalization, the auto/auto[0] was not same as auto_convolve/auto_convolve[0]

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  • \$\begingroup\$ Please can you check the equation. Why do you have \$\frac{1}{t_{max}-t}\$ twice in the second line of the equation and not just simplify to \$\frac{1}{(t_{max}-t)^2}\$. Additionally please use parentheses in situations like the above it's unclear what is inside the Sigma, is that a nested Sigma? \$\endgroup\$
    – Peilonrayz
    Jun 4, 2020 at 22:07
  • \$\begingroup\$ It's because there are two averages in the second term. I tried putting parenthesis, but it showed error: Your post appears to contain code that is not properly formatted as code. \$\endgroup\$ Jun 5, 2020 at 1:50
  • \$\begingroup\$ The second term in expression of \$\chi(t)\$ is \$\bigg(\frac{1}{t_{max}-t}\sum\limits_{t'=0}^{t_{max}-t}m(t')\bigg)\bigg(\frac{1}{t_{max}-t}\sum\limits_{t'=0}^{t_{max}-t}m(t'+t)\bigg)\$. \$\endgroup\$ Jun 5, 2020 at 3:20

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