If the correlation function decays exponentially

$$\phi (t) \sim \exp{(-t/\tau)}$$

then the exponential correlation time can be computed as the average

$$\tau_{\mathrm{exp}} = -\langle \frac{t}{log|\frac{\phi(t)}{\phi(0)}|} \rangle.$$

If the decay is exponential, then

$$\int_0^{\infty} dt \phi(t) = \int_0^{\infty} dt \phi(0)\exp{(-t/\tau)} = \tau \phi(0),$$

which suggests another measure of correlation

$$\tau_{\mathrm{int}} = \sum_k \frac{\phi(k)}{\phi(0)},$$

called the integrated correlation time.