# Green-Kubo relation for viscosity

I want to compute the shear viscosity of a system using Green-Kubo relation. I have the full stress-tensor evolution $$P_{ij}(t)$$ and the system is isotropic so I can write

$$\eta = \frac{V}{k_B T} \int_0^\infty < P_{xy}(0) \cdot P_{xy}(t') > \cdot dt'$$

I thought that $$< P_{ij}(0) \cdot P_{ij}(t) >$$ was the time autocorrelation function for the non-diagonal components of the stress tensor, but it cannot be because acf is normalized and for something with units it not make any sense to me... I thought in autocovariance, but I don't know because I check the value with another method and both values not match... or maybe I'm doing something wrong with autocovariance...

However, I would like to know an explicit expression for $$< P_{ij}(0) \cdot P_{ij}(t) >$$ and the stadistical name of this magnitude, not the physics name "time autocorrelation function" because it is a little confuse and I cannot fully understand this formula

The autocorrelation function doesn't need to be normalized, see the definition here. It also must have units of $$[P^2]$$, or $$N^2\cdot m^{-4}$$ since $$\eta$$ has to have units of $$N\cdot s\cdot m^{-2}$$.

To actually calculate this, I'd try using scipy's correlate function (linked) and check out this answer for how to calculate the autocorrelation with it.

• So, when I compute the acf I have to use stress-tensor in Pa? Maybe I'm not doing this right, but to compute this acf I'm trying to use python an the function from statsmodels.tsa.stattools import acf (statsmodels.org/dev/generated/…), this acf is always in between 0 and 1, independence of the units, that's why I talk about normalization ... Jun 22, 2021 at 8:06