# Onsager's hypothesis: why is true that correlations decay with increasing time?

I am studying from Chandler's book (Introduction to Modern Statistical Mechanics) the fluctuation-dissipation theorem. Before introducing it, the book states something without really demonstrating it. It says that the time correlation function of fluctuations goes to zero when the time difference between the two instants that appear in the formula goes to infinity, namely:

$$C(t)=\int dxf(x)\delta A(t';x)\delta A(t'+t;x) \rightarrow0 \quad as \quad t\rightarrow +\infty$$

where $$\delta A(t)$$ is the fluctuation at time $$t$$ of $$A(t)$$ from the time-independent average value $$\langle A\rangle$$, and $$f(x)$$ is the probability density function of phase space points. This comes from stating that for large $$t$$ we have $$\langle \delta A(t')\delta A(t'+t)\rangle\rightarrow\langle\delta A(t')\rangle\langle\delta A(t'+t)\rangle$$ I cannot justify this last statement. Where does it come from? Is it an axiom?

## 1 Answer

At equilibrium, one would expect fluctuations $$\delta A$$ at a time $$t'$$ and fluctuations at a time $$t'+t$$ would become statistically independent when $$t \rightarrow + \infty$$.

The average of a product of statistically independent variables is the product of the averages.

• Thanks. This independence is a postulate based on a reasonable argument, correct? Apr 22 at 7:14
• @SalvatoreManfrediD It is a reasonable guess for many equilibrium systems. A more formal justification is related to the possibility of proving that the dynamics of the system is mixing (see en.wikipedia.org/wiki/Mixing_(mathematics) ), at least in art of the accessible phase space. Apr 22 at 11:53