Stochastic Processes - Mathematical framework

References:

• Our lord and saviour David Tong’s notes: Stochastic Processes.pdf
• A good text on application of Langevin dynamics and computer simulations: Pastor - 1994 - Techniques and Applications of Langevin Dynamics Simulations.pdf

Time dependent correlation functions:

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Transclude of 2025-05-29#stochastic-process

Essence of time correlation functions:

The essence of correlation function is that it tells how much of a quantity/property/system/etc. $A$ at a given time $t$ are affected by the value of that quantity at an earlier time $t^{\prime }$. This essence is captured by the way correlation function is defined. To see how much the quantity $A$ is correlated over time at two different timesteps $t$ and $t^{\prime }$, the correlation function:

$$C_{AA}(t-t^{\prime })=⟨A(t)A(t^{\prime })⟩$$

For physical quantities such as velocity of a gas molecule in an open system, it would make sense that the correlation at two different timesteps would be larger if timesteps are close to each other ($t-t^{\prime }\ll 0)$ (Not sure what’s the physical significance of the value of correlation function for $\tau =0$ being equal to the mean square value $⟨A^2⟩$), and over bigger time difference, the correlation is less, and it tends to zero as the time goes on ($t$ get’s bigger compared to $t^{\prime }$):

$$C_{AA}\to 0,\text{ for }t-t^{\prime }\gg 0$$

This is exactly the point of correlation and the math in page=1 is just the formal proof of these statements.

Key takeaway from page=3:

• Noise ($\vec{f}(t)$) is uncorrelated $⟹$ $⟨f_i(t)f_j(t^{\prime })⟩=0$ ($\forall t^{\prime }-t\gg {\tau }_{col}$, the direction of the force)