Let \(u(t)\) be a unit vector parallel to the rod in rotational Brownian motion. Show that the time correlation function \(\left\langle(u(t) \cdot u(0))^{2}\right\rangle\) is given by $$ \left\langle(u(t) \cdot u(0))^{2}\right\rangle=\frac{2}{3} e^{-6 D_{r} t}+\frac{1}{3} $$ Explain why the correlation time of \(\left\langle(u(t) \cdot u(0))^{2}\right\rangle\) is shorter than that of \(\langle\boldsymbol{u}(t) \cdot \boldsymbol{u}(0)\rangle\).

The time correlation function helps us understand how a system keeps memory of its initial state over time. For rotational Brownian motion, we observe how the unit vector parallel to a rod, denoted as \,\(u(t)\), changes direction randomly over time. Specifically, the time correlation function \,\(\left\langle(u(t) \cdot u(0))^{2}\right\rangle \) shows how the square of the dot product of vectors at times \,\(t\) and \,\(0)\) evolves.
This function tells us how similar the direction of \,\(u(t)\) is to \,\(u(0)\) as time passes. For rotational Brownian motion, the correlation function decays exponentially, signifying that the memory of the initial direction is lost gradually.
Specifically, the correlation function for rotational Brownian motion is given by: \,\( \left\langle(u(t) \cdot u(0))^{2}\right\rangle=\frac{2}{3} e^{-6 D_{r} t}+\frac{1}{3} \).
This equation shows both an exponential decay term and a constant term. The constant term \,\(\frac{1}{3}\) represents the long-term behavior after the directional memory is lost, while the term with \,\(e^{-6 D_r t}\) represents the decaying memory with time.

The rotational diffusion coefficient, \,\(D_{r}\), quantifies the rate at which the orientation of the unit vector in rotational Brownian motion changes with time.
It plays a crucial role in determining how fast or slow the rotational Brownian motion occurs. The larger the value of \,\(D_{r}\), the faster the randomization of the directional vector \,\(u(t)\). As a result, the correlation function decays faster when \,\(D_{r}\) is larger.
In the context of the given correlation function, the rate at which the correlation decays is expressed by the term \,\(e^{-6 D_{r} t}\). This connection indicates that the exponential decay of our time correlation function is directly influenced by the value of \,\(D_{r}\).
By understanding \,\(D_{r}\), we can infer how quickly the orientation will lose its initial correlation, which is key for applications in understanding molecular rotations and dynamics in various mediums.

Exponential decay describes how a quantity decreases rapidly over time. In our case, the correlation functions exhibit exponential decay signified by the term \,\(e^{-6 D_{r} t}\).
This means that as time \,\(t\) increases, the value of \,\(e^{-6 D_{r} t}\) becomes smaller, making the term \,\(\left\langle(u(t) \cdot u(0))^{2}\right\rangle\) approach the constant \,\(\frac{1}{3}\) value.
Understanding exponential decay helps us appreciate how quickly the system forgets its initial state. It shows that shortly after starting, the direction of the vector \,\(u(t)\) bears little resemblance to its initial direction \,\(u(0)\).
This rapid decline phase is encapsulated in the time correlation function, showing the most significant changes occur initially and then level off.