Consider a dilute solution composed of a polar molecule solute in a nonpolar solvent (polar molecules are molecules with a permanent electric dipole). The electric polarization \(P(t)\) of this fluid is driven out of equilibrium by an electric field \(E(t)\) and obeys the Langevin equation of motion $$ \frac{d P(t)}{\mathrm{d} t}+5 P(t)=\xi(t)+5 a E(t) $$ where \(\mathcal{E}(t)\) is a delta-correlated white noise (due to background magnetic fluctuations) with zero mean \((\mathcal{\xi}(t)\rangle=0\), and \(a\) is a constant. (a) Compute the linear response function \(K(t) .\) (b) Compute the dynamic susceptibility \(X(\omega)\) and find its limit as \(\omega \rightarrow 0\), where \(1 / 2 C\left(Q(0)^{2}\right\rangle_{T}=\frac{1}{2} k_{B T}\) from the equipartition theorem.

In the context of physics, **polarization** refers to the orientation of the dipole moments within a material. Specifically, for molecules with a permanent electric dipole, polarization quantifies the extent to which the dipoles align under an external electric field.

Polar molecules, such as water, inherently possess an electric dipole moment. When an external electric field is applied, these dipoles tend to align themselves with the field, contributing to the overall **electric polarization**. This process can be mathematically described using the Langevin equation. In our specific scenario, the polarization of a dilute solution of polar molecules in a nonpolar solvent is given by the equation:

\[\frac{d P(t)}{\text{d} t} + 5 P(t) = \xi(t) + 5 a E(t)\].

Here:

• **P(t)** represents the electric polarization at time **t**.
• **E(t)** is the applied electric field.
• **\xi(t)** is a term representing background magnetic fluctuations treated as white noise.
• **a** is a constant related to the material properties.

This equation tells us how the polarization, **P(t)**, evolves over time under the influence of an electric field **E(t)** and random fluctuations **\xi(t)**.

The **Linear Response Function (K(t))** is a measure of how a system responds to an external perturbation. In our case, it tells us how the electric polarization **P(t)** responds to the applied electric field **E(t)**.

To compute **K(t)**, we consider the deterministic part of the Langevin equation, ignoring the noise term **\xi(t)**:
\[ \frac{d P(t)}{\text{d} t} + 5 P(t) = 5 a E(t) \]
By taking the Fourier transform of both sides, we move from the time domain to the frequency domain. This helps simplify the equation, allowing us to solve for **K(t)**:

\[i\omega \tilde{P}(\tilde{\omega}) + 5 \tilde{P}(\tilde{\omega}) = 5a \tilde{E}(\tilde{\omega}) \].
Solving for \tilde{P}(\tilde{\omega}) gives us:
\[ \tilde{P}(\tilde{\omega}) = \frac{5a \tilde{E}(\tilde{\omega})}{i\omega + 5} \].
The Fourier transform of **K(t)**, denoted as \tilde{K}(\tilde{\omega}), relates to **P(t)** and **E(t)** via:
\[ \tilde{K}(\tilde{\omega}) = \frac{5a}{i\omega + 5} \].
To find **K(t)** in the time domain, we take the inverse Fourier transform:

\[ K(t) = 5a e^{-5t} \theta(t) \],
where **\theta(t)** is the Heaviside step function, which ensures the function is zero for **t < 0** and 1 for **t >= 0**.

Simply put, this function describes how quickly the polarization returns to equilibrium after a sudden change in the electric field.

The **Dynamic Susceptibility (X(\tilde{\omega}))** of a system characterizes how it reacts to changes in an external field over different frequencies. It’s essentially the frequency domain equivalent of the **Linear Response Function**.

In our case, the dynamic susceptibility is the Fourier transform of the linear response function **K(t)**. From the previous computation, we have:

\[X(\omega) = \tilde{K}(\tilde{\omega}) = \frac{5a}{i\omega + 5} \].
This tells us how the system's polarization responds to varying frequencies of the applied electric field. When you want to understand the response at a specific frequency, this function gives you that information.

To find the behavior as the frequency approaches zero (the static limit), we evaluate:
\[\text{Lim}_{\omega \to 0} X(\tilde{\omega}) = \frac{5a}{i\omega + 5} = a \].
This result means that at static conditions (very low frequencies), the response is simply proportional to the constant **a**, which describes the material's intrinsic properties.

Key points to remember about dynamic susceptibility:

• It provides insights into how a system responds to varying frequencies.
• At static equilibrium conditions (\omega \to 0), the susceptibility simplifies to a constant.
• It is derived from the linear response function, bridging the time and frequency domains.

This concept is pivotal in understanding the behavior of materials under varying external influences.