Response Function: Generalized Susceptibility

The goal of this section is to introduce the concept of a response function$\chi(t,t')$, or generalized susceptibility, for the description of non-equilibrium disturbances.

According to Eqs.(438) and (439), the response to a perturbation $\Delta H$ in the linear regime (i.e., a perturbation that is linear in the field $f$) is

$$\Delta A(t,f) \approx f \beta \langle \delta A(t) \delta A(0) \rangle,$$ (440)

where $\Delta A(t,f) = \overline{A}(t) - \langle A \rangle$. Therefore,

$$\Delta A(t,\lambda f) = \lambda \Delta A(t, f).$$ (441)

The most general expression for a macroscopic response that is consistent with Eq. (441) is

$$\Delta A(t, f) = \int_{-\infty}^{\infty} d t' \chi(t, t') f(t') = \sum_j \chi(t,t_j) f(t_j) dt_j,$$ (442)

where $dt_j=(t_{j+1}-t_j)$ is the time increment, $f(t)$ is the external perturbational field and $\chi(t,t')$ is the response function.
Note that the name response function is due to the fact that $\chi(t,t_0)$ is equal to the response at time t when the perturbational field is an impulse at time $t_0$ (i.e., $\chi(t,t_0)=\Delta A(t,\delta(t-t_0))$). In addition, any other perturbational field can always be written as a linear superposition of impulses since any function can be expanded as a linear combination of delta functions.

Another important observation is that according to Eq. (442),

$$\chi (t, t_j) = \frac{\partial \Delta \bar{A} (t)}{\partial f(t_j)},$$ (443)

Therefore, $\chi (t, t_j)$ defines the first nonvanishing term in an expansion of $\Delta A(t,f)$ in powers of $f(t)$. The response function $\chi(t,t')$ is thus also called generalized susceptibility, since the electric susceptibility $\chi_e$ defines the first nonvanishing term in an expansion of the polarization $\vec{P}$ (i.e., the total electric dipole moment per unit volume) in powers of the electric field $\vec{E}$ as follows,

$$\vec{P} = \chi_e \vec{E}.$$ (444)

In analogy to the electric susceptibility $\chi_e$, that is a property of the unperturbed system, the response function $\chi (t, t_j)$ is a function of the unperturbed system at equilibrium. In order to show this important aspect of the response function, consider that the perturbational field represented by the following diagram:
 

which is defined as follows:

$$f(t)= \begin{cases}f, \hspace{.2cm} \text{when} \hspace{.2cm} t\leq 0,\\ 0, \hspace{.2cm} \text{otherwise}. \end{cases}$$ (445)

The response at time $t > 0$ is

$$\Delta A(t) = f \int_{-\infty}^0 dt' \chi(t-t') = -f \int_{\infty}^t dt'' \chi(t''),$$ (446)

where $t''=t-t'$, $dt''=-dt'$. Here,  we have assumed that $\chi(t,t')=\chi(t-t')$.
 

Therefore,

$$\frac{\partial \Delta A(t)}{\partial t} = - f \chi(t),$$ (447)

since, according to the second fundamental theorem of calculus ,

$$\chi(t) = \frac{\partial}{\partial t} \int_{a}^{t} dt' \chi(t'),$$ (448)

where $a$ is an arbitrary constant value. Substituting Eq. (440) into Eq. (447), we obtain

$$\chi(t) = - \beta \frac{d}{dt} <\delta A(0) \delta A(t)>,$$ (449)

where $t > 0$.