Suceptibility
Given a free energy functional,
Examples:
For \(t'
Fourier transforming this
A system with inversion symmetry,
Then,
C.f. the AC suceptibility
Obtaining the work,
Then,
Note:
Example: Calculate the current through a wire.
Suppose we have
Suppose we have an order parameter
Compare if we have a Schrodinger equation and a propogator:
Static Susceptibility
Removing the time dependence.
We will get:
For the uniform case, thermodynamically,
Fluctuation-Dissipation Theorem
Cases:
Starting from forced equilibrium at
From our response,
For the ideal gas,
And is zero for
Note that
Then we get the FD theorem in frequency domain for classical systems,
Then,
The QM version,
Starting point is QM version of linear response, Kudo formula,
Kramers-Kronig Relations
We have, and abbreviate,
Let
Then,
Since
Alternatively,