A small perturbation act on a system in equilibrium may produce an effect related to this external field, which can be treat as a linear response up to first order approximation.
We assume the system is in equilibrium with Hamiltonian before time . The perturbation is turned on after , and the Hamiltonian becomes
where is the external field which coupled with the quantum operator . The Shrodinger is thus
and we seek a solution in the form
where , by taking the above form into the Schordinger equation, satisfy
with the help of Dyson series, we can get
The expectation value of at stare is thus
At finite temperature, the average of is
In the first order approximation, we can still use the unperturbed density operator, . Using 7 in the above equation, we can get
Define the dynamic susceptibility as
Considering if and for , the response is thus
In fact, the dynamic susceptibility has time translation symmetry, that is
which can be easily proved with the trace property , take the term for example,
is obviously the convolution of and . After Fourier transformation, it will give
That’s why the response is called linear.
 Fetter, Alexander L., and John Dirk Walecka. Quantum theory of many-particle systems. Courier Corporation, 2012.
 Mazenko, Gene F. Nonequilibrium statistical mechanics. John Wiley & Sons, 2008.