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

(1)

where is the external field which coupled with the quantum operator . The Shrodinger is thus

(2)

and we seek a solution in the form

(3)

where , by taking the above form into the Schordinger equation, satisfy

(4)

with the help of Dyson series, we can get

(5)

The expectation value of at stare is thus

(6)

The fist term is just the equilibrium value, while the LHS is nonequilibrium value, thus

(7)

At finite temperature, the average of is

(8)

In the first order approximation, we can still use the unperturbed density operator, . Using 7 in the above equation, we can get

(9)

Define the dynamic susceptibility as

(10)

Considering if and for , the response is thus

(11)

In fact, the dynamic susceptibility has time translation symmetry, that is

(12)

which can be easily proved with the trace property , take the term for example,

(13)

The response

(14)

is obviously the convolution of and . After Fourier transformation, it will give

(15)

That’s why the response is called linear.

#### Refferences

[1] Fetter, Alexander L., and John Dirk Walecka. Quantum theory of many-particle systems. Courier Corporation, 2012.

[2] Mazenko, Gene F. *Nonequilibrium statistical mechanics*. John Wiley & Sons, 2008.