Semilinear response

Applications

The original motivation for introducing SLRT was the study of mesoscopic conductance

.

The term SLRT has been coined in

where it has been applied to the calculation of energy absorption by metallic grains.

Later the theory has been applied for analysing the rate of heating of atoms in vibrating traps

.

Definition of semilinear response

Consider a system that is driven by a source

f

(

t

)

{\displaystyle f(t)}

that has

a power spectrum

S

~

(

ω

)

{\displaystyle {\tilde {S}}(\omega )}

. The latter is defined

as the Fourier transform of

⟨

f

(

t

)

f

(

0

)

⟩

{\displaystyle \langle f(t)f(0)\rangle }

.

In linear response theory (LRT) the driving source induces a steady state

which is only slightly different from the equilibrium state.

In such circumstances the response (

G

{\displaystyle G}

) is a linear functional of the power spectrum:

G

=

G

[

S

~

(

ω

)

]

=

∫

−

∞

∞

η

(

ω

)

S

~

(

ω

)

d

ω

{\displaystyle G=G[{\tilde {S}}(\omega )]=\int _{-\infty }^{\infty }\eta (\omega ){\tilde {S}}(\omega )\,d\omega }

In the traditional LRT context

G

{\displaystyle G}

represents the rate of heating,

and

η

(

ω

)

{\displaystyle \eta (\omega )}

can be defined as the absorption coefficient.

Whenever such relation applies

[

A

]

S

~

(

ω

)

↦

λ

S

~

(

ω

)

implies

G

↦

λ

G

{\displaystyle [A]\ \ \ \ {\tilde {S}}(\omega )\mapsto \lambda {\tilde {S}}(\omega ){\text{ implies }}G\mapsto \lambda G}

[

B

]

S

~

(

ω

)

↦

S

~

1

(

ω

)

+

S

~

2

(

ω

)

implies

G

↦

G

1

+

G

2

{\displaystyle [B]\ \ \ \ {\tilde {S}}(\omega )\mapsto {\tilde {S}}_{1}(\omega )+{\tilde {S}}_{2}(\omega ){\text{ implies }}G\mapsto G_{1}+G_{2}}

If the driving is very strong the response becomes non-linear, meaning that both properties [A] and [B] do not hold. But there is a class of systems whose response becomes semi-linear, i.e. the first property [A] still holds, but not [B].