Semilinear response
Semi-linear response theory (SLRT) is an extension of linear response theory (LRT) for mesoscopic circumstances: LRT applies if the driven transitions are much weaker/slower than the environmental relaxation/dephasing effect, while SLRT assumes the opposite conditions.
SLRT uses a resistor network analogy (see illustration) in order to calculate the rate of energy absorption: The driving induces transitions between energy levels, and connected sequences of transitions are essential in order to have a non-vanishing result, as in the theory of percolation.
The original motivation for introducing SLRT was the study of mesosopic conductance [1] [2] [3] .
[4] The term SLRT has been coined in [5] 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 .
) is a linear functional of the power spectrum: In the traditional LRT context
Whenever such relation applies 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].
SLRT applies whenever the driving is strong enough such that relaxation to the steady state is slow compared with the driven dynamics.
Yet one assumes that the system can be modeled as a resistor network, mathematically expressed as
stands for the usual electrical engineering calculation of a two terminal conductance of a given resistor network.
Resistor network calculation is manifestly semi-linear because it satisfies
represent Fermi-golden-rule transition rates between energy levels.
If only neighboring levels are coupled, serial addition implies which is manifestly semi-linear.
Results for sparse networks, that are encountered in the analysis of weakly chaotic driven systems, are more interesting and can be obtained using a generalized variable range hopping (VRH) scheme.
