Lectures on Theoretical Physics
Linear Response Theory
Peter Hertel
University of Osnabr
uck, Germany
The theory of linear response to perturbations of the equilibrium state, or linear
response theory, is the subject of this series of lectures.
Ordinary matter, if left alone, will sooner or later attain an equilibrium state. This
equilibrium state depends on the temperature of the environment and on external
parameters. External parameters may be the region of space within which a certain
number of particles are conĜned, mechanical stress, or the strength of an external
electric or magnetic Ĝeld.
If temperature or the external parameters change slowly enough, the system can
attain the new equilibrium state practically instantaneously, and we speak of a
reversible process. On the other hand, if the external parameters vary so rapidly
that the system has no chance to adapt, it remains
away
from equilibrium, and we
speak of irreversibility.
The most important application is optics. There is a medium which is exposed to
an electromagnetic wave. The electric Ĝeld changes so rapidly that matter within
a region of micrometer dimensions cannot react instantaneously, it responds with
retardation. We shall work out the retarded response in linear approximation.
There are quite a few general and important results which hold irrespective of
a particular Hamiltonian, such as the Kramers-Kronig relations, the ∞uctuation-
dissipation theorem, the second law of thermodynamics, and Onsager's relation.
We discuss various electro- and magnetooptic eĈects, such as the Pockels eĈect,
the Faraday eĈect, the Kerr eĈect, and the Cotton-Mouton eĈect.
We also treat spatial dispersion, or optical activity and indicate how the theory is
to be developed further in order to handle the non-linear response as well. |