  What is meant by continuum lowering and ionization potential depression (IPD)
in a Coulomb system depends upon precisely what question is being asked. It is
shown that equilibrium (equation-of-state) phenomena and non-equilibrium
dynamical processes like photoionization are characterised by different values
of the IPD. In the former, the ionization potential of an atom embedded in
matter is the difference in the free energy of the many-body system between
states of thermodynamic equilibrium differing by the ionization state of just
one atom. Typically, this energy is less than that required to ionize the same
atom in vacuo. Probably, the best known example of such an IPD is that of
Stewart and Pyatt (SP). However, it is a common misconception that this formula
should apply directly to the energy of a photon causing photoionization, since
this is a local adiabatic process that occurs in the absence of a response from
the surrounding plasma. To achieve the prescribed final equilibrium state,
additional energy, in the form of heat and work, is transferred between the
atom and its surroundings. This additional relaxation energy is sufficient to
explain the discrepancy between recent spectroscopic measurements of IPD in
dense plasmas and the predictions of the SP formula. This paper provides a
detailed account of an analytical approach to calculating thermodynamic and
spectroscopic (adiabatic) IPDs in multicomponent Coulomb systems of arbitrary
coupling strength. The ramifications are carefully examined in order to
elucidate the roles of the various IPD forms. A formulation in terms of free
energy leads to an analytical equation of state (EoS) that is thermodynamically
self-consistent, provided that the bound and free electrons are dynamically
separable. Of the various proposed formulae, the Spectroscopic (adiabatic) IPD
gives the most consistent agreement with spectroscopic measurements.
