14.3 Echoes 413damping mechanism and raises some interesting questions about how Landau dampingre-
lates to entropy. This arises from consideration of the following two apparently conflicting
observations:
- Wave damping should destroy the information content of the wave by converting or-
dered motion into heat. Thus, wave damping should increase the entropy of the sys-
tem. - The collisionless Vlasov equation conserves entropy. This is because collisions are
the agent that increases randomness and hence entropy. A collisionless system is in
principle completely deterministic, so that the future of the system can be predicted
with complete precision simply by integrating the system of equations forward in time.
This entropy-conserving property of the collisionless Vlasov equation can be seen by
direct calculation of the rate of change of the entropy,
dS
dt=
d
dt∫
dx∫
dvf(x,v,t)lnf(x,v,t)=
∫
dx∫
dv(lnf+1)∂f
∂t= −∫
dx∫
dv(lnf+1)(
v
∂f
∂x+
q
mE
∂f
∂v)
= −
∫
dx∫
dv(
v∂
∂x+
q
mE
∂
∂v)
(flnf−f)= 0 (14.81)
since
∫
dv
∂f
∂v= 0,
∫
dx
∂f
∂x=0
∫
dv∂
∂v(flnf−f) = 0,∫
dx∂
∂x(flnf−f)=0. (14.82)So, what really happens– does Landau damping increase entropy or not? The answer
goes right to the heart of what is meant by entropy. In particular, it should be recalled
that entropy is defined as the natural logarithm of the number of microscopicstates corre-
sponding to a given macroscopic state. The concept “macroscopic state”presumes there
exist macroscopic states which are (i) composed of different microscopic states and (ii) for
all intents and purposes indistinguishable. Collisions would cause the systemto continu-
ously evolve through all the various microscopic states and an observer of the macroscopic
system would not be able to distinguish one of these microscopic states from another.
The paradox is resolved because the concept of many microscopic states mapping toa
single macroscopic state fails for Landau damping since the physical system for the Landau
damping problem is actually in just one well-defined state which can be calculated. The
macroscopic state therefore maps to just one microscopic state and so thesystem does not
continuously and randomly evolve through a sequence of microscopic states.
Landau damping does not involve turning ordered information into heat. Instead, macro-
scopically ordered information is turned into microscopically ordered information. The in-
formation is still there, but is encoded in a macroscopically invisible form. A good analogy