2.4 Two-fluid equations 43By now, it is obvious thatfcould be the velocity distribution function in which casef(v)dv
is just the number of particles in the group having velocity betweenvandv+dv.Since the
peg groups correspond to different velocity ranges coordinates, having more dimensions
just means having more groups and so for three dimensions the entropy generalizes to
S=−
∫
dvf(v)lnf(v). (2.43)If the distribution function depends on position as well, this corresponds to still more peg
groups, and so a distribution function which depends on both velocity and position will
have the entropy
S=−∫
dx∫
dvf(x,v)lnf(x,v). (2.44)2.4.2 Effect of collisions on entropy
The highest entropy state is the most likely state of the system because the highest entropy
state has the most number of microscopic states corresponding to the macroscopic state.
Collisions (or other forms of randomization) will take some prescribed initial microscopic
state where phase-space positions of all particles are individually specified and scramble
these positions to give a new microscopic state. The new scrambled statecould be any
microscopic state, but is most likely to be a member of the class of microscopic states
belonging to the highest entropy macroscopic state. Thus, any randomization process such
as collisions will cause the system to evolve towards the maximum entropy macroscopic
state.
An important shortcoming of this argument is that it neglects any conservation rela-
tions that have to be satisfied. To see this, note that the expression for entropy could be
maximized if all the particles are put in one group, in which caseC=N!,which is the
largest possible value forC.Thus, the maximum entropy configuration ofNplasma parti-
cles corresponds to all the particles having the same velocity. However, this would assign
a specific energy to the system which would in general differ from the energy of the initial
microstate. This maximum entropy state is therefore not accessible inisolated system, be-
cause energy would not be conserved if the system changed from its initialmicrostate to
the maximum entropy state.
Thus, a qualification must be added to the argument. Randomizing processeswill
scramble the system to attain the state of maximum entropy stateconsistentwith any con-
straints placed on the system. Examples of such constraints would be the requirements that
the total system energy and the total number of particles must be conserved. Wetherefore
re-formulate the problem as: given an isolated system withNparticles in a fixed volumeV
and initial average energy per particle〈E〉,what is the maximum entropy state consistent
with conservation of energy and conservation of number of particles? Thisis a variational
problem because the goal is to maximizeSsubject to the constraint that bothNandN〈E〉
are fixed. The method of Lagrange multipliers can then be used to take into account these
constraints. Using this method the variational problem becomes
δS−λ 1 δN−λ 2 δ(N〈E〉) = 0 (2.45)whereλ 1 andλ 2 are as-yet undetermined Lagrange multipliers. The number of particles