QuantumPhysics.dvi

introduce Lagrange multipliers for the constant energy condition, as well as for the condition of
unit trace ofρ. Thus, we extremize, as a function ofρ, the quantity

−Tr

(

ρlnρ

)

−βTr(ρH)−γTr(ρ) (16.28)

under the condition that

Tr(ρ) = 1 (16.29)

The corresponding equation reads,

Trδρ

(

−lnρ−I−βH+γI

)

= 0 (16.30)

The operator in parentheses is self-adjoint, and the variationsδρofρare also self-adjoint. Thus, if
the above variational equation is to hold for arbitrary self-adjoint variationsδρ, we must have

lnρ=−βH+ constant×I (16.31)

Enforcing now the unit trace condition guarantees that we recover (16.25), and thatβis indeed
related to temperature as given above.

16.5.1 Generalized equilibrium ensembles

The derivation of the Boltzmann weights and associated density matrix corresponds to the canonical
ensemble, in which only the energy of the system is kept constant. In the grand canonical ensemble,
both the energy and the number of particles in the system is kept constant. More generally, we
consider an ensemble in which the ensemble average of a number of commuting observablesAi,
i= 1,···,K is kept constant. To compute the associated density operator ρof this ensemble,
we extremize with respect to variations inρthe entropy, under the constraint that the ensemble
averages Tr(ρAi) are kept constant. Using again Lagrange multipliersβi,i= 1,···,K, we extremize

−Tr

(

ρlnρ

)

−

∑K

i=1

βiTr(ρAi) (16.32)

Upon enforcing the normalization Tr(ρ) = 1, this gives,

ρ =

1

Z

exp

{

−

∑K

i=1

βiAi

}

Z = Tr

(

exp

{

−

∑K

i=1

βiAi

})

(16.33)

In the grand canonical ensemble, for example, these quantities are

ρ =

1

Z

e−βH−μN

Z = Tr

(

e−βH−μN

)

(16.34)

whereNis the number operator andμis the chemical potential. Other observables whose ensemble
averages are often kept fixed in this way are electric charge,baryon number, electron number etc.