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The canonical ensemble

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The canonical ensemble. Q = -Q R. Consider system at constant temperature and volume. adiabatic wall. System . Heat Reservoir R. T=const. We have shown in thermodynamics that system with (T,V)=const . in equilibrium is at a minimum of the Helmholtz free energy, F.

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slide1
The canonical ensemble

Q = -QR

Consider system at constant temperature and volume

adiabatic wall

System

Heat Reservoir R

T=const.

We have shown in thermodynamics that

system with (T,V)=const. in equilibrium is at

a minimum of the Helmholtz free energy, F

(T=const, V=const.)

slide2
We use a similar approach now in deriving density function and partition function

System can exchange energy with the heat reservoir:

Find maximum of S under the constraint that average (internal) energy is given

under constraints

found by maximizing

Using again Lagrange multiplier technique

slide3
Partition function of the

canonical ensemble

with

Next we show

@ V,N constant

From the constraint

With the equilibrium distribution

back into the entropy expression

slide4
With

and

@ V,N constant

Using

we find

With

Gives meaning to the Lagrange parameter

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