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Ch3. The Canonical Ensemble

• Microcanonical ensemble: describes isolated systems of known energy. The system does not exchange energy with any external system so that (N,V,E) are fixed.• Macrostate is determined by (N, V, E)• Distinct microstate number (N,V,E) or

(N,V,E;) so that entropy S=k ln(N,V,E) or S=k ln(N,V,E;)

• Microcanonical ensemble 0

,

constpq If E-/2 H(q,p)

E+/2otherwise

0/,,, EVN – allowed volume in phase space;0 –volume of one microstate (~h3)

The Canonical Ensemble

• Describes a system of known temperature, rather than known energy. The energy is variable due toi exchange of energy with external system at common T.

• Macrostate is determined by (N, V, T). T is common between the system and reservoir.

systemHeat reservoir

exchange E

Fixed T at equilibrium

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The probability Pr

Pr is the probability that a system, at any time t, is found to be in one of the states with energy value Er.

rrr

rr

rrrr

PEE

Pwith

PPkPkS

1

lnln

The entropy of the system is given by

The energy E can be any value from 0 to infinity. Pr is the probability that E=Er.

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Pr in microcanonical ensemble

ln1

ln1

ln

11

,,/1

kkPPkS

P

EVNP

EfixedEEE

rrrr

r rr

r

r

Each mocrostate is equally accessible

3.1 Equilibrium between a system and a heat reservoir

Consider a system A immersed in a very large heat reservoir A’. They are thermal equilibrium with common T at time t.

System A: in a state of Er Reservoir A’: in a microstate of Er’ Composite system A(0) (=A+A’): Conservation of energy

A(Er;T)

A’(Er’;T)

Er+Er’=E(0) = const

’(Er’) – number of reservoir’s states with given energy value of Er’

Pr – Probability that system A is found to have E=Er.Pr ~ ’(Er’) = ’(E(0) -Er)=e-E

r

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Remark Normalization

rr

r

kTE

kTE

r

Pwith

e

eP

r

r

1

)/(

)/(

• When a system is in thermal equilibrium with external reservoir, its energy Er is exchanged with reservoir.

• The system energy Er can be any values from 0 to infinity.

• The probability that the system has an energy E=Er is Pr given by …

=kT

3.2 A system in the canonical ensemble

An ensemble: N identical system (i=1,2, …N) sharing a total energy E=NU.

U =E/N is the average energy per system in the ensemble.

The number of different ways of E distributes among N members according to the mode {nr} (distribution set) W{nr}~

The average number of systems having energy Er is <nr> ~

The most probable distribution set {nr*} – to make W{n*r} maximum

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Canonical distribution For a single system, the probability that the system has energy E=Er is Pr

rr

r

E

Er

r

Pwith

e

enP

r

r

1

=kT

3.3 Physical significance of the various statistical quantities in the canonical ensemble

The canonical distribution- the probability that the system has energy E=Er

),( TVQ

e

e

enP

N

E

r

E

Er

r

r

r

r

=kT

Partition function of the system with (N,V,T)

r

kTE

r

EN

rr eeTVQ /),(

The average energy of the system with (N,V,T)

),(ln TVQeEEU Nr

Err

r

Physical significance in the canonical ensemble

Helmholtz free energy

),(ln TVQkTTSUA N

Entropy

),(ln),(lnln),,( TVQkTVQkPkTVNS NNr

Specific heat Gibbs free energy Pressure Chemical potential

Example – single quantum oscillator

The state of a single oscillator

,....3,2,1,0),2/1( nnn

Partition function (N=1)

Average energy of one oscillator The average number of quantum The entropy