1

Various ensembles in statistical mechanics and the derivation

of thermodynamic properties.

To make the connection between statistical mechanics and thermodynamics, average

quantities are calculated over a (very large) ensemble of individual systems. The

ensemble itself is always a so-called isolated entity. It contains a specific total number of

particles, it has a precise total volume and a precise total energy. In what follows each

system comprising the ensemble is assumed to be in a particular energy eigenstate jψ ,

and has a particular number of particles jN , a specific energy jE and a specific volume

jV . The possible energies of a particular system are not completely arbitrary: Given the

volume and number of particles in the system, jE is an eigenvalue of the Schrödinger

equation, and depends on jN and jV . ( , )j j j jE E N V= . The states ( , )j j jN Vψ form a

linearly independent basis of energy eigenstates of the Hamiltonian ˆ ( , )j jH N V . This

dependence on ,j jV N is implicit in what follows. The label j summarizes all of the

characteristics of an individual system and it means that it corresponds to a particular

, , ,j j j jV E N ψ . Specifying j defines everything, and this is referred as the “state of the

system”. The basic unknowns are the probabilities to find a system in a particular state.

These probabilities are provided by the partition function. In what follows below, a

(general) partition function O, depending on variables , ,...x y , is defined as a sum over

relative probabilities

O(x, y,...) = !Pj

j! (x, y,...) (1)

and the associated normalized probabilities to find a system of type j (or in state j) in the

ensemble are then given by

Pj (x, y,...) =

!Pj

O(x, y,...) (2)

Knowledge of the probabilities allows one to calculate ensemble averages, notably

2

j jj

j jj

j jj

E P E U

N P N N

V P V V

= =

= =

= =

∑

∑

∑

(3)

In addition the probabilities define the entropy of any type of ensemble as

lnj jj

k P P S− =∑ (4)

The above provides a recipe to calculate ensemble averages, and thermodynamical

quantities. It is not implied that each system in an ensemble is indeed described by an

eigenfunction of the Hamiltonian. The argument is that if we make this assumption, and

calculate the averages in the prescribed fashion, we obtain agreement with the laws of

thermodynamics. The understanding of the precise physical nature of a large number of

molecules in accordance with time-dependent quantum mechanics is a non-trivial

problem. One would like to be able demonstrate that the wave function of a large system

of interacting molecules (in the gas phase for example) evolves in time such that average

values of molecular quantities follow the laws of statistical mechanics, and quickly

become more or less independent of time. On the most fundamental level statistical

mechanics would be expected to derive from quantum mechanics and the time-dependent

Schrödinger equation. There is nothing in the time-dependent Schrödinger equation to

suggest that systems are to be eigenstates of the Hamiltonian. Rather, the argument is

made that thermodynamic properties, for systems in equilibrium, are independent of time.

This is achieved by taking systems to be described by stationary states, i.e. eigenfunctions

of the Hamiltonian, and taking an average to obtain ensemble properties. It will be clear

that the properties calculated in this way, indeed will be independent of time, even if we

would evolve the ensemble in time. In reality, systems do fluctuate and show a time-

dependence. For most intents and purposes their thermodynamic properties are

independent of time, however, and this is described by the ensembles we will consider. It

is good to point out that in an actual experiment we have one system, and this system

itself attains thermodynamic equilibrium. In statistical mechanics the system is replicated

many times, and we calculate averages over the replicas, assuming they are each

described by an eigenstate of the Hamiltonian. This representation of the situation is

3

clearly quite different from the actual situation, and it is a bit of a miracle why this all

works. Let us leave these (poorly understood) fundamental questions behind and return to

the derivation of thermodynamic properties along the conventional lines of statistical

mechanics.

The ensemble can consist of isolated systems, meaning the volume, energy and number

of particles is the same for each element in the ensemble, , ,j j jV V N N E U j= = = ∀ .

This is called the microcanonical ensemble. Another widely used ensemble is the

canonical ensemble, in which each system is closed, meaning it is allowed to exchange

energy, but not matter with neighbouring systems. In a canonical ensemble each system

has the same number of particles jN N= and volume jV V= , but the energy is specific

for each state and is denoted as jE . This is the ensemble discussed in Metiu. In the grand

canonical ensemble the individual systems can differ in both the number of particles jN

and their energies jE , but the volume is fixed, jV V= . We will discuss two more

ensembles, one in which only the number of particles is fixed, while the volume and

energy can vary. In the generalized ensemble, all extensive variables, , ,j j jV N E can vary.

The partition functions either depend on the constant=average value for V, N and/or U, or

they depend on an intensive variable that is a Lagrange multiplier associated with the

constraint that the total number of particles, the total volume or the total energy of the

complete ensemble is constant. The thermodynamic variable conjugate to preserving total

energy is the temperature T. The variable associated with preserving the total volume is

the pressure p, while the variable associated with the number of particles is the chemical

potential µ . Therefore, any partition function “O” has independent variables as follows

( , , )O U orT V or p N or µ , (5)

with the actual choice of variables depending on the extensive variables U, V, N that are

kept constant in each system in the ensemble. The partition functions then relate to a

thermodynamic potential that has precisely the same natural variables as the ensemble,

for example, we have seen already ( , , ) ln ( , , )A T V N kT Q T V N= − . We will find other

4

similar relations. From thermodynamics we know that the Helmholtz free energy is the

suitable thermodynamical potential to consider when a system is kept at constant

temperature through a reservoir. Here the idea is similar. The other systems in the

ensemble act as the heat reservoir, allowing a redistribution of energy, and providing a

resulting partition function that depends on the temperature T, that might be viewed as a

controllable parameter, or, equivalently, a variable in the partition function. The

probabilities of the most likely distribution depend on the value of T, which was used as a

Lagrange multiplier required to keep the total energy constant (as would be the case when

we consider a system in contact with a heat reservoir: the total energy of system +

reservoir would remain constant). We will see that the various types of ensembles we can

create map precisely to the kind of thermodynamic potentials that are generated by

Legendre transformations in thermodynamics.

Thermodynamic properties can be derived from any of these ensembles, and the final

results are equivalent. For example, Metiu discusses the results for the canonical

ensemble and this provides all thermodynamic properties. This feature of statistical

mechanics is reflected in the fact that the thermodynamic potentials, when viewed as

functions of their proper ‘natural variables’ all yield complete thermodynamic

information. We will discuss the derivations in a unified context for the various

ensembles, as it puts the theory in a general framework. It more clearly shows what is

involved, and what freedoms exist to derive the results. Moreover, to derive certain

results in statistical mechanics it may be far more convenient to use a particular

ensemble, as the mathematics is ‘easy’, or even feasible, only for certain ensembles. So it

is good to know about the existence of various ensembles. They are part of the tricks of

the trade. The fact that various ensembles in statistical mechanics lead to basically

identical results hinges from a physical perspective on the fact that even if one allows

volumes, particle numbers and energies to vary per system, the fluctuations around the

mean are very small, for large enough individual systems. This is well known from

experience. For example, we expect temperature, pressure and density only to vary very

little in a macroscopic system in equilibrium.

5

Let me sketch the basic procedure for any type of ensemble, and provide the knowledge

that we assume as input. This summary will be somewhat abstract at first reading, but as

we go through examples, I think this summary may prove useful to you. In the

derivations below I assume that entropy is given by the basic formula

lnj jj

S k P P= − ∑ (6)

The quantities temperature, pressure and chemical potential are defined through the

partial derivatives of entropy, as it was done in the notes on thermodynamics: hence,

, , ,

1 , ,V N U N U V

S S p SU T V T N T

µ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞≡ ≡ ≡ −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (7)

We will restrict ourselves to one component systems here. For all of the ensembles the

procedure is that we maximize entropy subject to the constraints imposed on the

ensemble

j j jj

j j jj

j j jj

E U j or E P U Lagrangemultiplier

V V j or V P V Lagrangemultiplier

N N j or N P N Lagrangemultiplier

β

α

γ

= ∀ = +

= ∀ = +

= ∀ = +

∑

∑

∑

(8)

The constraints indicate that we can create 32 8= different ensembles in principle. From

the maximization procedure for the constrained entropy we will find an expression for the

partition function of the type

; /j j jj

Q P P P Q= =∑ % % , (9)

where the jP% will be simple exponential factors that correspond to unnormalized

probabilities. The logarithm of Q will be found to be related to particular chemical

potentials that have the same natural variables as the variables in Q. We will find that

maximizing the entropy subject to constraints is equivalent to minimizing (or sometimes

maximizing) the chemical potential. We will see that from the procedure we will either

obtain a relation between the Lagrange multiplier and intensive thermodynamic variables

(in particular: 1 ; ,pkT kT kT

µβ α γ= = = − ), or we will find an explicit expression for the

intensive variables as an average over a mechanical variable, e.g.

6

,j jj j

j jN V

E Ep P P

V Nµ

∂ ∂⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∑ ∑ (10)

Let us now discuss the various ensembles and establish the connections to

thermodynamical quantities.

1. Microcanonical ensemble

In the microcanonical ensemble all systems defining the ensemble have an identical

energy, volume and number of particles, each element of the ensemble is itself an isolated

system. The partition function is simply the number of (linearly independent) quantum

states, and is written as

( , , )N V EΩ (11)

The probability to find the system in a particular state is given by 1( , , )jP N V E

=Ω

,

which is the same for every state in the ensemble. This is precisely the fundamental

postulate of statistical mechanics: In an isolated system each possible state is equally

likely. The thermodynamic identification proceeds through Boltzmann’s fundamental

law,

ln ( , , )S k N V E= Ω (12)

It is well known that in thermodynamics the condition for a spontaneous process in an

isolated system is that entropy increases. Or: In a spontaneous process the logarithm of

the corresponding partition function increases. The variables of the partition function are

N, V, E, and these are also the natural variables of S in thermodynamics. Identifying E

with U as usual,

(from )dU pdS dV dN dU TdS pdV dNT T T

µ µ= + − = − + (13)

Hence

7

ln

, , , ,,

, , ,

, , ,

ln 1 1 ln;

ln 1 ; ...

ln 1 ; ...

N V N V N V N VN V

N T N T N T

T V T V T V

S eE k U kT E E E kT

S p pV k V kT V kT

SN k N kT N

µ

Ω⎛ ⎞∂ Ω ∂ ∂Ω ∂ ∂ Ω Ω⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = =Ω =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠∂ Ω ∂ ∂Ω⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = =Ω⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠∂ Ω ∂ ∂Ω⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = − = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ kT

µ−Ω

(14)

To unify the microcanonical ensemble with the treatment of other ensembles (see below),

we can proceed alternatively as follows. The partition function for the microscopic

ensemble is defined by finding the most likely distribution for the probabilities by

maximizing

1, 1,

ln ( 1)j j jj jP P Pλ

= Ω = Ω

− − −∑ ∑ (15)

This yields the partition function 1( , , ),( , , )jN V E PN V E

Ω =Ω

. In the most likely

distribution of the ensemble each state occurs equally likely. We can identify

1 1ln ln ln lnj j jj j

S k k k P k P PP

= Ω = = − = −Ω∑ ∑ . (16)

We will verify below that for every possible ensemble

lnj jj

S k P P= − ∑ , (17)

and for every possible ensemble precisely this quantity is maximized under additional

constraints depending on the particulars of the ensemble. At equilibrium the most likely

distribution is reached (within fluctuations), and this is precisely what is meant by stating

that entropy reaches a maximum at equilibrium. The (constrained) maximum of the

quantity lnj jj

k P P− ∑ defines the most likely distribution, and the most likely

distribution defines all thermodynamic quantities, as it is overwhelmingly more likely

than any other distribution, and it is the only distribution that needs to be taken into

account to define the average for the large ensembles under consideration.

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2. Canonical ensemble.

In the canonical ensemble the individual systems in the ensemble can exchange energy,

but they all have the same number of particles and volume. In thermodynamic language,

each element of the ensemble is a closed system. The total energy of the ensemble is

conserved (or a constant). This provides a Lagrange multiplier, β , associated with the

constraint of energy conservation. The partition function is derived from the most likely

distribution that preserves the average total energy U: i.e. maximize

ln ( ) ( 1)j j j j jj j jP P P E U Pβ λ− − − − −∑ ∑ ∑ , (18)

from which one obtains , 1j j jj jP E U P= =∑ ∑ (stationarity w.r.t. andβ λ )

Slightly more conveniently, we can maximize the unnormalized probabilities, and define

the partition function accordingly. Hence maximize

ln ( )

; ; /j

j j j jj j

Ej j j j

j

P P P E U

P e Q P P P Qβ

β

−

− − −

→ = = =

∑ ∑

∑

% % % %

% % % (19)

where we used the general result from the previous set of notes. Carrying out the

maximization the partition function for the canonical ensemble is given by

( , , ) ; /j jE Ej

jQ N V e P e Qβ ββ − −= =∑ (20)

where j runs over the possible states in the ensemble (see further notes, point 1, for

further discussion). The expression for entropy is hence given by:

ln ( ) ln

ln ln[ ]j

j j j j jj j j

E

j

S k P P k E P k Q P

kU k Q kU k e β

β

β β −

≡ − = − − +

= + = +

∑ ∑ ∑

∑ (21)

To establish the connection with thermodynamics consider the partial derivatives

9

,

,

,

1 1

1 1

1 1

j

j

N V

Ej j jj j

j j jU N

Ej j jj j

j j jU V

S kT U kT

E E Ep S k e P p PT V Q V T V V

E E ES k e P PT N Q N T N N

β

β

β β

β

µ β µ

−

−

∂⎛ ⎞= = → =⎜ ⎟∂⎝ ⎠∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂⎛ ⎞= = − = − → = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂⎛ ⎞− = = − = − → =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑ ∑ ∑

∑ ∑ ∑

(22)

Hence from the partial derivatives we obtain a definition for β , but we also find

expressions for the intensive variables pressure and chemical potential as average values.

Substituting the expression for β in the relation for entropy (Eqn. 21 ) we obtain

/ ln

ln ( , , ) ( , , )S U T k Q

U TS kT Q N V T A N V TdA dU TdS SdT SdT pdV dNµ

= +→ − = − ≡= − − = − − +

(23)

The natural variables for Q are N, V, T, and these are also the natural variables of the

corresponding characteristic function A, the Helmholtz free energy. Since at equilibrium

the entropy takes on a maximum value under the constraint that U is constant, it follows

that the Helmoltz free energy attains a minimum at equilibrium for a system in contact

with a heat reservoir that keeps a constant temperature in the system.

In summary

/ /( , , ) ; /j jE kT E KTj

jQ N V T e P e Q− −= =∑

( , , ) ln ( , , )A N V T kT Q N V T= − (24)

dA SdT pdV dNµ= − − +

and

, ,

, ,

, ,

lnln

ln

ln

N V N V

N T N T

V T V T

A QS k Q kTT T

A Qp kTV V

A QkTN N

µ

∂ ∂⎛ ⎞ ⎛ ⎞= − = +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂⎛ ⎞ ⎛ ⎞= − =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂⎛ ⎞ ⎛ ⎞= = −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

(25)

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3. Grand canonical ensemble.

In the grand canonical ensemble the individual systems in the ensemble can exchange

both energy and matter, hence the number of particles per individual system in the

ensemble can change, while the volume is still the same for every element in the

ensemble. Each element in the ensemble is an open system. The total energy and the total

number of particles of the ensemble is conserved. This provides a Lagrange multiplier, β ,

associated with energy conservation, and a Lagrange multiplier,γ associated with

particle number conservation. This partition function is derived as the most likely

distribution that preserves the total energy and the number of particles: i.e. maximize

ln ( ) ( ) ( 1)

... ; , /j j

j j j j j j jj j j j

E Nj j j j

j

P P P E U P N N P

P e e Z P P P Zβ γ

β γ λ

− −

− − − − − − −

→ → = = =

∑ ∑ ∑ ∑

∑% % %. (26)

And we also find ; , 1j j j j jj j j

U P E N P N P= = =∑ ∑ ∑ from the stationarity condition with

respect to the Lagrange multipliers. The partition function for the grand canonical

ensemble is given by

( , , ) ; /j j j jE N E Nj

jZ V e e P e e Zβ γ β γβ γ − − − −= =∑ (27)

The expression for entropy is hence given by

ln ln

ln

j j j j j j jj j j j

S k P P k E P k N P Z P

k U k N k Z

β γ

β γ

= − = + +

= + +

∑ ∑ ∑ ∑ (28)

where as before, the sum over j runs over the accessible states in the ensemble.

To provide the connection with thermodynamics:

11

,

,

,

1 1

1 [ ]

( )1 [ ]

j j

N V

U V

E Nj j

jU N

j j j jj j

j j

S kT U kT

S kT N kT

E Np S k e eT V Z V V

E N E NP p P

T V V V

β γ

β β

µ µγ γ

β γ

µµ

− −

∂⎛ ⎞= = → =⎜ ⎟∂⎝ ⎠∂⎛ ⎞− = = → = −⎜ ⎟∂⎝ ⎠

∂ ∂⎛ ⎞ ⎛ ⎞∂⎛ ⎞= = − −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠∂ ∂ ∂ −⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= − − → = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑

∑ ∑

(29)

Hence we obtain expressions for the Lagrange multipliers, but also a different expression

for pressure for the grand canonical ensemble. Substituting the expressions for the

Lagrange multipliers in the expression for S (Eqn. 28), we find

/ / lnln ( , , )

( )

S U T N T k ZU TS N kT Z V T

U TS G pV

µµ µ

= − +− − = −

= − − = − (30)

where the identification N Gµ = is made (see further notes 2). Let me also note that this

slightly unusual thermodynamic potential has been discussed in the notes “Fundamental

Equilibrium Thermodynamics”. The natural variables for the grand canonical partition

function are µ , V, T, and these are also the natural variables of the characteristic function

(pV). From pV N TS Uµ= + − we readily derive

( )d pV Nd dN TdS SdT TdS pdV dN

SdT pdV Ndµ µ µ

µ= + + + − + −= + +

(31)

Summarizing / / / /( , , ) ; /j j j jE kT N kT E kT N kT

jj

Z T V e e P e e Zµ µµ − −= =∑

( ) ln ( , , )pV kT Z V Tµ= (32)

( )d pV SdT pdV Ndµ= + +

and therefore

12

, ,

, ,

, ,

( ) lnln

( ) ln

( ) ln

V V

T T

V T V T

pV ZS k Z kTT T

pV Zp kTV V

pV ZN kT

µ µ

µ µ

µ µ

∂ ∂⎛ ⎞ ⎛ ⎞= = +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂= =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

(33)

Since entropy takes on a maximum at thermodynamic equilibrium under constraints

of constant U and N, it follows that ( )pV also takes on a maximum, and can only

increase in a spontaneous process in which the temperature and chemical potential

remain constant through interactions with suitable reservoirs.

4. Isobaric-Isothermal ensemble.

In this ensemble the individual systems in the ensemble can exchange energy, and the

volume in individual systems can adjust, while the particle number is constant in each

system. The total energy of the ensemble, and the total volume of the ensemble is

conserved. This provides Lagrange multiplier, β , associated with energy conservation,

(and temperature). The second Lagrange multiplier, α , associated with the preservation

of the total volume of the ensemble, is naturally associated with pressure, as this is

equilibrated as the volumes adjust to equilibrium. This partition function can be derived

by maximizing the most likely distribution under the constraints that total volume and

total energy are conserved, i.e. maximize

ln ( ) ( ) ( 1)

... ; ( , , ) , /j j

j j j j j j jj j j j

E Vj j j j

j

P P P E U PV V P

P e e N P P Pβ α

β α λ

β α− −

− − − − − − −

→ → = Δ = = Δ

∑ ∑ ∑ ∑

∑% % % (34)

The partition function for the isobaric-isothermal canonical ensemble is given by

( , , ) ; /j j j jE V E Vj

jN e e P e eβ α β αα β − − − −Δ = = Δ∑ (35)

The expression for entropy is then given by

ln ln

ln

j j j j j jj j j

S k P P k E P k V P k

k U k V k

β α

β α

= − = + + Δ

= + + Δ

∑ ∑ ∑ (36)

13

To provide the connection with thermodynamics:

,

,

,

1 1

1 [ ]

( )1 [ ]

j j

N V

U N

E Vj j

jU V

j j j jj j

j j

S kT U kT

p S pkT V kT

E VS k e eT N N N

E V E pVp P P

T N N N

β α

β β

α α

µ β α

µ

− −

∂⎛ ⎞= = → =⎜ ⎟∂⎝ ⎠∂⎛ ⎞= = → =⎜ ⎟∂⎝ ⎠

∂ ∂⎛ ⎞ ⎛ ⎞∂⎛ ⎞− = = − −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ Δ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠∂ ∂ ∂ +⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= − + → =⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑

∑ ∑

(37)

Substituting the expressions for the Lagrange multipliers, we find

/ / lnln ( , , ) ( , , )

S U T pV T kU pV TS kT p T N G p T N

= + + Δ+ − = − Δ ≡

(38)

Hence, the natural variables for the isobaric-isothermal partition function are N, p, T, and

these are also the natural variables of the characteristic function, the Gibbs free energy G.

A process is spontaneous under the conditions of this ensemble (i.e. p and T remain

constant through interaction with a pressure and temperature reservoir) if the free energy

decreases, again in direct relation to maximizing entropy under the constraints of

preserving total energy and volume. In terms of the natural variables, the partition

function is given by / / / /( , , ) ; /j j j jE kT pV kT E kT pV kT

jj

N p T e e P e e− − − −Δ = = Δ∑

( , , ) ln ( , , )G p T N kT p T N= − Δ (39)

dG SdT Vdp dNµ= − + +

and

, ,

, ,

, ,

lnln

ln

ln

N p N p

N T N T

p T p T

GS k kTT T

GV kTp p

G kTN N

µ

∂ ∂ Δ⎛ ⎞ ⎛ ⎞= − = Δ +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂ Δ= = −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂ Δ⎛ ⎞ ⎛ ⎞= = −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

(40)

14

5. The generalized ensemble.

Each system in this ensemble can have a variable number of particles, volume and

energy. The ensemble is constrained such that the total number of particles, total volume

and total energy is conserved. This provides three Lagrange multipliers, denoted , ,α β γ ,

which will be identified with the intensive variables pressure, temperature and chemical

potential respectively. As before this partition function can be derived by maximizing the

most likely distribution under the constraints that total volume, total number of particles

and total energy are conserved, i.e. maximize

ln ( ) ( ) ( )

... ; ( , , ) , /j j j

j j j j j j j jj j j j

E V Nj j j j

j

P P P E U PV V P N N

P e e e Y P P P Nβ α γ

β α γ

α β γ− − −

− − − − − − −

→ → = = =

∑ ∑ ∑ ∑

∑% % % (41)

and we also find

, ,j j j j j jj j j

U P E V PV N P N= = =∑ ∑ ∑

The partition function denoted Y is given by

( , , ) ; /j j j j j jV E N V E Nj

jY e e e P e e e Yα β γ α β γα β γ − − − − − −= =∑ (42)

The expression for entropy is then given by

ln ln

ln

j j j j j j j jj j j j

S k P P k E P k V P k N P Y

kU kV kN k Y

β α γ

β α γ

= − = + + +

= + + +

∑ ∑ ∑ ∑ (43)

To provide the connection with thermodynamics:

,

,

,

1 1

N V

U N

U V

S kT U kT

p S pkT V kT

S kT N kT

β β

α α

µ µγ γ

∂⎛ ⎞= = → =⎜ ⎟∂⎝ ⎠∂⎛ ⎞= = → =⎜ ⎟∂⎝ ⎠∂⎛ ⎞− = = → = −⎜ ⎟∂⎝ ⎠

(44)

and, substituting the expressions for the Lagrange multipliers in the equation for S we

find

15

/ / / ln

0 ln1

S U T pV T N T k YU pV TS N U pV TS G kT Y

Y

µµ

= + − ++ − − = + − − = = −

=

ln ( , , ) ... 0kT Y T p U TS pV Nµ µ− = − + − = = (45)

From this we can derive

( ln )0

d kT Y dU TdS SdT pdV Vdp dN NdSdT Vdp Nd

µ µµ

− = − − + + − −= − + − =

(46)

using the usual expression for dU. This indicates that the thermodynamic variables

, ,p T µ cannot be varied independently, and the above relation is precisely the Gibbs-

Duhem relation. The results for the generalized ensemble are a bit different from the

other ensembles, and the interpretation of the results is not entirely straightforward.

Further discussion can be found in an advanced book, “Statistical mechanics” by Terell

Hill, Chapter 3. Let us write the partition function

/ / / / / /( , , ) 1;j j j j j jE kT pV kT N kT E kT pV kT N kTj

jY T p e e e P e e eµ µµ − − − −= = =∑ (47)

This indicates that the generalized partition function is simply the normalization

condition on the probabilities, and it apparently does not have to be calculated! Despite

this seemingly attractive feature, or perhaps because of it, it is stated in the literature there

is not much use for the generalized partition function (except that it provides the very

useful Gibbs-Duhem equation).

6. Concise summary of all possible ensembles.

From the general ensemble we can deduce the general relation of the partition function to

the Lagrange multipliers and then the thermodynamic quantities. In this section we will

refer to each partition function as ( , , )Y T orU porV or Nµ , depending on the variables

, ,j j jE V N that are held constant in the specific ensemble. We have the most general form

of the probabilities

/

ln ln

j j jE V Nj

j jj

P e e e Y

S k P P kU kV kN k Y

β α γ

β α γ

− − −=

= − = + + +∑ (48)

16

Making the identifications 1/ , / , /kT p kT kTβ α γ µ= = = − this can be written as

ln 0kT Y U pV TS Nµ− = + − − = (49)

This formula can be taken to be more general in the sense that -TS is always present, U

is present if jE is variable, pV is present if the jV are variable in the ensemble, while

Nµ− is present if jN is variable. This can be checked for the actual derivations we did

for the various ensembles. Hence we can make the following 8 types of ensembles

Variable

quantities

Partition function Characteristic

function

Name of

ensembl

e

Thermodynamic Differential

from statistical mechanics

none ln ( , , )kT Y U V N TS Micro-

canonica

l

/ ( / ) ( / )dS dU T p T dV T dNµ= + −

jE - ln ( , , )kT Y T V N TS-U=A canonica

l

dA SdT pdV dNµ= − − +

jV ln ( , , )kT Y U p N TS-pV=U-G ? ( ) ... ... ...d TS pV dU dp dN− = + +

jN - ln ( , , )kT Y U V µ TS Nµ+

=U+pV=H

? ... ... ...dH dU dV dµ= + + !

,j jE V - ln ( , , )kT Y T p N U TS pV G− + = Isobaric-

isotherm

al

dG SdT Vdp dNµ= − + +

,j jE N ln ( , , )kT Y T V µ TS U N pVµ− + = Grand-

canonica

l

( )d pV SdT pdV Ndµ= − + +

,j jV N - ln ( , , )kT Y U p µ TS pV N Uµ− − + = ? 0 0dU dU dp dµ= + + !

, ,j j jE V N ln ( , , )kT Y T p µ 0

TS U pV Nµ− − +=

General 0 SdT Vdp Ndµ= − + −

Gibbs Duhem

17

From the above table, it will be clear that not all results are equally obvious, and not all

cases have their correspondence in commonly known functions of thermodynamics. In

particular the enthalpy function looks a little suspect, as the natural variables appear to U,

V and µ . The interested reader may want to verify the results in this table.

7. Further notes.

1. In the literature one often finds expressions for the partition function in which the sums

are performed not over individual states but over energy levels. To derive equations, it is

much better to start from equations in terms of states. To show the dangers, let us try it

the other way, using sums over levels. Let us consider the example of the canonical

partition function and write

( , ) ( , )1( , , ) ( , , ) ; ( ) ( , , )j j

j

E N V E N Vj j j

EQ N V N V E e P E N V E e

Qβ ββ − −= Ω = Ω∑

and evaluate the derivative of lnQ with respect to V. One might be tempted to write

,

ln 1 ( , , ) ( )j

j j

Ej jj j

E ET N

E EQ N V E e P E pV Q V V

ββ β β−∂ ∂∂⎛ ⎞ = − Ω = − =⎜ ⎟∂ ∂ ∂⎝ ⎠∑ ∑

The end result is correct, but the mathematics is suspect as the number of states

( , , )jN V EΩ (i.e. the partition function of the microcanonical ensemble) explicitly

depends on V and on ( , )jE N V , and its derivatives are not taken into account. Moreover,

it assumes that all states of a given energy have the same derivative with respect to

volume, which is not necessarily true, (certainly if other types of derivatives would be

considered). Instead of the above, following simple rules of mathematics, I would have to

evaluate

,

ln

( , , ) ( , , )1 1 1( , , ) j j j

j j j

T N

E E Ej j j jj

E E E j

QV

E N V E N V E EN V E e e e

Q V Q V Q E Vβ β ββ − − −

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠∂ ∂Ω ∂Ω ∂

− Ω + +∂ ∂ ∂ ∂∑ ∑ ∑

18

and using the derivatives of Ω as defined in the microcanonical ensemble (see subsection

1), one obtains

1 1 1 1( , , ) ( , , ) ( , , )j j j

j j j

E E Ej jj j j

E E E

E EpN V E e N V E e N V E eQ V kT Q kT Q V

p

β β ββ

β

− − −∂ ∂= − Ω + Ω + Ω

∂ ∂

=

∑ ∑ ∑It follows that taking the derivative of Ω into account, the two terms indeed properly

cancel, such that the proper result is obtained. It still means that the first way of deriving

the result is essentially wrong, if no mention is made of this cancellation. Moreover, the

above identification of the derivatives of ( , , )jN V EΩ for arbitrary energies is dubious,

and it would probably be more appropriate to write for the last two terms

( )1 1 1( , , ) ( , , ) 0( ) ( )

j j

j j

E Ej jj j

E Ej j

p E EN V E e N V E e

Q kT E Q kT E Vβ β− −∂

Ω + Ω =∂∑ ∑

such that cancellation occurs for each energy level jE separately. It is clear that the

above derivation, using sums over energy levels is full of danger. There is an easier

solution. In the development discussed in these notes, we always assume a sum over

individual states, such that

( , )( , , ) jE N V

jQ N V e ββ −=∑

ln 1 j

j

Ej jj

j E

E EQ e P pV Q V V

ββ β β−∂ ∂∂ = − = − =∂ ∂ ∂∑ ∑

There are no ambiguities, and the mathematics is straightforward.

2. There is an interesting perspective to see that G Nµ= . Since ( , , )G N p T and N are

both extensive variables (they scale linearly with the size of the system), we can write

( , , ) ( , )G N p T N g p T= , where ( , )g p T is independent of the size of the system. Hence

,

( , ) ( , , )p T

G g p T G N p T NN

µ µ∂⎛ ⎞= = → =⎜ ⎟∂⎝ ⎠

or, the chemical potential is the Gibbs free energy per particle.