Chapter 6 Basic Methods & Results of Statistical Mechanics

Historical Introduction

• Statistical Mechanics developed by Maxwell, Boltzman, Clausius, Gibbs.

• Question: If we have individual molecules – how can there be a pressure, enthalpy, etc?

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Maxwell

Key Concept In Statistical MechanicsIdea: Macroscopic properties are a thermal average of microscopic properties.

• Replace the system with a set of systems "identical" to the first and average over all of the systems. We call the set of systems

“The Statistical Ensemble”.• Identical Systems means that they are all in the same

thermodynamic state • To do any calculations we have to first

Choose an Ensemble!

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Common Statistical Ensembles• Micro Canonical Ensemble: Isolated Systems. • Canonical Ensemble: Systems with a fixed number

of molecules in equilibrium with a heat bath.

• Grand Canonical Ensemble: Systems in equilibrium with a source a heat bath which is also a source of molecules. Their chemical potential is fixed.

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All Thermodynamic Properties Can Be Calculated With Any Ensemble

We choose the one most convenient.

For gases: PVT properties – canonical ensemble

Vapor-liquid equilibrium – grand canonical ensemble.

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Properties Of The Canonical and the Grand Canonical Ensemble

Gibbs showed that the ensemble average was equivalent to a state average

F F pnn

n

(6.10)

Pn=the probability that the system is in a configuration (state) n.

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Properties of the Canonical Ensemble:

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p g eQnn

U

canonN

n

(6.11)

The Grand Canonical Ensemble:

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p g eQnn

E

grand

n

(6.12)

with:

E U Nn n n

(6.13)

Partition Functions

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Qcanon

N =canonical partition function Qgrand= grand canonical partition function

Q g ecanonN

nn

Un

Q g egrand nn

E n

(6.15)

(6.16)

Partition Functions• If you know the volume, temperature, and the

energy levels of the system you can calculate the partition function.

• If you know T and the partition function you can calculate all other thermodynamic properties.

Thus, stat mech provides a link between quantum and thermo. If you know the energy levels you can calculate partition functions and then calculate thermodynamic properties.

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• Partition functions easily calculate from the properties of the molecules in the system (i.e. energy levels, atomic masses etc).

• Convenient thermodynamic variables. If you know the properties of all of the molecules, you can calculate the partition functions.

• Can then calculate any thermodynamic property of the system.

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Thermal Averages with Partition Functions

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nB n

n n

pS k p Lng

NB canonA k TLn(Q )

( )LnQ UcanonN

(6.40)

(6.59)

(6.60)

(6.61)

NNcanon

B B canonV,N V,N

LnQAS=- =k T +k LnQT T

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Ncanon

BT,N T,N

LnQAP =k TV V

(6.62)

Ncanon

BT,V T,V

LnQA k TN N

(6.63)

grandB B grand

V, V,

LnQPVS k T +k Ln(Q )T dT

(6.64)

(6.65)

grandB

T,V T,V

LnQPVΝ= =k Tμ μ

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Canonical Ensemble Partition Function ZStarting from the fundamental postulate of equal a prioriprobabilities, the following are obtained:i. the results of classical thermodynamics, plus their

statistical underpinnings;ii. the means of calculating the thermodynamic parameters

(U, H, F, G, S ) from a single statistical parameter, the partition function Z (or Q), which may be obtained from the energy-level scheme for a quantum system.

The partition function for a quantum system in contact with a heat bath is

Z = i exp(– εi /kT), where εi is the energy of the i’th state.

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The partition function for a quantum system in contact with a heat bath is Z = i exp(– εi /kT), where εi is the energy of the i’th state.

The connection to the macroscopic thermodynamicfunction S is through the microscopic parameter Ω

(or ω), which is known as thermodynamic degeneracy or statistical weight, and gives the number of microstates in a given macrostate.

The connection between them, known as Boltzmann’s principle, is S = k lnω.

(S = k lnΩ is carved on Boltzmann’s tombstone).

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Relation of Z to Macroscopic ParametersSummary of results to be obtained in this section

<U> = – ∂(lnZ)/∂β = – (1/Z)(∂Z/∂β),CV = <(ΔU)2>/kT2,

where β = 1/kT, with k = Boltzmann’s constant.S = kβ<U> + k lnZ ,

where <U> = U for a very large system.F = U – TS = – kT lnZ,

• From dF = S dT – PdV, we obtain S = – (∂F/∂T)V and P = – (∂F/∂V)T .

Also, G = F + PV = PV – kT lnZ.H = U + PV = PV – ∂(lnZ)/∂β.

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Systems of N Particles of the Same Species• Z = zN for distinguishable particles (e.g. solids); Z = zN/N for indistinguishable particles (e.g.fluids).

<u> = – ∂(lnz)/∂β = – (1/z)(∂z/∂β), U = N<u>.cV = <(Δu)2>/kT2, CV = NcV, CP = NcP.

Distinguishable particles: F = Nf = – kT ln zN = – NkT lnz.Since F = U – TS, so that S = (U – F)/T or S = – (∂F/∂T)V.

Indistinguishable particles: F = – kT ln(zN/N)

= – kT [ln(zN) – ln N] = – NkT [ln(z/N) – 1],Since for very large N, Stirling’s theorem gives ln N! = N lnN – N.

Also, S = – (∂F/∂T)V and P = (∂F/∂V)T as before.

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Mean Energies and Heat Capacities• Equations obtained from Z = r exp (– Er),

where = 1/kT. U = rprEr/rpr = – (ln Z)/ = – (1/Z) Z/ . U2 = rprEr

2/rpr = (1/Z) 2Z/2. Un = rprEr

n/rpr = (–1)n(1/Z) nZ/n. (ΔU)2 = U2 – (U)2 = 2lnZ/2 or – U/ . • CV = U/T = U/ . d/dT = – k2. U/, or CV = k2 (ΔU)2 = (ΔU)2/kT2;

i.e. (ΔU)2 = kT2CV .NotesSince (ΔU)2 ≥ 0, (i) CV ≥ 0, (ii) U/T ≥ 0.

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Entropy and Probability• Consider an ensemble of n replicas of a system.• If the probability of finding a member in the state r is pr, the

number of systems that would be found in the r’th state is nr = n pr, if n is large.

• The statistical weight of the ensemble Ωn (n1 systems are in state 1, etc.), is Ωn = n/(n1 n2…nr..), so that Sn = k ln n – k r ln nr.

• From Stirling’s theorem, ln n ≈ n ln n – n, r ln nr ≈ r nr ln nr – n.

Thus Sn = k {n ln n – r nr ln nr} = k {n ln n – r nr ln n – r nr ln pr},

so that Sn = – k r nr ln pr = – kn r pr ln pr .

For a single system, S = Sn/n ; i.e. S = – k r pr ln pr .

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Ensembles 1A microcanonical ensemble is a large number of identicalisolated systems. The thermodynamic degeneracy may be written as ω(U, V, N).From the fundamental postulate, the probability of finding thesystem in the state r is pr = 1/ω.

Thus, S = – k r pr ln pr = k r (1/ω) ln ω

= (k/ω) ln ω r1 = k ln ω.

Statistical parameter: ω(U, V, N).Thermodynamic parameter: S(U, V, N) [T dS = dU – PdV + μdN].Connection: S = k ln ω. Equilibrium condition: S Smax.

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Ensembles 2A canonical ensemble consists of a large number of identicallyprepared systems, which are in thermal contact with a heatreservoir at temperature T.The probability pr of finding the system in the state r is given by

the Boltzmann distribution: pr = exp(– Er)/Z, where Z = r exp(–Er), and = 1/kT.

Now S = – k r pr ln pr = – k r [exp(–Er)/Z] ln[exp(–Er)/Z]

= – (k/Z) r exp(–Er) {ln exp(–Er) – ln Z}

= (k/Z) rEr exp(–Er) + (k lnZ)/Z . rexp(–Er),

so that S = k U + k lnZ = k lnZ + kU.

Thus, S(T, V, N) = k lnZ + U/T and F = U – TS = – kT lnZ.

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Ensembles 3 S(T, V, N) = k lnZ + U/T , F = U – TS = – kT lnZ.Statistical parameter: Z(T, V, N).Thermodynamic parameter: F(T, V, N). Connection: F = – kT ln Z. Equilibrium condition: F Fmin.

A grand canonical ensemble is a large number of identicalsystems, which interact diffusively with a particle reservoir.

Each system is described by a grand partition function,

G(T, V, μ) = N{r(μN – EN,r)},

where N refers to the number of particles and r to the set of statesassociated with a given value of N.

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Statistical Ensembles• Classical phase space is 6N variables (pi, qi) with a Hamiltonian

function H(q,p,t).

• We may know a few constants of motion such as energy, number of particles, volume, ...

• The most fundamental way to understand the foundation of statistical mechanics is by using quantum mechanics:– In a finite system, there are a countable number of states with

various properties, e.g. energy Ei.– For each energy interval we can define the density of states.

g(E)dE = exp(S(E)/kB) dE, where S(E) is the entropy.– If all we know is the energy, we have to assume that each state

in the interval is equally likely. (Maybe we know the p or another property)

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Environment• Perhaps the system is isolated. No contact with outside

world. This is appropriate to describe a cluster in vacuum.• Or we have a heat bath: replace surrounding system with

heat bath. All the heat bath does is occasionally shuffle the system by exchanging energy, particles, momentum,…..

The only distribution consistent with a heat bath is a canonical distribution:

( , )Prob( , ) /H q pq p dqdp e Z

See online notes/PDF derivation

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Statistical ensembles• (E, V, N) microcanonical, constant volume• (T, V, N) canonical, constant volume• (T, P N) canonical, constant pressure• (T, V , μ) grand canonical (variable particle number)• Which is best? It depends on:

– the question you are asking– the simulation method: MC or MD (MC better for phase

transitions)– your code.

• Lots of work in recent years on various ensembles (later).

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Maxwell-Boltzmann Distribution

• Z=partition function. Defined so that probability is normalized.

• Quantum expression• Also Z= exp(-β F), F=free energy (more convenient since F

is extensive)• Classically: H(q,p) = V(q)+ Σi p2

i /2mi

• Then the momentum integrals can be performed. One has simply an uncorrelated Gaussian (Maxwell) distribution of momentum.

( , )Prob( , ) e / !H q pq p dqdp N Z

Z exp( E i )

Microcanonical ensembleMicrocanonical ensembleE, V and N fixedS = kB lnW(E,V,N)

Canonical ensembleCanonical ensembleT, V and N fixedF = kBT lnZ(T,V,N)

Grand canonical ensembleGrand canonical ensembleT, V and fixed = kBT ln (T,V,)