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Chapter 6 Basic Methods & Results of Statistical Mechanics

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Chapter 6Basic Methods & Results of Statistical Mechanics

Maxwell

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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Idea: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 Systemsmeans that they are all in the same thermodynamic state
- To do any calculations we have to first
Choose an Ensemble!

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- 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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We choose the one most convenient.

For gases: PVT properties – canonical ensemble

Vapor-liquid equilibrium – grand canonical ensemble.

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6

(6.11)

7

(6.12)

with:

(6.13)

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(6.15)

(6.16)

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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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(6.40)

(6.59)

(6.60)

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(6.61)

(6.62)

(6.65)

(6.63)

(6.64)

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Starting from the fundamental postulate ofequal a priori

probabilities, the following are obtained:

the results of classical thermodynamics, plus their statistical underpinnings;

the means of calculating the thermodynamic parameters (U, H, F, G, S ) from a single statistical parameter, the partition functionZ (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 thermodynamic

function 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 lnΩis carved on Boltzmann’s tombstone).

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Summary 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)Vand P = – (∂F/∂V)T .

Also, G = F + PV = PV – kT lnZ.

H = U + PV = PV – ∂(lnZ)/∂β.

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Z = zNfor distinguishable particles (e.g. solids);

Z = zN/Nfor 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)Vand P = (∂F/∂V)Tas before.

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Equations obtained from Z = rexp (– Er), where = 1/kT.

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

U2 = rprEr2/rpr = (1/Z) 2Z/2.

Un = rprErn/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.

Notes

Since (ΔU)2 ≥ 0, (i) CV≥ 0, (ii) U/T ≥ 0.

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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 thatSn= k ln n – k r ln nr.

From Stirling’s theorem,

ln n ≈ n ln n – n, r ln nr ≈ rnr ln nr – n.

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

so thatSn= – k r nr ln pr= – kn rpr ln pr.

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

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A microcanonical ensembleis a large number of identical

isolated systems.

The thermodynamic degeneracy may be written asω(U, V, N).

From the fundamental postulate, the probability of finding the

system in the state r ispr = 1/ω.

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

= (k/ω) ln ωr1 = 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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Acanonical ensembleconsists of a large number of identically

prepared systems, which are in thermal contact with a heat

reservoir 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 = rexp(–Er), and = 1/kT.

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

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

= (k/Z) rErexp(–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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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 ensembleis a large number of identical

systems, which interact diffusively with a particle reservoir.

Each system is described by agrand 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 states

associated with a given value of N.

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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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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:

See online notes/PDF derivation

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(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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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 p2i /2mi

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

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Microcanonical ensemble

E, V and N fixed

S = kB lnW(E,V,N)

Canonical ensemble

T, V and N fixed

F = -kBT lnZ(T,V,N)

Grand canonical ensemble

T, V and m fixed

F = -kBT ln (T,V,m)