1. Fluctuations are important because the number of particles in a system is much less than Avogadro’s number;

2. Importance of the surface properties. Thermodynamic quantities no longer scale with the number of atoms in a system becuase the energy associated with the surface may be a significant fraction of the total.

Nanosystems generally contain too many atoms to be thought Nanosystems generally contain too many atoms to be thought of as simple mechanical systems, but too few to be described by of as simple mechanical systems, but too few to be described by bulk properties.bulk properties.

Equilibrium thermodynamic properties are well defined because fluctuations are negligible in large (N=1023) systems.

Microscopic view of Microscopic view of bulk propertiesbulk properties

P(1,2)=P(1)P(2).

E(1,2)=E(1)+E(2.).

)](Eexp[Const)(P 11

)](Eexp[Const)(P 22

,

TkB

1

kB = 1.381·10-23 J·K-1 = 8.62·10-5 eV·K-1

Normalized Boltzmann distribution

The Boltzmann DistributionThe Boltzmann Distribution

Copyright (c) Stuart Lindsay 2008

Values for kTValues for kT

• kBT at room temperature (300K) = 4.14 10-21J

• = 25 meV (Much smaller than most bonds)

• = 0.6 Kcal·mol-1

• kBT at room temperature in terms of force· distance = 4.14 pN·nm

- molecular motors produce forces ca. ten times more over nm distances

Copyright (c) Stuart Lindsay 2008

Normalized Boltzmann distributionNormalized Boltzmann distribution

)exp(1

)(Tk

E

ZrP

B

r

r B

r

Tk

EZ exp partition functionpartition function

In case of degeneracy g(r):

)exp()(

)(Tk

E

Z

rgEP

B

rr

The partition function enumerates all the states as a function of energy, so all the equilibrium properties of a system can be derived from it.

Copyright (c) Stuart Lindsay 2008

• Entropy is proportional to the number of ways (statistical weight, ) a given macrostate r can occur.

)(ln)( rkrS B

Copyright (c) Stuart Lindsay 2008

The degeneracy of a state leads naturally to a statistical definition of EntropyEntropy:

In terms of the probability of the rth state:

r

B rprpkS )(ln)(

The Equipartition TheoremThe Equipartition Theorem

• Average thermal energy of e.g., a harmonic oscillator

• For a classical system (all energies allowed) replace the Boltzmann sum with an integral and calculate the product of E and p(E), e.g. for potential energy:

22

2

1

2

1xxmE

dx

Tk

x

dxTk

xx

dxxEPxx

B

B

)2

exp(

2exp

2

1

))((2

1

2

12

22

22

Copyright (c) Stuart Lindsay 2008

• With a change of variables and a standard integral we find

• Similarly

• The thermal average of any quantity that appears in the classical Hamiltonian as a quadratic term is

Tkx B2

1

2

1 2

Tkxm B2

1

2

1 2

TkB2

1 Equipartition Equipartition theoremtheorem

The Equipartition theorem assumes that all degrees of freedom are in equilibrium with the heat bath and are independent.

However, it takes coupling between the degrees of freedom to ‘spread’ the thermal energy out evenly and this requires a non linear response.

Ex. It takes an anharmonic potential to couple vibrational and translational degres of freedom (V-T energy transfer).

Thermodynamics and Statistical mechanicsThermodynamics and Statistical mechanics

• Thermodynamic potentials (“Free Energies”) can be minimized to obtain the equilibrium properties of a system.

Copyright (c) Stuart Lindsay 2008

0S

TSEA

TSPVEG

0A

0G

Isolated system

Closed system (V,T)

Closed system (p,T)

Thermodynamic potentials in terms of Thermodynamic potentials in terms of partition functionspartition functions

r B

r

Tk

EZ exp

r

B )r(pln)r(pkS

)Tk

Eexp(

Z)r(p

B

r1

T

EZkNVTS B ln),,(

TSEA

)N,V,T(ZlnTk)N,V,T(A B

Grand Canonical ensembleGrand Canonical ensemble

Each system is enclosed in a container whose walls are both heat conducting and permeable to the passage of molecules.→ transport of matter allowed, N variable

V,T,μ V,T,μ V,T,μ

V,T,μ V,T,μ V,T,μ

V,T,μ V,T,μ V,T,μ

N r

Nra A

aaNrNr = number of systems in the ensemble that contain N molecules

and are in the state r.The set of occupation numbers {aNr} is a distribution.

Each possible distribution must satisfy the balance equations:

Number of systems in the ensemble

N r

NrNr Ea Ε Total energy of the ensemble

N r

Nr Na N Total number of molecules

For any possible distribution, the number of states is given by:

N r NrNr !a

!})a({W

A

The distribution that maximizes W is:

N)V(ENr eee}*a{ Nr

N rN)V(E ee

eNr

AkT

1

kT

NTSPVEG

is the chemical potentialchemical potential

NrNr

Nr

ENexp}*a{P

A

Nr

NrENVTZ exp),,(

Gibbs DistributionGibbs Distribution

Grand Partition Grand Partition functionfunction

),,(ln VTTkG B

Nr

NrENexp),V,T(Z

Summing over r states, it is possible to write:

N

N

N

kT

N

)T,V,N(Qe)T,V,N(Q),V,T(Z

Canonical partition function

kTe

lnkTactivityactivity

Ideal Gas: Z for one free particle (N=1)Ideal Gas: Z for one free particle (N=1)

222222

22 zyx kkkmm

E k

L

nk

L

nk

L

nk z

zy

yx

x

2,

2,

2

zyx nnn

zyx nnnEVTZ,,

),,(exp)1,,(

2

2

3

2

28

4

dkVkdkVk

dn

For large systems:

dkTmk

kk

VVTZ

B

0

222

2 2exp

2)1,,(

VTmk

VTZ B2

3

22)1,,(

2

3

22

TmkB

Quantum concentration (one particle per

wavelength3)

The de Broglie wavelength for a free particle is:

2

12

22

Tmkk Bz,y,x

So, one particle occupies a quantum volume of about λ3:

2

32

3 2

TmkB

Ex. Quantum volume for a free electron at 300K

3326

3

2

2131

683

22

791097

1014410119

1011312 2

nmm.

..

.

TmkB

NN

rr VTZ

NE

NNVTZ )1,,(

!

1)exp(

!

1),,(

For N non-interacting particles:

A sphere of a radius of 2.7nm!A sphere of a radius of 2.7nm!

Quantum statisticsQuantum statistics

N

N

N

kT

N

)T,V,N(Qe)T,V,N(Q),V,T(Z

Expliciting Q(N,V,T) (i.e. energy distribution):

}n{

n

j

E

k

i iij ee)T,V,N(Q

Ej(N,V) = energy states available to a system containing N moleculesεk= molecular quantum statesnk= number of molecules in the kth molecular state when the system energy is Ej.

k

kkj nE k

knN

N

n

}n{

n

N }n{

nN i ii

k

i i

k

i ii ee),V,T(Z

max, max,

iii

k

i

n

n

n

n i

nn

N }n{ i

e...e1

1

2

20 0

This last passage originates from the fact that we are summing over all values of N and that nk ranges over all possible values.

i

n

n

nn

n

n

n

nn max,i

i

ii

max, max,

e......ee00 0

1

1

2

2

2211

Fermi-Dirac statistics: nFermi-Dirac statistics: nii=0 or 1, n=0 or 1, ni,maxi,max=1=1

i

FDieZ 1

iT,VT,V

FD i

i

e

eZlnZlnkTN

1

Bose-Einstein statistics: nBose-Einstein statistics: nii=0, 1, 2,… n=0, 1, 2,… nmaxmax==∞∞

ii n

n

BEi

i

ii eeZ1

0

1

Where we used:

j

j

j )x(x0

11

iT,VT,V

BE i

i

e

eZlnZlnkTN

1

i

FDieZ 1

i

BEieZ

11

iFD i

i

e

eN

1

iBE i

i

e

eN

1

k

k

kkk k

k

e

enNE

1

k

kelnkTpV 1

+ = FD+ = FD

- = BE- = BE

Classical limit (Classical limit (λ→λ→0):0):

At the classical limit (high temperatures or low density) the number of available molecular quantum states is much greater than the number of particles.

The average number of molecules in any state is very small (nk→0, λ→0).

Thermodynamically:

(T fixed)V

N 0 fixed)

V

N(T

iMBBEorFD

ieNNlim

0

Maxwell-Boltzmann distribution

ieni

Summing over i:

ii i

ien qeNi

i

q

e

N

n ii

i i

ieE

Quantum gassesQuantum gasses

.....3,2,1,0rnBosonsBosons

1,0rnFermionsFermionsNn

rr

...,

221121

21

............exp),,(nn

iii nnnnnnVTZ

Using the Grand partition function:

Z

nnnnnnnnPP iii

Nr

............exp...),( 221121

21

With the Gibbs distribution

We consider just an ideal gas (non-interacting particles) now subject to restrictions on how states are counted:

Writing numerator and denominator as products:

i

ii

i Z

nnnp

exp....),(

121

Single particle distribution

Fermi-Dirac (FD) statisticsFermi-Dirac (FD) statistics: ni=1 or 0

iiZ exp1

in i

iiiiii exp

expexp)n(pnn

1

10

1exp

1

iin

Fermi Dirac thermal Fermi Dirac thermal average occupationaverage occupation

The chemical potential at The chemical potential at T=0 is called the Fermi T=0 is called the Fermi energy.energy.

The electronic properties of most conductors are dominated by quantum statistics.

Ex. Fermi energy of Na is 3.24 eV.

For εi=μ:

For metals is several eVs!!

2

1in

eV.)K(kT 0250300 K,.

.).n(T i 80038

0250

24350

• Summing Zi from n = 0 to :

1exp

1

iin <i

Copyright (c) Stuart Lindsay 2008

Bose-Einstein (BE) statisticsBose-Einstein (BE) statistics: ni=0, 1, 2, 3….

As ε approaches μ in the BE distribution, the occupation number approaches infinity, i.e. bosons condense into one quantum state at very low temperatures (Bose condensation).

Phonons are bosons with no chemical potential (μ=0), so that the occupation number goes to zero as temperature approaches zero.

μ=0