Classical Statistical Mechanicsin the Canonical Ensemble

Classical Statistical Mechanics

1. The Equipartition Theorem

2. The Classical Ideal Gasa. Kinetic Theoryb. Maxwell-Boltzmann Distribution

The Equipartition TheoremValid in Classical Statistical Mechanics

ONLY!!!“Each degree of freedom in a system of particles contributes (½)kBT to the

thermal average energy of the system.”Note: 1. This is valid only if each term in the

classical energy is proportional either a momentum squared or a coordinate squared.

2. The possible degrees of freedom are those associated with translation, rotation &vibration of

the system’s molecules.

Classical Ideal Gas

kTvmKE2

3

2

1 2

For this system, it’s easy to show that The Temperature is related to the average kinetic energy. For one molecule moving with velocity v, in 3 dimensions this takes the form:

kTvmvmvm zyx 2

1

2

1

2

1

2

1 222

Further, for each degree of freedom, it can be shown that

The Boltzmann Distribution:Define

The Energy Distribution Function(Number Density) nV(E):

•This is defined so that nV(E) dE the number ofmolecules per unit volume with energy between E and E + dE.

The Canonical Probability Function P(E):•This is defined so that P(E) dE the probability to find aparticular molecule between E and E + dE

kT

eP

kTE

E

)(

kTV enn EE 0)(

Z

Equipartition

22mvKE EkT

evP

kTmv 22

)(

Simple Harmonic Oscillator

kTvmKE x 2

1

2

1 2

kTPE2

1

kTPEKE EE

Free ParticleZ

Thermal Averaged Values

0

0

)(

)(

EE

EEE

EE

dP

dP

Average Energy:

Average Velocity:

0

0

)(

)(

dvvP

dvvvP

vv

1)()(00

dvvPdP EEOf course:

Kinetic Theory of Gases &

The Equipartition Theorem

Classical Kinetic Theory Results• The kinetic energy of individual particles is

related to the gas temperature as:(½)mv2 = (3/2) kBT

Here, v is the thermal average velocity.

• There is a wide range of energies (& speeds) that varies with temperature:

Boltzmann Distribution of Energy

The Kinetic Molecular Model for Ideal GasesThe Kinetic Molecular Model for Ideal Gases

• The gas consists of large number of small individual particles with negligible size.

• Particles are in constant random motion & collisions.• No forces are exerted between molecules.• From the Equipartition Theorem,

The Gas Kinetic Energy is Proportional to the Temperature in Kelvin.

TRKE 2

3

Maxwell-Boltzmann Velocity DistributionMaxwell-Boltzmann Velocity Distribution

•The Canonical Ensemble gives a distribution of

molecules in terms of Speed/Velocity, & Energy.

•The One-Dimensional Velocity Distribution in

the x-direction (ux) has the form:

x

TkumdueA

N

dN x

/2

1 2

x

TkumdueA

N

dN x

/2

1 2

Low T

High T

Maxwell-Boltzmann DistributionMaxwell-Boltzmann Distribution3D Velocity Distribution: a (½)[m/(kBT)]

xau

uD

duea

N

dNx

x

2

1

In Cartesian Coordinates:

zyxuuua

D

dududuea

N

dN zyx

][

2/3

3

222

yau

uD

duea

N

dN y

y

2

1 zau

uD

duea

N

dNz

z

2

1

•Change to spherical coordinates: Reshape thebox into a sphere in velocity space of the samevolume with radius u .

V = (4/3) u3 with u2 = ux2 + uy

2 + uz2

dV = dux duy duz = 4 u2 du

222/3

3

4/ ua

D

euadu

NdN

Maxwell-Boltzmann Speed DistributionMaxwell-Boltzmann Speed Distribution

3D Maxwell-Boltzmann Speed Distribution3D Maxwell-Boltzmann Speed Distribution

Low T

High T

3D Maxwell-Boltzmann Speed Distribution3D Maxwell-Boltzmann Speed Distribution

Convert the velocity-distribution into an energy-distribution:

= (½)mu2, d = mu du

222/3

3

4/ ua

uD

euadu

NdN

kT

D

ekTd

NdN

2/1

2/3

3

12/

Velocity Values from the M-B DistributionVelocity Values from the M-B Distribution

• urms = root mean square velocity

• uavg = average speed

• ump = most probable velocity

x

naverage

n

N

dNxx )(

Comparison of Velocity ValuesComparison of Velocity Values

Ratio in Terms of :

urms uavg ump

1.73 1.60 1.41

m

kT3

m

kT

8

m

kT2

-6 -4 -2 0 2 4 60.00

0.25

0.50

0.75

1.00

T3

/2ex

p(v

2 /T)

v

T = 1 T = 2 T = 4 T = 8 T = 16

Maxwell-Boltzmann Maxwell-Boltzmann VelocityVelocity Distribution Distribution

-6 -4 -2 0 2 4 60.00

0.25

0.50

0.75

1.00

exp(v

2 /T)

v

T = 1 T = 2 T = 4 T = 8 T = 16

3/ 2 2

3/ 2 2

exp / 2

exp / 2i i

f v T mv kT

f v T mv kT

Maxwell-BoltzmannMaxwell-Boltzmann SpeedSpeed DistributionDistribution

3/ 2

2 24 exp / 22

mf v v mv kT

kT

3/ 2

2 24 exp / 2 ; ( ) ( )2

mf v v mv kT N v N f v

kT

1/ 2 1/ 2 1/ 22 8 3

;m rms

kT kT kTv v v

m m m

Maxwell-BoltzmannMaxwell-Boltzmann SpeedSpeed DistributionDistribution

The Probability Density Function The Probability Density Function •The random motions of the molecules can becharacterized by a probability distribution function. •Since the velocity directions are uniformly distributed, wecan reduce the problem to a speed distribution functionwhich is isotropic. •Let f(v)dv be the fractional number of molecules in thespeed range from v to v + dv.•A probability distribution function has to satisfy the

condition

0

1f v dv

0

v vf v dv

2 2

0v v f v dv

2

rmsv v

•We can then use the distributionfunction to compute the averagebehavior of the molecules:

Some Other Examples of theSome Other Examples of theEquipartion TheoremEquipartion Theorem

2 2 1 12 2

2 2 1 12 2

2 2 2 32

2 2 1 1, 2 2

1 1

2 21 1

2 21

21

2average, or r.m.s. value

LC B B

HO B B

trans x y z B

rot dia x y B B

E C V L i k T k T

E k x m v k T k T

E m v v v k T

E I k T k T

LC Circuit:

Harmonic Oscillator:

Free Particle in 3 D:

Rotating Rigid Body :