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Ch2. Elements of Ensemble Theory

Ensemble: An ensemble is a large collection of systems in different microstates for the same macrostate (N,V,E) of the given system.1) An ensemble element has the same

macrostate as the original system (N,V,E), but is in one of all possible microstates.

2) A statistical system is in a given macrostate (N,V,E), at any time t, is equally likely to be in any one of a distinct microstate.

Ensemble theory: the ensemble-averaged behavior of a given system is identical with the time-averaged behavior.

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2.1 Phase space of a classical system

Consider a classical system consisting of N-particles, each described by (xi,vi) at time t.

A microstate at time t is (x1,x2,…,xN; v1,v2,…,vN), or

(q1,q2,…,q3N; p1,p2,…,p3N), or

(qi, pi) - position and momentum, i=1,2,…..,3N

Phase space: 6N-dimension space of (qi, pi). pi

qi

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Representative point

Representative point: a microstate (qi,pi) of the given system is represented as a point in phase space.

An ensemble is a very large collection of points in phase space . The probability that the microstate is found in region A is the ratio of the number of ensemble points in A to the total number of points in the ensemble .

pi

qi

AP(A)=

Number of points in A

Number of points in

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Hamilton’s equations The system undergoes a continuous change

in phase space as time passes by

i

iii

i

iii

q

pqHp

p

pqHq

,

,

Trajectory evolution and velocity vector v Hamiltonian

i=1,2,…..,3N

N

jiji

N

ii

N

i

iii rrrU

m

ppqH

1

3

1

2

2,

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Hypersurfce

Hypersurface is the trajectory region of phase space if the total energy of the system is E, or (E-/2, E+/2).

H(qi,pi) = E

• e.g. One dimensional harmonic oscillator

Hypershell (E-/2, E+/2). qi

piH(qi,pi) = E E+/2

E-/2

H(qi,pi) = (½)kq2 + (1/2m)p2 =E

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Ensemble average For a given physical quantity f(q, p), which

may be different for systems in different microstates,

where

d3Nq d3Np – volume element in phase space

(q,p;t) – density function of microstates

pdqdtpq

pdqdtpqpqff

NN

NN

33

33

;,

;,,

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Microstate probability density The number of representative points in the

volume element (d3Nq d3Np) around point (q,p) is given by

(q,p;t) d3Nqd3Np

0t

Microstate probability density: C(q,p;t)

Stationary ensemble system: (q,p) does not explicitly depend on time t. <f> will be independent of time.

pdqdtpqC NN 33;,

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2.2 Liouville’s theorem and its consequences

The equation of continuity

N

ii

ii

i

ii

pp

qq

v

pqvwhere

vt

3

1

),(

0

At any point in phase space, the density function (qi,pi;t) satisfies

03

1

N

ii

ii

i

pp

qqt

So,

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Liouville’s theorem

03

1

N

ii

ii

i

pp

qqtdt

d

0,

3

1

Htdt

d

q

H

pp

H

qtdt

d N

i iiii

From above

Use Hamilton’s equations

N

i iiii q

H

pp

H

qH

3

1

,where

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Consequences

0,03

1

N

i iiii q

H

pp

H

qH

t

pdqdd

dpqff

NN 33

,1

For thermal equilibrium

• One solution of stationary ensemble

constpq ,

where

(Uniform distribution over all possible microstates)

Volume element on phase space

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Consequences-cont.

0,03

1

N

i iiii q

H

pp

H

qH

t

kTpqHpq /,exp,

pqHpq ,,

• Another solution of stationary ensemble

A natural choice in Canonical ensemble is

0,3

1

N

i iiii q

H

p

H

Hp

H

q

H

HH

satisfying

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2.3 The microcanonical ensemble

Microcanonical ensemble is a collection of systems for which the density function r is, at all time, given by

0

,

constpq If E-/2 H(q,p)

E+/2otherwise

qi

piH(qi,pi) = E E+/2

E-/2

In phase space, the representative points of the microcanonical ensemble have a choice to lie anywhere within a “hypershell” defined by the condition

E-/2 H(q,p) E+/2

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Microcanonical ensemble and thermodynamics

– the number of microstates accessible; – allowed region in phase space;0 – fundamental volume equivalent to one microstate

0

0

lnln

/,,,

kkS

EVN

qi

piH(qi,pi) = E E+/2

E-/2

Microcanonical ensemble describes isolated sysstems of known energy. The system does not exchange energy with any external system so that (N,V,E) are fixed.

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Example – one particle in 3-D motion

amilton: H(q,p) = (px2+py

2+pz2)/(2m)

32

0

2

/24/

24

hmEV

mEVdpdpdpdxdydz zyx

Microcanonical ensemble

0

,

constpq If E-/2 H(q,p) E+/2

otherwise

py

pzH(qi,pi) = E

E+/2E-/2

px

Fundamental volume, 0~h3

Accessible volume

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2.4 Examples1. Classical ideal gas of N particles

a) particles are confined in physical volume V; b) total energy of the system lies between E-/2 and E+/2.

2. Single particle a) particles are confined in physical volume V; b) total energy of the system lies between E-/2 and E+/2.

3. One-dimensional harmonic oscillator

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2.4 Examples1. Classical ideal gas of N particles

Particles are confined in physical volume V The total energy of system lies between (E-/2, E+/2)

N

i

iN

i

iE

m

pE

NN

Em

pE

Vq

NN pdVpdqd3

1

2

3

1

2 2/2

2/

3

2/2

2/

33

N

i

iii m

ppqH

3

1

2

2,Hamiltonian

V

Volume of phase space accessible to representative points of microstates

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Examples-ideal gas

mERRN

ydRV NN

Ry

NN

N

ii

2,!2/

2/

0

3

3

1

22

!12/3

2

2

22

!2/3

3

2/3

2

132/3

)2/(2)2/(2

3

3

1

2

N

mEV

E

E

mmE

N

NVydV

NN

NNN

EmyEm

NN

N

ii

where

2

2,

!2/1

2/

E

mdRdRR

N

NRdV N

N

N

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Examples-ideal gas

The fundamental volume: Nh30

• The multiplicity (microstate number)

* A representative point (q,p) in phase space has a volume of uncertainty, for N particle, we have 3N (qi,pi) so,

~Nh30

!12/3

2 2/3

30

N

mE

h

V

E

NN

NN

mE

h

VN

2

3

3

4lnln

2/3

3

and

NkN

mE

h

VNk

kEVNS

2

3

3

4ln

ln,,2/3

3

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Example-single free particle2. Classical ideal gas of 1 particles

Particle confined in physical volume V The total energy lies between (E-/2, E+/2)

2321321 3

4)()(

223

22

21

PVdpdpdpdqdqdq

PpppVq

2222

2

1

2, zyx ppp

mm

ppqH Hamiltonian

Volume of phase space with p< P=sqrt(2mE) for a given energy E

V

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Examples-single particle

dm

h

Vd

d

dda 2/12/3

30

22)(

• The number of microstates with momentum lying btw p and p+dp,

dpph

Vdp

dp

ddppg 2

30

4)(

• The number of microstates of a free particle with energy lying btw and +d,

where 30

2

~,2

hm

p

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Example-One-dimensional simple harmonic oscillator

3. Harmonic oscillator

22

2

1,/

sin

cos

AmEmkwith

tAmqmp

tAq

22

2

1

2

1, p

mkqpqH Hamiltonian

Solution for space coordinate and momentum coordinate

Where k – spring constant m – mass of oscillating particle

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Example-One-dimensional simple harmonic oscillator

The phase space trajectory of representative point (q,p) is determined by

12/2

2

1

2

1,

2

2

2

22

mE

p

mE

qor

Epm

kqpqH

With restriction of E to

p

q

E-/2 H(q,p) E+/2

• The “volume” of accessible in phase space

2

2/2/2

)()(2/),(2/

v

EpqHE

v EEdpdq

E

mEm

EpqAv

2

22

200

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Example-One-dimensional simple harmonic oscillator

If the area of one microstate is 0~h

0

vE

The number of microstates (eigenstates) for a harmonic oscillator with energy btw E-/2 and E+/2 is given by

lnln kEkS

So, entropy

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Problem 3.5 For a collection of N 3-D quantum harmonic oscillators of frequency w and total energy E, compute the entropy S and temperature T.