ch2. elements of ensemble theory
DESCRIPTION
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. An ensemble element has the same macrostate as the original system (N,V,E), but is in one of all possible microstates. - PowerPoint PPT PresentationTRANSCRIPT
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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
3
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
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)
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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/
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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.