Download Phase Space Phase Space

Phase Space
Phase Space pN
t2
2
t1
1
Phase space   p1 , p 2 , p 3 ,..., p N , r1 , r1 , r3 ,..., rN 
rN
Phase Orbit
Iff the
h Hamiltonian
il i off the
h system is
i denoted
d
d bby H(q,p),
( ) the
h motion
i off phase
h
point can be along the phase orbit and is determined by the canonical
equation of motion
H
qi 
 pi
 H
pi  
 qi

P

(i=11,2....s)
(i
2 s)
(1 1)
(1.1)
H ( q, p )  E
Phase Orbit
Constant energy surface
(1.2)
Therefore the phase orbit must lie
on a surface of constant energy
(ergodic
ergodic surface
surface).
H(q,p)=E
2
 - space and 
-space
space
Let us define  - space as phase space of one particle (atom or molecule).
molecule) The
macrosystem phase space (-space)
space is equal to the sum of  - spaces
spaces.
The set of possible microstates can be presented by continues set of phase
points. Every point can move by itself along it’s own phase orbit. The overall
picture of this movement possesses certain interesting features, which are best
appreciated in terms of what we call a density function (q,p;t).
(q p;t)
This function is defined in such a way that at any time t, the number of
representative
p
points
p
in the ’volume element’ ((d3Nq d3Np) around the p
point
(q,p) of the phase space is given by the product (q,p;t) d3Nq d3Np.
Clearly, the density function (q,p;t) symbolizes the manner in which the
members of the ensemble are distributed over various possible microstates at
various instants of time.
3
Function of Statistical Distribution
Let us suppose that the probability of system detection in the volume
ddpdqdp1.... dps dq1..... dqs near point (p,q) equal dw (p,q)= (q,p)d.
The function of statistical distribution  (density function) of the system
over microstates in the case of nonequilibrium systems is also depends on
time. The statistical average of a given dynamical physical quantity f(p,q) is
equal
q 
 f 

f ( p, q )  (q, p; t )d 3 N qd 3 N p
3N
3N

(
q
,
p
;
t
)
d
qd
p

(1.3)
The right
g ’’phase
p
portrait’’
p
off the system
y
can be described byy the set off points
p
distributed in phase space with the density . This number can be considered
as the description of great (number of points) number of systems each of
which has the same structure as the system under observation copies of such
system at particular time, which are by themselves existing in admissible
4
microstates
Statistical Ensemble
The number
Th
b
off macroscopically
i ll identical
id ti l systems
t
di t ib t d along
distributed
l
admissible microstates with density  defined as statistical ensemble.
ensemble A
statistical ensembles are defined and named by the distribution function
which characterizes it. The statistical average value have the same
meaning as the ensemble average value.
An ensemble
A
bl is
i said
id to be
b stationary
i
if  does
d
not depend
d
d explicitly
li i l
on time, i.e. at all times

 0
t
(1 4)
(1.4)
Clearly, for such an ensemble the average value <f> of any physical quantity
Clearly
f(p,q) will be independent of time
time. Naturally, then, a stationary ensemble
qualifies to represent a system in equilibrium. To determine the circumstances
under
d which
hi h Eq.
E (1.4)
(1 4) can hold,
h ld we have
h
to make
k a rather
h study
d off the
h
movement of the representative points in the phase space.
5
Lioville’s theorem and its consequences
q
Consider
C
id an arbitrary
bit
" l
"volume"
"  in
i the
th relevant
l
t region
i off the
th phase
h
space
and let the "surface” enclosing this volume increases with time is given by

t
  d 
(1.5)
where
h
d(d3Nq d3Np)
p).
). On
O the
th other
th hand,
h d the
th nett rate
t att which
hi h the
th
representative points ‘’flow’’ out of the volume  (across the bounding
surface ) is given by

 ρ(ν  n )dσ
(1.6)
σ
here v is the vector of the representative points in the region of the surface
n̂
element d, while
is the (outward)
unit vector normal to this element. By
the
h divergence
di
theorem,
h
(1 6) can be
(1.6)
b written
i
as
6
Statistics of Multiparticle Systems in Thermodynamic Equilibrium
Statistics of Multiparticle Systems in Thermodynamic Equilibrium
The macroscopic thermodynamic parameters, X = (V,P,T,…), are
macroscopically observable quantities that are, in principle, functions of the
canonical variables, i.e.
fi  fi ( p1 , p2 ,..., ps , q1 , q2 ,..., qs ), i  1, 2,..., n, n
s
( f1 , f 2 ,..., f n )  (V , P, T ,...))  X

X  X ( p1 , p2 ,..., ps , q1 , q2 ,..., qs )
However, the specification of all the macroparameters X does not determine a
unique microstate,
pi  pi ( X ), qi  qi ( X )
• Consequently, on the basis of macroscopic measurements, one can make
only statistical statements about the values of the microscopic variables.
Statistical Description of Mechanical Systems
Statistical Description of Mechanical Systems
Statistical description of mechanical systems is utilized for multi-particle
problems, where individual solutions for all the constitutive atoms are not
affordable,, or necessary.
y Statistical description
p
can be used to reproduce
p
averaged macroscopic parameters and properties of the system.
Comparison of objecti
objectives
es of the deterministic and statistical approaches:
Deterministic particle dynamics Statistical mechanics
Provides the phase vector, as
a function of time Q(t),
Q(t) based
on the vector of initial
conditions Q(0)
Provides the time-dependent
probability density to observe
the phase vector Q, w(Q,t),
based on the initial value
w(Q,0)
(Q 0)
Statistical Description of Mechanical Systems
From the contemporary point of view, statistical mechanics can be regarded as a
hierarchical multiscale method, which eliminates the atomistic degrees of
freedom, while establishing a deterministic mapping from the atomic to
macroscale variables, and a probabilistic mapping from the macroscale to the
atomic variables:
Microstates
Macrostates
Xk
(p,q)
k
deterministic
conformity
probabilistic
conformityy
Distribution Function
Though the specification of a macrostate Xi cannot determine the microstate
(p,q)i = (p1,p2,…,ps; q1,q2,…,qs)i, a probability density w of all the microstates
can be found,
w  p1 , p2 ,..., ps ; q1 , q2 ,..., qs ; t 
or abbreviated:
w( p, q, t )
The probability of finding the system in a given phase volume G:
W (G, t )   w( p, q, t )dpdq
G
The normalization condition:

( p ,q )
w( p, q, t )dpdq  1
Statistical Ensemble
Within the statistical description, the motion of one single system with given
initial conditions is not considered; thus, p(t), q(t) are not sought.
Instead, the motion of a whole set of phase points, representing the collection of
possible states of the given system.
Such a set of phase points is called a phase space ensemble.
If each point in the phase space is considered as a random quantity with
a particular probability ascribed to every possible state (i.e. a probability density
w(p,q,t) is introduced in the phase space), the relevant phase space ensemble is
called a statistical ensemble.
p
t = t2: G2
G – volume in the phase
space, occupied by the
statistical ensemble.
t = t1: G1
q
Statistical Averaging
Statistical average (expectation) of an arbitrary physical quantity F(p,q),
F(p q) is given
most generally by the ensemble average,
F (t ) 

F ( p, q) w( p, q, t )dpdq
( p ,q )
The root-mean-square fluctuation (standard deviation):
( F ) 
F F
2
The curve representing the real motion (the experimental curve) will mostly
proceed within the band of width 2Δ(F)
F
True value
F  (F )
F
t
For some standard equilibrium systems, thermodynamic parameters can be
g pphase space
p
integral.
g
This approach
pp
is discussed below.
obtained, usingg a single
Ergodic Hypothesis and the Time Average
Evaluation of the ensemble average (previous slide) requires the knowledge of the
distribution function w for a system of interest.
Alternatively, the statistical average can be obtained by utilizing the ergodic
hypothesis in the form,
form
F F
Here, the right-hand
right hand side is the time average
(in practice, time t is chosen finite, though as large as possible)
1 t
F   F  p( ), q( )  d , t  
t 0
This approach requires F as a function of the generalized coordinates.
Some examples
Internal energy:
U  H ( p, q )
Temperature:
T
2 Ek
kB
T2 C
dU
Change of entropy: S  
  V dT
U1 T
T1 T
vl
G diff
Gas
diffusion
i constant:
t t D
3
U2
Here,
Ek  mean kinetic energy per degree of freedom
v  mean velocity of molecules
l  mean free path of gas particles
Law of Motion of a Statistical Ensemble
Law of Motion of a Statistical Ensemble
A statistical ensemble is described by the probability density in phase space,
w(p,q,t). It is important to know how to find w(p,q,t) at an arbitrary time t, when the
initial function w(p,q,0)
w(p q 0) at the time t = 0 is given
given.
In other words, the equation of motion satisfied by the function w(p,q,t) is needed.
p
w(p,q,0)
w(p,q,t1)
w(p q t2)
w(p,q,t
Γ1
Γ0
Γ2
q
The motion of of an ensemble in phase space may be considered as the motion of a
phase space fluid in analogy to the motion of an ordinary fluid in a 3D space.
space
Liouville’s theorem claims that
0  1   2  ...
Due to Liouville’s theorem, the following equation of motion holds
2s
 H w H w 
w

 [ H , w],
[ H , w]   
 (Poisson bracket)
t

q

p

p

q
i 1 
i
i
i
i 
Equilibrium Statistical Ensemble: Ergodic Hypothesis
For a system in a state of thermodynamic equilibrium the probability density in
phase space must not depend explicitly on time,
w
0
t
Thus, the equation of motion for an equilibrium statistical ensemble reads
[ H , w]  0
A direct solution of this equation is not tractable.
Therefore the ergodic hypothesis
Therefore,
h pothesis (in a more general form) is utilized:
tili ed: the
probability density in phase space at equilibrium depends only on the total
energy:
w( p , q )    H ( p , q , a ) 
Notes: the Hamiltonian gives the total energy required; the Hamiltonian may depend
on the values of external parameters a = (a1, a2,…),
) besides the phase vector X.
X
This distribution function satisfies the equilibrium equation of motion, because
E
Exercise:
i Check
Ch k the
h above
b
equality.
li
[ H ,  ( H )]  0


Ensemble Method
Ensemble ? : Infinite number of mental replica of the system of interest
the system of interest
Large Reservoir (const.T) All the ensemble members have the
All
th
bl
b h
th
Same N,V.T
EEnergies can be exchanged i
b
h
d
but molecules cannot.
Current N = 20 but N  infinity Two postulates
Two postulates •
L
Long time average = Ensemble average at N 
ti
E
bl
t N  infinity i fi it
time
E1
•
E2
E3
E4
E5
In an ensemble , the systems of enembles are distributed uniformly (equal probability or frequency)
probability or frequency) –  Ergodic Hypothesis –  Principle of equal a priori probability Averaging Method
Averaging Method
• Probability of observing particular quantum state i Pi 
ni
 ni
i
• Ensemble average of a dynamic property
 E   Ei Pi
i
• Time average and ensemble average
U  lim  Ei ti  lim  Ei Pi
 
n 
i
Calculation of Probability in Ensemble
Calculation of Probability in Ensemble
• Several methods are available – Method of Undetermined multiplier p
–:
–:
Maximization of Weight ‐ Most probable distribution • Weight N!
N!
W

n1!n2 !n3!...  ni !
i
Ensembles
•
•
•
•
Micro‐canonical ensemble: E,V,N
Canonical ensemble: T,V,N
Canonical ensemble: T,V,N
Constant pressure ensemble: T,P,N
Grand‐canonical ensemble: T,V,μ
21
Canonical Ensemble:
The canonical ensemble occurs when a system with fixed V and N
and the system is at constant temperature (connected to an infinite
heat bath).
In the canonical ensemble, the probability of each microstate is
proportional to exp (- βEm).
βEm)
E1
E2  E  E1
Systems 1 and 2 are weakly coupled
s ch that they
such
the can exchange
e change energ
energy.
What will be E1?
  E1 , E  E1   1  E1    2  E  E1 
BA: each configuration is equally probable; but the number of
states that give an energy E1 is not know
know.
23
  E1 , E  E1   1  E1    2  E  E1 
ln   E1 , E  E1   ln 1  E1   ln  2  E  E1 
  ln
l   E1 , E  E1  
0


E1

 N1 ,V1
 

Energy is conserved!
dE1=-dE2

  ln 1 E1 
  ln 2 E  E1 




E1
E1

 N ,V 
N
1
1
0
2 ,V2
  ln 1  E1  
  ln  2  E  E1  




E1
E2

 N1 ,V1 
 N2 ,V2
This can be seen as
an equilibrium
condition
  ln   E  
 

E

 N ,V
1   2
24
Entropy and number of configurations
fi
ti
Conjecture:
S  ln 
25
S  k B ln   E 
With kB= 1.380662 10-23 J/K
In thermodynamics, the absolute (Kelvin)
temperature scale was defined such that
1
 S 

 
 E  N ,V T
n
dE  TdS-pdV
p   i dN i
i 1
But we define:
  ln   E  
 

E

 N ,V
26
And this gives the “statistical”
statistical definition of temperature:
  ln   E  
1
 kB 

T
E

 N ,V
In short:
Entropy and temperature are both related
to the fact that we can COUNT states.
Basic assumption:
1 leads to an equilibrium condition: equal temperatures
1.
2. leads to a maximum of entropy
y
3. leads to the third law of thermodynamics
27
Number of configurations
g
How large is ?
•For macroscopic systems
systems, super
super-astronomically
astronomically large
large.
•For instance, for a glass of water at room temperature:
  10
210
25
•Macroscopic
p deviations from the second law of thermodynamics
y
are not
forbidden, but they are extremely unlikely.
28
Canonical ensemble
1/kBT
Consider a small system that can exchange heat with a big rese
Ei
 ln 
ln   E  Ei   ln   E  
Ei 

E
E  Ei
l
ln
  E  Ei 
E
Hence, the probability to find Ei:
P  Ei  
  E  Ei 

exp   Ei k BT 
 E  E  
j
j
Ei

k BT
j
exp   E j k BT 
P  Ei   exp   Ei k BT 
29
Boltzmann distribution
Averaging Method
Averaging Method
• Probability of observing particular quantum state i Pi 
ni
 ni
i
• Ensemble average of a dynamic property
 E   Ei Pi
i
• Time average and ensemble average
U  lim  Ei ti  lim  Ei Pi
 
n 
i
Thermodynamics
y
What
at iss the
t e average
a e age e
energy
e gy o
of tthe
e syste
system?
E
E exp    E 

  E PE  
 exp   E 
 ln  exp
e p   E 

i
i
i
i
i
i
j
j
i

Compare:
Co
pa e
F T

 1 T

E

i

 ln QN ,V ,T

Hence:
F
  ln QN ,V ,T
k BT
31

Canonical Partition Function
Q  e
j
 E j
The Boltzmann Distribution


Task : Find the dominating configuration for given N and total energy E.
i
N d
l
E
 Find Max. W which satisfies ; N   ni
 dn
Et   Ei ni
d
 E dn
i
i
i
0
i
i
i
i
0
Method of Undetermined
Method of Undetermined Multipliers
p

Maximum weight , W 
Recall the method to find min, max of a function
Recall the method to find min, max of a function…
…
d ln W  0
  ln W 

  0
 dni 

Method of undetermined multiplier : 
Constraints should be multiplied by a constant and added to the main variation equation. Method of undetermined multipliers
  ln W 
dni    dni    Ei dni
d ln W   
i  dni 
i
i
  ln W 

    Ei dni  0
  
i  dni 

  ln W 

    Ei  0
 dni 
ln W  N ln N   ni ln ni
 (n j ln n j )
  ln W  N ln N

 

ni
ni
j
 ni 
N ln N  N 
1  N 
 ln N  N  
  ln N  1
 
ni
N  ni 
 ni 

j
 (n j ln n j )
ni
 n j 
1


  
ln n j  n j 

nj
j 
 ni 
ni
 ln W
 (ln ni  1)  (ln N  1)   ln
ni
N
 n j 

  ln ni  1
 ni 
 ln
ni
   Ei  0
N
ni
 e  Ei
N
N   n j  Ne  e
j

e 
 E j
j
1
e
 E j
j
Boltzmann Distribution
ni
e  Ei
Pi  
N  e  E j
j
(Probability function for energy distribution)
Canonical Partition Function

Boltzmann Distribution
ni
e  Ei
e  Ei
Pi  

N  e  E j
Q
j

Canonical Partition Function
Q  e
j
 E j
Canonical Distribution: Preliminary Issues
Canonical Distribution: Preliminary Issues
One important preliminary issue related to the use of Gibbs’ canonical distribution is
the additivity of the Hamiltonian of a mechanical system.
Structure of the Hamiltonian of an atomic system:
H  E kin (p1 , p 2 ,..., p N )  U (r1 , r2 ,..., rN )
p i2

  W1 (ri )   W2 (ri , r j )   W3 (ri , r j , rk )  ...
i 2m
i
i , j i
i , j i , k  j
Here, ki
H
kinetic
ti energy andd the
th one-body
b d potential
t ti l are additive,
dditi i.e.
i they
th can be
b
expanded into the components, each corresponding to one particle in the system:
E kin (p1 , p 2 ,..., p N )  E kin (p1 )  E kin (p 2 )  ...
 W (r )  W (r )  W (r )  ...
1
i
1
1
1
2
i
Two body and higher order potentials are non-additive (function Q2 does not exist)
Two-body
exist),
W2 ( rij )  W | ri  r j | 
 W (r , r )  ...  Q (r )  ...  Q (r )  ...,
i , j i
2
i
j
2
i
2
j
Canonical Distribution: Preliminary Issues
Thus if the inter
Thus,
inter-particle
particle interaction is negligible
negligible,
W2  W3  ...  E kin  W1
the system is described by an additive Hamiltonian,
H   hi ,
i
p i2
hi 
 W1 (ri )
2mi
Here H is the total Hamiltonian,
Here,
Hamiltonian and hi is the one
one-particle
particle Hamiltonian.
Hamiltonian
• For the statistical description, it is sufficient that this requirement holds for the averaged
quantities only.
• The multi-body components, W>1, cannot be completely excluded from the physical
consideration, as they are responsible for heat transfer and establishing the thermodynamic
equilibrium between constitutive parts of the total system.
• A micromodel with small averaged contributions to the total energy due to particle-particle
interactions is called the ideal gas.
Example: particles
i l in
i a circular
i l cavity.
i Statistically
S i i ll averagedd value
l W2 is
i small:
ll
3 particles:
W2
0 049 E tot
0.049
5 particles:
W2
0 026 E tot
0.026
Canonical Distribution
Suppose that system under investigation Σ1 is in
thermal contact and thermal equilibrium with a
much larger system Σ2 that serve as the
thermostat, or “heat
heat bath
bath” at the temperature T.
From the microscopic point of view, both Σ1 and
Σ2 are mechanical systems whose states are
described byy the pphase vectors ((sets of canonical
variables X1 and X2). The entire system Σ1+Σ2 is
adiabatically isolated, and therefore the
microcanonical distribution is applicable to Σ1+Σ2,
w( p1 , q1 ; p2 , q2 ) 
Thermostat T
N1
Σ1
1
  E  H ( p1 , q1 ; p2 , q2 ) 
( E )
Assume N1 and N2 are number of particles in Σ1
and Σ2 respectively. Provided that N1 << N2, the
Gibbs’ canonical distribution applies to Σ1:
1 
w( p, q )  e
Z
H ( p ,q )
kT
N2
Σ2
( p, q )  ( p1 , q1 )
Canonical Distribution: Partition Function
Canonical Distribution: Partition Function
Thermostat T
The normalization factor Z for the canonical
distribution called the integral over states or
partition function is computed as

1
Z
e
3N

(2 ) N ! ( p ,q )
H ( p ,q ,a )
kT
N1
Σ1
dpdq
N2
Σ2
H ( p ,r )

1

e kT dp1...dp N dr1...drN
3N

(2 ) N ! (p ,r )
Before
B
f
th
the normalization,
li ti
thi iintegral
this
t
l represents
t th
the statistically
t ti ti ll averagedd phase
h
volume occupied by the canonical ensemble.
The total energy for the canonical ensemble is not fixed,
fixed and,
and in principle,
principle it may
occur arbitrary in the range from – to (for the infinitely large thermostat, N2
).
Partition Function and Thermodynamic Properties
Partition Function and Thermodynamic Properties
The partition function Z is the major computational characteristic of the
canonical ensemble. The knowledge of Z allows computing thermodynamic
pparameters of the closed isothermal system
y
(a
(
V,, external pparameter):
)
Free energy:
(relates to mechanical work)
Entropy
(variety of microstates)
Pressure
Internal energy
F (T ,V )  kT ln Z

 F 


S      k  ln Z  T ln Z 
T
 T V



 F 
P      kT
ln Z
V
 V T

U  F  TS  kT
ln Z
T
2
These are the major results in terms of practical calculations over canonical ensembles.
Class exercise: check the last three above formulas with the the method of thermodynamic potentials,
using
i the
h first
fi formula
f
l for
f the
h free
f energy.
Thermodynamic Properties and Canonical Ensemble
Internal Energy
1
U  E   Ei Pi   Ei e  Ei
Q i ( qs )
i
 Q 

    Ei e  Ei
i ( qs )
   N ,V
  ln Q 
1  Q 
  

U   
Q    N ,V
   N ,V
Thermodynamic Properties and Canonical Ensemble
Pressure at i state
Small Adiabatic expansion of system
dx
(wi ) N  Pi dV  Fi dx
( dE i ) N   Fi dx   Pi dV  wi
 E i 
Pi  

 V  N
dV
dEi
Fi
V
Ei
Thermo recall (2)
First law of thermodynamics
dE  TdS  pdV
Helmholtz Free energy:
F  E  TS
dF   SdT  pdV
F T

 1 T

1 F
F

F


F

T

T
T 1 T

 F  TS  E
46
We have assume quantum mechanics (discrete states) but
we are interested in the classical limit
2



p
1

N
N
N 
i
 i exp   Ei   h3 N !  dp dr exp   i 2m  U r  
 
i
 
1
phase space
p
(p
(particle in a box))
 Volume of p
3
h
1
Particles are indistinguishable
g

N!
Integration over the momenta can be carried out for most system
 
2
2







p
p


N
i
 dp exp   i 2mi      dp exp   2m 


 
3N
 2 m 





3
N
2
47
Define de Broglie wave length:
h 


 2 m 
2
1
2
Partition function:
Q  N ,V , T  
1
 
N
N



d
r
exp

U
r


3 N N ! 
48
Example: ideal gas
1
Q  N ,V , T  
dr exp   U  r  

 N!
N
N
3N
N
V
 3 N  dr N 1  3 N
 N!
 N!
1
Free energy:
 VN 
 F   ln  3 N 
  N !
N
 N ln
l   N ln
l    N ln
l  3  N ln
l 
V 
3
Pressur
e:
N
 F 
P  
 
 V T V
Energy:
  F  3 N  3
E 
 Nk BT

     2
49
Ideal gas (2)
g ( )
Chemical potential:
 F 
i = 
 N i  T ,V , N j
 N
 F  N ln   N ln  
V 
3
  ln  3  ln   1
 IG  

 0  ln
l 
50
Ensembles
•
•
•
•
Micro‐canonical ensemble: E,V,N
Canonical ensemble: T,V,N
Canonical ensemble: T,V,N
Constant pressure ensemble: T,P,N
Grand‐canonical ensemble: T,V,μ
51
Summary:
Canonical ensemble (N,V,T)
l
bl (
)
Partition function:
Q  N ,V , T  
1
 
N
N


d
r
exp


U
r


3 N N ! 
Probability to find a particular configuration
P     exp   U    
F
Free
energy
 F   ln QN ,V ,T
52
Thermodynamic Properties and Canonical Ensemble
Pressure
Pressure at Pressure
at
quantum state i
Probability
P  P   Pi Pi
i
P
1
1
 E i   E i
 E i

P
e

 e


i
Q i
Q i  V  N

 E i    E i
  ln Q 


 e



 V   , N Q i  V  N
P
1   ln Q 


  ln V 
Equation of State in Statistical Mechanics
Thermodynamic Properties and Canonical Ensemble
Entropy
dU   q rev   wrev
dU  d (  E i Pi )   E i dPi   Pi dE i
i
i
i
 E i 
dV   wrev
i Pi ddE i  i Pi  V  dV   PdV
N
1
 Ei dPi   ( ln Pi dPi  ln Q  dPi )

i

i
1
ln P dP


i
i
i
i
Thermodynamic Properties and Canonical Ensemble
Entropy
 Pi dEi  
i
1
ln P dP  q


i
i
rev
i
qrev  d ( Pi ln Pi )  TdS

i
qrev  d (U  ln Q)  TdS
S  k ln Q  U / T  S 0
S  k ln Q  U / T
S  k  Pi ln P  k  ln P 
i
The only function that links heat (path integral) and state property is TEMPERATURE.   1 / kT
Summary of Thermodynamic Properties in Canonical Ensemble
 ln Q
)V , N
 ln T
 ln Q


)V , N 
S  k  ln Q  (
 ln T


U  kT (
l Q
 ln
l Q
  ln

H  kT  (
)V , N  (
)T , N 
 ln V
  ln T

A   kT ln Q
 ln Q


)T , N 
G   kT  ln Q  (
 ln V


  ln Q 

 i  kT 
 N i T ,V , N
j i
All thermodynamic properties
Can be obtained from PARTITION FUNCTION
FUNCTION”
“PARTITION
Ensembles
•
•
•
•
Micro‐canonical ensemble: E,V,N
Canonical ensemble: T,V,N
Canonical ensemble: T,V,N
Constant pressure ensemble: T,P,N
Grand‐canonical ensemble: T,V,μ
57
Constant pressure simulations: N PT
N,P,T ensemble
bl
E  Ei ,
Vi , Ei
V  Vi
1/kBT
p/kBT
Consider a small system that can exchange volume
and energy with a big reservoir
  ln  
  ln  
ln  V  Vi , E  Ei   ln  V , E   
 Ei  
 Vi  
 E V
 V  E
  E  Ei , V  Vi 
Ei
pVi
ln


  E ,V 
k BT k B T
Hence, the probability to find Ei,Vi:
P  Ei , Vi  

  E  Ei , V  Vi 
j ,k
  E  E j ,V  Vk 
 exp     Ei  pVi  
exp     Ei  pVi  

 j ,k exp   E j  pVk 
58
Thermo recall (4)
First law of thermodynamics
dE  TdS  pdV + i i dN i
Hence
1  S 
=

T  E V , N
and
d
p
 S 

 
 V T , N T
59
N,P,T ensemble (2)
, ,
( )
In the classical limit, the p
partition function becomes
Q  N , P, T  
1
 
N
N


dV
exp


PV
d
r
exp


U
r


3N



 N!
The probability to find a particular configuration:



r N ,V
 
P r N , V  exp    PV  U r N 


60
Grand‐canonical simulations: μ,V,T ensemble
VT
bl
E  Ei ,
N i , Ei
N  Ni
1/kBT
-μ/kBT
Consider a small system that can exchange particles
and energy with a big reservoir
  ln  
  ln  
ln   N  N i , E  Ei   ln   N , E   
 Ei  
 Ni  
 E  N
 N  E
  E  Ei , N  N i 
i N i
Ei
ln


  E, N 
k BT k B T
Hence, the probability to find Ei,Ni:
P  Ei , N i  

  E  Ei , N  N i 
j ,k
  E  E j , N  Nk 
 exp     Ei  i N i  


exp     Ei  i N i  
j ,k
exp     E j  k N k  
61
Thermo recall (5)
First law of thermodynamics
dE  TdS  pdV + i i dN i
Hence
1  S 
=

T  E V , N
and
d
 S 
i

 
T
 N i T ,V
62
μ, ,
μ,V,T ensemble (2)
( )
In the classical limit, the p
partition function becomes

Q   ,V , T   
N 1
exp  
 N
 
N
N


r
U
r

d
exp

3N



 N!
The probability to find a particular configuration:


N,rN
 
P N , r N  expp   N  U r N 
63
Classical Statistical Mechanics
Classical Statistical Mechanics
• It is not easy to derive all the partition functions using quantum mechanics
• Classical mechanics can be used with negligible error when energy difference between energy levels (Ei) are smaller
energy difference between energy levels (Ei) are smaller thank kT. • However, vibration and electronic states cannot be treated with classical mechanics. (The energy spacings are order of kT)
with classical mechanics. (The energy spacings are order of kT) Phase Space
Phase Space •
Recall Hamiltonian of Newtonian Mechanics
Recall Hamiltonian of Newtonian Mechanics
H (r N , p N )  KE(kinetic energy)  PE(potential energy)
H (r N , p N )  
i
pi
U (r1 , r2 ,..., rN )
2mi
 H 

  p i
 ri 
 H 

  ri
 p i 
•
•
Instead of taking replica of systems (ensemble members), use abstract p
p
compose of momentum space and position space (6N)
p
p
p
p
( )
‘phase space’
 Average of infinite phase space Ensemble Average
Ensemble Average
U  lim
 
1


0
E ()d  lim  P N () E ()d
P N ()d
n 
Fraction of Ensemble members in this range ( d)
(d)
Using similar technique used for
B l
Boltzmann distribution
di ib i
P N (  ) d 
exp( H / kT )d
 ... exp( H / kT )d
Canonical Partition Function
Canonical Partition Function
Phase Integral
T   ... exp( H / kT )d
Canonical Partition Function
Q  c  ... exp( H / kT )d
Match between Quantum and Classical Mechanics
c  lim
T 
 exp( E / kT )
i
i
 ... exp( H / kT )d
c
1
N!h NF
For rigorous derivation see Hill, Chap.6
Canonical Partition Function in Classical Mechanics
1
Q 
... exp( H / kT )d
NF  
N !h
Example : Translational Partition Function for an Ideal Gas
H (r N , p N )  KE(kinetic energy)  PE(potential energy)
H (r N , p N )  
i
3N
H 
i
pi
U (r1 , r2 ,..., rN )
2mi
No potential energy, 3 dimensional
space.
2
pi
2mi
pi2
1
Q
... exp(
)dp1...dp N dr1...drN
3N  
N !h
i 2mi
1

N !h 3 N
3N

 V

p
)dp    dr1dr2 dr3 
  exp(
2mi
 
 0

1  2mkT 

N !  h 2 
3N / 2
VN
N
Semi Classical Partition Function
Semi‐Classical Partition Function
The energy of a molecule is distributed in different modes
‐ Vibration, Rotation (Internal : depends only on T)
‐ Translation (External : depends on T and V) Assumption 1
Hamiltonian operator can be separated into two parts (internal + center of mass motion) CM
int
H op  H op
 H op
EiCM  Eiint
EiCM
Eiint
)   exp(
) exp(
)
Q   exp(
kT
kT
kT
Q  QCM ( N , V , T )Qint ( N , T )
Semi Classical Partition Function
Semi‐Classical Partition Function
• Internal parts are density independent and most of the components have the same value with ideal gases. Qint ( N ,  , T )  Qint ( N ,0, T )
• For solids and polymeric molecules, this assumption is not valid any more
assumption is not valid any more.
Semi‐‐Classical Partition Function
Semi
Classical Partition Function
Assumption 2
For T > 50K, classical approximation can be used for translational part.
H CM  
pix2  piy2y  piz2
2m
i
 U (r1 , r2 ,..., r3 N )
pix2  piy2  piz2
1
Q
... exp(
)dp 3 N  ... (U / kT )dr 3 N
3N  
N !h
2mkT
i
3 N
Z

N!
1/ 2
 h2 

  
2

mkT


Configuration
Integral
Z   ... (U / kT )dr1dr2 ...dr3 N
Q 
1
Q int   3 N Z
N!
For non‐central forces For non‐
((orientation effect) )
1
Z  

   d
N
 ... (U / kT )dr dr ...dr
1
2
3N
d1...d N
Canonical Ensembles
After adoption
Af
d i off the
h ergodic
di hypothesis,
h
h i it
i then
h remains
i to determine
d
i the
h actuall
form of the function φ(H). This function depends on the type of the
thermodynamic system under consideration, i.e. on the character of the interaction
b t
between
the
th system
t andd the
th external
t
l bodies.
b di
We will consider canonical ensembles of two types of systems:
1) Adiabatically isolated systems that have no contact with the surroundings and
have a specified energy E.
• The corresponding statistical ensemble is referred to as the microcanonical
ensemble, and the distribution function – microcanonical distribution.
2) Closed isothermal systems that are in contact and thermal equilibrium with an
external thermostat of a given temperature T.
• The corresponding statistical ensemble is referred to as the canonical
ensemble,, and the distribution function – Gibbs’ canonical distribution.
Both systems do not exchange particles with the environment.
Microcanonical Distribution
For an adiabatically
F
di b ti ll isolated
i l t d system
t with
ith constant
t t external
t
l parameters,
t
a, the
th total
t t l
energy cannot vary. Therefore, only such microstates X can occur, for which
H ( p, q, a )  E  constant
This implies (δ – Dirac’s delta function)
w( p, q )   E  H ( p, q, a ) 
and finally:
w( p, q ) 
1
  E  H ( p, q, a ) 
( E , a )
E, a
(P,T,V,…)
where Ω is the normalization factor,
( E , a ) 
p q
   E  H ( p, q, a)  dpdq
( p ,q )
Within the microcanonical ensemble, all the energetically allowed microstates have
an equal probability to occur.
Microcanonical Distribution: Integral Over States
Thee normalization
o
o factor
c o Ω iss ggiven
ve by
 ( E , a ) 
( E , a )  

 E a
where
h Γ iis the
h integral
i
over states,
or phase integral:
(E , a) 

Phase volume
1
dpdq
d d
(2  ) f N N ! H ( p , q, a )  E
1
  E  H (p , q )  dp1 ...d p N d q1 ...dq N
(2  ) f N N ! ( p,q )
 0, x  0
,  - Planck's constant,
1 x0
 1,
 ( x)  
j - number of DOF per particle
Γ(E,a) represents the normalized phase volume, enclosed within the
hypersurface of given energy determined by the equation H(X,a) = E.
Phase integral Γ is a dimensionless quantity.
Thus the normalization factor Ω shows the rate at which the phase volume
varies
aries due
d e to a change of total energy
energ at fixed
fi ed external
e ternal parameters.
parameters
Microcanonical Distribution: Integral Over States
Microcanonical Distribution: Integral Over States
The integral over states is a major calculation characteristic of the microcanonical
ensemble. The knowledge of Γ allows computing thermodynamic parameters of the
closed
l d adiabatic
di b i system:
1
S  k ln
l ,
 S 
T 
 ,
 E 
P
1
  


( E , V )  V  E
(These are the major results in terms of practical calculations over microcanonical ensembles.)
Summary:
micro‐canonical ensemble (N,V,E)
i
i l
bl (
)
Partition function:
 
1
Q  N , V , E   3 N  dp N dr N  H p N ,r N  E
h N!


Probability to find a particular configuration
P   1
F
Free
energy
 S  ln QN ,V , E
78
Microcanonical Ensemble: Illustrative Examples
Microcanonical Ensemble: Illustrative Examples
We will consider one-dimensional illustrative examples of computing the phase
integral, entropy and temperature for microcanonical ensembles:
Spring-mass harmonic oscillator
Pendulum (non-harmonic oscillator)
We will use the Hamiltonian equations of motion to get the phase space trajectory,
andd then
th evaluate
l t the
th phase
h
integral.
i t
l
Harmonic Oscillator: Hamiltonian
Hamiltonian: general form
H  T ( p)  U ( x)
Kinetic energy
p2
T
2m
Potential energy
k x2
U
2
k
x
Potential energy is a quadratic function of the coordinate (displacement form the
equilibrium position)
The total Hamiltonian
m
p2 k x2
H ( p, x ) 

2m
2
Parameters:
m  1021 kg,
k k  25  10 21 N/m
N/
Harmonic Oscillator: Equations of Motion and Solution
H ilt i and
Hamiltonian
d equations
ti
motion:
ti
H
H
p2 k x2
H ( x, p ) 

 x 
, p  

2m
2
p
x
Initial conditions (m, m/s):
x(0)  0.2, x (0)  0
x (0)  0.
0.4,, x (0)  0
x(0)  0.6, x (0)  0
Parameters:
m  10 21 kg, k  25  10 21 N/m
x 
p
, p   k x
m
Total energy:
Harmonic Oscillator: Total Energy
Harmonic
Oscillator: Total Energy
E  H ( x, p )  Const (at any x(t ), p (t ))
E1
E2
x(0)  0.2
x (0)  0
x (0)  0.4
x (0)  0
E3
x (0)  0.6
x (0)  0
E1  0.5  1021 J, E2  2.0 1021 J, E3  4.5 1021 J
Phase integral:
( E ) 
1
2 
A( E ),
Harmonic Oscillator: Phase Integral
Harmonic
Oscillator: Phase Integral
 x p N N
A( E )     E  H ( x , p )  dxdp
    E  H (i x , j p ) 
2 
( x, p)
 x  2 xmax / N  step for x ,  p  2 pmax / N  step for p ,
i0 j 0
N  number of integration steps
A1
A2
1  0.95 1012
 2  3.80 1012
3  8.5
8.54 10
012
For the harmonic oscillator, phase
volume grows linearly with the
i
increase
off totall energy.
A3
Harmonic Oscillator: Entropy and Temperature
Entropy:
S  k ln  ,
k  1.38  10 23 J/K
S1  3.8
3.81 10022 J/K
J/
S 2  4.00 1022 J/K
S3  4.111022 J/K
Temperature:
 S 
T 

 E 
1
We perturb the initial conditions (on 0.1% or less) and compute new values E and S.
The temperature is computed then, as
T
1

 S S 
2 kin
(
T

E )
benchmark:
 

k
EE
T1  36.3 K
T2  145.1 K
T3  326.5 K
Pendulum: Total Energy
Total energy:
p2
E  H ( , p ) 
 mgl cos   Const (at any  (t ), p (t ))
2
2ml
E1
 (0)  0.3
 (0)  0
E2
 (0)  1.8
 (0)  0
E3
 (0)  3.12
3 12
 (0)  0
p   l - angular
g
momentum
E1  1.87  1021 J, E2  0.45  1021 J, E3  1.96 1021 J
Phase integral:
( E ) 
1
2 
A( E ),
Pendulum: Phase Integral
Pendulum:
Phase Integral
 p N N
A( E )     E  H ( , p )  d  dp
    E  H (i  , j p ) 
2 
( , p )
i0 j 0
   2 xmax / N  step for  ,  p  2 pmax / N  step for p ,
N  number of steps
A1
A2
1  0.4 1012
 2  11.2 1012
3  21.4
.  10
012
For the pendulum, phase volume
grows NON-linearly with the increase
off totall energy at large
l
amplitudes.
li d
A3
Pendulum: Entropy and Temperature
Entropy:
S  k ln  ,
k  1.38  10 23 J/K
S1  3.68  10
022 J/
J/K
S 2  4.15  1022 J/K
S3  4.24  1022 J/K
Temperature:
 S 
T 


E


1
We perturb the initial conditions (on 0.1% or less) and compute new values E and S.
The temperature is computed then, as
T
1

 S S 
2 kin

(
benchmark:
T
E )
 

k
EE
T1  6.3
6 3K
T2  153.8 K
T3  96.0 K
Summary of the Statistical Method: Microcanonical Distribution
1.
Analyze the physical model; justify applicability of the microcanonical
distribution.
2
2.
M d l individual
Model
i di id l particles
ti l andd boundaries.
b
d i
3.
Model interaction between particles and between particles and boundaries.
4.
Set up initial conditions and solve for the deterministic trajectories (MD).
5.
Compute two values of the total energy and the phase integral – for the
original
i i l andd perturbed
t b d initial
i iti l conditions.
diti
6.
Using the method of thermodynamic parameters, compute entropy,
temperature
p
and other thermodynamic
y
parameters.
p
If possible
p
compare
p the
obtained value of temperature with benchmark values.
1
S  k ln ,
7.
 S 
T 
 ,
 E 
P
1
  


( E , V )  V  E
If required, accomplish an extended analysis of macroscopic properties (e.g.
functions T(E), S(E), S(T), etc.) by repeating the steps 4-7.
Free Energy and Isothermal Processes
Free energy, also
F
l Helmholtz
H l h lt potential
t ti l is
i off importance
i
t
for
f the
th description
d
i ti off
isothermal processes. It is defined as the difference between internal energy and
the product of temperature and entropy.
F  U  TS
Since free energy is a thermodynamic potential,
potential the function F(T,V,N,…)
F(T V N )
guarantees the full knowledge of all thermodynamic quantities.
Physical content of free energy: the change of the free energy dF of a system at
constant temperature, represents the work accomplished by, or over, the system.
Indeed,
dF  dU  TdS  SdT
 dU  TdS   W 
  SdT   W
dF   W
Isothermal processes tend to a minimum of free energy, i.e. due to the definition,
simultaneously to a minimum of internal energy and maximum of entropy.