What defines the granular nature of our Universe?

2

346.626 10

kgm

sdeBroglieh

mv mv

-24 -10

-35

for an atom, 10

f

~1

or a person 60

0

~ 10

~

~

Kgm

s

Kgm

s

m

m

mv

mv

Our classical “behavior” vs the atomic “quantum” characteristics

are a consequence of the absolute size of hSo far, we “solved” the Q.M. problem and then count all states to get the partition function Q. While trying to count states, we invoke the classical limit and

Can we obtain Q assuming classical behavior for the H?

What do you expect to be “different” between the 2 answers?

So far, we “solved” the Q.M. problem and then count all states to

get the partition function Q. While trying to count states, we

invoke the classical limit and

Can we obtain Q assuming classical behavior for the H?

We can obtain the states energies by solving the classical Hamiltonian equations

where j=1,2,...3N

j jq p

qj j

H Hp

We count the states by analogy to the quantum treatment…

QM treatment classical treatment

e e

classq e dpdq

p,qH

2

21 conjugated pair p,q

1Harmonic Oscillator: ,

2 2 osc

pp q k q

mH

vib,class

,q

p qe dp dq

-

H

2

21

2 2 osc

pk q

mkT kTe dp dq-

2

21

2 2vib,classq

oscp

k qmkT kTe dp e dq

- -

121

2vib,class

2q . 2

osc

kTconst mkT

k

122

we use1

2

bxe dx

b0

12

. 2osc

mconst kT

k

vib,classq .

kTconst

osc

where we used 1

2

m

k

vib,QMqkT

h

1const

h

2 2 2

Translation kinetic energy: ,2

x y zp p pp q

mH

trans,class 3 conjugated variables pairs

,q x y z x y z

p qe dp dp dp dq dq dq

-

H

2 2 2

2trans,class

-

q . x y zp p p

mkTx y z x y zconst e dp dp dp dq dq dq

-

Volume

22 2

2 2 2trans,classq .

yx zpp p

mkT mkT mkTx y zconst e dp e dp e d Vp

- - -

122

we use1

2

bxe dx

b0

3

2trans,classq . 2const mkT V

32

trans,QM 2

2q

mkTV

h

1const 3h

22

2

1Rotations ,

2 sin

pp q p

IH

rot,class 2 conjugated variables pairs

,q

p qe dp dp d d

H

22

22

rot,class

0 0

12 sin

q

pp

kTe dp dp d d

-

I

2rot,classq .8const kTI

2

rot,QM

8q

kT2

Ih

1

const 2h

vib

3trans

2rot

q 1 conjugated variables pairs

q 3 conjugated variables pairs

q 2 conjugated variables pairs

h

h

h

Comparing the high T limit of the QM Stat.Mech. with classical

Stat. Mech. we infer the constants

We are establishing a “correspondence principle” between

QM and Classical Statistical Mechanics

classical treatmentQM treatment

,

1

1 p q

j jj

sq e e dp dq

H

sh

# of conjugated pairs p,qs

,

1

for a system with N indistinguishable partic

1

les !

!

N

p q

class j jj

N

s

qQ

dp dq

N

Q eN

H

sh

Consider N interacting particles:

Each particle has s degrees of freedom

j

j

coordinates q

momenta p

s N

s N

EVERYTHINGknowing and we know about the syst.

we can predict a trajectory

j jq p

Construct a space with 2 dimensions,

2 axes (one for each

aSP CE

nd )A

j j

s N

sPHASE

N q p

in fully desc rone point ph ibes the syase s stem p e ac at t

evolution of the complete system = evolution phase pof oint

where j=1,2,...sN

j jq p

qj j

H Hp

j jintegration of these eqs. q t t p

a setMicro of canonical identi Ense cal sm ystems ble: A

each identical system one phase point

Equal a priori probability principle

each point is equally probable

there must be a point for each and every set of ,

consistent with N,V,E

p q

in a surface of constant E, density of points is uniform

j j

evolution of each system is independent of the others

in phase space, each point's trajectory is independent

q t t evolves independently p

for a given instant in time density number , ,

, , = # systems with between

between q

t f p q t

f p q t p p and p dp

q q and dq

integrating over the whole phase space

gives the total number of systems, , f p q t dpdq A

Ensemble average of property ,

, , , 1, , , ,

, ,

p q

p q f p q t dpdqp q p q f p q t dpdq

f p q t dpdq A

Gibbs postulate:

Ensemble average of property hermodynamic value prop.T

Liouville equation

1 2 1 2

1 2 1 2

consider a small volume defined by

V

around the phase point , ,... , , ,...n n

n n

p p p q q q

p p p q q q

#points inside VN

N N’

#pts entering through one face # pts exiting through another face?

q1q1q1 q1+q1

1

2 1 2

1# pts entering at q

at a given time , ,n nt

dqf p q t q q p p p

dt

1 1 1 2

1 1 2 2

1 1# pts exiting at

at a given time , , ,.., , ,., , n

n n

q q

t dq p q q q qf p q q q q t q q p

dt

1 1 1# pts entering at q # pts exiting at

at a given time at a given time the net flow is given by - q q

t t

1

2 1 2

1 1 1 2

1 1 2 2

, ,

, , ,.., , ,., ,

n n

n

n n

dqf p q t q q p p p

dt

dq p q q q qf p q q q q t q q p

dt

1 1 1 2 1 1 1 2 2

, , , , ,., , , , ,..n n n

f p q t q f p q q q q t q p q q q q q q p

1

1

1

1 1 2 1

11 1 2 1 1 1

11

, ,., ,, , ,., , , ,., ,

for small q,

, , ,., , , ,., ,

n

n n

f p q q tq

qf p q q q q t f p q q t

qq p q q q q t q p q q t q

n n q

1 1 1 2 1 1 1 2

1 1

1 1 1 1 1 1 1

1 1 1 1

f=f(p,q,t)

, , , , ,., , , , ,..

n n

where

f p q t q f p q q q q t q p q q q q

q f f qfq fq f q q q q q

q q q q

1

1 1 1

1 1

q ff q q q

q q

1 1 1 2 1 1 1

2

2 2

1

1 1

1 1

1 this is the net flow in q d

, , , , ,., , , , ,.

irection

.

n n n

n

f p q t q f p q q q q t q p q q

q p

q

q

q q q p

q ff q q

q q

2

1

1

1

1 1

1 1 this is the net flow in q direction

in a similar way we can obtain the net flow in p direction

n

p p qp f

f pp p

p

Net flow in all directions

since V is the volume in phase space (generated by all )

the net flow in all directions is

qp axes

d N

dt

we can simplify the equation by recalling the eqs. of motion

2 1

1 1

1 1

1 1 1 1

11

n n

n

j

p p q qp f q f

f f q pp p q q

d Np

dt

j jq p

p qj j

H H

j

j

jq

q

p

pj

2 2

and j

j

j

j

q

q

p

q p p p qj j j j

H H

V

1

, ,j

j j

n

jj

f fq

p q

f p q tp

t

Nd

Vdt

the change in density with time

around point (p,q) is

Full derivative of f(p,q,t)

1

it is a partial derivative

because ( , , )

, , j

j j

n

jj t

f fq

p p qq f

f p q tp

t

on the other hand, the full derivative is

, , j

j j

j jj

f f fq

t p qdf p q t dt dp dq

, ,j j

j

j jj

dp dqf f fq

t p dt q dt

df p q t

dt

j

j j

j jj

f f fq

t p qp q

1j

j j

j

j j

n

j j jj j

f fq

f

p qpq

q

fpp q

, ,0

df p q t

dt

Constant density conceptthe density around any moving phase point

is constant for that trajectory

ot the point has coordinates ,

t the point has coordinates ,o op q

p q , ; , ;o o of p q t f p q t

, ;

, ; and are functions of ,o o

o o

o o

p p p q t

q q p q tp q p q

Volume around p,q

NN

Volume around po,qo

with N in the surface

o at t , V around and has phase points on its surfaceo op q N

o at t +dt, points propagated, to a new V around and

(different shape).

in initial volume final volume

p q

N N

no trajectory can cross the surface of that volume because

if 2 trajectories cross (they are at the same in the same

phase space position), then they must remain the same.

Since the process is govern

t

by which cannot yield two

different evolutions trajectories do not cross

H

, ; , ; and # pts inside volume is constant

shape changes, but volume is constanto o of p q t f p q t

o o p q p q at all times