Transcript BEZOUT IDENTITIES WITH INEQUALITY CONSTRAINTS

STATISTICAL
THERMODYNAMICS
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
Tel (65) 874-2749
Fax (65) 779-5452
SECOND LAW REVISITED
Lord Kelvin : A transformation whose only final
result is to transform into work heat extracted from a
source which is at the same temperature throughout is
impossible
Rudolph Clausius : A transformation whose only
final result is to transfer heat from a body at a given
temperature to a body at a higher temperature is
impossible (this principle implies the previous one)
Martian Skeptic : What temperature ?
SECOND LAW REVISITED
Definition : Body A has higher temperature than body
B (A  B ) if, when we bring them into thermal
contact, heat flows from A to B. Body A has the same
temperature as B ( A  B ) if, when we bring them
into thermal contact, no heat flows from 
A to B and no
heat flows from B to A. ( [A] := {U | U A} )
Enrico Fermi : (Clausius Reformulated) If heat
flows by conduction from a body A to another
body B, then a transformation whose only final
result is to transfer heat from B to A is impossible.
SECOND LAW REVISITED
Sadi Carnot : If A  B then we can use
a reversible engine to absorb a quantity
Q A  0 of heat from A, surrender a
quantity of heat Q A  Q B  0 to B,
and perform a quantity of work W
Carnot & Kelvin implies
 0.
Clausius, and
Q A Q B  f ([A], [B]), and for B  C,
f ([A], [B]) f ([B], [C])  f ([A], [C])
SECOND LAW REVISITED
Definition : Absolute Thermodynamic Temperature
Choose a body D
If A  D
then T( A )  f ([ A ], [ D ])
AD
then T( A )  1
DA
then T( A )  1 f ([ D ], [ A ])
In a reversible process
QA
T( A )  Q B
T( B )  0
SECOND LAW REVISITED
System : Cylinder that has a movable piston and
contains a fixed amount of homogeneous fluid
States (Macroscopic) : Region in positive quadrant
of the (V = volume, T = temperature) plane.
Functions (on region) : V, T, p = pressure
Paths (in region) : Oriented curves 
Differential Forms : can be integrated over paths
WORK
W(  ) 
HEAT


pdV
Q(  ) 


?
SECOND LAW REVISITED
Definition : Entropy Function S
Choose a state ( V1 , T1 )
Define S(V, T)  Q( 1 ) T1
where 1  2 joins ( V1 , T1 ) to (V, T),
1 is isothermal , and 2 is adiabatic
(by thermal equilibrium and by thermal isolation)
1   2 i s
Carnot' s cyle
2
and Q(  ) 


TdS
FIRST LAW REVISITED
There exists an (internal energy) function U
such that
TdS - pdV  dU
Therefore
TdS 
U
T
dT 
 U

 V


p

dV
FIRST & SECOND LAWS COMBINED
1 U
S

;
T T T

1  U
S
 
 p
V T  V

Therefore, the basic (but powerful) calculus identity
 S
 S

V T T V
Yields (after some tedious but straightforward algebra)
p
U
T
-p
V
T
IDEAL GAS LAW
(Chemists) Boyl, Gay-Lussac, Avogardo
p(V, T)  n R g / V
n  amount of gas in moles
R  ideal gas constant
(8.314
joules / degree Kelvin)
g  ideal gas temperature in Kelvin
(water freezes at 373.16 degrees)
JOULE’S GAS EXPANSION EXPERIMENT
We substitute the expression for p
(given by the ideal gas law) to obtain

nR  d g
U

 g
T
V
V  dT

and observe that the outcome of
Joule’s gas expansion experiment
U
0
V

Tg
IDEAL GAS LAW
(Physicists)
p(V, T)  N k T
V
N  number of molecules of gas
( 6 . 0225  10
23
molecules
/ mole)
k  Boltzmann’s constant
(1 . 38  10
 23
joules / deg _Kelvin)
GAS THERMODYNAMICS
Experimental Result : (dilute gases)
dU
 Nk (   1)
1
dT
Therefore
TV
 1
&

p V constant on adiabatic paths
S  Nk (   1)
1
ln( T / T1 )  Nk ln( V / V1 )
GAS KINETICS
Monatomic dilute gas, m = molecular mass
U
N

1
m
2

2
3
1
2
2
m x
average kinetic
energy / molecule
 N   A x t 
F  t  
 2 m  x

2
 V 

F
1
p 
 N (   1)
m2
V
2
A
kT  (   1)
1
2
2
m

 
5
3
GAS KINETICS
 
Photon gases E  c  momentum
3
Maxwell Equipartition of Energy
 m 1 1  m 2  2
(1   2 )  

m1  m 2

kinetic energy
in each direction
  1

1
4




kT
2
2
degrees of freedom
0
EQUIPARTITION
Number of ways of partitioning N objects into
m bins with relative frequencies (probabilities)
p 1 , p 2 ,..., p m is C  N ! ( p 1 N )!... ( p m N )!
Stirling’s formula ( ln N !  N ln N - N ) yields
ln C  N H ( p 1 ,..., p m ) where H ( p 1 ,..., p m )
denotes Shannon’s information-theoretic entropy
H ( p 1 ,..., p m )   p 1 ln p 1    p m ln p m 
EQUIPARTITION
If the bins correspond to energies, then
H ( p 1 ,..., p m ), and therefore (nearly) C,
is maximized, subject to an energy constraint
E  N ( p 1 E 1    p m E m ), by the Gibbs
distribution p i  exp(  E i kT ) Z ( T )
Therefore
1 T  dS dE
and free energy
S  k ln C
E  TS   NkT ln Z ( T )
2
Maxwell dist. prob(x,  )  exp (-m  / 2 kT )
THIRD LAW
Nernst : The entropy of every system at absolute
zero can always be taken equal to zero
inherently quantum mechanical
discrete microstates, a quart bottle of air has about
10
24
molecules
& 1 0 10
1 bit of i nformation
22
microstate s
 kT ln 2 energy
Maxwell’s demon : may he rest in peace
Time’s arrow : probably forward ???
REFERENCES
V. Ambegaokar, Reasoning about Luck
H. Baeyer, Warmth Disperses and Time Passes
F. Faurote, The How and Why of the Automobile
E. Fermi, Thermodynamics
R. Feynman, Lectures on Physics, Volume 1
REFERENCES
H. S. Green and T. Triffet, Sources of Consciousness,
The Biophysical and Computational Basis of Thought
K. Huang, Statistical Mechanics
N. Hurt and R. Hermann, Quantum Statistical
Mechanics and Lie Group Harmonic Analysis
C. Shannon and W. Weaver, The Mathematical
Theory of Communication