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Chapter 6 Fundamental principles
of statistical physics
§1 Statistical laws of macroscopic matter
Statistical laws
 In classic physics, the motion of a single particle
will obey Newton’s law. If the initial position and
velocity are known, we can predict its position at
any time by solving the Newton equation of
motions.
 A macroscopic body has a large number of particles.
For example, there are about 6×1023 molecules for
1 mol gas at standard conditions. The macroscopic
properties (pressure, specific heat, phase transition,
etc.) are the average of motions of many molecules.
 Within the framework of classic mechanics, one needs
to solve Newton’s equations of 6×1023 interacting
molecules. This is obviously not possible and also
unnecessary. The reason is that the presence of a large
number of particles causes new types of regularity
(statistical law). These statistical laws cannot in any
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way be reduced to purely mechanical laws. The
macroscopic properties can be determined by such
new types of regularity (For example, the speed
distribution of gas molecule)
Statistical average
The macroscopic quantities (observable) are defined as
statistical mean values of the corresponding microscopic
n
quantities: u    i ui
i 1
The (statistical) ensemble is a collection of identical
replicas of a given physical system with distinct
microstates.
 many systems with identical macroscopic properties
 each microstate is distinct
 the ensemble average → the statistical average
1 T
 the time average u  lim 0 udt
T  T
 the principle of statistic equivalency: the time
average = the ensemble average
The ergodicity hypothesis
The fluctuations
u  u  u
2
The mean square fluctuations
The relative fluctuations
( u) 2  u 2  u
(u) 2 u
3
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§ 2 Quantum properties of microscopic
particles

The wave-particle duality and the uncertainty
principle
E  h  

Wave
p
functions,
h

 k
qp  h
operators
and
Schrödinger’s
equations
(1) The state of a physical system is characterized by a
wave function
(2) All physical quantities correspond to observable
operators
(3) The motion of a physical system satisfies
Schrödinger’s equation

Some examples
(1) Translations: free particles
(2) Vibrations: Harmonic oscillator
(3) Rotations: free rotator
(4) Spin: intrinsic properties
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
The identity of particles
Bosons with integer spin have symmetric wave functions
and disobey Pauli’s exclusion principle.
Fermions with half-integer spin have anti-symmetric
wave functions and obey Pauli’s exclusion principle.
Correspondence law: for a particle with f degree of
freedom, each possible quantum state corresponds to a
d
phase volume of hf  i   f
h
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§ 3 Thermodynamic
Boltzmann relation
probability
and
The thermodynamic probability is the number of
microstates included in a given macrostate. It is an
indication of disorder.
The principle of equal probability: For an isolated system,
each microstate appears with the same probability
The equilibrium state has the maximum probability
W  Wmax
Boltzman relation: S  k ln W

S is the entropy defined in thermodynamics

isolated system

the principle of increase of entropy
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§4 Liouville’s theorem
Description
of
state
of
motions:
phase
space,
representative point and phase trajectory
 particle:  space with f generalized coordinates qi
and f generalized momentum pi
 system:
 space
with
s
(=Nf)
generalized
coordinates qi and s (=Nf) generalized momentum pi
s
 phase volume d   dqi dpi
i 1
 Hamilton’s equation of motion
qi 
H
pi
 d  1
p i  
H
qi
u   ud
s
( d   dqi dpi )
i 1
  is probability density （ or the distribution
function）
 can be understood as the density of representative
points
 In general,  depends on the energy    (E )
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It follows that the distribution function is constant along
the phase trajectories. This is Louville’s theorem.
d 
   (qi , pi , t )  C

  , H   0
dt
t
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