Transcript Document

Lecture 1
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Introduction to statistical mechanics.
The macroscopic and the microscopic states.
Equilibrium and observation time.
Equilibrium and molecular motion.
Relaxation time.
Local equilibrium.
Phase space of a classical system.
Statistical ensemble.
Liouville’s theorem.
Density matrix in statistical mechanics and its
properties.
 Liouville’s-Neiman equation.
1
Introduction to statistical mechanics.
From the seventeenth century onward it was realized that
material systems could often be described by a small
number of descriptive parameters that were related to
one another in simple lawlike ways.
These parameters referred to geometric, dynamical and
thermal properties of matter.
Typical of the laws was the ideal gas law that related
product of pressure and volume of a gas to the
temperature of the gas.
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Bernoulli (1738)
Joule (1851)
Krönig (1856)
Clausius (1857)
C. Maxwell (1860)
L. Boltzmann (1871)
J. Loschmidt (1876)
H. Poincaré (1890)
J. Gibbs (1902)
Planck (1900)
Langevin (1908)
Compton (1923)
Smoluchowski (1906)
Bose (1924)
Debye (1912)
T. Ehrenfest
Einstein (1905)
Pauli (1925)
Thomas (1927)
Dirac (1927)
Fermi (1926)
Landau (1927)
3
Microscopic and macroscopic states
The main aim of this course is the investigation of general properties of the
macroscopic systems with a large number of degrees of dynamically
freedom (with N ~ 1020 particles for example).
From the mechanical point of view, such systems are very complicated. But
in the usual case only a few physical parameters, say temperature, the
pressure and the density, are measured, by means of which the ’’state’’ of
the system is specified.
A state defined in this cruder manner is called a macroscopic state or
thermodynamic state. On the other hand, from a dynamical point of view,
each state of a system can be defined, at least in principle, as precisely as
possible by specifying all of the dynamical variables of the system. Such a
state is called a microscopic state.
4
Averaging
The physical quantities observed in the macroscopic state are the result
of these variables averaging in the warrantable microscopic states. The
statistical hypothesis about the microscopic state distribution is
required for the correct averaging.
To find the right method of averaging is the fundamental principle of
the statistical method for investigation of macroscopic systems.
The derivation of general physical lows from the experimental results
without consideration of the atomic-molecular structure is the main
principle of thermodynamic approach.
5
Zero Low of Thermodynamics
One of the main significant points in thermodynamics (some times they
call it the zero low of thermodynamics) is the conclusion that every
enclosure (isolated from others) system in time come into the
equilibrium state where all the physical parameters characterizing the
system are not changing in time. The process of equilibrium setting is
called the relaxation process of the system and the time of this process
is the relaxation time.
Equilibrium means that the separate parts of the system (subsystems)
are also in the state of internal equilibrium (if one will isolate them
nothing will happen with them). The are also in equilibrium with each
other- no exchange by energy and particles between them.
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Local equilibrium
Local equilibrium means that the system is consist
from the subsystems, that by themselves are in the
state of internal equilibrium but there is no any
equilibrium between the subsystems.
The number of macroscopic parameters is increasing
with digression of the system from the total
equilibrium
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Classical phase system
Let (q1, q2 ..... qs) be the generalized coordinates of a system
with i degrees of freedom and (p1 p2..... ps) their conjugate
moment. A microscopic state of the system is defined by
specifying the values of (q1, q2 ..... qs, p1 p2..... ps).
The 2s-dimensional space constructed from these 2s variables
as the coordinates in the phase space of the system. Each
point in the phase space (phase point) corresponds to a
microscopic state. Therefore the microscopic states in classical
statistical mechanics make a continuous set of points in phase
space.
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Phase Orbit
If the Hamiltonian of the system is denoted by H(q,p),
the motion of phase point can be along the phase orbit
and is determined by the canonical equation of motion
H
qi 
 pi
 H
pi  
 qi


(i=1,2....s)
H ( q, p )  E
P
(1.1)
(1.2)
Phase Orbit
Constant energy surface
Therefore the phase orbit must
lie on a surface of constant
energy (ergodic surface).
H(q,p)=E
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 - space and -space
Let us define  - space as phase space of one particle (atom or molecule).
The macrosystem phase space (-space) is equal to the sum of  - 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).
This function is defined in such a way that at any time t, the number of
representative points in the ’volume element’ (d3Nq d3Np) around the 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.
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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:
 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 ’’phase portrait’’ of the system can be described by the set of
points 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 microstates
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Statistical Ensemble
The number of macroscopically identical systems distributed along
admissible microstates with density  defined as statistical 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 is said to be stationary if  does not depend explicitly
on time, i.e. at all times

0
t
(1.4)
Clearly, for such an ensemble the average value <f> of any physical
quantity f(p,q) will be independent of time. Naturally, then, a stationary
ensemble qualifies to represent a system in equilibrium. To determine the
circumstances under which Eq. (1.4) can hold, we have to make a rather
study of the movement of the representative points in the phase space.
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Lioville’s theorem and its consequences
Consider an arbitrary "volume"  in the relevant region of the phase
space and let the "surface” enclosing this volume increases with time is
given by

t
  d
(1.5)
where d(d3Nq d3Np). On the other hand, the net rate at which the
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
ˆ is the (outward) unit vector normal to this
surface element d, while n
element. By the divergence theorem, (1.6) can be written as
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 div ( v )d
(1.7)
where the operation of divergence means the following:

 


div( v )    (  qi ) 
(  pi )
pi
i 1  qi

3N
(1.8)
In view of the fact that there are no "sources" or "sinks" in the phase
space and hence the total number of representative points must be
conserved, we have , by (1.5) and (1.7)


 div( v )d    t   d


or
 

   t  div( v )d  0
t
  d
(1.9)
(1.10)
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The necessary and sufficient condition that the volume integral (1.10)
vanish for arbitrary volumes  is that the integrated must vanish
everywhere in the relevant region of the phase space. Thus, we must
have

 div ( v )  0
t
(1.11)
which is the equation of continuity for the swarm of the representative
points. This equation means that ensemble of the phase points moving
with time as a flow of liquid without sources or sinks.
Combining (1.8) and (1.11), we obtain

 


div( v )    (  qi ) 
(  pi )
pi
i 1  qi

3N
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



      

q
 

p


i
  
qi 
pi     

0

 t i 1   qi
 pi 
 pi 
i 1   qi


3N
3N
(1.12)
The last group of terms vanishes identically because the equation of
motion, we have for all i,


 qi  H (qi , pi )  H (qi , pi )
 p



 qi
 qi  pi
 qi  pi
 pi
2
2
(1.13)
From (1.12), taking into account (1.13) we can easily get the Liouville
equation
 ρ 3N   ρ   ρ    ρ
  
qi 
pi  
 ρ,H   0
 t i 1   qi
 pi   t
(1.14)
where {,H} the Poisson bracket.
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Further, since (qi,pi;t), the remaining terms in (1.12) may be
combined to give the «total» time derivative of . Thus we finally have
d  

  , H   0
dt  t
(1.15)
Equation (1.15) embodies the so-called Liouville’s theorem.
According to this theorem (q0,p0;t0)=(q,p;t) or for the equilibrium
system (q0,p0)= (q,p), that means the distribution function is the integral
of motion. One can formulate the Liouville’s theorem as a principle of
phase volume maintenance.
p
t
 t= 0
0
q
17
Density matrix in statistical mechanics

The microstates in quantum theory will be characterized by a H
(common) Hamiltonian, which may be denoted by the operator. At time
t the physical state of the various systems will be characterized by the
correspondent wave functions (ri,t), where the ri, denote the position
coordinates relevant to the system under study.
Let k(ri,t), denote the (normalized) wave function characterizing the
physical state in which the k-th system of the ensemble happens to be at
time t ; naturally, k=1,2....N. The time variation of the function k(t) will
be determined by the Schredinger equation
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
H  k (t )  i k (t )
(1.16)
Introducing a complete set of orthonormal functions n, the wave
functions k(t) may be written as
 k (t )   ank (t ) n
(1.17)
n
ank (t )    n k (t ) d
(1.18)
here, n* denotes the complex conjugate of n while d denotes the
volume element of the coordinate space of the given system.
Obviously enough, the physical state of the k-th system can be
described equally well in terms of the coefficients . The time variation
of these coefficients will be given by
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iank (t )  i   n* k (t ) d    n*H  k (t ) d
=
* 
n H 

m

k
am (t ) m d
=  Hnmamk (t )

(1.19)
m
where
Hnm   n*H  md
(1.20)
k
The physical significance of the coefficients a n (t ) is evident from eqn.
(1.17). They are the probability amplitudes for the k-th system of the
ensemble to be in the respective states n; to be practical the number
2
k
a n ( t ) represents the probability that a measurement at time t finds
the k-th system of the ensemble to be in particular state n. Clearly, we
must have
20

2
k
an (t )
1
(for all k)
(1.21)
n

We now introduce the density operator  (t ) as defined by the matrix
elements (density matrix)


1 N k
 mn (t )   am (t )ank* (t )
N k 1
(1.22)
clearly, the matrix element mn(t) is the ensemble average of the
quantity am(t)an*(t) which, as a rule, varies from member to member in
the ensemble. In particular, the diagonal element mn(t) is the ensemble
average of the probability, a nk ( t ) 2 the latter itself being a (quantummechanical) average.
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Equation of Motion for the Density Matrix mn(t)
Thus, we are concerned here with a double averaging process - once
due to the probabilistic aspect of the wave functions and again due
to the statistical aspect of the ensemble!!
The quantity mn(t) now represents the probability that a system,
chosen at random from the ensemble, at time t, is found to be in the
2
state n. In view of (1.21) and (1.22) we have
 ank (t )  1
n
 nn  1
n
 mn (t ) 


1 N k
am (t )ank * (t )

N k 1
(1.23)
Let us determine the equation of motion for the density matrix mn(t).
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1
i mn (t ) 
N

N
k 1


i a mk (t )ank * (t )  amk (t )a nk * (t )

 k*


k
k
* k*
  Hml al (t ) an (t )  am (t )  Hnl al (t ) 
k 1   l

 l

=   Hml ln (t )  ml (t ) Hln 
1
=
N
N
l
= (H    H ) mn
(1.24)
Here, use has been made of the fact that, in view of the Hermitian
ˆ H*nl=Hln. Using the commutator notation,
character of the operator, H
Eq.(1.24) may be written as
  i  
 H,  0
t 
 
(1.25)
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This equation Liouville-Neiman is the quantum-mechanical
analogue of the classical equation Liouville.
Some properties of density matrix:
•Density operator is Hermitian, +=
-
•The density operator is normalized
•Diagonal elements of density matrix are non negative   0
•Represent the probability of physical values  nn  1
n
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