Download Lecture 2 - Department of Applied Physics

Lecture 2
 The microcanonical ensemble.
 Quantum states and the phase space.
 Some paradoxes in statistical physics.
 Ergodic hypothesis.
 Quasi-ergodic systems.
 Some model systems in statistical physics
1
The microcanonical ensemble
When a system consisting of a great number of particles
(more generally a system having a great number of
degrees of freedom) is isolated for a long time from its
environment, it will finally reach a thermal equilibrium
state.
In this case ensemble is defined by the number of
molecules N, the volume V and the energy E.
The energy of the system is constant, so that it is
presumed to be fixed at the value E with certain
allowance .
It means that we specify a range of energy values, say
from (E - ) to (E+).
With a specified macrostate, a choice still remains for the
systems of the ensemble to be in any one of a large
number of possible microstates.
2
In the phase space, correspondingly, the representative
points of the ensemble have a choice to lie anywhere
within a "hypershell" defined by the condition
1
1
(E   )  H (q , p)  (E   )
2
2
(2.1)
The volume of the phase space enclosed within the shell is
given by '
   d  (d 3 N q d 3 N p)
(2.2)
where the primed integration extends only over that part
of the phase space which conforms to the condition (2.1).
It is clear that  will be a function of the parameters N,V,E
and .
3
Now the microcanonical ensemble is a collection of
systems for which the density function  is, at all times,
given
1
1 
 (q , p)  const if (E -  )  H (q , p)  (E +  ) 
2
2 

=0
otherwice

(2.3)
If (q,p)=const, it means that we are dealing with a swarm
of representative points uniformly distributed over the
relevant region of the phase space (outside the relevant
region  is identically zero).
4
Physically it corresponds to an ensemble of systems, which
at all times are uniformly distributed over all possible
microstates. In this case the ensemble average can be
written in the following way
 f 
1
 
f (q , p)d
(2.4)
where  denotes the total «volume» of the accessible
region of the phase space.
Clearly, in this case, any member of the ensemble is equally
likely to be in any one of the various possible microstates,
inasmuch as any representative point in the swarm is
equally likely to be in the neighborhood of any phase point
in the allowed region of the phase space.
5
This statement is usually referred to as the «postulate of equal and
a priory probabilities» for the various possible microstates (or for
the various volume elements in the allowed region of the phase
space); the corresponding ensemble is referred to as the
microcanonical ensemble.
Let us try to clarify the physical meaning of the ensemble average <f>,
as given by (2.4). Since the ensemble under study is a stationary one,
the ensemble average of any physical quantity f must be independent
of time; accordingly, taking a time average thereof will not produce any
new result.
Thus
<f> the ensemble average of f
= the time average of (the ensemble average of f).
Now the, the processes of time averaging and ensemble
averaging are completely independent processes, so the
order in which they are performed may be reversed
without causing any change in the value of <f>. Thus
6
<f>= the ensemble average of (the time average of f).
Now, the time average of any physical quantity whatsoever,
taken over a reasonably long interval of time, must be the
same for every member of the ensemble, for after all, we
are dealing with only the mental copies of a given system.
Therefore, taking an ensemble average thereof should be
inconsequential and we may write
<f>= the long-time average of f.
where the latter may be taken over any member of the
ensemble. We further observe that the long time average of
a physical quantity is all one obtains by making
measurements of that quantity on a given system;
therefore, it should be identified with the value one expects
to obtain through experiment. Thus we finally have
7
 f  f exp
(2.5)
This brings us to the most important result: the ensemble
average of any physical quantity f is identical with the
value one expects to obtain on making an appropriate
measurement on the given system
The next that we have to establish is the connection
between the mechanics and microcanonical ensemble and
the thermodynamics of the member systems.
To do this we observe that there exists a direct
correspondence between the various microstates of the
given system and the various locations in the phase space.
The volume  (of the allowed region of the phase space)
is, therefore, a direct measure of the multiplicity  of the
microstates obtaining in the system.
8
To establish a numerical correspondence between  and
, we must discover a fundamental volume 0, which
could be regarded as "equivalent to one microstate". Once
this is done, we can right away conclude that,
asymptotically

(2.6)
Γ 
0
The thermodynamics of the system would then go through
the relationship

S ( N ,V , E )  k ln   k ln  
 0 
(2.7)
The basic problem then consists in determining 0. From
the dimensional considerations (see eqn.2.2), 0 must be in
the nature of an “angular momentum raised to the power
3N”.
    d   (d 3 N q d 3 N p)
9
To determine it exactly, we consider in the sequel certain
simplified systems, both from the point of view of the
phase space and from the point of view of the distribution
of quantum states. We find that
 0  ( ) 3 N
(2.8)
where  is the Planck constant.
In quantum theory the construction of the microcanonical
ensemble is based on the same postulate of equal a priori
probabilities for the various accessible states. Accordingly,
the density matrix mn (which, in the energy representation,
must be a diagonal matrix) will be of the form
 mn   n  mn
with
(2.9)
10
1/ g ( E )
n  
0
for each of the accesible states
for all other states
(2.10)
where g(E) is the degree of energy level degeneration (or
degree of levels degeneration), which energies are in the
interval of ( E  12 ), ( E  12 ) microcanonical ensemble gives
the statistical interpretation of Equilibrium State.
Some paradoxes of statistical physics.
Accordingly to the zero low of thermodynamics the
relaxation process of the system to equilibrium is
irreversible. If the Liouville equation will be a simple (real)
motion equation of the macroscopic system, this point has
to taken into account.
11
However, the Liouville equation is generally equivalent to
the ordinary motion equations and transferred to them in
the case of «pure» microstates and that is why it reversible
in time. Moreover, the limited motion has the periodical
cycles (Poincare cycles), that is bringing us to the
contradiction with the equilibrium. Such contradictions are
formulated as paradoxes.
Paradox of reversibility (Poincare, Zermilo) The
mechanical conservative system in finite motion is passing
the state as much as close to the initial state (as much as
far from equilibrium)
Paradox of convertibility (Loshmidt) as a result of time
reversibility (or particle velocities) the system returned to
the initial (not necessary equilibrium) state.
12
Poincare Recurrence Theorem (1890 - 1897)
If you play bridge long enough you will eventually be dealt
any grand-slam hand, not once but several times. A similar
thing is true for mechanical systems governed by Newton's
laws, as the French mathematician Henri Poincare (18541912) showed with his recurrence theorem in 1890: if the
system has a fixed total energy that restricts its dynamics
to bounded subsets of its phase space, the system will
eventually return as closely as you like to any given initial
set of molecular positions and velocities. If the entropy is
determined by these variables, then it must also return to
its original value, so if it increases during one period of
time it must decrease during another.
13
This apparent contradiction between the behavior of a
deterministic mechanical system of particles and the
Second Law of Thermodynamics became known as the
"Recurrence Paradox." It was used by the German
mathematician Ernst Zermelo in 1896 to attack the
mechanistic worldview. He argued that the Second Law is
an absolute truth, so any theory that leads to predictions
inconsistent with it must be false. This refutation would
apply not only to the kinetic theory of gases but to any
theory based on the assumption that matter is composed
of particles moving in accordance with the laws of
mechanics.
14
Boltzmann had previously denied the possibility of such
recurrences and might have continued to deny their
certainty by rejecting the determinism postulated in the
Poincare-Zermelo argument. Instead, he admitted quite
frankly that recurrences are completely consistent with the
statistical viewpoint, as the card-game analogy suggests;
they are fluctuations, which are almost certain to occur if
you wait long enough. So determinism leads to the same
qualitative consequence that would be expected from a
random sequence of states! In either case the recurrence
time is so inconceivably long that our failure to observe it
cannot constitute an objection to the theory.
15
Loschmidt's paradox states that if there is a motion of a
system that leads to a steady decrease of H (increase of
entropy) with time, then there is certainly another allowed
state of motion of the system, found by time reversal, in
which H must increase.
This puts the time reversal symmetry of (almost) all
known low-level fundamental physical processes at odds
with the second law of thermodynamics which describes
the behaviour of macroscopic systems. Both of these are
well-accepted principles in physics, with sound
observational and theoretical support, yet they seem to be
in conflict; hence the paradox.
16
May be that are not paradoxes and is a real situation. One
of the indirect supports of this assertion is the example
with «spin echo». It is known that the changing of the
direction of magnet particle motion under the influence of
magnetic fields leads in some time to the non-Equilibrium
State where the particles are gathered.
In any case we have to take into consideration the
Boltzmann’s answer: «Long time one have wait...» and «try to
turn them back...». It means here that the prolongation of
Poincare cycle of the big systems much more then the age
of macrocosm, and it is almost impossible to reverse the
velocities of the molecules without changing there
positions.
All that statements means that in order to base the
statements of statistical mechanics one have to develop
the way of obtaining the irreversible equations from the
Liouville equation. It can be reached by some
17
simplification of macroscopic system description.
Ergodic Hypothesis
The measured physical quantities are the time average
values in one system
T
1
 f   f ( p(t ), q(t )) dt
T0
(2.11)
At comparatively long times T (longer then relaxation
times) <f> is equilibrium value of the physical quantity f .
In this case one can write:
T
1
lim   f ( p(t ), q(t )) dt   f ( p(t ), q(t )) 0 dpdq
T 
T0

(2.12)
where 0 is the equilibrium distribution function. This
statement for the closed systems with microcanonical
distribution 0 embodies the so-called ergodic hypothesis,
which was first introduced by Boltzmann (1871).
18
According to this hypothesis, the trajectory of a
representative point passes, in the course of time, through
each and every point of the relevant region of the phase
space.
A little reflection, however, shows that the statement as
such cannot be strictly true; we better replace it by the socalled quasi-ergodic hypothesis, according to which the
trajectory of a representative point traverses, in the course
of time, any neighborhood of any point of the relevant
region. An ergodic system behaves accordingly to the
ergodic hypothesis.
19
Some model systems of statistical mechanics
Most of the macroscopic systems properties are connected
with the great number of degrees of dynamical freedom
only.
In order to study of these properties it will be very useful
to use the simplest model systems, that allowed there
detail consideration on the base of classical or quantum
mechanics.
If we will consider the system where the interaction
between the particles is so small that we can neglect it in
the calculation of energy spectrum, this system can be
defined as an ideal model system.
20
Possible energetic levels are defined by the energy
spectrum of independent particles. The model systems
that are usually used in statistical physics are the following:
•
The system of the non interactive particles in the box
with the ideal reflected walls (ideal gases)
•
The system of non interactive spins in the external
magnetic field (ideal paramagnetic)
•
The systems of non-interactive oscillators with specified
oscillation frequency (Einstein model of oscillation in
solids).
21
Ideal spin system (classical consideration)
Let us consider an ideal spin 1/2 system of N
independent particles, each bearing a magnetic moment 
which may directed either parallel or antiparallel to an
external magnetic field B. The energy of each particle is
E=B, according to the orientation of the magnetic
moment.
B
Fig.2.1
The system of N spins equal to 1/2.
Every arrow shows the direction of magnetic moment.
22
Let us calculate the probability distribution of the total
magnetic moment M of the system in the absence of the
magnetic field.
We know of course that the average value of M in this
conditions is zero, but we are interested in the
probability distribution w(M).
In zero magnetic fields the projection of each moment is
equally likely to be .
We are interested in the number of arrangements, which
result in ½ (N+n) moments being positive and ½ (N-n)
moments being negative.
The problem for B=0 is essentially identical with the
problem of the random walk, in one dimension, the steps
taken as of equal length.
23
We observed first that the (normalized) probability of a
given specific sequence of particles is
 1
 
 2
N
as for each individual moment there is a probability 1/2
that it will take the orientation required by the assignment,
and there are N particles required to have their spins
ordered in the specific sequence.
We mean by a specified sequence that, for example,
particle A should be up, particle B up, particle C down...
However, there are a number of different ways in which
we can satisfy the weaker requirement that any ½ (N+n) of
the particles one way and the remainder ½ (N-n) point the
other way.
24
We note that N particles can be ordered among themselves
in N! ways. There are N ways of selecting the first particle
to be drawn, N-1 ways of selecting the second, and so on,
to give N! for the number of ways of ordering the particles.
Many of these N! ways do not give independent
distinguishable arrangements into groups of ½ (N+n) and ½
(N-n) particles.
Interchanges of the ½ (N+n) particles purely among
themselves lead to nothing new, and there are [ ½ (N+n)]!
such interchanges.
Similarly the [ ½ (N-n)]! Interchanges of the (N-n) particles
do not give new arrangements. Thus the total number W(n)
of independent arrangements or sequences giving a net
N!
moment M=n is W (n) 
1
 1

(
N

n
)
!
(
N

n
)
 2
  2
!
(2.13)
25
and the probability w(M) of a net moment M=n is
obtained on multiplying (2.13) by the probability
(1/2)N of specific sequence giving
1
N!
w(n)  w(n )  ( ) N
2 1
 1

(
N

n
)
!
(
N

n
)
 2
  2
!
(2.14)
For large values of the factorials we may use Stirling’s
approximation
x! 


2x x x e  x
lnx!  21 ln2x + xlnx - x = 21 ln2 +(x + 21 )lnx - x
(2.15)
(2.16)
26
Thus (2.14) gives, on taking the ln of both sides and
using (2.16)
1
1

ln w( M )   N ln 2  ln 2   N   ln N 

2
2
1
N
 ( N  n  1) ln 
2
2
n  1
N

1

(
N

n

1
)
ln


2

N   2

n 

1




N  
(2.17)
For n<<N we use the series expansion
n
n
n2

ln1     
.......
2


N
N 2N
(2.18)
and so, after some transformations and collecting terms we
have
1
1
n2
1  N  n 2
ln w( M )   ln N  ln 2  ln 2 
  ln

2
2
2N
2  2  2N
(2.19)
27
and so
1
n2
 2 2N
e
 2
w( M )   
 N 
(2.20)
We see from this result that the magnetization has a
Guassian distribution about the value zero. Thus the
average value of the magnetization in the absence of a
filed is zero. The probability distribution has its maximum
at this point, so the most probable value coincides with the
average value.
28
Ideal spin system (the quantum consideration)
The spectrum of one spin can be determined by the
Schredinger equation H1= with H1=-B=SB, where
 is the magnet moment of the particle,  is the
giromagnetic relation and B the permanent external
magnetic field directed along the axes z.
The spectrum consist from two nondegradated energy
levels 1/2 (in units  B), that are related to the stationary
states (orbits in the case of individual particles) oriented
along and the field and the opposite direction.
Figure 2.2 Spectrum of spin ½
29
The spectrum of all the system consist of N+1 equidistant
levels, that are state symmetrically around zero (figure
2.3).
The minimal energy -N/2 () is corresponds to the state
with orientation of all spins against the field. The inversion
of any spin increase the energy on one unit. The level m
corresponds to the state where (N/2)+m spins are already
inverse and the rate of degradation of that level is equal
Figure 2.3 Spectrum of the system
consisting of N spin S=1/2
30
g ( N , m) 
N!
1
 1

(
N

m
)
!
(
N

m
)
 2
  2
 !
(2.23)
When m <<N this relation is transferred to the Gauss
distribution. It can be obtained taking ln g or from direct
using of Stirling formula.
1
 m2
 2 2N
e
 2
g ( N , m)   
 N 
(2.24)
The width of the distribution m/N is decreasing with
increasing of N . It decreases according to 1/N.
31