The microcanonical ensemble'

1
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

2
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.
3
In the phase space, correspondingly, the
representative points of the ensemble have a
choice to lie anywhere within a "hypershell"
defined by the condition
(2.1)
The volume of the phase space enclosed within the
shell is given by
(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 ?.
4
Now the microcanonical ensemble is a collection
of systems for which the density function ? is,
at all times, given
(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).
5
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
(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.
6
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 ltfgt, 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 ltfgt? 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 ltfgt. Thus
7
ltfgt 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
ltfgt 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
8
(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.
9
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)
The thermodynamics of the system would then go
through the relationship
(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.
10
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
(2.8)
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
(2.9)
with
11
(2.10)
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.
12
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.
13
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 (1854-1912) 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.
14
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.
15
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.
16
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.
17
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 Boltzmanns 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 simplification of macroscopic
system description.
18
Ergodic Hypothesis
The measured physical quantities are the time
average values in one system
(2.11)
At comparatively long times T (longer then
relaxation times) ltfgt is equilibrium value of the
physical quantity f . In this case one can write
(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).
19
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 so-called 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.
20
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.
21
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).

22
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.
Fig.2.1 The system of N spins equal to 1/2.
Every arrow shows the direction of magnetic
moment.
23
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 (Nn) moments
being positive and ½ (N-n) moments being
negative. The problem for B0 is essentially
identical with the problem of the random walk, in
one dimension, the steps taken as of equal
length.
24
We observed first that the (normalized)
probability of a given specific sequence of
particles is
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 (Nn) of the particles one
way and the remainder ½ (N-n) point the other
way.
25
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 (Nn) and ½ (N-n) particles.
Interchanges of the ½ (Nn) particles purely
among themselves lead to nothing new, and there
are ½ (Nn)! 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 moment Mn? is
(2.13)
26
and the probability w(M) of a net moment Mn? is
obtained on multiplying (2.13) by the
probability (1/2)N of specific sequence giving
(2.14)
For large values of the factorials we may use
Stirlings approximation
(2.15)
(2.16)
27
Thus (2.14) gives, on taking the ln of both sides
and using (2.16)
(2.17)
For nltltN we use the series expansion
(2.18)
and so, after some transformations and collecting
terms we have
(2.19)
28
and so
(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.
29
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
30
The spectrum of all the system consist of N1
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 S1/2
31
(2.23)
When m ltltN this relation is transferred to the
Gauss distribution. It can be obtained taking ln
g or from direct using of Stirling formula.
(2.24)
The width of the distribution ??m?/N is
decreasing with increasing of N . It decreases
according to 1/?N.