Download Part III - TTU Physics

Section 2.3: The Basic (Fundamental)
Postulate of Statistical Mechanics
•Definition: ISOLATED SYSTEM
≡ A system which has no interaction
of any kind with the “outside world”
• This is clearly an idealization! Such a system has No
Exchange of Energy (or particles) with the outside world.
 The laws of mechanics tell us that the
total energy E of this system is conserved.
E ≡ Constant
• So, an isolated system is one for which
Total Energy is Conserved.
• Consider an isolated system. The total
energy E is constant & the system is
characterized by this energy. So
All microstates accessible to it
MUST have this energy E.
• For many particle systems, there are usually a
HUGE number of microstates with the same energy.
Question: What is the probability of
finding the system in any one of
these accessible states?
•Before answering this, lets define:
“Equilibrium”
A System in Equilibrium
is one for which the
Macroscopic Parameters
characterizing it are
Independent of Time.
Isolated System in Equilibrium:
•In the absence of any experimental
data on some specific system
properties, all we can really say
about this system is that it must be
in one of it’s accessible states (with
that energy). If this is all we know,
We can “handwave” the following:
“Handwave”:
•There is nothing in the laws of
mechanics (classical or quantum) which
would lead us to suspect that,
for an ensemble of similarly
prepared systems,
we should find the system in some (or
any one) of it’s accessible microstates
more frequently than in any of the others.
 It seems reasonable to
ASSUME that the system is
EQUALLY LIKELY
TO BE FOUND IN ANY ONE
OF IT’S ACCESSIBLE STATES
• In (equilibrium) statistical mechanics, we make
this assumption & elevate it to the level of a
POSTULATE.
THE FUNDAMENTAL (!)
(BASIC) POSTULATE OF
(equilibrium) Statistical Mechanics:
“An isolated system in equilibrium
is equally likely to be found in any
one of it’s accessible microstates”
• This is sometimes called the
“Postulate of Equal à-priori Probabilities”
• This is the basic postulate (& the only postulate)
of equilibrium statistical mechanics.
• I want to emphasize the
Vital Importance of this in
Statistical Mechanics!
• I can’t stress enough that
This is the Fundamental, Basic (&
only) Postulate of equilibrium
statistical mechanics.
“An isolated system in equilibrium
is equally likely to be found in any
one of it’s accessible microstates”
“An isolated system in
equilibrium is equally likely
to be found in any one of it’s
accessible microstates”
• This is sometimes called the
“Postulate of Equal
à-priori Probabilities”
• With this postulate, we can (& will) derive
ALL of
1. Classical Thermodynamics,
2. Classical Statistical Mechanics,
3. Quantum Statistical Mechanics.
• It is reasonable & it doesn’t contradict any laws of
classical or quantum mechanics. But, is it valid &
is it true? To answer this, remember that
Physics is an Experimental
science!
• The only practical way to see if this hypothesis is valid is
to develop a theory based on it & then to remember that
Physics is an Experimental Science
• Whether the Fundamental postulate is valid or not
can only be decided by
Comparing the predictions of a
theory based on it with
experimental data.
• A HUGE quantity of data taken over 250+ years exists!
None of this data has been found to be in disagreement
with the theory based on this postulate.
So, lets accept it & continue on.
Simple Examples: Example 1
• Back to the example of 3 spins, an isolated system
in equilibrium. Suppose that the total energy is
measured as: E ≡ - μH.
• We’ve seen that the only 3 possible system states
consistent with this energy are:
(+,+,-) (+,-,+) (-,+,+)
 The Fundamental Postulate says
that, when the system is in equilibrium,
it is equally likely (with probability = ⅓) to
be in any one of these 3 states.
Summary
• 3 spins, isolated & in equilibrium. Total energy:
E ≡ - μH.
• 3 possible system states consistent with this energy:
(+,+,-) (+,-,+) (-,+,+)
 The Fundamental Postulate tells us it is
equally likely (with probability = ⅓) to be in
any one of these 3 states.
• This probability is about the system, NOT about
individual spins. Under these conditions, it is obviously
NOT equally likely that an individual spin is “up” & “down”.
It is twice as likely for a given spin to be “up” as “down”.
Example 2
• Consider N (~ 1024) spins, each with spin = ½. Put the
system in an external magnetic field H. The total energy
is measured & found to be: E ≡ - μH.
• This is similar to the 3 spin system, but now there are a
HUGE number of accessible states. The number of
accessible states is equal to the number of possible ways
for the energy of N spins to add up to - μH.
 The Fundamental Postulate says
that, when the system is in equilibrium,
it is equally likely to be in any one of
these HUGE numbers of states.
Example 3
• Classical Illustration: Consider a 1-dimensional,
classical, simple harmonic oscillator mass m, with spring
constant κ, position x & momentum p. Total energy:
E = ½(p2)/(m) + ½κx2
(1)
• E is determined by the initial conditions. If the
oscillator is isolated, E is conserved. How do we find
the number of accessible states for this oscillator?
p
• Consider the (x,p) phase space.
In that space, E = constant, so
(1) is the equation of an ellipse:
x
E = Constant
• If we knew the oscillator energy E exactly, the
accessible states would be the points on the
ellipse. In practice, we never know the energy
exactly! There is always an experimental error δE.
δE ≡ Uncertainty in the Energy.
We always assume: |δE| <<< |E|
• For the geometrical picture in the x-p plane,
this means that the energy is somewhere
between 2 ellipses, one corresponding to E
& the other corresponding to E + δE.
δE ≡ Uncertainty in the energy.
Always: |δE| <<< |E|
• In the x-p plane, the energy is somewhere between the
2 ellipses, one corresponding to E & the other
corresponding to E + δE. See the figure:
# accessible states ≡ # phase
space cells between 2 ellipses
≡ (A/ho) A ≡ area between
ellipses & ho ≡ qp
• In general, there are many cells in the phase space area
between the ellipses (ho is “small”). So, there are a
A HUGE NUMBER of accessible microstates
for the oscillator with energy between E & E + δE.
• That is, there are many possible values of (x,p) for a
set of oscillators in an ensemble of such oscillators.
The Fundamental Postulate
of Statistical Mechanics:
 All possible values of (x,p) with energy
between E & E + δE are equally likely.
• Stated another way, ANY CELL in phase space
between the ellipses is equally likely.
Approach to Equilibrium
The Fundamental Postulate
of Statistical Mechanics:
“An Isolated system in Equilibrium
is equally likely to be in any one
of it’s accessible micro states.”
• Suppose that we know that, in a certain situation, a
particular system is NOT equally likely to be in
any one of it’s accessible states.
•Is this a violation of the
Fundamental Postulate?
NO!!
• But, in this situation we can use the
Fundamental Postulate to infer that either:
.
NO!!
• But, in this situation we can use the
Fundamental Postulate to infer that either:
1. The system is NOT ISOLATED
NO!!
• But, in this situation we can use the
Fundamental Postulate to infer that either:
1. The system is NOT ISOLATED
or
2. The system is NOT IN EQUILIBRIUM
NO!!
• But, in this situation we can use the
Fundamental Postulate to infer that either:
1. The system is NOT ISOLATED
or
2. The system is NOT IN EQUILIBRIUM
• In this course, we’ll spend most of our time
discussing item 1. That is, we’ll discuss systems
which are not isolated. Now, here, we’ll very
BRIEFLY discuss item 2. That is, we’ll briefly
discuss systems which are not in equilibrium.
Non-Equilibrium Statistical Mechanics:
This is still a subject of research in the
21st Century. It is sometimes called
Irreversible Statistical Mechanics
• If a system is not in equilibrium, we expect
the situation to be a time-dependent one.
•That is, the average values of various
macroscopic parameters will be
TIME-DEPENDENT.
• Suppose, at time t = 0, an ISOLATED system is
known to be in only a subset of the states accessible
to it. There are no restrictions which would then
prevent the system from being found in ANY ONE
of it’s accessible states at some time t > 0 later.
• Therefore, it is very improbable that the system
will remain in this subset of its accessible states.
•That is,
The parameters characterizing
the system will change with time
until an equilibrium situation is reached.
What Will Happen?
• Due to interactions between the particles
The system parameters will
change with time.
What Will Happen?
• Due to interactions between the particles
The system parameters will
change with time.
• It will make transitions between its various accessible states.
After a long time, we would expect an ensemble of similar
systems to be uniformly distributed over the accessible states.
What Will Happen?
• Due to interactions between the particles
The system parameters will
change with time.
• It will make transitions between its various accessible states.
After a long time, we would expect an ensemble of similar
systems to be uniformly distributed over the accessible states.
• That is, equilibrium will be reached if we wait “long
enough”. After that time, the system will be equally likely
to be found in any one of it’s accessible states.
What Will Happen?
• Due to interactions between the particles
The system parameters will
change with time.
• It will make transitions between its various accessible states.
After a long time, we would expect an ensemble of similar
systems to be uniformly distributed over the accessible states.
• That is, equilibrium will be reached if we wait “long
enough”. After that time, the system will be equally likely
to be found in any one of it’s accessible states.
How long is “long enough”?
What Will Happen?
• Due to interactions between the particles
The system parameters will
change with time.
• It will make transitions between its various accessible states.
After a long time, we would expect an ensemble of similar
systems to be uniformly distributed over the accessible states.
• That is, equilibrium will be reached if we wait “long
enough”. After that time, the system will be equally likely
to be found in any one of it’s accessible states.
How long is “long enough”?
• Depends on the system. It could be femtoseconds,
nanoseconds, centuries, or billions of years!
A principle of Non-Equilibrium
Statistical Mechanics:
“All isolated systems will,
after a ‘sufficient time’,
approach equilibrium”
≡ “Boltzmann’s HTheorem”.
Example 1
• Consider the 3 spin system in an external
magnetic field again. Suppose we know that the
total energy is E = -μH. Suppose that we prepare
the system so it is in the state (+,+,-)
Example 1
• Consider the 3 spin system in an external
magnetic field again. Suppose we know that the
total energy is E = -μH. Suppose that we prepare
the system so it is in the state (+,+,-)
• Recall that this is only 1 of the 3 accessible states
consistent with this energy. Now allow some
“small” interactions between the spins.
Example 1
• Consider the 3 spin system in an external
magnetic field again. Suppose we know that the
total energy is E = -μH. Suppose that we prepare
the system so it is in the state (+,+,-)
• Recall that this is only 1 of the 3 accessible states
consistent with this energy. Now allow some “small”
interactions between the spins.
• These can “flip” them.
 We expect that, after a long enough time, an
ensemble of similar systems will be found with
equal probability in any one of it’s 3 accessible states:
(+,+,-), (+,-,+), (-,+,+)
• Example 2: Consider a gas in a container, divided by a
partition into 2 equal volumes V. It is in equilibrium &
confined by the partition to the left side. See Figure.
Gas
Vacuum
• Example 2: Consider a gas in a container, divided by a
partition into 2 equal volumes V. It is in equilibrium &
confined by the partition to the left side. See Figure.
Gas
Vacuum
• Remove the Partition. The new situation is
clearly NOT an equilibrium one. All accessible states
in the right side are NOT filled. Now, wait some time.
• Example 2: Consider a gas in a container, divided by a
partition into 2 equal volumes V. It is in equilibrium &
confined by the partition to the left side. See Figure.
Gas
Vacuum
• Remove the Partition. The new situation is
clearly NOT an equilibrium one. All accessible states
in the right side are NOT filled. Now, wait some time.
• As a result of collisions between molecules,
They’ll eventually distribute themselves
uniformly over the entire volume 2V.