1. Section 2.3: The Basic (Fundamental) Postulate of Statistical Mechanics

2. Total Energy is Conserved.
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.

3. All microstates accessible to it
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.

4. All microstates accessible to it
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.

5. All microstates accessible to it MUST have this energy E.
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?

6. Before answering this, lets define:
“Equilibrium”

7. “Equilibrium” A System in Equilibrium
Before answering this, lets define:
“Equilibrium”
A System in Equilibrium
is one for which the Macroscopic Parameters characterizing it are
Independent of Time.

8. Isolated System in Equilibrium: We can “handwave” the following:
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:

9. for an ensemble of similarly prepared systems,
“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.

10. OF IT’S ACCESSIBLE STATES
 It seems reasonable to ASSUME that the system is
EQUALLY LIKELY
TO BE FOUND IN ANY ONE
OF IT’S ACCESSIBLE STATES

11. OF IT’S ACCESSIBLE STATES
 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.

12. (equilibrium) Statistical Mechanics:
THE FUNDAMENTAL (!)
(BASIC) POSTULATE OF
(equilibrium) Statistical Mechanics:

13. (equilibrium) Statistical Mechanics:
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”

14. (BASIC) POSTULATE OF THE FUNDAMENTAL (!)
(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”

15. (BASIC) POSTULATE OF THE FUNDAMENTAL (!)
(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.

16. Vital Importance of this in Statistical Mechanics!
I want to emphasize the
Vital Importance of this in
Statistical Mechanics!

17. Vital Importance of this in 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.

18. Vital Importance of this in 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”

19. à-priori Probabilities”
“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”

20. Physics is an Experimental science!
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!

21. Physics is an Experimental Science So, lets accept it & continue on.
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.

22. 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:
(+,+,-) (+,-,+) (-,+,+)

23. 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.

24. Summary (+,+,-) (+,-,+) (-,+,+)
3 possible system states with energy
E ≡ - μH.
(+,+,-) (+,-,+) (-,+,+)
 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”.

25. 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.

26. 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.

27. 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.

28. E = ½(p2)/(m) + ½κx2 (1) 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) + ½κx (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?

29. E = ½(p2)/(m) + ½κx2 (1) E = Constant Example 3 p x
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) + ½κx (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?
Consider the (x,p) phase space.
In that space, E = constant, so
(1) is the equation of an ellipse: p x
E = Constant

30. δE ≡ Uncertainty in the Energy.
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|

31. δE ≡ Uncertainty in the Energy.
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.

32. 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

33. The Fundamental Postulate of Statistical Mechanics:
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.

34. Approach to Equilibrium

35. 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.”

36. 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.

37. 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?

38. NO!! But, in this situation we can use the
Fundamental Postulate to infer that either:

39. NO!! But, in this situation we can use the
Fundamental Postulate to infer that either:
1. The system is NOT ISOLATED

40. 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

41. 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.

42. time-dependent. Irreversible Statistical Mechanics
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.

43. 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.

44. 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.

45. The parameters characterizing the system will change with time
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, due to interactions between system particles
The parameters characterizing the system will change with time
until an equilibrium situation is reached.

46. The system parameters will
What Will Happen?
Due to interactions between the particles
The system parameters will
change with time.

47. The system parameters will
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.

48. The system parameters will
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.

49. The system parameters will 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”?

50. The system parameters will
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!

51. ≡ “Boltzmann’s H-Theorem”.
A principle of Non-Equilibrium
Statistical Mechanics:
“All isolated systems will, after a ‘sufficient time’, approach equilibrium”
≡ “Boltzmann’s H-Theorem”.

52. 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 (+,+,-)

53. 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.

54. equal probability in any one of it’s 3 accessible states:
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:
(+,+,-), (+,-,+), (-,+,+)

55. 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

56. 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.

57. 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 & come to a new equilibrium.