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

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

2. 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. Consider an isolated system. The total energy E is constant & the system is characterized by this energy. So All microstates accessible to it MUSThave this energy E.

4. Consider an isolated system. The total energy E is constant & the system is characterized by this energy. So All microstates accessible to it MUSThave this energy E. • For many particle systems, there are usually aHUGEnumber of microstates with the same energy.

5. Consider an isolated system. The total energy E is constant & the system is characterized by this energy. So All microstates accessible to it MUSThave this energy E. • For many particle systems, there are usually aHUGEnumber 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. Before answering this, lets define: “Equilibrium” A System in Equilibrium is one for which the Macroscopic Parameterscharacterizing it are Independent of Time.

8. 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:

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

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

13. THE FUNDAMENTAL (!) (BASIC)POSTULATE OF (equilibrium)Statistical Mechanics: “An isolated system in equilibrium isequally likely to be found in any one of it’s accessible microstates”

14. THE FUNDAMENTAL (!) (BASIC)POSTULATE OF (equilibrium)Statistical Mechanics: “An isolated system in equilibrium isequally likely to be found in any one of it’s accessible microstates” • This is sometimes called the “Postulate of Equalà-prioriProbabilities”

15. THE FUNDAMENTAL (!) (BASIC)POSTULATE OF (equilibrium)Statistical Mechanics: “An isolated system in equilibrium isequally likely to be found in any one of it’s accessible microstates” • This is sometimes called the “Postulate of Equalà-prioriProbabilities” • This is the basic postulate (& the only postulate) of equilibrium statistical mechanics.

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

17. I want to emphasize the Vital Importance of this in Statistical Mechanics! • I can’t stress enough that This is the Fundamental, Basic(& only) Postulateof equilibrium statistical mechanics.

18. I want to emphasize the Vital Importance of this in Statistical Mechanics! • I can’t stress enough that This is the Fundamental, Basic(& only) Postulateof equilibrium statistical mechanics. “An isolated system in equilibrium isequally likely to be found inany one of it’s accessible microstates”

19. “An isolated system in equilibrium isequally likely to be found in any one of it’s accessible microstates” • This is sometimes called the “Postulate of Equal à-prioriProbabilities”

20. 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. 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 HUGEquantity 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 oneof 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 NOTequally 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. 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?

29. 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? • Consider the (x,p) phase space. In that space, E = constant, so (1) is the equation of an ellipse: p x E = Constant

30. 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. 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. In general, there are many cells in the phase space area between the ellipses (hois “small”). So, there are a A HUGE NUMBER of accessible microstates for the oscillator with energy betweenE &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 Isolatedsystem in Equilibrium is equally likely to be in any one of it’s accessible micro states.”

36. The Fundamental Postulate of Statistical Mechanics: “An Isolatedsystem 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 Isolatedsystem 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. 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-dependentone. • That is, the average values of various macroscopic parameters will be time-dependent.

43. Suppose, at time t = 0, anISOLATEDsystem 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, anISOLATEDsystem 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. Suppose, at time t = 0, anISOLATEDsystem 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. What Will Happen? • Due to interactions between the particles The system parameters will change with time.

47. 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. 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. 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. 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!