Section 2.2: Statistical Ensemble

1. Section 2.2: Statistical Ensemble A typical episode from the on-line comic series “PhD Comics” by Jorge Cham There are also movies! “PhD Movie I” &“PhD Movie II”

2. As we know, Statistical Mechanics deals with the behavior of systems with a large number of particles. • Because the number of particles is so huge, we give up trying to keep track of individual particles. • We can’t solve Schrödinger’s Equation in closed form for helium (4 particles), so what hope do we have of solving it for the gas molecules in this room (10f particles) ?? • Statistical Mechanics handles many particles by Calculating the Most Probable Behavior of the System as a Whole rather than by being concerned with the behavior of individual particles.

3. In Statistical Mechanics • We assume that the more ways there are to • arrange the particles to give a particular distribution • of energies, the more probable that distribution is.

4. In Statistical Mechanics • We assume that the more ways there are to • arrange the particles to give a particular distribution • of energies, the more probable that distribution is. Example: 6 energy units, 3 particles to give it to

5. 3 2 1 4 1 1 3 1 2 1 41 2 1 3 1 1 4 2 3 1 3 ways 1 2 3 1 3 2 6 ways • In Statistical Mechanics • We assume that the more ways there are to • arrange the particles to give a particular distribution • of energies, the more probable that distribution is. Example: 6 energy units, 3 particles to give it to Color Codes for Energy Units ≡ 1 ≡ 2 ≡ 3 ≡ 4 Most Probable Distribution

6. Another Example • Assuming that All Energy Distributions are Equally Probable • If E = 5 and N = 5 then 3 possible configurations are: 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 • All possible configurations have equal probability, but • the possible number of ways (weight) is different for each.

7. The Dominant Configuration • For a large number of molecules & a large number of energy levels, there is a Dominant Configuration. • In the probability distribution, the weight of the dominant configuration is much larger than the weight of the other configurations. Weight of the Dominant Configuration Weights Wi {ni}  Configurations

8. The Dominant Configuration If E = 5 and N = 5 then 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 W = 1 = (5!/5!) W = 20 = (5!/3!) W = 5 = (5!/4!) • The difference in the W’s becomes larger as N increases! • In molecular systems (N~1023) considering the most dominant • configuration is certainly enough to calculate averages.

9. The Principle of Equal à-prioriProbabilities • Statistical thermodynamics is based on the fundamental hypothesisof Equal à-prioriProbabilities. • That is, all possible configurations of a given systemwhich satisfy the given boundary conditions such as temperature, volume and number of particles, are equally likely to occur.OR The system is equally likely to be found in any one of its accessible microstates.

10. Example • Consider again the orientations of three unconstrained & distinguishable spin-1/2 particles. • What is the probability that 2 are spin up & 1 is down at any instant?

11. Solution Of the eight possible spin configurations for the system: ↑↑↑, ↑↑↓, ↑↓↑, ↓↑↑, ↑↓↓, ↓↑↓, ↓↓↑, ↓↓↓

12. Solution Of the eight possible spin configurations for the system: ↑↑↑, ↑↑↓, ↑↓↑, ↓↑↑, ↑↓↓, ↓↑↓, ↓↓↑, ↓↓↓ • The second, third, & fourth make up the subset "two up and one down".

13. Solution Of the eight possible spin configurations for the system: ↑↑↑, ↑↑↓, ↑↓↑, ↓↑↑, ↑↓↓, ↓↑↓, ↓↓↑, ↓↓↓ • The second, third, & fourth make up the subset "two up and one down". Therefore, The probability of occurrence of this particular configuration is P = 3/8

14. In principle, the problem of a many particle system is completely deterministic: • If we specify the many particle wavefunctionΨ(state) of the system (or the classical phase space cell) at time t = 0, we can determine Ψ for all other times t by solving The Time-Dependent Schrödinger Equation & from Ψ(t) we can calculate all observable quantities. • Or, classically, if we specify the positions & momenta of all particles at time t = 0, we can predict the future behavior of the system by solving The Coupled Many Particle Newton’s 2nd Law Equations of Motion.

15. Generally, we usually don’t have such a complete specification of the system available. • We need f quantum numbers, but f ≈ 1024! • Actually, we usually aren’t interested in such a complete microscopic description anyway. Instead, we’re interested in predictions of MACROSCOPIC properties.  We use Probability & Statistics.  To do this we need the concept of an ENSEMBLE.

16. A Statistical Ensembleis a LARGE number (≡ N) of identically prepared systems. • In general, the systems of this ensemble will be in different states & thus will have different macroscopic properties. We ask for the probability that a given macroscopic parameter will have a certain value. A Goal or Aim of Statistical Mechanics is to Predict this Probability.

17. Example • Consider the spin problem again. Let The System Have N = 3 Particles, fixed in position, each with spin = ½  Each spin is either “up” (↑, m = ½) or “down” (↓, m = -½). • Each particle has a vector magnetic moment μ. • The projection of μalong a “z-axis” is either: μz = μ, for spin “up” or μz = -μ, for spin “down”

18. Put this system into an External Magnetic Field H. • Classical E&M tells us that a particle with magnetic moment μin an external field H has energy: ε = - μH • Combine this with the Quantum Mechanical result: This tells us that each particle has 2 possible energies: ε+ ≡ - μHfor spin “up” ε- ≡ μHfor spin “down” So, for 3 particles, the Microstateof the system is specified by specifying each m = ½  There are(2)3 = 8 Possible Microtates!!

19. Possible States of a 3 Spin System • Given that we know no other information about this • system, all we can say about it is that • It has Equal Probability of Being Found • in Any One of These 8 States.

20. However, if (as is often the case in real problems) we have a partial knowledge of the system (say, from experiment), then, we know that The system can be only in any one of the states which are COMPATIBLE with our knowledge. (That is, it can only be in one of it’saccessible states)

21. However, if (as is often the case in real problems) we have a partial knowledge of the system (say, from experiment), then, we know that The system can be only in any one of the states which are COMPATIBLE with our knowledge. (That is, it can only be in one of it’saccessible states) “States Accessible to the System” ≡ those states which are compatible with all of the knowledge we have about the system. Its important to use all of the information that we have about the system!

22. Example • For our 3 spin system, suppose that we measure the total system energy & we find E ≡ - μH

23. Example • For our 3 spin system, suppose that we measure the total system energy & we find • E ≡ - μH • This additional information limits the states which are accessible to the system. Clearly, from the table, out of the 8 states, only 3 are compatible with this knowledge.

24. Example • For our 3 spin system, suppose that we measure the total system energy & we find • E ≡ - μH • This additional information limits the states which are accessible to the system. Clearly, from the table, out of the 8 states, only 3 are compatible with this knowledge. •  The system must be in one of the 3 states: • (+,+,-) (+,-,+) (-,+,+)

25. Example • For our 3 spin system, suppose that we measure the total system energy & we find • E ≡ - μH • This additional information limits the states which are accessible to the system. Clearly, from the table, out of the 8 states, only 3 are compatible with this knowledge. •  The system must be in one of the 3 states: • (+,+,-) (+,-,+) (-,+,+) • By the fundamental postulate, it is equally likely to be in any one of these 3 states