Classical Statistical Mechanics in the Canonical Ensemble

1. Classical Statistical Mechanicsin the Canonical Ensemble

2. The Equipartition Theorem is Valid in Classical Stat. Mech. ONLY!!!

3. Classical Statistical Mechanics 1. The Equipartition Theorem 2. The Classical Ideal Gas a. Kinetic Theory b. Maxwell-BoltzmannDistribution

4. The Equipartition Theoremin Classical Statistical Mechanics (ONLY!) It states: “Each degree of freedom in a system of particles contributes(½)kBTto the thermal average energyĒ of the system.”

5. The Equipartition Theorem is Valid in Classical Stat. Mech. ONLY!!! “Each degree of freedom in a system of particles contributes(½)kBTto the thermal average energyĒ of the system.”Note: 1. This is strictly valid only if each term in the classical energy is proportional to a momentum (p) squared or to a coordinate (q) squared. 2. The degrees of freedom are associated with translation, rotation & vibration of the system’s molecules.

6. In the Classical Cannonical • Ensemble, it is straighforward to • show that • The average energy of a particle per independent degree of freedom (½)kBT. • We just finished an outline of the proof

7. The Boltzmann (or Maxwell-Boltzmann) Distribution • Start with the Canonical Ensemble Probability FunctionP(E): • This is defined so that P(E) dE • probability to find a particular molecule • betweenE & E + dEhas the form: Z  Partition Function Z

8. The Boltzmann Distribution • Canonical Ensemble Probability • FunctionP(E): Z • Define: Energy Distribution Function •  Number Density nV(E): • Defined so that  nV(E) dE Number of • molecules per unit volume with energy • between E& E + dE

9. Examples: Equipartition of Energy in Classical Statistical Mechanics Free Particle (One dimension): Z

10. Equipartition Theorem Examples LC Circuit: 1 d Harmonic Oscillator:

11. Equipartition Theorem Examples Free Particle in 3 Dimensions: Rotating Rigid Body:

12. 1d Simple Harmonic Oscillator

13. Classical Ideal Monatomic Gas • For this system, it’s easy to show • thatthe Temperature T is related • to theaveragekinetic energy. For • 1 molecule moving with velocity v • in 3 d, equipartition takes the form: • For each degree of freedom, it’s easy to show:

14. Classical Statistical Mechanics: Canonical Ensemble Averages Probability Function: Z • P(E) dEprobability to find a • particular molecule betweenE & E + dE Normalization:

15. So: Z Average Energy: Average Velocity:

16. Classical Kinetic Theory Results • We just saw that, from the Equipartition Theorem, the kinetic energy of each particle in an ideal gas is related to the gas temperature as: <E> = (½)mv2 = (3/2)kBT (1) v is the thermal average velocity. • Canonical Ensemble Probability Function: Z • In this form, P(E) is known as the • Maxwell-Boltzmann Energy Distribution

17. Z • Using <E> = (½)mv2 = (3/2)kBTalong with P(E), the Probability Distribution of Energy Ecan be converted into a • Probability Distribution of Velocity P(v) • This has the form: • P(v) = C exp[- (½)m(v)2/(kT)] • In this form, P(v) is known as the • Maxwell-Boltzmann Velocity Distribution

18. Kinetic Molecular Model for Ideal GasesDue originally to Maxwell & Boltzmann Assumptions The gas consists of large number of individual point particles (zero volume). Particles are in constant random motion & collisions. No forces are exerted between molecules. Equipartition Theorem: Gas Average Kinetic Energy is Proportional to the Temperature in Kelvin.

19. Maxwell-Boltzmann Velocity Distribution • The Canonical Ensemble gives a distribution • of molecules in terms of Speed/Velocity or • Energy. • The 1-Dimensional Velocity Distribution • in the x-direction (ux) has the form:

20. Maxwell-Boltzmann Velocity Distribution High T Low T

21. 3D Maxwell-Boltzmann Velocity Distribution a  (½)[m/(kBT)] In Cartesian Coordinates:

22. Maxwell-Boltzmann Speed Distribution • Change to spherical coordinates in Velocity • Space. Reshape the box into a sphere in • velocity space of the same volume with radius u. • V = (4/3) u3with u2 = ux2 + uy2 + uz2 • dV = duxduyduz = 4  u2 du

23. 3D Maxwell-Boltzmann Speed Distribution Low T High T

24. Maxwell-Boltzmann Speed Distribution • Convert the speed-distribution into an energy • distribution:  = (½)mu2, d = mu du

25. Some Important Velocity Values from the M-B Distribution • urms = root mean square (rms) velocity • uavg = average speed • ump = most probable velocity

26. Comparison of Velocity Values

27. Maxwell-Boltzmann Velocity Distribution Z Z

28. Maxwell-BoltzmannSpeedDistribution

29. Maxwell-BoltzmannSpeedDistribution

30. The Probability Density Function • Random motions of the molecules can be • characterized by a probability distribution • function. Since the velocity directions are • uniformly distributed, the problem reduces to a • speed distribution. • The function f(v)dvis isotropic. f(v)dv • fractional number of mol eculesin the speed range • from v to v + dv. Of course, a probability • distribution function has to satisfy the condition:

31. The Probability Density Function • We can use the distribution function to • compute the average behavior of the • molecules: