Chapter 6: Basic Methods & Results of Statistical Mechanics

1. Chapter 6:Basic Methods & Resultsof Statistical Mechanics

2. Key Concepts In Statistical Mechanics Idea:Macroscopic properties are a thermal average of microscopic properties. • Replace the system with a set of systems "identical" to the first and average over all of the systems. We call the set of systems “The Statistical Ensemble”. • Identical Systems means that they are all in the same thermodynamic state. • To do any calculations we have to first Choose an Ensemble!

3. The Most Common Statistical Ensembles: 1. The Micro-Canonical Ensemble: Isolated Systems:Constant Energy E. Nothing happens!  Not Interesting! 3

4. The Most Common Statistical Ensembles: 1. The Micro-Canonical Ensemble: Isolated Systems:Constant Energy E. Nothing happens!  Not Interesting! 2. The Canonical Ensemble: Systems with a fixed number N of molecules In equilibrium with a Heat Reservoir(Heat Bath). 4

5. The Most Common Statistical Ensembles: 1. The Micro-Canonical Ensemble: Isolated Systems:Constant Energy E. Nothing happens!  Not Interesting! 2. The Canonical Ensemble: Systems with a fixed number N of molecules In equilibrium with a Heat Reservoir(Heat Bath). 3. The Grand Canonical Ensemble: Systemsin equilibrium with a Heat Bath which is also a Source of Molecules. Their chemical potential is fixed.

6. All Thermodynamic Properties Can Be Calculated With Any Ensemble Choose the most convenient one for a particular problem. For Gases:PVT properties use The Canonical Ensemble For Systems which Exchange Particles: Such asVapor-Liquid Equilibrium use The Grand Canonical Ensemble

7. J. Willard Gibbswas the first to show that An Ensemble Average is Equal to a Thermodynamic Average: That is, for a given property F, The Thermodynamic Average can be formally expressed as: F  nFnPn Fn  Value of F in state (configuration) n Pn  Probability of the system being in state (configuration) n. Properties of The Canonical& Grand Canonical Ensembles

8. Canonical Ensemble Probabilities QNcanon “Canonical Partition Function” gn  Degeneracy of state n Note that most texts use the notation “Z”for the partition function!

9. Grand Canonical Ensemble Probabilities: Qgrand “Grand Canonical Partition Function” or “Grand Partition Function” gn  Degeneracy of state n, μ  “Chemical Potential” Note that most texts use the notation “ZG”for the Grand Partition Function!

10. Partition Functions • Ifthe volume,V, the temperatureT, & the energy levelsEn, of a system are known, in principle The Partition Function Z can be calculated. • Ifthe partition function Zis known, it can be used To Calculate All Thermodynamic Properties. • So, in this way, Statistical Mechanics provides a direct linkbetween Microscopic Quantum Mechanics& Classical Macroscopic Thermodynamics.

11. Canonical Ensemble Partition Function Z Starting from the fundamental postulate of equal a priori probabilities,the following are obtained: ALL RESULTS of Classical Thermodynamics, plus their statistical underpinnings; A MEANS OF CALCULATINGthe thermodynamic variables (E, H, F, G, S ) from a single statistical parameter, the partition function Z(or Q), which may be obtained from the energy-levels of a quantum system. The partition function for a quantum system in equilibrium with a heat reservoir is defined as W Where εi is the energy of the i’thstate. Z i exp(- εi/kBT)

12. Partition Function for a Quantum System in Contact with a Heat Reservoir: , F εi = Energy of the i’th state. The connection to the macroscopic entropy function S is through the microscopic parameterΩ, which, as we already know, is the number of microstates in a given macrostate. The connection between them, as discussed in previous chapters, is Z i exp(- εi/kBT) S = kBln Ω. 12

13. Relationship of Z to Macroscopic Parameters Summary for the Canonical Ensemble Partition Function Z: (Derivations are in the book!) Internal Energy: Ē  E = - ∂(lnZ)/∂β <ΔE)2> = [∂2(lnZ)/∂β2] β = 1/(kBT),kB =Boltzmann’s constantt. Entropy: S = kBβĒ+ kBlnZ An important, frequently used result!

14. Summary for the Canonical Ensemble Partition Function Z: Helmholtz Free Energy F = E – TS = – (kBT)lnZ and dF = S dT – PdV, so S = – (∂F/∂T)V, P = – (∂F/∂V)T Gibbs Free Energy G = F + PV = PV – kBT lnZ. Enthalpy H = E + PV = PV – ∂(lnZ)/∂β

15. Canonical Ensemble:Heat Capacity & Other Properties Partition Function: Z = nexp (-En),  = 1/(kT)

16. Canonical Ensemble:Heat Capacity & Other Properties Partition Function: Z = nexp (-En),  = 1/(kT) Mean Energy: Ē = – (ln Z)/ = - (1/Z)Z/

17. Canonical Ensemble:Heat Capacity & Other Properties Partition Function: Z = nexp (-En),  = 1/(kT) Mean Energy: Ē = – (ln Z)/ = - (1/Z)Z/ Mean Squared Energy: E2 = rprEr2/rpr = (1/Z)2Z/2.

18. Canonical Ensemble:Heat Capacity & Other Properties Partition Function: Z = nexp (-En),  = 1/(kT) Mean Energy: Ē = – (ln Z)/ = - (1/Z)Z/ Mean Squared Energy: E2 = rprEr2/rpr = (1/Z)2Z/2. nth Moment: En = rprErn/rpr = (-1)n(1/Z) nZ/n

19. Canonical Ensemble:Heat Capacity & Other Properties Partition Function: Z = nexp (-En),  = 1/(kT) Mean Energy: Ē = – (ln Z)/ = - (1/Z)Z/ Mean Squared Energy: E2 = rprEr2/rpr = (1/Z)2Z/2. nth Moment: En = rprErn/rpr = (-1)n(1/Z) nZ/n Mean Square Deviation: (ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/ .

20. Canonical Ensemble:Constant Volume Heat Capacity CV = Ē/T = (Ē/)(d/dT) = - k2Ē/

21. Canonical Ensemble:Constant Volume Heat Capacity CV = Ē/T = (Ē/)(d/dT) = - k2Ē/ using results for the Mean Square Deviation: (ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/

22. Canonical Ensemble:Constant Volume Heat Capacity CV = Ē/T = (Ē/)(d/dT) = - k2Ē/ using results for the Mean Square Deviation: (ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/ CV can be re-written as: CV = k2(ΔE)2 = (ΔE)2/kBT2

23. Canonical Ensemble:Constant Volume Heat Capacity CV = Ē/T = (Ē/)(d/dT) = - k2Ē/ using results for the Mean Square Deviation: (ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/ CV can be re-written as: CV = k2(ΔE)2 = (ΔE)2/kBT2 so that: (ΔE)2 = kBT2CV

24. Canonical Ensemble:Constant Volume Heat Capacity CV = Ē/T = (Ē/)(d/dT) = - k2Ē/ using results for the Mean Square Deviation: (ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/ CV can be re-written as: CV = k2(ΔE)2 = (ΔE)2/kBT2 so that: (ΔE)2 = kBT2CV Note that, since(ΔE)2 ≥ 0 (i) CV≥ 0 and(ii) Ē/T ≥ 0.

25. Ensembles in ClassicalStatistical Mechanics As we’ve seen, classical phase space for a system with f degrees of freedom is f generalized coordinates & f generalized momenta (qi,pi). The classical mechanics problem is done in the Hamiltonian formulation with a Hamiltonian energy function H(q,p). There may also be a few constants of motion such as energy, number of particles, volume, ...

26. The Canonical Distribution in Classical Statistical Mechanics • The Partition Function • has the form: • Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT) • A 6N Dimensional Integral! • This assumes that we have already solved the classicalmechanics problem for each particle in the system so that we know the total energy E for the Nparticles as a function of all positions ri& momenta pi. • E  E(r1,r2,r3,…rN,p1,p2,p3,…pN)

27. CLASSICAL Statistical Mechanics: • Let A ≡any measurable, macroscopic quantity. The thermodynamic average of A ≡<A>. This is what is measured. Use probability theory to calculate <A> : P(E) ≡ e[-E/(kBT)]/Z <A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E) Another 6N Dimensional Integral!