Chapter 6: Basic Methods & Results of Statistical Mechanics

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! 2

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

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 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). 3. The Grand Canonical Ensemble: Systems in equilibrium with a Heat Bath which is also a Source of Molecules. Their chemical potential is fixed. 5

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 as Vapor-Liquid Equilibrium The Grand Canonical Ensemble 6

F  nFnPn The Thermodynamic Average Properties of The Canonical & Grand Canonical Ensembles J. Willard Gibbs was 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. 7

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

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! 9

The Partition Function Z Statistical Mechanics Partition Functions If the volume, V, the temperature T, & the energy levels En, of a system are known, in principle The Partition Function Z can be calculated. If the partition function Z is known, it can be used To Calculate All Thermodynamic Properties. So, in this way, Statistical Mechanics provides a direct link between Microscopic Quantum Mechanics & Classical Macroscopic Thermodynamics. 10

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 CALCULATING the 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’th state. Z  i exp(- εi/kBT) 11

εi = Energy of the i’th state. 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 parameter Ω, 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 12

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! 13

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 14

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

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

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

Canonical Ensemble: Heat Capacity & Other Properties Partition Function: Z = n exp (-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 18

Canonical Ensemble: Heat Capacity & Other Properties Partition Function: Z = n exp (-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/ . 19

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

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/ 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/ CV can be re-written as: CV = k2(ΔE)2 = (ΔE)2/kBT2 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 so that: (ΔE)2 = kBT2CV 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 Note that, since (ΔE)2 ≥ 0 (i) CV ≥ 0 and (ii) Ē/T ≥ 0. 24

Ensembles in Classical Statistical 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, ... 25

The Partition Function 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 classical mechanics problem for each particle in the system so that we know the total energy E for the N particles as a function of all positions ri & momenta pi. E  E(r1,r2,r3,…rN,p1,p2,p3,…pN) 26

P(E) ≡ e[-E/(kBT)]/Z 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!