## 8 044 Lecture Notes Chapter 6 Statistical Mechanics At-Free PDF

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6 1 Derivation of the Canonical Ensemble, In Chapter 4 we studied the statistical mechanics of an isolated system This meant fixed. From some fundamental principles really postulates we developed an algorithm for cal. culating which turns out not to be so practical as you ll have seen e g if you thought about. the random 2 state systems on pset 6,1 Model the system. 2 Count microstates for given E E V N,3 from derive thermodynamics by S kB ln. 4 from derive microscopic information by p X, The fixed energy constraint makes the counting difficult in all but the simplest problems. the ones we ve done Fixing the temperature happens to be easier to analyze in practice. Consider a system 1 which is not isolated but rather. is in thermal contact with a heat bath 2 and therefore. held at fixed temperature T equal to that of the bath. An ensemble of such systems is called the canonical en. semble I don t know why,heat bath T, The fact that T is fixed means E is not energy can be.
exchanged between the system in question and the reservoir. Assume that 1 2 together are isolated with fixed energy Etotal E1 E2 Then we. can apply the microcanonical ensemble to 1 2 Note that 1 could be itself macroscopic. it just has to have a much smaller CV than 2 in which case we can learn about its. thermodynamics Alternatively 1 could be microscopic with just a few degrees of freedom. like one square of a grid of 2 state systems in which case we can ask for the probability. that it is ON, Consider a specific microstate A of 1 with energy E1. Q What the equilibrium probability that system 1 is in state A We can apply the method. of Chapter 4 to 1 2,Prob system 1 is in state A p p1 q1. z 1 2 Etotal, specific values for all vars of system 1 in state A. of microstates of system 2 with energy Etotal E1, total of microstates of 1 2 with energy Etotal E1 E2 fixed. 2 Etotal E1,1 2 Etotal,Take logs for smoother variation.
kB ln p p1 q1 S2 Etotal E1 S1 2 Etotal, So far what we ve done would be valid even if systems 1 and 2 were of the same size. But now let s use the fact that 2 is a reservoir by recognizing that E1 Etotal We should. Taylor expand in small E1,kB ln p p1 q1,S2 Etotal E1 E2 S2 E2 E2 Etotal O E1 S1 2 Etotal. where T is the temperature of the bath 2 Here we are using the fact that 2 is a reservoir. in saying that its temperature remains T even if we ignore the contribution of E1 to Etotal. kB ln p p1 q1 S2 Etotal S1 2 Etotal O E12,independent of microstate of 1. H1 here is the energy of system 1 in the specified microstate A 1. H1 p1 q1 S2 Etotal Stotal Etotal,p p1 q1 e kB T,Boltzmann factor C indep of microstate of 1. End of Lecture 14,H is for Hamiltonian It s not the enthalpy.
Q why can we ignore the O E12 terms in the Taylor expansion. A because they become small very rapidly as we make the reservoir larger and we can. make the reservoir as large as we want In particular the next term of the Taylor expansion. of S2 Etotal E1 is,What s this last term,2 S take one derivative. 1 chain rule 1 T def of CV 1 1,E E T T E T CV, This CV is the heat capacity of the reservoir the defining property of the reservoir is the. hugeness of its heat capacity So the biggest term we are ignoring is of magnitude. and we should compare this to the last term we keep which is ET1 The term we are ignoring. becomes smaller and smaller as we add degrees of freedom to the reservoir e g it would go. like 1 N if it were comprised of N ideal gas atoms whereas E1 T does not. Boltzmann factor,In particular,p p1 q1 e T,Energy scale is set by kB T. Recall that in the ensemble with fixed energy we didn t ever compare microstates with. different energies, Microstates with high low energy are less more probable. This last statement is NOT the same as higher energy is less probable Suppose. there is some set of microstates of 1 with the same energy E1 Then. p 1 is in some state with energy E1 e BT degeneracy. of microstates with energy E1, This last factor called the density of states can contain a lot of physics It is the.
number of microstates of system 1 with energy E1 also known as. 1 E1 eS1 E1 kB,Notice that it depends on E1,Partition function. Missing the normalization of the probability distribution. 1 dp1 dq1 p p1 q1,all microstates of system 1,dp1 dq1 e H1 p1 q1 kB T C. the thing above determined by this equation,e H1 p1 q1 kB T e H1 p1 q1 kB T. Z dp1 dq1 e H1 p1 q1 kB T, This quantity which enters our consciousness as a normalization factor. Z dp1 dq1 e H1 p1 q1 kB T partition function, is called the partition function and it is the central object in the canonical ensemble Z is.
for Zustandssumme German for state sum, To recap our answer for the equilibrium probability distribution at fixed temperature is. 1 H1 p1 q1 kB T,p p1 q1 e Boltzmann distribution, This is the probability that system 1 is in the microstate labelled by p1 q1 when it is. in contact with a heat bath at temperature T and in equilibrium We derived this by. applying the microcanonical ensemble to system 1 plus the heat bath. Fixed E1 microcanonical chapter 4, Constrained integral that counts microstates with fixed E1. Integrand 1,Limits of integration tricky,Fixed T canonical chapter 6. Suitably weighted integral over all microstates of 1. Integrand is e H not 1 not just counting, Limits of integration straightforward Integrate over everything including some very un.
likely states, Note for a system with discrete states labelled i 1 2. p system 1 is in a specific microstate i e B,with Z e BT. i all states of the system,Thermodynamics from the partition function. So we already have the microscopic info we ve found p microstate at fixed T We haven t. yet assumed that 1 is thermodynamically large Next suppose 1 is also macroscopic. and let s learn to extract its thermodynamics from the partition function Z. Compare expressions for Z,e S2 Etotal S1 2 Etotal kB. Does the RHS really depend on system 2,S1 2 Etotal S1 hE1 i S2 hE2 i.
mean E1 in equilibrium mean E2 in equilibrium,S2 Etotal S2 Etotal hE1 i hE1 i E hE i. S1 hE1 i T1, Everything having to do with 2 has disappeared and we ve written the answer in terms of. thermodynamic variables of 1 So we have two true expressions for Z. H1 kB T k1 S1 hE1 i T1,Z e and Z e B,microstates of 1. the left one involving only microscopic information about 1 and the right one involving. only thermodynamics, Bridge between thermodynamics and canonical ensemble. We can now drop the notational baggage we ve been carrying around the subscripts and. the h is since everything refers to system 1 in thermal equilibrium We ve written Z in. terms of the thermodynamic variables of the system of interest 1. k 1 T E T S k FT, where I remind you that F T V N E T S is the Helmholtz free energy.
dF SdT P dV 2, This is the bridge between microscopic stuff Z and thermodynamics F in the canonical. F kB T ln Z T V N,All of thermodynamics follows from this. Previously ch 4 count find find S kB ln take derivatives. Now ch 6 compute Z find F kB T ln Z then from 2 we have. E F T S H E P V G H T S,A simpler way to get E,T 2 T V kB ln Z ln Z V. This concludes the derivation of the canonical ensemble The canonical ensemble is the. primary tool of the practicing statistical mechanic What to remember from Chapter 4 i e. the most important application of the microcanonical ensemble how to derive the canonical. Next a warning about a common misconception then an important special case Then. many examples the rest of 8 044, Probability for a fixed microstate vs probability for a fixed energy. For a system in equilibrium at fixed temperature T we have. p system is in microstate A Z 1 e EA kB T, However the probability distribution for the energy of the system.
p E is NOT proportional to e E kB T, Rather the dependence of this quantity on the energy must also include a degeneracy factor. p E Z 1 e E kB T degeneracy, For a discrete system the quantity in brackets is the number of microstates with energy. E the degeneracy This depends on E Because of this factor p E can have a completely. different dependence on E,If E is continuous then,p E Z 1 e E kB T D E 3. p E dE prob E energy E dE,D E dE the number of states with E energy E dE. D E is called the density of states Our first example below will be the ideal gas which. will illustrate this nicely,Putting things on top of other things.
Suppose the system separates into parts with independent degrees of freedom. q p qa pa qb pb, that is to specify a state of the system we have to specify a state of part a and a state of. part b and,H q p Ha qa pa Hb qb pb, so that there are no interactions between the parts Then the Boltzmann factor is a product. e H q p kB T e Ha qa pa kB T e Hb qb pb kB T,p q p p qa pa p qb pb. The variables qa pa are statistically independent of qb pb In this circumstance. Ha qa pa kB T,Z dqa dpa e dqb dpb e Hb qb pb kB T,F kB T ln Z kB T ln Za ln Zb Fa Fb. As a result all the thermodynamic variables that are extensive add. Next two special cases, N identical but distinguishable non interacting systems.
e g N atoms in a crystal distinguishable by their locations. or e g N 2 state systems distinguishable by their locations To specify the state of the. whole system we must specify the state of each atom If they don t interact. H H1 H2 HN,Z Z1 Z2 ZN, But since the systems are identical all the Hs are the same function of their respective. coordinates So therefore are all the Z integrals, This is an expression for the partition function of a potentially ginormous collection of N. non interacting systems in terms of the partition function for just one microscopic system. It follows that,F N F1 S N S1 are extensive,N indistinguishable non interacting subsystems. e g N atoms in an ideal gas, Claim here the consequence of indistinguishablity is. where Z1 partition function for one subsystem A direct canonical ensemble derivation of. this statement requires more QM than we want to use here Let s check that it is the same. prescription we derived microcanonically in Chapter 4. Z F N kT ln Z1 kT ln N,S S we would get without the 1 N kB ln N.
we would get without the 1 N, So this is the same prescription as we gave in Chapter 4. 6 2 Monatomic classical ideal gas at fixed T, Consider N non interacting atoms in a volume V We re going to treat them classically. Your further progress in statistical mechanics will be marked by a steady modification of. the adjectives from this phrase monatomic means we ignore internal degrees of freedom. of the particles we ll fix that in Chapter 6 6 classical means the wavefunctions of the. particles don t overlap much we ll fix that in Chapter 9 ideal means the particles don t. interact with each other that s a job for 8 08 or 8 333. H q p H x1 y1 z1 x2 y2 z2 px1 py1 pz1 px2 py2 pz2,pi 2 Hi pi. In treating the system classically we are assuming that the quantum wave functions of the. particles do not overlap much Much more on this assumption and what happens when it. breaks down in Chapter 9,Let s find Z1,p2x p2y p2z. dxdpx dydpy dzdpz px2mk y z, The factors of h make the integral dimensionless This constant doesn t affect the thermo.
dynamics and it cancels out of probabilities The factor will return in our later discussion. of the breakdown of the classical treatment At that time we ll show that this is the right. factor when the classical system arises from the classical limit of a quantum system. Z1 3 Lx Ly Lz dp e 2mkT,3 2 mkT 3 2,dimensionless volume z. mkB T mass energy mass time2,length 2 X,h2 energy2 time2 energy. This length is,2 mkB T 2 mkB T 2mEthermal momentumthermal. the thermal de Broglie wavelength In terms of which. ZN V N 3N VN,Thermodynamics from Z,F kB T ln Z,3N 2 mkB T. kB T N ln N N N ln V ln,V 3 3 2 mkB,N kB T ln ln T 1 ln.
Recall dF SdT P dV,P P V N kB T X,F V 3 3 2 mkB 3,S N kB ln ln T 1 ln 2. T V N 2 2 h 2,V 3 5 3 2 mkB,N kB ln ln T ln X,N 2 2 2 h2. E F T S N kB T X, Success we ve reproduced all the thermodynamics of the ideal gas. The hardest thing we had to do here was a Gaussian integral over one p variable This is. easier than the microcanonical ensemble where I had to quote some formula for volumes of. End of Lecture 15, Fixed T equilibrium distributions of microscopic vars in ideal gas. Think of system 1 as a single atom with location r x y z and momentum p. px py pz At fixed temperature T the distribution for these variables is the Boltmann. distribution,pB r p e BT,One small annoyance Note the normalization.
So this pB is actually dimensionless To get the usual normalization for the probability. distribution remove the h3,pB 1 1 2 2mk T,p r p 3 e p B. h V 2 mkB T, What s the momentum distribution Squash the probability mountain in r. d rp r p e p BT, This was what we got from the microcanonical ensemble. Distributions for speeds and energies, From this you can work out the distributions for various related random variables using the. functions of a random variable technology For example the distribution for the velocity. p v e 2 m v kB T, The velocity distribution is gaussian and isotropic.
6 1 Derivation of the Canonical Ensemble In Chapter 4 we studied the statistical mechanics of an isolated system This meant xed E V N From some fundamental principles really postulates we developed an algorithm for cal