Partition Functions. Chris Clark July 18, 2006

Transcription

1 Patition Functions Chis Clak July 18, Intoduction Patition functions ae useful because it is easy to deive expectation values of paametes of the system fom them. Below is a list of the mao examples. E = ln(z p = kt ln(z S = k ln(z kβ ln(z C V = kβ 2 2 ln(z 2 F = kt ln(z α = ln(z N µ = kt ln(z N You can see that evey paamete can be expessed in tems of ln(z and this is the simplest expession fo most paametes. Be caeful when deiving the expession fo pessue, it is not the deivative of E as is often the case. p = ( E S ( E T,V,N We will deive these in a systematic way in this pape. 1

2 2 Boltzmann Facto and Patition Function Conside a system A in themal contact with a esevoi at tempeatue T with combined enegy E o. et subscipt efe to the esevoi, o efe to the total system and no subscipt efe to the system A. 1 P = ω (E o E ω o (E o = es(eo E/k e So(Eo/k By expanding the entopy in powes of a small enegy, the top exponent becomes S (E o E = S (E o E + E E = S (E o E + (E E S E = S (E o E + (E E /T Eo E=E And by the additivity of entopy the bottom exponent can be witten as Theefoe S o (E o = S(E + S o (E o E P = es(eo E/k+(E E/kT e S(E/k+S(Eo E/k = e(e E/kT e S(E/k = e (E T S/kT e E/kT = e βf e βe So when we apply the nomalization condition P = 1 e βf e βe = 1 we see that the quantity defined by Z e βe which we call the patition function, satisfies Z = e βf The patition function is so named accoding to Wikipedia because it encodes how the pobabilities ae patitioned among the diffeent micostates, 1 This section is fom Callen Page

3 based on thei individual enegies. The lette Z stands fo Zustandssumme, which is Geman fo sum of states. It is woth noting that the sum must be taken ove all quantum states and not ove all allowed enegies since thee may be degeneacies. It is extemely impotant to undestand that the whole patition function fomalism is built on the assumption of a canonical ensemble, which means that the system is in contact with a themal esevoi. Theefoe, if you want to use the patition function at all, then you must be sue that you system is in contact with some kind of themal esevoi, unless if the system emains in themal equilibium at a constant tempeatue. In that case the esevoi would have no effect and so it is not necessay. Themal esevois come in many foms. Fo example electomagnetic adiation can act as a themal esevoi, howeve usually themal inteactions occu via paticle collisions. 3 Deived Quantities Z is always a function of the complete set T, V, N, so you cannot detemine these paametes fom Z. You do not need to wite the notation fo keeping T,V, o N constant when taking deivatives of Z because these paametes ae aleady assumed constant. Since these paametes ae a complete set, no othe paametes may be held constant o the system would be ovedetemined. That is why we could not find the pessue by taking the volume deivative of enegy we could not hold entopy constant. Of the fou types of enegy (E, F, G, H, the Helmholtz fee enegy F is the one that consides T and V as independent paametes. Note that all fou conside N a constant paamete. Theefoe only in tems of F can we ceate a fundamental elation with Z. This fundamental elation 2 is Z e βe = e βf So what is the best way to emembe all of the expessions fo quantities deived fom the patition function? Memoize this fundamental elation and how to use the Helmholtz fee enegy to deive the est. Remembe that F = E T S and stat fom the fundamental themodynamic elation. 2 This is fom Callen Pg 352 de = T ds pdv de = d(t S SdT pdv d(e T S = df = SdT pdv ( ( df = dt + dv V T ( ( S = and p = V T 3

4 et s stating deiving the elations. We automatically get Using the new patial deivatives, F = kt ln(z (T ln(z S = k = k ln(z + kt ln(z = k ln(z + kβ ln(z ( Now we can get µ fom µ = Then α = βµ so, p = kt ln(z N T,V,N µ = kt ln(z N α = ln(z N We can even get the enegy if we use some tickey. E = F + T S = F 1 kβ = F + β = (βf = ln(z And finally the heat capacity at constant volume comes fom C V = ( E C V = kβ 2 2 ln(z 2 4 Pobability Explanation et s look at this fom anothe angle. The patition function fomalism woks because it poduces the pobabilities of states to make an expectation value summation. 3 P = Ce βe = e βe Similaly fo E 2, e βe e βe E = P E = Z E = 1 Z = 1 ( e βe 1 Z = Z Z 3 This section taken fom Reif Pg 212 E 2 = 1 Z 2 Z 2 = e βe Z e βe E = ln(z V. 4

5 This can be ewitten as E 2 = ( 1 Z Z + 1 Z 2 Theefoe the fluctuation in enegy is 5 Geneal Paamete ( 2 Z = E + E2 ( E 2 = E 2 E 2 = E = 2 ln(z 2 If the micostate enegies all depend on a paamete λ by whee E (0 E = E (0 + λa does not depend on λ, then the expectation value of A is A = 1 Z = 1 1 Z β Ae β(e(0 +λa λ e β(e (0 +λa = kt ln(z λ You can also be eally ticky and use this to find the expectation values of quantities that ae not in the Hamiltonian by atificially adding them in and setting λ 0 at the end. 4 6 Momentum Integal Appoximation Since momentum is found in evey Hamiltonian that efes to a paticle, it is vey common that you have to sum ove momentum states in a patition function. The eason that the sum is not infinite is because momentum is always quantized inside any containe. In themodynamics we always have a volume V, which tells us the size of ou containe. We want to appoximate the sum ove momentum states with an integal, which will be much easie to handle. Howeve, in ode to deive the integal, we will have to assume the containe is a cubical box. As it tuns out, the esult is independent of the shape of the containe, even though the only evidence I have is an equation fom classical themodynamics. p = hk = h (πnx 2 + ( πny 2 ( πnz This section came fom Wikipedia s aticle: Patition Function 5

6 = hπ n 2 x + n 2 y + n 2 z So the patition function fo the simple Hamiltonian H = p 2 /2m is Z = e βe = e βp2 /2m = e β h2 π 2 (n x 2 /2m2 3 [ ] 3 = e β h2 π 2 n 2 x /2m2 dn x 0 [ ( ] 3 = e βp2 x /2m d 0 hπ p x = V [ ] 3 π 3 h 3 e βp2 x /2m dp x 0 = V 1 2π 3 h 3 e β(p2 x +p2 y +p2 z /2m dp x dp y dp z In the last line the momentum takes on its classical meaning as the components of the momentum vecto, wheeas befoe it was a positive quantity based on the quantum wave function. This esult is tue fo abitay shaped containes because the classical expession fo the patition function is 5 Z = 1 h 3 e βh dx dy dz dp x dp y dp z 7 Gand Canonical Patition Function Micocanonical Ensemble - ensemble of isolated systems, can be used to deive the expession fo entopy Canonical Ensemble - ensemble of systems that can exchange themal enegy with a esevoi, can be used to deive the Boltzmann facto and the patition function fomalism Gand Canonical Ensemble - ensemble of systems that can exchange both themal enegy and paticles with a esevoi 8 Quantum Statistics I think that you ae supposed to use the same expession fo the patition function when dealing with quantum statistics. 5 This is Callen (16.68 o Reif