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Grand-Canonical Ensembles

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Grand-Canonical Ensembles Grand-canonical ensembles As we know, we are at the point where we can deal with almost any classical problem (see below), but for quantum systems we still cannot deal with problems where the translational degrees of freedom are described quantum mechanically and particles can interchange their locations – in such cases we can write the expression for the canonical partition function, but because of the restriction on the occupation numbers we simply cannot calculate it! (see end of previous write-up). Even for classical systems, we do not know how to deal with problems where the number of particles is not fixed (open systems). For example, suppose we have a surface on which certain types of atoms can be adsorbed (trapped). The surface is in contact with a gas containing these atoms, and depending on conditions some will stick to the surface while other become free and go into the gas. Suppose we are interested only in the properties of the surface (for instance, the average number of trapped atoms as a function of temperature). Since the numbers of atoms on the surface varies, this is an open system and we still do not know how to solve this problem. So for these reasons we need to introduce grand-canonical ensembles. This will finally allow us to study quantum ideal gases (our main goal for this course). As we expect, the results we’ll obtain at high temperatures will agree with the classical predictions we already have, however, as we will see, the low-temperature quantum behavior is really interesting and worth the effort! Like we did for canonical ensembles, I’ll introduce the formalism for classical problems first, and then we’ll generalize to quantum systems. So consider an open system in contact with a thermal and particle reservoir. system: E,V,N,T,p, µ µ reservoir: ERRRRR , V , N ,T , p , R Figure 1: Model of an open thermodynamic system. The system is in contact with a thermal and particle reservoir (i.e., both energy and particles get exchanged between the two, but E + ER = ET = const, N + NR = NT = const). The reservoir is assumed to be much bigger than the system, E ≪ ER,N ≪ NR. In equilibrium T = TR,µ = µR. As we know (see thermo review), the macrostate of such a system will be characterized by its temperature T (equal to that of the reservoir), its volume V and its chemical potential µ (which will also be equal to that of the reservoir) – again, I am using a classical gas as a typical example. We call an ensemble of very many copies of our open system, all prepared in the same equilibrium macrostate T,V,µ, a grandcanonical ensemble. As always, our goal is to find out the relation- ships that hold at equilibrium between the macroscopic variables, i.e. in this case to find out how U,S,p, hNi,... depend on T,V,µ. Note that because the number of particles is no longer fixed, we can only speak about the ensemble average hNi (average number of particles in the container for 1 given values of T,V,µ) from now on. 1 Classical grand-canonical ensemble As was the case for the canonical ensemble, our goal is to find the density of probability ρg.c.(N, q, p) to find the system in a given microstate – once we know this, we can compute any ensemble average and answer any question about the properties of the system. Note that since the number of microsystems (atoms or whatever may be the case) that are inside the system varies, we will specify N explicitly from now on: a microstate is characterized by how many microsystems are in the system in that microstate, N, and for each for these microsystems we need f generalized coordinates and f generalized momenta to describe its behavior, so we have a total of 2Nf microscopic variables q, p. So we need to figure out ρg.c.(N, q, p). We will follow the reasoning we used for canonical ensembles – have a quick look over that and you’ll see the parallels. In fact, I’m going to copy and paste text from there, making only the appropriate changes here and there. First, we reduce the problem to one that we already know how to solve, by noting that the total system = system + reservoir is isolated, and so we can use microcanonical statistics for it. The R R microstate of the total system is characterized by the generalized coordinates N, q, p; NR, q , p , where the latter describe the microsystems that are inside the reservoir. Clearly, N +NR = NT is the total number of microsystems in the total system, which is a constant. So in fact the total system’s R R microstate is characterized by: N, q, p; NT − N, q , p . R R R R Its Hamiltonian is HT (N, q, p; NR, q , p ) = H(q, p)+ HR(q , p ) – here I won’t write the dependence of the Hamiltonian on N explicitly, but we know it’s there: for example, in the kinetic energy we sum over the energies of the microsystems inside the system, and the number of terms in the sum is N, i.e. how many microsystems are contributing to the energy in that particular microstate. Based on the microcanonical ensemble results, we know that the density of probability to find the R R total system in the microstate N, q, p; NT − N, q , p is: 1 R R R R Ω (E ,δE ,V,V ,N ) if ET ≤HT (N, q, p; NT − N, q , p ) ≤ ET + δET ρmc(N, q, p; NT −N, q , p )= T T T R T ( 0 otherwise where ΩT (ET ,δET ,V,VR,NT ) is the multiplicity for the total system. This is the same as saying that the total probability to find the system in a microstate with N microsystems that are located between q, q + dq and p, p + dp AND the reservoir to have the remaining NT − N microsystems located between qR, qR + dqR and pR, pR + dpR is: R R dqdpdq dp R R ρmc(N, q, p; NT − N, q , p ). Nf+(NT −N)fR GN GNT −N h Note that even though the microsystems are the same, they may have different number of degrees of freedom in the system and the reservoir – think about the previous example with atoms adsorbed on a surface. It is very likely that we need different degrees of freedom to describe the state of an atom when on the surface (inside the system) than when it’s in the gas (the reservoir). However, the total number of atoms on the surface plus inside the gas is fixed, and that’s all that matters. We do not care/need to know what the reservoir is doing, all we want to know is the probability that the system is in a microstate which has N microsystems between q, q + dq and p, p + dp. To 2 find that, we must simply sum up over all the possible reservoir microstates, while keeping the system in the desired microstate. Therefore R R dqdp dqdpdq dp R R ρg.c.(N, q, p) = ρmc(N, q, p; NT − N, q , p ), Nf Res Nf+(NT −N)fR GN h Z GN GNT −N h where the integral is over all the reservoir’s degrees of freedom. After simplifying, we have: R R dq dp R R ρg.c.(N, q, p)= ρmc(N, q, p; NT − N, q , p ) Res (NT −N)fR Z GNT −N h 1 dqRdpR = N N f E ≤H(q,p)+H (qR,pR)≤E +δE ( T − ) R ΩT Z T R T T GNT −N h 1 since ρmc = when this condition is satisfied, and zero otherwise. However, we can rewrite the ΩT condition as: 1 dqRdpR q p ρg.c.(N, , )= N N f → E −H(q,p)≤H (qR,pR)≤E −H(q,p)+δE ( T − ) R ΩT Z T R T T GNT −N h ΩR(ET −H(q, p),δET ,VR,NT − N) ρg.c.(N, q, p)= ΩT (ET ,δET ,V,VR,NT ) since the integral is by definition just the multiplicity for the reservoir to be in a macrostate of energy ET −H(q, p), with VR and NT −N microsystems. If you have a quick look at the canonical ensemble derivation, you’ll see that so far everything went very similarly, except that we kept track carefully of how many microsystems are where. Using the link between the entropy of the reservoir and its multiplicity SR = kB ln ΩR (because the reservoir is so big and insensitive to what the system does, all microcanonical formulae valid for an isolated system apply to the reservoir), we then have: ∂SR ∂SR SR(ET ,δET ,VR,NT )−H(q,p) −N SR(ET −H(q,p),δET ,VR,NT −N) ∂ER ∂NR k k ΩR(ET −H(q, p),δET ,VR,NT − N)= e B ≈ e B where I used the fact that the energy of the system H(q, p) ≪ ET and N ≪ NT and performed a Taylor expansion. So the appearance of the second derivative is where the difference between canonical and grandcanonical ensembles shows up. But we know that at equilibrium ∂S 1 1 ∂S µ µ R = = ; R = − R = − ∂ER TR T ∂NR TR T so we find the major result: 1 ρ (N, q, p)= e−β[H(q,p)−µN] (1) g.c.
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