Grand Canonical Formalism

Grand Canonical Formalism Grand Canonical Ensebmle For the gases of ideal Bosons and Fermions each single-particle mode behaves almost like an independent subsystem, with the only reservation that the total number of particles is conserved. Consider a macroscopic system. In this case the global constraint on the total number of particles should effect the statistics of the occupation numbers of each single-particle mode in some generic way insensitive to the details of the (infinitesimally small) interactions. Indeed, if we pick up some single-particle mode, then the rest of a macroscopic system will play the role of a heat bath plus particle reservoir. The situation is analogous to that of the so-called microcanonical ensemble for a macroscopic system, when the whole system behaves like a heat bath with respect to its smaller subsystems. And this gives us an idea of how to proceed with a macroscopic system. If we attach our macroscopic system to a heat bath plus proper particle reservoir which keeps the average number of particles in our system equal to the given value N, then practically nothing will change in the statistics of our system, because with respect to smaller subsystems of a macroscopic system the situation of heat bath plus proper particle reservoir already existed in the microcanonical ensemble. The trick with the particle reservoir is identical to the trick of using canonical ensemble (synonym to Gibbs distribution) for an isolated system. In the latter case, the trick allows us to lift the strict constraint on the total energy, in the former case we lift both the total energy and the total number of particles constraints. The distribution of the states of a system attached to the heat bath plus particle reservoir is called Grand canonical ensemble, or Gibbs distribution with variable number of particles. What is the explicit form of this distribution? To derive it we just slightly generalize the argument that led us to the Gibbs distribution. We recall that if the total number of particles is fixed, then the probability w n,n to find the system in the energy eigenstate with the energy eigenvalue E n,n can depend only on the energy itself, and not on the other parameters characterizing the state. [Note that now the subscript of the state involves N as a parameter. That is (n, N) labels the n-th eigenstate of the system of N particles.] Suppose we have two different eigenstates of the same energy, but with different numbers of particles: E n1,n 1 = E n2,n 2, N 1 N 2. (1) Should w n1,n 1 and w n2,n 2 coincide? Generally speaking, they should not, since we are dealing with a special heat bath for the sake of briefness, we adopt a convention that a heat bath always implies a particle reservoir, that conserves the total number of particles. This global constraint renders the states with the same energy but different numbers of particles non-equivalent, because our heat bath cares about how many particles are there in our system (and, in particular, does not want this number to be significantly different from its average value). Hence, we conclude that w n,n is a function of two variables, energy and the number of particles: w n,n f(e n,n, N). (2) Consider two independent (non-interacting with each other, but interacting with the heat bath) systems, A and B, and formally treat them as a united system AB. The energy and the number of particles are additive quantities. Hence, for the united system, for which each eigenstate can be labelled by the four numbers: n A, N A, n B, and N B, we have E (AB) ; = E (A), (3) 1

Macro system Heat bath Canonical ensemble (Gibbs distribution) Macro system Microcanonical ensemble (Isolated system) Macro system Heat bath + particle reservoir Grand canonical ensemble (Gibbs distribution with variable number of particles) Figure 1: Canonical, Microcanonical, and Grand canonical ensembles. All the three are equivalent in the macroscopic limit, as for a smaller sub-system the rest of a macroscopic system is equivalent to a heat bath + particle reservoir. N (AB) = N A + N B. (4) Correspondingly, the distribution function for the system AB should have the form w (AB) ; = f AB (E (A), N A + N B ). (5) On the other hand, the two systems are independent, and w (AB) ; has to be equal to the product of individual distributions for the two systems: w (AB) ; = w (A) w (B), (6) where We thus get the relation w (A) w (B) = f A (E (A), N A ), (7) = f B (E (B), N B ). (8) f AB (E (A), N A + N B ) = f A (E (A), N A ) f B (E (B), N B ) (9) that should be valid for any two systems weakly coupled to one and the same heat bath. Mathematically, the only function that satisfies Eq. (9) is the exponential f A (E, N) f B (E, N) f AB (E, N) e βe γn. (10) 2

The parameter β is already known to us as the inverse temperature. The parameter γ is normally written in the form γ = µ/t, (11) where µ is called chemical potential. We arrive at the Gibbs distribution with variable number of particles: w n,n e E n,n µn T. (12) The equilibrium heat bath is characterized by two different parameters, temperature and chemical potential. Talking of an isolated macroscopic system, we see that along with temperature its state can be characterized by one more intensive (=system-size-independent) variable: chemical potential. To normalize the distribution (12), we introduce the partition function Z = n,n e E n,n µn T (13) and write w n,n = Z 1 e E n,n µn T. (14) Grand Canonical Formalism The Grand canonical formalism is very close to the Canonical one. The partition function Z Z(T, µ, V ) plays the central part through the so-called Grand canonical potential Ω = T ln Z, (15) which is a direct analog of the Helmholtz free energy. Another crucial role is played by entropy S = n,n w n,n ln w n,n. (16) It is useful to note the the distribution (12) is identical to the Gibbs distribution of an effective system with the energy, E n,n = E n,n µn, (17) in which N is understood and treated as just one of the quantum numbers characterizing eigenstates of the system. This allows us to employ the results derived previously for the Canonical distribution. Namely, we have the following basic set of relations w n,n = e (Ω E n,n )/T, (18) E = n,n E n,n w n,n E µ N. (19) [Below for briefness we omit the averaging brackets when this does not lead to a confusion.] E = Ω + T S, (20) S = T µ,v 3. (21)

Relation (20) implies E = Ω + T S + µn, (22) where the average number of particles, N, now is not an external parameter, but a certain function: N = N(T, µ, V ). This function is readily related to Ω(T, µ, V ): N =. (23) µ T,V Problem 39. Derive the above relation. The thermodynamic pressure is defined as the average of the microscopic pressure: P = E n,n. (24) This average is related to Ω by the observation that E n,n E n,n. (25) Indeed, the number N in E n,n = E n,n µn not to be confused with its average (!) is just a quantum number which by no means can depend on V. Then, we write P = E n,n, (26) which reduces the problem to the one we have already solved for Canonical distribution: E ( ) n,n Ω =, (27) λ λ µ,t and thus P = µ,t In a macroscopic system, the relation is even more simple:. (28) P = Ω V. (29) The point is that in the functional dependence Ω = Ω(T, µ, V ) the volume is the only extensive (=macroscopic) variable, and, taking into account that in a macroscopic system Ω should be proportional to the total number of degrees of freedom, we conclude that Ω(T, µ, V ) = V f(t, µ), where f is some function. From (28) it follows that this function is nothing else than minus pressure. Note that Eqs. (23) and (29) yield P (µ, T ) n =, (30) µ where n = N/V is the number density (not to be confused with the energy eigenvalue subscript). Hence, in the Grand canonical formalism the equation of state, P = P (n, T ), comes in the parametric form { P = P (µ, T ), (31) n = P (µ, T )/ µ, where the chemical potential plays the role of parameter. 4

Differentials. The relations (21), (23), and (28) imply that the differential of Ω reads dω = S dt P dv Ndµ. (32) Combining this with the differential from Eq. (22), we find de = T ds P dv + µ dn (33) and, in particular, see that µ = ( ) E N S,V. (34) Problem 40. Argue that at T = 0 µ = ( ) E0 N V where E 0 is the groundstate energy of N particles in the volume V. (T = 0), (35) 5