Unit 4.- The grand canonical ensemble PDF

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Teoría de la asignatura en inglés. Profesor José Manuel Romero Enrique....

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Unit 4 The grand canonical ensemble.

4.1

The grand-canonical ensemble

Let’s consider two systems A1 and A2 in contact, so they can exchange not only energy but also particles. The combined system A1 ∪ A2 is isolated. We discussed in Section 1.6.5 of Unit 1 how to obtain the probability distribution of one of the systems, for example, A1 , from the microcanonical ensemble for the combined system. The marginal probability density of the system A1 to be in a microstate (q, p) ≡ (q1 , . . . , qf1 , p1 , . . . , pf1 ) with N1 = {N1,1 , . . . , Ng,1 } particles, where f1 is the number of degrees of freedom of A1 , is given by ρ1 (q, p; N1 ) =

hf1 (

1 Ω2 (E − H1 (q, p); N − N1 ) N i,1 !) Ω(E; N ) i=1

Qg

(4.1)

where we have introduced the corrections due to the indistinguishability between Ni,1 particles of the same species i (i = 1, . . . , g) in the system A1 to avoid the Gibbs’s paradox, and Ω is defined by Eq. (1.147). On the other hand, the probability density for E1 and N1 reads ω(E1 ; N1 ) =

Ω1 (E1 ; N1 )Ω2 (E − E1 ; N − N1 ) Ω(E; N )

1

(4.2)

2

The grand canonical ensemble.

We know that ω(E1 ; N1 ) has a very sharp maximum at E1 = E˜1 and N1 = N˜1 , which are obtained from the conditions   ∂ ln Ω2 (E2 , N2 ) ∂ ln Ω1 (E1 , N1 )  = (4.3)     ∂E2 ∂E1 ˜1 ˜1 ˜1 ,N2 =N − N ˜1 ,N1 =N E2 =E−E E1 =E    ∂ ln Ω2 (E2 , N2 )  ∂ ln Ω1 (E1 , N1 )  = (4.4)     ∂Ni,2 ∂Ni,1 ˜ ˜ ˜ ˜ E1 =E1 ,N1 =N1

E2 =E−E1 ,N2 =N − N1

or equivalently that β1 = β2 = β, where β = 1/(kB T ), and that (µi )1 = (µi )2 for i = 1, . . . , g, where µi is the chemical potential of the i species. Thus, this condition is equivalent to the fact that the temperatures T1 and T2 of both systems must be the same T1 = T2 = T , as well as the chemical potentials (µi )1 = (µi )2 = µi . Now let’s consider that A2 is much larger than A1 , in the sense that the number of degrees of freedom of A2 , f2 , is much larger than f1 . By using arguments similar to those explained in the Unit 2, in the typical microstates of the combined system H1 (q, p) ≪ H2 (Q, P ) ≈ H1 (q, p) + H2 (Q, P ) = E

,

Ni,1 ≪ Ni,2 ≈ Ni

(4.5)

This fact allows us to Taylor expand ln Ω2 (E −H1 (q, p), N −N1 ) around H1 (q, p) = 0 and Ni,1 = 0. Truncating the expansion in the linear terms, we get that   ∂ ln Ω2 (E2 , N2 ) ln Ω2 (E − H1 (q, p), N − N1 ) ≈ ln Ω2 (E, N ) − H1 (q, p) ∂E2 E2 =E,N2 =N  g  X ∂ ln Ω2 (E2 , N2 ) − Ni,1 ∂N i,2 E = E,N = N 2 2 i=1 = ln Ω2 (E) − βH1 (q, p) +

g X

βµi Ni,1

(4.6)

i=1

where β and µi are defined for A2 , but by condition Eqs. (4.3) and (4.4) they are also the values for the system A1 . Strictly speaking, the definition of β and µi for the A2 system correspond to the evaluation of the derivative at E˜2 and N˜i,2 , but these values do not differ significatively (in macroscopic terms) from E and Ni , so we can neglect the effect of the variation of E2 and N2 in the evaluation of β2 = β and (µi )2 = µi . For this reason, the system A2 is said to act as a thermal and particles bath or reservoir for A1 . Substituting Eq. (4.6) into Eq. (4.1), we get that g Ω2 (E; N ) −βH1 (q,p)+Pi=1 βµi Ni,1 e hf1 ( i=1 Ni,1 !) Ω(E; N ) 1 −βH1 (q,p)+Pg βµiNi,1 1 i=1 ≡ f Qg e h 1 ( i=1 Ni,1 !) Q

ρ1 (q, p) =

1 Qg

(4.7)

3

The grand canonical ensemble.

where Q is the grand-canonical partition function, which can be obtained from the normalization condition of Eq. (4.7) as Q ≡ Q(T, X1 , µ) =

∞ X

...

Ng,1 =0

N1,1 =0 ∞ X

=

N1,1 =0

∞ X

...

∞ X

Pg

e i=1 βµiNi,1 Q hf1 gi=1 Ni,1 ! Pg

e

i=1

βµi Ni,1

Z

dqdpe−βH1 (q,p)

Z(T, X1 , Ni,1 )

(4.8)

Ng,1 =0

where µ ≡ {µ1 , . . . , µg } and Z is the canonical partition function. This expression can be extended in a straightforward way for systems with a discrete energy spectrum. We see that the partition function depends on the temperature T and chemical potentials µ, as well as external extensive parameters X1 = {X1,1 , . . .} of system A1 . With these definitions, the statistical properties of system A1 only depend on A2 via the factors β and µ. As a consequence, we will focus only on the system A1 , which will be in thermal equilibrium with the thermal and particles bath characterized by a temperature T and chemical potentials µ, but keeping constant the values of X1 of the system A1 . Hereafter we will drop the subscript 1 which identifies the system of interest. So, the macrostate of the system A is determined by the values of T , µ and X, and the corresponding Gibbs’s ensemble for this situation is the grand-canonical ensemble. The set of microstates compatible with this macrostate are those with the same values of X than the macrostate regardless their microscopic energy and the number of particles of each species. On the other hand, the probability distribution of microstates is given by Eq. (4.7), which depends on the additional parameters β = 1/(kB T ) and µ. Note that, as it is a function of the energy of our system, it is a stationary solution of Liouville’s equation (as the probability distribution of the microcanonical ensemble). In this ensemble, the averages of a microscopically defined quantity A(q, p; N ) (which provide their macroscopic values for the grand-canonical macrostate) are given by A =

∞ X

N1,1 =0

=

...

Z ∞ X

Ng,1 =0

N1,1 =0 . . . P∞ N1,1 =0

P∞

dqdpA(q, p; N )ρ(q, p; N )

Pg R dqdpA(q, p)e−βH (q,p)+ i=1 βµiNi Ng,1 =0 Pg R P∞ . . . Ng,1 =0 dqdpe−βH (q,p)+ i=1 βµiNi

P∞

(4.9)

Sometimes it is useful to write the expressions in terms of αi ≡ −βµi , as well as in terms of the fugacity zi = exp(βµi ) ≡ exp(−αi ).

4

The grand canonical ensemble.

4.2

Evaluation of thermodynamic properties in the canonical ensemble

The internal energy of the system is defined as E ≡ H(q, p) =

∞ X

...

∞ X

Ng =0

N1 =0

Pg

e− i=1 αiNi Qg hf ( i=1 Ni !) Q

Z

dqdpH(q, p)e−βH (q,p)

This property can be evaluated by noticing that Pg   Z ∞ ∞ X X ∂e−βH (q,p) e− i=1 αiNi ∂Q dqdp Q ... = g ∂β ∂β X,α N =0 N =0 hf ( i=1 Ni !) g 1 P g Z ∞ ∞ X X e− i=1 αiNi Q ... =− dqdpH(q, p)e−βH (q,p) f ( g N !) h i i=1 N =0 N =0 1

(4.10)

(4.11)

g

Thus, Eq. (4.10) can be recast as     1 ∂Q ∂ ln Q E=− =− ∂β Q ∂β X,α X,α

(4.12)

Note that this derivative is taken for constant α, not for constant µ. The variance 2 of the energy is σ 2E = (H (q, p) − E )2 = H(q, p)2 − E . To calculate this quantity, we evaluate the derivative of E with respect to β  2    !   ∂ ∂ ln Q 1 ∂Q ∂E =− =− ∂β Q ∂β X,α ∂β 2 ∂β X,α X,α X,α "   2 # 2  1 ∂ Q 1 ∂Q (4.13) =− − 2 2 Q ∂β X,α Q ∂β X,α 2

The second term in the last bracket is easily identified with E . On the other hand, the first term can be obtained by differentiating Eq. (4.11) 

∂ 2Q ∂β 2



X,α

P

Z g ∞ X ∂e−βH (q,p) e− i=1 αiNi Q dqdpH(q, p) = − ... hf ( gi=1 Ni !) ∂β N1 =0 Ng =0 Pg Z ∞ ∞ X X e− i=1 αiNi Q dqdp(H (q, p))2 e−βH (q,p) (4.14) = ... g f h ( i=1 Ni !) ∞ X

N1 =0

Ng =0

5

The grand canonical ensemble.

Thus, the first term in the last bracket of Eq. (4.13) can be identified as H(q, p)2 . So,     ∂E ∂E 2 2 2 = kB T = kB T 2 CX,α (4.15) σE = H(q, p)2 − E = − ∂T X,α ∂β X,α where CX,α is the heat capacity for X and α constants. As in the canonical case, the ratio between σE and E scales as q f˜ 1 σE ∼q ∼ (4.16) ˜ E f f˜

where f˜ is the most-probable number of degrees of freedom of the system. This result states that the mode (i.e. the most probable value of the energy) E˜ will coincide macroscopically with the mean value E, since their difference is of order of σE ≪ E. In this sense, for macroscopic systems the canonical ensemble probability distribution converges to the microcanonical probability distribution for an energy E = E, which on the other hand it is the energy at which the (microcanonical) temperature is the same as the one in the thermal and particles bath. Again this is a special case of the equivalence between equilibrium ensembles. In the grand-canonical ensemble, the number of particles is a fluctuating quantity. The mean value of the number of particles of species i can be expressed as: Pg   Z ∞ ∞ X X 1 ∂Q e− i=1 αiNi −βH (q,p) =− Ni f Qg ... Ni = dqdpe Q ∂αi β,X,αj6=i h ( i=1 Ni !) Q Ng =0 N1 =0       ∂ ln Q ∂ ln Q ∂ ln Q (4.17) =− = kB T = k B T zi ∂αi β,X,αj6=i ∂µi β,X,µj6=i ∂zi β,X,zj6=i

Note that



∂E ∂αi



β,X,αj6=i

∂ 2 ln Q =− = ∂αi ∂β



∂Ni ∂β



(4.18)

α,X

We can obtain the variance of Ni proceeding in a similar way as we did for the energy variance. So       2 2 ∂N ∂N ∂ ln Q i i (∆Ni )2 ≡ Ni2 − N i = =− = kB T ∂αi2 β,X,αj6=i ∂αi β,X,αj6=i ∂µi β,X,µj6=i (4.19)

6

The grand canonical ensemble. As ∂N i /∂µi ∼ N i ∼ f˜, we get that q f˜ σ Ni 1 ∼q ∼ Ni f˜ f˜

(4.20)

so in the typical microstates the mean number of particles does not differ from its mean value. However, this result requires a finite ∂N i /∂µi , and there are situations where this is not true. Although we still have not set the connection between the Statistical Physics and the Thermodynamics in the grand-canonical ensemble, we will make use of some known results from Thermodynamics. Let’s consider a hydrostatic monocomponent system. Taking into account the Gibbs-Duhem relationship: −SdT + V dp − N dµ = 0

(4.21)

we know that, for any quantity A,     ∂A N ∂A = V ∂µ T ∂p T so



∂N ∂µ



T,V

N = V



∂N ∂p



=N

T,V

(4.22) 

∂n ∂p



(4.23)

T,V

where n = N /V . Now, in the last derivative, we can drop the condition of differentiating for constant V since n is a function of T and p. Alternatively, we can instead make this derivative for constant N , so  Thus

∂N ∂µ



T,V

=N

 −1    2 ∂V ∂n N ∂V 2 =N =− 2 ∂p T,N V ∂p T,N ∂p T,N



2 σN 2

=

nkB T κT

N N where κT is the isothermal compressibility   1 ∂V κT = − V ∂p T,N

(4.24)

(4.25)

(4.26)

which must be q a positive quantity. If κT is finite, our previous estimation for σN /N ∼ √ 1/ N ∼ 1/ f˜ is valid (note that κT is an intensive quantity). However there are

7

The grand canonical ensemble.

conditions for which κT → +∞, such as at the liquid-vapour critical point. In this case, density fluctuations are much larger than off-critical situations, and this has macroscopic effects in phenomena such as the critical opalescence. Something similar happens for the energy fluctuations, since CV,α also diverges at the critical point. However, it can be shown that at the critical point κT ∼ V x (a similar scaling can be obtained for CV,α), where x = γ/(3ν) (γ and ν are critical exponents) is a number smaller than 1, so the equivalence between ensembles is still valid. Another quantities of interest are the generalized forces, which can be obtained as ∂H (4.27) Yα≡− ∂Xα which can also be obtained as a derivative of the partition function     ∂Q 1 ∂ ln Q 1 1 (4.28) = Yα= β ∂Xα T,µ,Xγ6=α β Q ∂Xα T,µ,Xγ6=α In the case that Xα is the volume V , its conjugated generalized force Y α is the pressure p. The results obtained up to know identify the grand-canonical partition function as a key quantity to get the thermodynamic properties of the system in the grandcanonical ensemble, analogous to the number of microstates or the phase volume in the microcanonical ensemble or the canonical partition function in the canonical ensemble. In the next section we will explore in more detail its importance and its connection with the Thermodynamics.

4.3

The grand-canonical partition function and its connection with the Thermodynamics

First, we will revisit the thermodynamics of open systems. In this case, the differential form of the First Law of the Thermodynamics for an infinitesimal reversible process takes the form dE = d¯Q − d¯W +

g X i=1

µi dNi = T dS −

X

Y αdXα +

α

g X

µi dN i

(4.29)

i=1

where the last term takes into account the energy change due to the variation of the number of particles. Let’s consider the Gibbs free energy or free enthalpy G X Y α Xα (4.30) G ≡ G(T, Y , N ) = E − T S + α

8

The grand canonical ensemble. Then dG = −SdT +

X

XαdY α +

α

g X

µi dN i

(4.31)

i=1

Consequently, the chemical potentials can be identified as   ∂G µi = ∂N i T,Y ,N j6=i

(4.32)

In particular, for monocomponent systems µ = G/N . Now, as G is a homogeneous function of order 0 on T and Y , and order 1 on N i , then G=

g X

µi N i

(4.33)

i=1

In order to give a thermodynamic interpretation to the grand-canonical partition function, let’s see how it changes when the temperature, the chemical potentials and the external extensive parameters vary quasi-statically. Then    X ∂ ln Q  ∂ ln Q dXα d ln Q = dβ + ∂β ∂X α X,α β,X γ6=α α  g  X ∂ ln Q + dαi ∂αi β,X,αj6=i i=1 = −Edβ + β

X α

Y αdXα −

g X

N i dαi

(4.34)

i=1

where inPthe last expression we used Eqs. (4.12), (4.17) and (4.28). If we add d(βE + gi=1 N i αi ) to both sides of Eq. (4.34), we get that ! g g X X X N i dαi Y αdXα − = −Edβ + β N i αi d ln Q + βE + α

i=1

i=1

+ βdE + Edβ +

g X

αi dN i +

= β

dE +

α

Y αdXα −

N i dαi

i=1

i=1

X

g X

g X i=1

µi dN i

!

=

dS (4.35) kB

9

The grand canonical ensemble.

where we used the expression of the first law of the thermodynamics for reversible processes. Thus, the entropy in the grand-canonical ensemble can be defined as ! g X N i µi S = kB ln Q + βE − β i=1

= kB = kB = kB

#!    g X ∂ ln Q ∂ ln Q ln Q − β − µi ∂β ∂αi β,X,αj6=i X,α i=1 !   ∂ ln Q ln Q − β ∂β X,µ    !  ∂ ln Q ∂(kB T ln Q) ln Q + T = ∂T ∂T X,µ X,µ "

(4.36)

and from this expression TS − E +

g X i=1

µi N i ≡

X

Y αXα = kB T ln Q

(4.37)

α

For hydrostatic systems, kB T ln Q = pV . The expression (4.37) can be related to the grand-canonical thermodynamic potential Φ, defined as Φ ≡ Φ(T, X, µ) = E − T S −

g X i=1

µi N i = F −

g X

µi N i

(4.38)

i=1

where F is the Helmholtz free energy. So, comparison with Eq. (4.37) leads to the identification X Y αXα = −kB T ln Q (4.39) Φ=− α

which, for a hydrostatic system, reduces to Φ = −pV . Note that Φ, as Q, is a function of T , µ and X. With the definition Eq. (4.38), Eqs. (4.17), (4.28) and (4.36) can be recast as       ∂Φ ∂Φ ∂Φ (4.40) , Ni = − , Yα=− S=− ∂µi T,X,µj6=i ∂Xα T,µ,Xγ6=α ∂T X,µ

in agreement with the thermodynamic relationships which arises from the fact that ! g X µi N i dΦ = d E − T S − i=1

= −SdT −

X α

Y αdXα −

g X i=1

N i dµi

(4.41)

10

The grand canonical ensemble.

The expression Eq. (4.37) is consistent with the results in the microcanonical ensemble. Note that ω(E; N ), in the grand-canonical ensemble, can be expressed as Ω(E; N )e−βE+β ω(E; N ) = Q

Pg

i=1

µi Ni

(4.42)

As in the canonical ensemble, this expression shows a sharp maximum for E = E˜ ˜ , so and N = N Pg g ˜ ˜ Y ˜ N˜ )e−β E+β XZ i=1 µi Ni Ω(E; dEω(E; N ) ≈ nσE 1= σ Ni (4.43) Q N i=1

˜ ≈ N , σE ≪ E and σN ≪ N i , we obtain that Taking into account that E˜ ≈ E, N i ln Q ≈ ln Ω(E; N ) − βE + β

g X i=1

µi N i =

g X S − βE + β µi N i kB i=1

(4.44)

where we used the microcanonical expression for the entropy. The additivity of Φ can be shown in a similar way as we did it for the entropy in Unit 1 and the Helmholtz free energy in Unit 2. We will consider the following cases of interactions between two subsystems A1 and A2 , with the combined system to be in contact with a thermal and particles bath characterized by a temperature T and chemical potentials µ. So, if X1 and X2 are kept constant, the grand-canonical partition function of the combined system can be written as Pg Z ∞ ∞ X X e− i=1 αi(Ni,1 +Ni,2 ) Qg Q ... Q= dqdpdQdP e−β(H1 (q,p)+H2 (Q,P )) f1 +f2 ( g N !) ( N !) h i,1 i,2 i=1 i=1 N =0 N =0 1,1

g,2

(4.45) where we assumed that the Hamiltonian of the combined system can be approximated as H(q, Q, p, P ; N1 , N2 ) = H1 (q, p; N1 ) + H2 (Q, P ; N2 ) (4.46)

Now, we notice that we can factorize the integral in Eq. (4.45) as   P Z ∞ ∞ − gi=1 αi Ni,1 X X e Qg Q =  dqdpe−βH1 (q,p)  ... f h 1 ( i=1 Ni,1 !) Ng,1 =0 N1,1 =1   P Z ∞ ∞ − gi=1 αi Ni,2 X X e Qg ×  dQdP e−βH2 (Q,P )  ... f h 2 ( i=1 Ni,2 !) N1,2 =1

= Q1 × Q2

Ng,2 =0

(4.47)

11

The grand canonical ensemble.

which is the factorization property of the partition functions of weakly interacting systems. This expression is exact and valid for any non-interacting systems, no matter how large they are. On the other hand, the result is the same regardless the systems A1 and A2 can exchange particles or not. This is expected because, even if A1 and A2 cannot exchange particles directly, they can do it through the particle reservoir. If we take the logarithm in both sides of Eq. (4.47) we get that ln Q = ln Q1 + ln Q2

(4.48)

which yields to the additivity of the grand-canonical thermodynamic potential Φ = Φ1 + Φ2 since both subsystems must be at the same temperate in equilibrium. Now let’s suppose that A1 and A2 can interact mechanically. By simplicity, we will consider that A1 and A2 are hydrostatic systems, so their volumes V1 and V2 can change, but with the constraint that V = V1 + V2 is constant. Analogously to the previous case, we can obtain the partition function of the combined system as Z V dV Q1 (T, V1 , µ)Q2 (T, V − V1 , µ) (4.49) Q(T, V, µ) = 0

Now we consider that both subsystems are macroscopic. Then, the integrand in this expression has a sharp maximum for V1 = V˜1 , so (4.50) Q(T, V, µ) ≈ Q1 (T, V˜1 , µ)Q2 (T, V − V˜1 , µ)nσV

˜ 1 . The value of V1 is obtained from the condition where n ∼ 1 and σV ≪ V   ∂ l...