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The Grand Canonical Ensemble University of Central Arkansas The Grand Canonical Ensemble Stephen R. Addison Directory ² Table of Contents ² Begin Article Copyright °c 2001 [email protected] Last Revision Date: April 10, 2001 Version 0.1 Table of Contents 1. Systems with Variable Particle Numbers 2. Review of the Ensembles 2.1. Microcanonical Ensemble 2.2. Canonical Ensemble 2.3. Grand Canonical Ensemble 3. Average Values on the Grand Canonical Ensemble 3.1. Average Number of Particles in a System 4. The Grand Canonical Ensemble and Thermodynamics 5. Legendre Transforms 5.1. Legendre Transforms for two variables 5.2. Helmholtz Free Energy as a Legendre Transform 6. Legendre Transforms and the Grand Canonical Ensem- ble 7. Solving Problems on the Grand Canonical Ensemble Section 1: Systems with Variable Particle Numbers 3 1. Systems with Variable Particle Numbers We have developed an expression for the partition function of an ideal gas. Toc JJ II J I Back J Doc Doc I Section 2: Review of the Ensembles 4 2. Review of the Ensembles 2.1. Microcanonical Ensemble The system is isolated. This is the ¯rst bridge or route between mechanics and thermodynamics, it is called the adiabatic bridge. E; V; N are ¯xed S = k ln ­(E; V; N) Toc JJ II J I Back J Doc Doc I Section 2: Review of the Ensembles 5 2.2. Canonical Ensemble System in contact with a heat bath. This is the second bridge between mechanics and thermodynamics, it is called the isothermal bridge. This bridge is more elegant and more easily crossed. T; V; N ¯xed, E fluctuates. F = ¡kT ln Z(T; V; N) Toc JJ II J I Back J Doc Doc I Section 2: Review of the Ensembles 6 2.3. Grand Canonical Ensemble System in contact with heat bath/paricle reservoir. This third bridge is called the open bridge. T; V; ¹ ¯xed, E and N fluctuate. ­ = ¡kT ln ¥ = ¡kT ln Z, where ­ is the grand potential and ¥(= Z), is the grand partition function. Toc JJ II J I Back J Doc Doc I Section 3: Average Values on the Grand Canonical Ensemble 7 3. Average Values on the Grand Canonical Ensemble For systems in thermal and di®usive contact with a reservoir, let x(N; r) be the value of x when the system has N particles and is in state Nr. The thermal average is thus x(N; r)e¯(N¹¡ENr ) N;r hxi = x(N; r)p = P Nr Z XN;r 3.1. Average Number of Particles in a System Ne¯(N¹¡ENr ) N;r hNi = P Z Toc JJ II J I Back J Doc Doc I Section 3: Average Values on the Grand Canonical Ensemble 8 We can evaluate this by di®erentiating Z. @Z = N¯e¯(N¹¡ENr ) @¹ XN;r @Z = ¯ Ne¯(N¹¡ENr ): @¹ XN;r Now we can rewrite our equation for hNi as ZhNi = Ne¯(N¹¡ENr ) XN;r and we see that @Z ¯ZhNi = @¹ 1 @Z hNi = ¯Z @¹ 1 @(ln Z) hNi = : ¯ @¹ Toc JJ II J I Back J Doc Doc I Section 3: Average Values on the Grand Canonical Ensemble 9 Now it is useful to introduce the absolute activity ¸ at this point. We de¯ne the absolute activity as ¸ = e¯¹: We use this to rewrite the expressions for Z and hNi. Z = e¯(N¹¡ENr ) XN Xr = e¯N¹e¡ENr X X = ¸N e¡ENr : X X So we can rewrite our our expression for hNi 1 @(ln Z) hNi = ; ¯ @¹ using ¸ = e¹¯. Starting from ¸ = e¹¯, we ¯nd @¸ = ¯e¹¯ = ¸¯: @¹ Toc JJ II J I Back J Doc Doc I Section 3: Average Values on the Grand Canonical Ensemble 10 So 1 @(ln Z) 1 @(ln Z) @¸ ¸¯ @(ln Z) @(ln Z) = = = ¸ ¯ @¹ ¯ @¸ @¹ ¯ @¸ @¸ This is a result of practical importance. We can ¯nd ¸ by matching hNi to N the actual number of particles in the system. In chemistry, it is common to introduce the fugacity at this point. The de¯nition is close to the de¯nition of the absolute activity, we de¯ne the fugacity f as ¹ f = e k : We will not make further use of the fugacity. In calculating the thermal properties of systems, the ensemble on which we choose to make the calculations is a matter of convenience. We choose the ensemble independently of the actual environment. This is possible because fluctuations tend to be small. Since N = const is mathematically awkward, the grand ensemble is often the most convenient approach. Toc JJ II J I Back J Doc Doc I Section 4: The Grand Canonical Ensemble and Thermodynamics 11 4. The Grand Canonical Ensemble and Ther- modynamics Starting from Z = e¯(N¹¡ENr ) XN;r and e¯(N¹¡ENr ) p = ; Nr Z we relate the grand canonical ensemble to thermodynamics using the Gibbs entropy formula S = ¡k pr ln pr: Xr In this case we write the Gibbs entropy formula as S = ¡k pNr ln pNr: XN;r Toc JJ II J I Back J Doc Doc I Section 4: The Grand Canonical Ensemble and Thermodynamics 12 Inserting the Gibbs distribution, we get e¯(N¹¡ENr ) S = ¡k p ln Nr µ Z ¶ XN;r 1 = ¡k p ln ln + ¯N¹ ¡ ¯E Nr µ µZ ¶ Nr¶ XN;r 1 = ¡k p ln ¡ k p ¯N¹ + k p ¯E Nr Z Nr Nr Nr XN;r XN;r XN;r 1 = ¡k ln p ¡ k¯¹ p N + k¯ E p Z Nr Nr Nr Nr XN;r XN;r XN;r S = k ln Z ¡ k¯¹hNi + k¯hEi: And, after rearranging S ln Z = + ¯¹hNi ¡ ¯hEi; k Toc JJ II J I Back J Doc Doc I Section 5: Legendre Transforms 13 which will normally be written as S ln Z = + ¯¹N ¡ ¯E; k where, as usual, we have written E = hEi and N = hNi: This expression can be further simpli¯ed. To simplify it we will make use of the Legendre transform. 5. Legendre Transforms Legendre transforms are widely used in mechanics and thermodynam- ics. Legendre transforms transform are most easily explained in one dimension. For each function f(x) de¯ne a new function Lf(z), the Legendre transform, as follows: De¯ne df z = dx Toc JJ II J I Back J Doc Doc I Section 5: Legendre Transforms 14 thus relating the new variable z to the old variable x. We require that d2f 6= 0 dx2 to guarantee that we can ¯nd the inverse function x(z). Hence we have a unique relation between x and z. The Legendre transform can be written as Lf(z) = zx(z) ¡ f (x(z)) So we simultaneously change our variable to the derivative and modify the function. Note that in physics the transform is often de¯ned as the negative of the function above (e.g. F = E ¡ T S) The mathematical de¯nition of the transform is \better" as there is no minus sign in the inverse transformation. But in thermodynamics we want to keep the meaning of the transformed function in terms of energy, and T S ¡ E would be awkward. Thus we have seen Legendre transforms but have not de¯ned them. That completes the theory as I'll present it, if you are interested you can easily ¯nd the complete theory worked out (usually in terms Toc JJ II J I Back J Doc Doc I Section 5: Legendre Transforms 15 of mappings) and explicit calculations of the inverse transform. I'm going to show you how to ¯gure it out practically for two variables. 5.1. Legendre Transforms for two variables Consider a function of two variables f(x; y) so that the di®erential of f is of the form @f @f df = dx + dy @x @y or df = udx + vdy @f where u = @x and v = @f@y. Our task is change the independent variables x; y to the independent variables u; y. The di®erential quan- tities will be expressed in terms of du and dy. to e®ect the change let g = f ¡ ux then dg = df ¡ udx ¡ xdu Toc JJ II J I Back J Doc Doc I Section 5: Legendre Transforms 16 but df = udx + vdy so dg = vdy ¡ xdu: This is the form we desire. The quantities x and v are now functions of u and y and are given by @g @g x = ¡ and v = : @u @y In this case, these are our inverse transforms. The transformation of the Lagrangian of mechanics to the Hamiltonian is an example of such a transformation.1 1See for example: Herbert Goldstein, Classical Mechanics, Addison-Wesley, 1950, Chapter 7 Toc JJ II J I Back J Doc Doc I Section 5: Legendre Transforms 17 5.2. Helmholtz Free Energy as a Legendre Trans- form We want to transform from E(S; V; N) to F (T; V; N so using Lf(z) = f (x(z)) ¡ zx(z) @E we know T = @S V;N so we write ¡ ¢ @E LE(N; V; T ) = E(N; V; S(T )) S @S or F = E ¡ T S: calculating the di®erential shows that we have changed the indepen- dent variable from S to T . Toc JJ II J I Back J Doc Doc I Section 6: Legendre Transforms and the Grand Canonical Ensemble 18 6. Legendre Transforms and the Grand Canon- ical Ensemble Recall we have found S ln Z = + ¯¹N ¡ ¯E k we can rewrite this as ¡kT ln Z = E ¡ T S ¡ ¹N: Rewriting this as ¡kT ln Z = F ¡ ¹N we recognize this as another Legendre transform, as we know that ¹ is the conjugate variable of N.
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