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PHY 471, Statistical Physics, 2007. Lecture 09. Grand Canonical Ensemble. Mahn-Soo Choi (Korea University). F. Reif, Fundamentals of Statistical and Thermal Physics (1965). Chapter 6. Thermal and Chemical Equilibrium of the “System” A with the “Bath” B. "Universe". "System".
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PHY 471, Statistical Physics, 2007 Lecture 09. Grand Canonical Ensemble Mahn-Soo Choi (Korea University) F. Reif, Fundamentals of Statistical and Thermal Physics (1965) Chapter 6.
Thermal and Chemical Equilibrium of the “System” A with the “Bath” B "Universe" "System" "Environment (Bath)" The probability PA (EA , NA ) that the “system” has the energy EA and the particle number NA ? The probability Pα for the “system” to be found in the microstate α?
Equilibrium Condition The probability distribution PA (EA , NA ): Γ(EA , NA |E , N ) Γ(E , N ) PA (EA , NA ) = ΓA (EA , NA )ΓB (E − EA , N − NA ) Γ(E , N ) = PA (EA , NA ) and hence log PA (EA , NA ) has a sharp peak at EA = E¯A and NA = N¯ A in equilibrium. PA(EA) ΓA(EA) ΓB (E − EA) EA
The equilibrium condition is thus ∂ ∂ EA 1 TA 1 TB ⇒ log PA =0 = EA =E¯A ∂ ∂ NA ⇒ µA = µB log PA =0 ¯ NA =NA
Grand Canonical Ensemble The probability Pα ΓB (E − Eα , N − Nα ) Γ(E , N ) Pα = kB log Pα ∂ SB ∂ EB ∂ SB ∂ NB ≈ SB (E , N ) − EA − NA EB =E NB =N µNα T Eα T = SB (E , N ) − + Canonical distribution Eα − µNα kB T 1 ZG exp − Pα = where ZG is the normalization constant: Eα − µNα kB T exp − . ZG = α
Grand Partition Function ZG ZG is more than just a normalization constant. From the expression for the entropy µN T E T S = −kB − Pα log Pα = + kB log ZG α −kB T log ZG = E − TS − µN = G the Gibbs free energy!
Thermodynamics with Grand Canonical Ensemble Partition function ZG G (T , µ, V , · · · ) = −kB T log ZG (T , µ, V , · · · ) dG = −SdT − Nd µ − PdV + · · · Thermodynamic quantities ∂ G ∂ T ∂ G ∂µ ∂ G ∂ V S = − , N = − , P = − , ···
More about the Entropy S kB = − Pα log Pα = (E − µN ) + log ZG α ∂ ∂β E − µN = −β log ZG ∂ ∂β S kB 1 − β = log ZG
More about Pressure The elementary consideration leads to ∂ G ∂ V P = − Since the Gibbs free energy is an extensive quantity, it should be directly proportional to V . It means that ∂ G ∂ V = constant. The constant should be the pressure P : G = −kB T log ZG = PV .
Summary of Ensembles Microcanonical ensemble Γ(E , N , V , · · · ) = δ(E − Eα ) α S (E , N , V , · · · ) = kB log Γ(E , N , V , · · · ) Canonical ensemble Eα kB T Z (T , N , V , · · · ) = exp − α A(E , N , V , · · · ) = −kB T log Z (T , N , V , · · · ) Grand canonical ensemble Eα − µNα kB T ZG (T , N , V , · · · ) = exp − α G (E , µ, V , · · · ) = −kB T log FG (T , µ, V , · · · )