1 / 10

PHY 471, Statistical Physics, 2007

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".

fox
Download Presentation

PHY 471, Statistical Physics, 2007

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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.

  2. 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?

  3. 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

  4. The equilibrium condition is thus ∂ ∂ EA 1 TA 1 TB ⇒ log PA =0 = EA =E¯A ∂ ∂ NA ⇒ µA = µB log PA =0 ¯ NA =NA

  5. 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

  6. 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!

  7. 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 = − , ···

  8. 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

  9. 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 .

  10. 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 , · · · )

More Related