(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble

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Veronica Goodwin
5 years ago
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(or e.g. E± E/2): Microcanonical ensemble & = /01Ω maximized Ω: fundamental statistical quantity for this ensemble Helmholtz free energy:! = # %&. (!

1 Recall from before: Internal energy (or Entropy): &, *, - (# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble & = /01Ω maximized Ω: fundamental statistical quantity for this ensemble Helmholtz free energy:! = # %&. (! = &(% )(* +,(- %, *, - Canonical ensemble! minimized in equilibrium 3 45/78 : :;<:<;=><1?0 =< :;<@?@>0>=A <B C=?=3, D>E31 F note, also derivable from maximum entropy G = IJKJLI M FN: F M //% : Partition Function. (fundamental statistical quantity for this ensemble) Boltzmann distribution New, Chapter 6 (sum is over all distinct states; same as microstates)

2 Other cases: Internal energy (or Entropy): 3# = *3+ %3& +.3/ #(+, &, /) +, &, / Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble + = 456Ω maximized Enthalpy:! = # + %&.!(+, %, /) Helmholtz free energy: ( = # *+. ((*, &, /) 3! = *3+ + &3% +.3/ 3( = +3* %3& +.3/ +, %, / *, &, / Canonical ensemble ( = 4*567 Gibbs free energy:,(*, %, /), = # *+ + %&. 3, = +3* + &3% +.3/ *, %, / Isobaric ensemble Grand Potential (homework): Φ = # *+./. Φ(*, &,.) Grand canonical ensemble Φ = 4*567 8

3 Canonical Ensemble: Heat bath = reservoir : So large its T doesn t change T 1 = constant Q= #$ T 2, V 2, N 2 Recall, S maximized (entire system; temperatures become equal); or, F minimized (system 2 only); by spontaneous processes, approaching equilibrium For 2 specific microstates in system 2: % & % ' = ( ) *(&)-) * (') // 0 = ( -12// 0 % 3 = 1 5 (-2 6//7 Probability of finding system in state i in equilibrium 5 = 8 =>? = 3 /@A 9:;:<9 3

4 Canonical Ensemble: Heat bath = reservoir : So large its T doesn t change T 1 = constant % & = 1 ( )*+,/./ Probability of finding system in state i in equilibrium Q= #$ T 2, V 2, N 2 :, <, %, ( = & & /89 Result: probabilistic interpretation of system 2 s energy, entropy, etc. Valid for small or large systems. Sum over microstates: Classical systems: integral over phase space Quantum 1-particle systems: sum over single-particle energy states. In general, sum over all eigenstates.

5 Some Results: Energy averaging! = # = % %& '() (equivalent to U in thermodynamic limit) Also can show, energy fluctuations vanish in thermodynamic limit. Works for finite system contacting reservoir, even though reservoir is infinitely large. Results for canonical ensemble approach those for microcanonical.

6 Some Results: Energy averaging! = # = % %& '() (equivalent to U in thermodynamic limit) Also can show, energy fluctuations vanish in thermodynamic limit. Works for finite system contacting reservoir, even though reservoir is infinitely large. Results for canonical ensemble approach those for microcanonical. Δ# +,- = #. #. //. #. = 1 ) %. %&. ) %. %&. ln ) = 1 %. ) ) %&. 1 %) ). %&. = #. #. Result: Δ# +,- # typical, not //. %! = 4! ~ 1 always 4 %& 6 0

7 spin-1/2 non-interacting paramagnet (canonical ensemble solution): E = ±!B per atom -> " =!$ % & % ( =!$ 2% & % % &,-. /0 % ( = *& + solving: " =!$%123h(!$/78) = > = * (?@ + AB +* &?@ + AB!$ = 2 cosh + 78 = 2 cosh G!$ = = > = > ; includes all cross terms. systems of distinguishable particles, non-interacting. C : = ;" ;8 = %7. 2!$/78, * -.//0 + * &-.//0, Recall, these are average values for the paramagnet system (not fixed as in microcanonical). U

8 spin-1/2 non-interacting paramagnet (microcanonical ensemble solution): E = ±!B per atom -> " =!$ % & % ( =!$ 2% & % % & %! * = +, -. % (! % &! % ( = 0& 1 23, large-n limit; 45 Boltzmann distribution S/k T solving: " =!$%67.h(!$/+;) C = = >" >; = %+, 2!$/+; 2 0 3,/ &3,/45 2 U

9 Some Results: Energy averaging! = # = % %& '() (equivalent to U in thermodynamic limit) Also can show, energy fluctuations vanish in thermodynamic limit. Results for canonical ensemble approach those for microcanonical in thermodynamic limit. Free energy, * = +,'() Entropy, - = + &! + '() = + & % '() + '() %& All state variables can be obtained from Partition function, Z Helmholtz free energy has a central role; simply defined in terms of Z

10 Other cases: Internal energy (or Entropy): 3# = *3+ %3& +.3/ #(+, &, /) +, &, / Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble + = 456Ω maximized Enthalpy:! = # + %&.!(+, %, /) Helmholtz free energy: ( = # *+. ((*, &, /) 3! = *3+ + &3% +.3/ 3( = +3* %3& +.3/ +, %, / *, &, / Canonical ensemble ( = 4*567 Gibbs free energy:,(*, %, /), = # *+ + %&. 3, = +3* + &3% +.3/ *, %, / Isobaric ensemble Grand Potential (homework): Φ = # *+./. Φ(*, &,.) Grand canonical ensemble Φ = 4*567 8

11 Equipartition theorem Classical systems (continuum not discrete energies) Works in cases having separable variables. Requires energy quadratic in position and/or momentum:! = #$ % Text notation (means variables q and p). Result: for any energy of this general form, easy to show, & = ' 2 )* from canonical partition function, (1/2)kT obtained for each degree of freedom f.

Phys Midterm. March 17

Phys Midterm. March 17 Phys 7230 Midterm March 17 Consider a spin 1/2 particle fixed in space in the presence of magnetic field H he energy E of such a system can take one of the two values given by E s = µhs, where µ is the