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Handout 9. NPT and Grand Canonical Ensembles

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Handout 9. NPT and Grand Canonical Ensembles ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 9. NPT and Grand Canonical Ensembles January 26, 2011 Contents 1 Summary of NVE and NVT ensembles 2 2 NPT ensemble 3 2.1 Equilibrium distribution . .4 2.2 Thermodynamic potential . .5 2.3 Volume fluctuations . .5 2.4 Ideal gas example . .6 3 Grand canonical ensemble 7 3.1 Equilibrium distribution . .8 3.2 Thermodynamic potential . .8 3.3 Number of particles fluctuations . .9 3.4 Ideal gas example . 10 3.5 Lattice gas model . 11 Reading Assignment: Reif x6.9. 1 1 Summary of NVE and NVT ensembles Let us start with a quick summary of the microcanonical (NVE) ensemble. It describes isolated systems with fixed number of particles N, volume V and energy E. • The microcanonical ensemble is described by a uniform distribution with two constant energy shells. • The connection with thermodynamics is made through Boltzmann's entropy formula: S = kB ln Ω, where Ω is the number of microscopic states consistent with thermody- namic (macroscopic) variables N; V; E. • Inverting S(N; V; E) we can obtain E(S; V; N). The other thermodynamic quantities are defined through partial derivatives. @E @E @E temperature T ≡ @S V;N , pressure p ≡ − @V S;N , chemical potential µ ≡ @N S;V Next, a quick summary of the canonical (NVT ) ensemble. It describes systems in contact with a thermostat at temperature T . As a result, the energy of the system no longer remain constant. The number of particles N and volume V remain fixed. • The canonical ensemble is described by Boltzmann's distribution. 1 −βH(fpig;fqig) ρ(fqig; fpig) = e (1) Z~ Z 3N ~ Y −βH(fpig;fqig) Z = dqi dpi e (2) i=1 2 • The connection with thermodynamics is made through the expression of Helmholtz free energy A(N; V; T ) through the partition function Z, 3N 1 Z Y A = −k T ln Z;Z = dq dp e−βH(fpig;fqig) (3) B N!h3N i i i=1 • The other thermodynamic quantities are defined through partial derivatives. @A @A @A entropy S ≡ @T V;N , pressure p ≡ − @V T;N , chemical potential µ ≡ @N T;V • The energy (Hamiltonian) of system is no longer conserved, but fluctuate around its average value. 1 @Z @ E ≡ hHi = − = − ln Z (4) Z @β @β @E (∆E)2 ≡ hH2i − hHi2 = − = k T 2C = Nk T 2c (5) @β B v B v Hence in the thermodynamic limit N ! 1, p 2 ∆E 1 ∆E = NkBT cv / p ! 0 (6) E N The difference between microcanonical (NVE) ensemble and canonical (NVT ) ensem- ble vanishes. 2 NPT ensemble The NPT ensemble is also called the isothermal-isobaric ensemble. It describes systems in contact with a thermostat at temperature T and a bariostat at pressure p. The system not only exchanges heat with the thermostat, it also exchange volume (and work) with the bariostat. The total number of particles N remains fixed. But the total energy E and volume V fluctuate at thermal equilibrium. Q: What is the statistical distribution ρ(fqig; fpig) at thermal equilibrium? Q: What is the microscopic expression for the thermodynamic potential? Approach: Consider system of interest + thermostat + bariostat all together as a closed system, which can be described using the microcanonical ensemble. 3 2.1 Equilibrium distribution Notice that in the (NPT ) ensemble, the probability distribution function must also include V as its variable, because the volume can (in principle) take any value at thermal equilibrium. ~ ρ(fqig; fpig;V ) / number of ways (Ω) the thermostat and the bariostat can rearrange them- selves to allow the system to have energy E = H(fqig; fpig) and volume V . Let S~ be the entropy of the thermostat + bariostat, then ! S~ Ω~ = exp (7) kB Let V0 and E0 be the total volume and total energy of the thermostat + bariostat + system of interest. Let V and E be the volume and energy of the system of interest. Then the volume and energy left for the thermostat + bariostat are, V0 − V and E0 − E, respectively. ~ ! ~ ! ~ ~ ~ ~ @S @S S(N;V0 − V; E0 − E) = S(N;V0;E0) − V − E (8) @V~ @E~ N;E N;V @S~ 1 We recognize ~ ≡ T where T is the temperature of the thermostat. @E N;V @S~ But what is ~ ? @V N;E This is the time to use the second type of Maxwell's relationship. ! ! ! @S~ @E~ @V~ · · = −1 (9) @E~ @V~ @S~ V;N S;N E;N ! 1 @V~ · (−p) · = −1 (10) T @S~ E;N ! @S~ p =) = (11) @V~ T N;E where p is the pressure of the bariostat. Therefore, p 1 S~(N;~ V;~ E~) = S~(N;V~ ;E ) − V − E (12) 0 0 T T E + pV Ω~ = const · exp − (13) kBT Therefore, the equilibrium distribution of the isothermal-isobaric (NPT ) ensemble is, 1 ρ(fq g; fp g;V ) = e−β[H(fqig;fpig)+pV ] (14) i i Ξ Z 1 Z 3N Y −β[H(fqig;fpig)+pV ] Ξ = dV dqi dpi e (15) 0 i=1 4 2.2 Thermodynamic potential By now, we would expect the normalization factor Ξ should be interpreted as a kind of partition function that will reveal us the fundamental equation of state. To find out the precise expression, we start with the Shanon entropy expression. (Notice here that V is an internal degree of freedom to be integrated over and p is an external variable.) X S = −kB pi ln pi i Z 1 Z 3N Y H(fqig; fpig) + pV = −k dV dq dp ρ(fq g; fp g;V ) · − ln Ξ~ B i i i i −k T 0 i=1 B 1 = (hHi + phV i) + k ln Ξ~ (16) T B 1 = (E + pV ) + k ln Ξ~ (17) T avg B Hence ~ −kBT ln Ξ = E − TS + pVavg ≡ G(N; T; P ) (18) This is the Gibbs free energy, which is the appropriate thermodynamic potential as a function of N; T; P ! So everything falls into the right places nicely. We just need to careful that the volume in thermodynamics is the ensemble average Vavg ≡ hV i, because in (N; T; P ) ensemble, V is not a constant. Of course, we still need to put in the quantum corrections 1=(N!h3N ), just as before. So the final expression for the Gibbs free energy and chemical potential µ is, µN = G(T; p; N) = −kBT ln Ξ (19) 3N 1 Z 1 Z Y Ξ(T; p; N) = dV dq dp p e−β[H(fqig;fpig)+pV ] (20) N!h3N i i i 0 i=1 Z 1 Ξ(T; p; N) = dV Z(T; V; N) e−βpV (21) 0 Therefore, Ξ(T; p; N) is the Laplace transform of the partition function Z(T; V; N) of the canonical ensemble! 2.3 Volume fluctuations To obtain the average of volume V and its higher moments, we can use the same trick as in the canonical ensemble and take derivatives of Ξ with respect to p. 5 1 @Ξ hV i = −k T B Ξ @p 1 @2Ξ hV 2i = (k T )2 B Ξ @p2 " # @V @ 1 @Ξ 1 @2Ξ 1 @Ξ2 − = k T = k T − @p B @p Ξ @p B Ξ @p2 Ξ2 @p " 2 2# 1 2 1 @ Ξ 1 @Ξ = (kBT ) 2 − kBT kBT Ξ @p Ξ @p 1 = (hV 2i − hV i2) kBT 1 ≡ (∆V )2 (22) kBT Define compressibility1 1 @V (∆V )2 βc ≡ − = (23) V @p T;N kBTV Then we have, 2 (∆V ) = kBT βcV (24) p ∆V = kBT βcV (25) ∆V 1 / p ! 0 as V ! 1 (26) V V In other words, in the thermodynamic limit (V ! 1), the relative fluctuation of volume is negligible and the difference between (NPT ) ensemble and (NVT ) ensemble vanishes. 2.4 Ideal gas example To describe ideal gas in the (NPT ) ensemble, in which the volume V can fluctuate, we introduce a potential function U(r;V ), which confines the partical position r within the volume V . Specifically, U(r;V ) = 0 if r lies inside volume V and U(r;V ) = +1 if r lies outside volume V . The Hamiltonian of the ideal gas can be written as, 3N N X p2 X H(fq g; fp g) = i + U(r ;V ) (27) i i 2m i i=1 j=1 1 Not to be confused with β ≡ 1=(kBT ). 6 Recall the ideal gas partition function in the (NVT ) ensemble. V N V N Z(T; V; N) = (2πmk T )3N=2 = (28) N!h3N B N!Λ3N p where Λ ≡ h= 2πmkBT is the thermal de Broglie wavelength. Z 1 Ξ(T; p; N) = dV · Z(T; V; N) · e−βpV 0 Z 1 1 N −βpV = 3N dV · V · e N!λ 0 Z 1 1 1 N −x = 3N N+1 dx · x · e N!λ (βp) 0 1 1 = N! N!λ3N (βp)N+1 k T N+1 1 = B (29) p Λ3N In the limit of N ! 1, k T N (2πmk T )3N=2 Ξ(T; p; N) ≈ B · B (30) p h3N The Gibbs free energy is k T (2πmk T )3=2 G(T; p; N) = −k ln Ξ = −Nk T ln B · B (31) B B p h3 This is consistent with Lecture Notes 6 Thermodynamics x3:2, G k T (2πmk T )3=2 µ = = −k T ln B · B (32) N B p h3 3 Grand canonical ensemble The grand canonical ensemble is also called the µV T ensemble. It describes systems in contact with a thermostat at temperature T and a particle reservoir that maintains the chemical potential µ.
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