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29 jul 2016 classical monatomic ideal gas . L10–1 Classical Monatomic Ideal Gas Equipartition Principle • Statement: The equipartition principle is a statement about classical systems with quadratic Hamiltonians. It states that any canonical variable ξ (a q or a p) that only appears in a quadratic additive term of the form κ ξ 2 in H, where κ is a constant, and whose range of values can be considered to be (−∞, ∞), contributes a term 21 kB T to the mean energy of the system, or 12 kB to the heat capacity. In particular, the mean energy of a system of N non-interacting particles (distinguishable or not), each of which contributes f quadratic terms in the positions and/or momenta to the total Hamiltonian is Ē = 21 f N kB T . • Proof: Suppose the Hamiltonian is of the form H = H̃ + κ ξ 2 , where κ and ξ are as described above and H̃ does not involve ξ. Then the partition function and mean energy can be calculated as Z Zc = dΩ e−βH(q,p) = Z̃ Z +∞ 2 dξ e−βκ ξ = Z̃ −∞ Γ r π , βκ Ē = − ∂ ln Zc = Ẽ + ∂β 1 2 ∂ ln β = Ẽ + 12 kB T , ∂β where Z̃ and Ẽ are the parts of Zc and Ē that come from the variables other than ξ, and we have used the p R +∞ 2 well-known Gaussian integral −∞ dx e−ax = π/a. Classical Monatomic Ideal Gas Thermodynamics • Setup: As a first example of application of the equipartition principle, consider a monatomic ideal gas. The Hamiltonian is a sum of terms from individual non-interacting particles; if the particles are single atoms and T is low enough that their internal, electronic or nuclear degrees of freedom areP not excited, then each N term in the Hamiltonian is just the particle’s translational kinetic energy, H(q, p) = i=1 p~i2 /2m. • Canonical partition function: For a single particle, using again the Gaussian integral, Z1 = 1 h3 Z V d3 x Z +∞ dp e−βp −∞ 2 /2m 3 = V h3 r π β/2m 3 = V , λ3 p where λ := h2 β/2πm is the thermal wavelength. For N particles, since the particles are non-interacting (H = H1 + ... + HN ) the partition function is the product of N single-particle ones. If the particles are identical, however, one must take into account the fact that any permutation of the N particles leads to an indistinguishable state, even classically. This Boltzmann statistics leads to the N -particle partition function Zc = 1 N VN Z1 = . N! N ! λ3N • Mean energy: We can obtain the mean energy directly from the equipartition principle, or as a result of a short calculation, 3N/2 ∂ V N 2πm Ē = − ln = 32 N kB T , ∂β N ! h2 β an expression that agrees with what we expect from the kinetic theory of ideal gases. • Helmholtz free energy: Using the partition function Zc above for a monatomic ideal gas, we find F = −kB T ln Zc = −kB T ln VN = −kB T (ln V N − ln N ! − ln λ3N ) . N ! λ3N This expression can now be used to obtain the pressure p, entropy S and chemical potential µ using definitions obtained from the fundamental identity of thermodynamics for F . 29 jul 2016 classical monatomic ideal gas . L10–2 • Pressure: The ideal gas pressure equation of state can be obtained from the free energy, d ∂F 1 = kB T p=− ln V N = N kB T , or pV = N kB T . ∂V T,N dV V (Alternatively, from S(E, V ) = kB ln Ω = kB ln(V N f (E)) and dS = δQ/T = (dE + pdV )/T we get that p/T = ∂S/∂V |E = N kB /V .) • Entropy: From the general expression for the entropy given by S = T −1 (Ē − F ), we get n h V 2πm 3/2 i o 5 + . S = 23 N kB + kB (ln V N − ln N ! − ln λ3N ) ≈ N kB ln 2 N h2 β This last expression, obtained using the Stirling approximation ln N ! ≈ N ln N − N , is the Sackur-Tetrode formula. Notice that, had we not had the N ! factor in the denominator of Z, the ln N ! term would have been absent in the first expression above (Ē would not have changed, nor the pressure equation of state). As a result, the entropy would not have been additive (extensive), leading to the Gibbs paradox. • Chemical potential: Using the expression of µ as a derivative of the Helmholtz free energy, we get ∂ ∂F V = , − kB T (ln V N − ln N ! − ln λ3N ) ≈ −kB T ln µ := ∂N T,V ∂N N λ3 where we have again approximated ln N ! using the Stirling formula. Alternatively, we can use G 1 µ= = (E − T S + pV ) . N N Second-Order Quantities and Fluctuations • Remark: As our system is in a canonical state, we were able to derive expressions for all thermodynamic variables above just using the partition function Zc instead of having to use the full ρ(q, p). Second-order thermodynamic quantities can also be calculated using ony Zc , because they are derivatives of first-order ones, and some fluctuations of phase-space functions can be reduced to second-order quantities. • Heat capacity: There is a simple expression for any system in a canonical state. Using (∂/∂β)Zc = −Zc Ē, i (∆E)2 1 ∂ Ē dβ ∂ 1 1 h 1 2 −βH Ē − CV = Ē) Z = tr (H e−βH ) = − − (−Z tr (H ) e = . c c ∂T V,N dT ∂β Zc kB T 2 Zc2 Zc kB T 2 This is a special case of a general result known as the fluctuation-dissipation theorem. Notice that it is always positive, CV > 0. For the classical monatomic ideal gas in particular, we can calculate CV = ∂ Ē/∂T = 32 N kB , as we already knew from kinetic theory. • Compressibility: As we also knew already, the isothermal compressibility is κt = −(1/v) (∂V /∂p)T,N = 1/p. (And again, like other response functions for any system in a canonical ensemble, it is always positive.) • Energy fluctuations: The variance of the distribution of values of the energy can be obtained from the general relation (∆E)2 = kB T 2 CV for any system in thermal equilibrium. Then, in our case of the p monatomic ideal gas, (∆E)2 = 23 N (kB T )2 = Ē 2 /( 32 N ), and the relative energy fluctuation is (∆E)/Ē = 2/(3N ). The Maxwell Speed Distribution A quantity for which we actually need to know the full distribution function ρ(q, p) is the Maxwell probability distribution of particle speeds. From the fact that the probability density that a particle have momentum p R is h−3 V dr ρ(r, p), we get that the probability density that the magnitude of the velocity be v is 3/2 Z Z Z 2 4π m3 v 2 m m3 v 2 −βH1 (r,p) sin θ dθ dφ dr ρ(r, p) = dr e = 4πv 2 e−mv /2kB T . f (v) = 3 3 h h Z1 2π kB T V V Reading • Pathria & Beale: Secs 3.9 (classical) and 8.2 (quantum). • Other textbooks: Halley, pp 150–154; Mattis & Swendsen, Sec 2.2; Plischke & Bergersen, —; Reif, Secs 6.3 (classical), 7.8 (quantum); Schwabl, Secs 6.3–6.4.