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PHYSICS 715
Problem Set 5 Due Friday, March 3, 2006
Reading: Landau and Lifshitz, Secs. 32, 74–
LD 14: The virial theorem and the equation of state of a neutral plasma.
A neutral plasma consists of N electrons with mass me and charge −e, labelled i = 1 ,... , N, and N ions with mass M and charge +e, labelled i = N + 1,... , 2 N, all confined at temperature T in an insulating box of volume V. The internal Hamiltonian for the system is
H =
∑^ N
i=
p^2 i 2 me
∑^2 N
i=N +
P^2 i 2 M
∑^2 N
i,j= i<j
eiej |xi − xj|
(Neutral species and more charged species could also be included.) Show that the equation of state for the plasma can be written as
P = nkT
( 1 +
ECoulomb 3 kT
) , where ECoulomb =
2 N
〈 ∑
i<j
eiej |xi − xj|
〉
is the average Coulomb interaction energy per particle in the exact distribution, and n = 2 N/V is the total number density of the particles. [Hint: use the general form of the virial theorem.] Obtain an explicit expression for ECoulomb in terms of electron-electron, ion-ion, and electron-ion interaction integrals, with all equivalent terms counted and combined as much as possible. Is the pressure increased or reduced by the interactions? Explain.
LD 15: The grand partition function Y(T, P, N), density fluctuations, and a mean- field description of critical opalescence. We can define a grand partition function Y for systems in which T and N are fixed, but in which V can vary (e.g., a small sample of a larger volume of gas) in a way similar to that in which we introduced the grand partition function Z. We calculate the canonical partition function Z(T, V, N) for a given V , multiply by a factor yV^ = e−βP V^ , sum (integrate) over all possible volumes, and define
Y(T, P, N) =
V 0
∫ (^) ∞
0
e−βP V^ Z(T, V, N)dV,
where V 0 is an arbitrary small volume included for dimensional reasons.
(a) It may be shown that G(T, P, N) = −kT ln Y is the Gibbs free energy of the system, G = E − T S + P V. Show that this identification is correct for the ideal monotonic gas by calculating Y and G explicitly, and showing that the appropriate derivatives of G give the correct values for the specific volume v = V /N, the entropy, and the chemical potential.
(b) Density fluctuations lead to the scattering of light in fluids, e.g., the blueness of the sky or critical opalescence near a liquid-gas phase transition, and a corresponding attenuation of the intensity of a beam of light with an attenuation coefficient
α =
8 π^3 3
λ^4
∣∣ ∣∣ ∣
∣∣ ∣∣ ∣
2 ∆V 2
V
, (Gaussian units)
(Einstein, 1910; see Jackson, Classical Electrodynamics, Sec. 10.2 D). Here is the dielectric constant of the medium, λ is the wavelength of the light, and ∆V 2 is the mean-square fluctuation ∆V 2 = 〈V 2 〉 − 〈V 〉^2.
Obtain a general expression for the fluctuation ∆V 2 in terms of derivatives of Y. Show that ∆V 2 = kT V κT , where κT is the isothermal compressibility of the medium, κT = − (^) V^1
( ∂V ∂P
) T
. (This is another example of the relation between fluctuations and the linear response of a system.) (c) Obtain an expression for κT in terms of the critical pressure and volume Pc, Vc, and the scaled variable (V − Vc)/Vc, for a van der Waals gas near the critical point on its critical isotherm T = Tc (see Landau and Lifshitz, Secs. 76, 84). [Hint: expand the expression for ∂P/∂V which follows from Eq. (84.5) in a Taylor series in powers of (V −Vc), and keep only the first nonzero term. How many derivatives should vanish at the critical point on the P −V diagram?] Estimate the absorption length ` = α−^1 for blue light (λ = 4. 5 × 10 −^5 cm) in CO 2 (Tc = 304 K, Pc = 72. 9 atm, nc = 8Pc/ 3 kTc) for (V − Vc)/Vc = 10−^2. = 1 + 4πnγ, where n = N/V and γ is the polarizability of the molecule. = 1.000985 for CO 2 at 273 K and 1 atm (but not at Tc, Pc !). Gaseous and liquid CO 2 are normally transparent. Comment—relative to your result—on the transparency near the critical point.
LD 16: Effect of binary collisions on the entropy and energy of a gas
(a) Derive expressions for the chemical potential, entropy, and total energy of a non- relativistic monatomic gas that take the effects of binary interactions through a potential V (r) into account to first order in the cluster integral b 2. Express the results in terms of N, T , and the number density n = N/V , and show that the changes from the results for the ideal gas are given by
∆E = NkT · nλ^3
( T
db 2 dT
− 32 b 2
) ,
∆S = Nk · nλ^3
( T
db 2 dT
− 12 b 2
) .
[Hints: Start with the cluster expansion for Ω = −kT ln Z. Solve the equation for μ or z by iteration correct to first order in b 2. Recall that E = −(∂ ln Z/∂β)V,z. Eliminate z and introduce N immediately in the expressions for E and S.]
