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By Mehran Kardar

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The Vlasov equation assumes the limit no λ3 ≫ 1, which requires that (kB T )3/2 1/2 no e3 ≫ 1, where ℓ is the interparticle spacing. e. the interaction energy is much less than the kinetic (thermal) energy. (f) Show that the characteristic relaxation time (τ ≈ λ/c) is temperature independent. What property of the plasma is it related to? • A characteristic time is obtained from τ∼ λ ∼ c kB T · no e2 m ∼ kB T 1 m ∼ , 2 no e ωp where ωp is the plasma frequency. ******** 5. g. Si or Ge), their conduction electrons can be thermally excited to move freely in the host lattice.

Averaging over them leads to ρ˜(q, p, t ≫ τ ) = 1 f (p), 2Q as (i) the density f (p) at each p is always the same, and (ii) all points along q ∈ [−Q, +Q] are equally likely. e. ρ˜ is stationary. ∂t m ∂q ******** 2. Evolution of entropy: The normalized ensemble density is a probability in the phase space Γ. This probability has an associated entropy S(t) = − dΓρ(Γ, t) ln ρ(Γ, t). (a) Show that if ρ(Γ, t) satisfies Liouville’s equation for a Hamiltonian H, dS/dt = 0. • A candidate “entropy” is defined by S(t) = − dΓρ(Γ, t) ln ρ(Γ, t) = − ln ρ(Γ, t) .

E. ℓ − Nℓ lim p(ℓ) = exp − N→∞ 2N σ 2 2 √ 1 2πN σ 2 , where ℓ and σ 2 are the mean and variance of ln r respectively. The product x is distributed according to the log-normal distribution ln(x) − N ℓ dℓ 1 p(x) = p(ℓ) = exp − dx x 2N σ 2 33 2 √ 1 2πN σ 2 . The probability that the first integer of x in a decimal representation is i is now obtained approximately as follows: 10q (i+1) pi = dxp(x). e. has q + 1 digits before the decimal point). Since the number is quite widely distributed, we then have to sum over possible magnitudes q.

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