8.044 STATISTICAL PHYSICS I Grand Canonical Ensemble Phase Transitions Kinetic Theory of Gases Language: Probability Canonical Ensemble 3rd Law 2nd Law Microcanonical Ensemble 1st Law Transport Processes 8.044 L1B1 PROBABILITY Random variable (ignorance and/or QM) Continuous, discrete, or mixed Probability density: p(x) ↔ px(ζ) Histogram (normalized) p(x) x PROB(ζ ≤ x < ζ + dζ) = px(ζ)dζ 8.044 L1B2 ⇒ px(ζ) ≥ 0, ∞ −∞ px(ζ)dζ = 1, PROB(a ≤ x < b) = b a px(ζ)dζ Cumulative probability: Px(ζ) ≡ ζ −∞ px(ζ )dζ d ⇒ px(ζ) = Px(ζ) dζ Either px(ζ) or Px(ζ) completely specifies the RV x. 8.044 L1B3 Example Physical adsorption of a gas most of the time ( when hot) z v n θ y x small fraction of the time (when hot) φ leaving the surface 8.044 L1B4 p( φ) p(v) v3 2σ4 1 2π e σ= p( θ) φ 2π 3σ π/2 1 τe θ τ kT m v p(t) 2sin( θ)cos( θ) v2 2σ2 t τ t 8.044 L1B5a p( φ) 1.0 p( θ) 1.0 p(v) P(φ) P(v) 1.0 2π φ P(θ) 3σ p(t) P(t) 1.0 π/2 θ v τ t 8.044 L1B5b z dΩ PROB = p(θ)dθ p(φ)dφ θ = 2 sin(θ) cos(θ)dθ(1/2π)dφ dθ y dΩ = sin(θ)dθdφ x φ dφ PROB/dΩ = (1/π) cos(θ) 8.044 L1B6 Example Atom escaping from a cavity Atom escapes after the n th A Hole wall encounter H )(1 − AH )n p(n) = ( A A A A Total T T n = 0, 1, 2, · · · 8.044 L1B7 pn(x) = ∞ H )(1 − AH )n δ(x − n) (A A A n=0 T T Called a geometric or a Bose-Einstein density Pn(x) p n(x) 0 2 4 x 0 2 4 x 8.044 L1B8 Example Mixed, t dependent RV p(x) Chemical adsorption e- t / τ Physical adsorption x Given: atom on bottom at t = 0 x 1 P(x) e- t / τ p(x) = e−t/τ δ(x)+(1−e−t/τ ) f (x) x 8.044 L1B9 Averages < f (x) >≡ ∞ p(x) f (x)p(x) dx std. dev. −∞ x <x> < x > is the mean < x2 > is the mean square < (x− < x >)2 > = < (x2 − 2x < x > + < x >2) > = < x2 > −2 < x >2 + < x >2 = < x2 > − < x >2 is the variance ≡ (standard deviation)2 8.044 L1B10 Gaussian Exponential p(x) p(t) 1/τ 2σ 0 p(x) = x0 √ 1 2πσ 2 x 2 /2σ 2 e −(x−x0 ) 0 p(t) = 1 τ =0 < x > = x0 Var(x) = σ 2 Controlled separately t τ e −t/τ t ≥ 0 t<0 < t >= τ Var(t) = τ 2 Determined by same parameter 8.044 L1B11 Example Mean free path <L> = = π/2 0 π/2 0 d n θ L=d/cos θ (d/ cos θ)p(θ) dθ (d/ cos θ) 2 sin θ cos θ dθ = 2d π/2 = 2d [0 π/2 0 sin θ dθ (− cos θ) = 2d 8.044 L1B12 Poisson density Events occur randomly along a line at a rate r per unit length L ∆x x p(1) → r∆x as ∆x → 0 Events are statistically independent 1 1 n −rL p(n) = (rL) e = < n >n e−<n> n! n! 8.044 L1B13 Examples of Poisson probability densities 0.6 0.25 0.5 <n> = 0.5 <n> = 2.3 0.2 0.4 0.15 0.3 0.1 0.2 0.05 0.1 1 2 4 3 2 5 4 6 8 10 30 40 50 0.07 0.12 <n> = 10 0.1 0.06 <n> = 30 0.05 0.08 0.04 0.06 0.03 0.04 0.02 0.02 0.01 5 10 15 20 25 30 10 20 8.044 L1B14 MIT OpenCourseWare http://ocw.mit.edu 8.044 Statistical Physics I Spring 2013 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.