STATISTICAL PHYSICS I 8.044

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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
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