STATISTICAL MECHANICS
PD Dr. Christian Holm
http://fias.uni-frankfurt.de/~simbio/Teaching
Email: [email protected]
Tel: 798-47505
Room Nr. 2.301
Motivation
Two key questions of modern science:
1. What is the microscope structure of matter( High Energy
Physics, Analytical Chemistry, Biochemistry, Genetics…)
2. How do properties of macroscopic systems follows from
their many microscopic constituents. (Statistical Mechanics,
analytical chemistry, cell biology)
• The aim of statistical physics is:
Explain macroscopic phenomena from microscopic laws!
Examples:
--The meaning of temperature, heat, entropy.
--Equation of state for gases, liquids, solids.
-> thermostatics)
--Transport phenomenon and irreversible processes
-> thermodynamics
Recent Successes:
•
•
•
•
How come that water boils at 100 C? Why is there a
gaseous, liquid, and solid phase? Why does a magnet
lose ist magnetization if one heats it up?
-> Theory of phase transitions
Why does liquid 4He lose ist viscosity below 2.2 K
-> Theory of superfluidity.
Why does copper conduct an electric current, but glas
does not?
-> Theory of electrical conductivity.
Where do the structures come from which spontaneously
form in microemulsions?
-> Theory of pattern formation
Methodology of Statistical Physics:
•
•
Appropriate treatment of non-linear and large numbers:
This is important for almost all branches of physics
Basic assumption
• The fundamental assumption of thermal physics is
that a closed system is equally likely to be in any of
its g accessible micro- states. All accessible microstates are assumed to be equally probable.
Accessible state
Most probable state
Wall
Equilibrium state does not
depend on time
Basic assumption
• The fundamental assumption of thermal
physics is that a closed system is equally
likely to be in any of its g accessible microstates. All accessible micro- states are
assumed to be equally probable. A microstate is accessible, if its properties are
compatible with the system parameters.
• A closed system will have constant energy
E, a constant number of particles N,
constant volume V, and constant values of
all external parameters that may influence
the system (gravity, electric fields, etc).
Toy models (binary spin system)
• Magnetic moment:
+/-m
⇑ B↑1↑2↑3↑4↑5↑6↑7↑8 •••↑N • Energy of a spin in
a magnetic field B:
E=-mB
Total number of micro-states:
21⋅ 4
2 ⋅2
24
⋅ ⋅3
⋅ 2 = 2N
N
2 Spins : (↑1 + ↓1)(↑2 + ↓2 ) =↑1↑2 + ↑1↓2 + ↓1↑2 + ↓1↓2
N Spins : (↑1 + ↓1 )(↑2 + ↓2 )...(↑N + ↓N ) = 2 N terms
“generating function for my state”
• Total magnetic moment:
N
M = ∑ mi
N: even
i=1
The values of M run from Nm, (N-2)m, (N-4)m, …2m,0, -2m,…-Nm,
In total there are (N+1) values for M
There are many more states than possible magnetizations!
(↑1 + ↓1)(↑2 + ↓2 ) =↑1↑2 + ↑1↓2 + ↓1↑2 + ↓1↓2
two states with magnetization 0 =>
Number of up-spins:
Number of down-spins:
+2m
0
1
N↑ = N + s
2
1
N↓ = N − s
2
0
-2m
N↓ − N↑ = 2s
Spin excess
s runs from -N/2, -N/2+1, …, N/2 and has (N+1 values)
Binominal expansion
(↑ + ↓) N = (x + y) N
N  
N N −k k N
N!
= ∑  x y =∑
x N −k y k
k
k= 0  
k= 0 (N − k)!k!
Introduce variable k =
Pascal's triangle
1
N−s
2
N
N!
x N −k y k
k = 0 ( N − k )! k!
=∑
−N / 2
1
1
N +s
N −s
N!
2
2
= ∑
x
y
1
1
s= N / 2
( N + s )!( N − s )!
244424
2 443
1
=
N /2
∑
s =− N / 2
N!
N↓! N↑!
1
424
3
g ( N , s ) multiplicity function
≅ number of states having
the same value of s
x
1
N +s
2
y
1
N −s
2
• Total number of states
N /2
N /2
∑ g ( N , s) = ∑ g ( N , s)1
s =− N / 2
N / 2+ s N / 2− s
s =− N / 2
Energy of the binary magnetic system:
U = −M ⋅ B
N
= − B ∑ mi
i =1
= −2smB
210 = 1024
Equal spacing of energy levels
1
=(1 + 1)
N
States of a Model System
• It is the number of micro-states that is
important in thermal physics, not the number
of energy levels. Two states at the same
energy must always be counted as two states,
not as one level.
• 1. Multiplicity function for a system of N
magnets with spin excess
• in the limit s/N << 1, with N>>1, we can
derive the Gaussian approximation
g ( N , s ) ≅ g ( N ,0)e
2
−2 s / N
with
1/ 2
 2 
N
g ( N ,0) = 
 2
 πN 
To show this is left as an exercise.... For the homework
sheet
When s2=1/2 N the value of
g is reduced to e-1.
s
1
=
The relative value
measures the the fractional width of the distribution
N
2N
Average Value:
• Let P(s) be a probability distribution function
f = ∑ f ( s) P( s)
∑ P( s) = 1
Normalization:
s
s
g(N,s) is not properly normalized, since
N
(
,
)
=
2
a use
g
N
s
∑
s
Let us calculate
approximation:
<s2>
using the Gauss
N /2
2 −2 s
s
∫ e
g (0, s )
s2 =
2
/N
where we replaced ∑ → ∫
−N / 2
N
2
1/ 2
 2  N
=
  
π
N

 2
1/ 2
 2  N
=
  
 πN   2 
3/ 2 ∞
∫ dxx e
2 − x2
−∞
3/ 2
g ( N , s)
P( N , s) =
2N
2s 2
N
, here we use x =
and sds = xdx
N
2
2
(π / 4)1/ 2
 2 N 3π  1
2

(
)
= 
=
N
or
for
the
mean
square
spin
excess
2
s
=N
3 
 πN 2 4  4
The root mean square spin excess:
(2s )2
= N
The fractional (relative) fluctuation in 2s is
defined as
For large values of N the fractional fluctuation
goes to zero! For N=1020 we find =10-10