Statistical Mechanics Notes Page 1 of 24
Mathematical Relations
1. Integrals
¥ 2 p
ò -¥
e -ax dx =
a
¥ (2n )!
2 p
ò -¥
x 2ne -ax dx =
n !22n
a 2n +1
. Obtained by differentiating both sides of
the equation with respect to a a number of times.
¥ ¥ 2 n!
ò
2
ò -¥
x 2n +1e -ax dx = 0 and
0
x 2n +1e -ax dx =
2an +1
¥ ¥
n! = ò0
x ne -x dx and G(n ) = ò 0
x n -1e -x = (n - 1)! for integer n.
2. Approximations
ln n ! » n ln n - n (Sitrling’s Approximation)
3. Series
N -1
a(1 - r N )
Geometric progressions – a å r n =
n =0 1-r
n n (a1 + an ) n éêë2a1 + (n - 1)d ùúû
Arithmetic progressions – å a1 + (n - 1)d = 2
=
2
n =1
Taylor Series
n(n - 1) 2 n(n - 1)(n - 2) 3
(1 + x )
n
o = 1 + nx + x + x +
2! 3!
x3 x5
o sin x = x - + +
3! 5!
x2 x4
o cos x = 1 - + -
2! 4!
x 3 2x 5
o tan x = x + + +
3! 15
x2 x3
o ln (1 + x ) = x - + +
2 3
© Daniel Guetta, 2008
, Statistical Mechanics Notes Page 2 of 24
x3 x5
o sinh(x ) = x - + +
6 120
x2 x4
o cosh(x ) = 1 + + +
2 24
x 3 2x 5
o tanh(x ) = x - + +
3 15
x 2 5x 4
o sech(x ) = 1 - + +
2 24
x 7x 3
o cosech(x ) = x -1 - + +
6 360
-1 x x3
o coth(x ) = x + - +
3 45
N
é1 - (x / N )ùûú = e -x , when N is very large.
ëê
4. Miscellaneous
The volume of a hypersphere radius r in D-dimensions is
2p D /2r D
VD =
G( 21 D + 1)
© Daniel Guetta, 2008
, Statistical Mechanics Notes Page 3 of 24
Useful Thermodynamics Results
1. Microcanonical distribution
w(E ) = Number of states with e < E
W(E ) = w ¢(E ) = Number of states with energy E
S = k ln W
-1
T = (¶S / ¶E )
C = (¶U / ¶T )
V
2. Canonical distribution
Z is the sum over all energies of e -bE .
2
The variance E 2 - E = ¶ 2 ln Z / ¶b 2
Define Z = e -bF (b ) – F is the Helmholtz Free Energy.
Quantities of interest
o U = -¶ ln Z / ¶b
o F = -kBT ln Z
o S = -(¶F / ¶T )V
o p = -(¶F / ¶V )T
o H = U + pV
o G = F + pV = H - TS
o CV = (¶U / ¶T )V
3. Grand Canonical distribution
The chemical potential is the amount of energy each new particle brings to
the system – in other words
æ ¶U ÷ö æ ¶F ö÷ æ ¶G ÷ö
m = ççç ÷÷ = çç ÷÷ = çç ÷÷
è ¶N ÷øS ,V çè ¶N ø÷V ,T èç ¶N ÷øp,T
For two systems in thermal equilibrium, T must be the same. For two
systems in particle-exchange equilibrium, m must be the same.
-bFG
We define the grand potential by Z grand = e
© Daniel Guetta, 2008
Mathematical Relations
1. Integrals
¥ 2 p
ò -¥
e -ax dx =
a
¥ (2n )!
2 p
ò -¥
x 2ne -ax dx =
n !22n
a 2n +1
. Obtained by differentiating both sides of
the equation with respect to a a number of times.
¥ ¥ 2 n!
ò
2
ò -¥
x 2n +1e -ax dx = 0 and
0
x 2n +1e -ax dx =
2an +1
¥ ¥
n! = ò0
x ne -x dx and G(n ) = ò 0
x n -1e -x = (n - 1)! for integer n.
2. Approximations
ln n ! » n ln n - n (Sitrling’s Approximation)
3. Series
N -1
a(1 - r N )
Geometric progressions – a å r n =
n =0 1-r
n n (a1 + an ) n éêë2a1 + (n - 1)d ùúû
Arithmetic progressions – å a1 + (n - 1)d = 2
=
2
n =1
Taylor Series
n(n - 1) 2 n(n - 1)(n - 2) 3
(1 + x )
n
o = 1 + nx + x + x +
2! 3!
x3 x5
o sin x = x - + +
3! 5!
x2 x4
o cos x = 1 - + -
2! 4!
x 3 2x 5
o tan x = x + + +
3! 15
x2 x3
o ln (1 + x ) = x - + +
2 3
© Daniel Guetta, 2008
, Statistical Mechanics Notes Page 2 of 24
x3 x5
o sinh(x ) = x - + +
6 120
x2 x4
o cosh(x ) = 1 + + +
2 24
x 3 2x 5
o tanh(x ) = x - + +
3 15
x 2 5x 4
o sech(x ) = 1 - + +
2 24
x 7x 3
o cosech(x ) = x -1 - + +
6 360
-1 x x3
o coth(x ) = x + - +
3 45
N
é1 - (x / N )ùûú = e -x , when N is very large.
ëê
4. Miscellaneous
The volume of a hypersphere radius r in D-dimensions is
2p D /2r D
VD =
G( 21 D + 1)
© Daniel Guetta, 2008
, Statistical Mechanics Notes Page 3 of 24
Useful Thermodynamics Results
1. Microcanonical distribution
w(E ) = Number of states with e < E
W(E ) = w ¢(E ) = Number of states with energy E
S = k ln W
-1
T = (¶S / ¶E )
C = (¶U / ¶T )
V
2. Canonical distribution
Z is the sum over all energies of e -bE .
2
The variance E 2 - E = ¶ 2 ln Z / ¶b 2
Define Z = e -bF (b ) – F is the Helmholtz Free Energy.
Quantities of interest
o U = -¶ ln Z / ¶b
o F = -kBT ln Z
o S = -(¶F / ¶T )V
o p = -(¶F / ¶V )T
o H = U + pV
o G = F + pV = H - TS
o CV = (¶U / ¶T )V
3. Grand Canonical distribution
The chemical potential is the amount of energy each new particle brings to
the system – in other words
æ ¶U ÷ö æ ¶F ö÷ æ ¶G ÷ö
m = ççç ÷÷ = çç ÷÷ = çç ÷÷
è ¶N ÷øS ,V çè ¶N ø÷V ,T èç ¶N ÷øp,T
For two systems in thermal equilibrium, T must be the same. For two
systems in particle-exchange equilibrium, m must be the same.
-bFG
We define the grand potential by Z grand = e
© Daniel Guetta, 2008
