PH-203 Cheat Sheet
Equations of State Maxwell Equations Classical Microcanonical Ensemble
P V = N kT (Ideal Gas Law) First Law Differential - Large, Known, Constant N
dU = T dS − P dV - Known V
2
(P + a NV )(V − N b) = N kT (Van der Waals - In Equilibrium ( Doesn’t Change in Time)
∂U - Systems are Isolated (U = E Constant and Known)
Gas) T = ∂S
∂U ———————————————————————
P = − ∂V
´
For a van der waals gas when a=b=0 we reco- Σ(E) = h3N1 N ! d3N q d3N p
ver the ideal gas law Helmholtz Differential
´
dF = −SdT − P dV d3N q = V N
Laws of Thermodynamics (If there’s no q dependence in Hamiltonian)
First Law P ∂F
= − ∂V
´ a a
dU = δQ − δw S = − ∂F da x = π2E2
∂T (a
2 )!
Second Law Gibbs Differential S = K ln (Σ)
δQ
dS = T dG = −SdT + V dP
Third Law Trig + Hyperbolic Functions
S = − ∂G
∂T
eit +e−it
lim (S) = 0 V = ∂G Cos(t) = 2
T →0 ∂P it −it
Sin(t) = e −e 2i
t −t
Thermodynamic Potentials Enthalpy Differential Cosh(t) = e +e 2
t −t
F = U − T S (Helmholtz Free Energy) dH = T dS + V dP Sinh(t) = e −e 2
∂
G = F + P V (Gibbs Free Energy) ∂t Sinh(t) = Cosh(t)
∂
H = U + P V = G + T S (Enthalpy) T = − ∂H ∂t Cosh(t) = Sinh(t)
∂S
V = − ∂H
∂P
Gaussian Integrals The Canonical Ensemble
´ ∞ −γx2 q -N particles in contact with a heat reservoir
e dx = πγ Equipartition Theorem
−∞
´ ∞ 2 −γx D -T fixed and Known
x e dx = γ23 Cv = γK
−∞ -N fixed and known
———————————————————————
Stirling Approximation D - Degrees Of Freedom
ρ(p, q) = 1
N !h3N
e−βH (Density Function)
ln (N !) ≈ N ln (N ) − N γ - Power Dependance from Hamiltonian 1
β=
´
KT
Zn = ρ(p, q)d3N p d3N q (Partition Function)
This is very important when doing ensemble For example
1
P3N 2 F = −KT ln (Zn )
calculations, keep it in this form to make the calcu- H = 2m pi
U = F + TS
lations easier later
Having this Hamiltonian shows us that D = 3N
and γ = 2
, Classical Grand Canonical Ensemble
-N is Not Known and Not Constant
-T is Known and Fixed
-µ is Known and Fixed
———————————————————————
∂F
µ= ∂N (µ is Chemical Potential)
Now we have introduced chemical potentials we
have to augment the thermodynamic potential
equations
dU = T dS − P dV + µdN
dG = −SdT + V dP + µdN
dF = −SdT − P dV + µdN
z = eβµ (Z is Fugacity)
Z = z N ZN (Grand Partition)
P
´
ZN = N !h13N d3N p d3N q e−βH (Classical Partition)
PV
ln (Z) = KT
∂
< N >= z ∂z ln (Z)
z - fugacity
Z - Grand Partition Function
ZN - Classical Canonical Partition Function
,Microcanonical Ensemble
26 June 2023 09:24
,
Equations of State Maxwell Equations Classical Microcanonical Ensemble
P V = N kT (Ideal Gas Law) First Law Differential - Large, Known, Constant N
dU = T dS − P dV - Known V
2
(P + a NV )(V − N b) = N kT (Van der Waals - In Equilibrium ( Doesn’t Change in Time)
∂U - Systems are Isolated (U = E Constant and Known)
Gas) T = ∂S
∂U ———————————————————————
P = − ∂V
´
For a van der waals gas when a=b=0 we reco- Σ(E) = h3N1 N ! d3N q d3N p
ver the ideal gas law Helmholtz Differential
´
dF = −SdT − P dV d3N q = V N
Laws of Thermodynamics (If there’s no q dependence in Hamiltonian)
First Law P ∂F
= − ∂V
´ a a
dU = δQ − δw S = − ∂F da x = π2E2
∂T (a
2 )!
Second Law Gibbs Differential S = K ln (Σ)
δQ
dS = T dG = −SdT + V dP
Third Law Trig + Hyperbolic Functions
S = − ∂G
∂T
eit +e−it
lim (S) = 0 V = ∂G Cos(t) = 2
T →0 ∂P it −it
Sin(t) = e −e 2i
t −t
Thermodynamic Potentials Enthalpy Differential Cosh(t) = e +e 2
t −t
F = U − T S (Helmholtz Free Energy) dH = T dS + V dP Sinh(t) = e −e 2
∂
G = F + P V (Gibbs Free Energy) ∂t Sinh(t) = Cosh(t)
∂
H = U + P V = G + T S (Enthalpy) T = − ∂H ∂t Cosh(t) = Sinh(t)
∂S
V = − ∂H
∂P
Gaussian Integrals The Canonical Ensemble
´ ∞ −γx2 q -N particles in contact with a heat reservoir
e dx = πγ Equipartition Theorem
−∞
´ ∞ 2 −γx D -T fixed and Known
x e dx = γ23 Cv = γK
−∞ -N fixed and known
———————————————————————
Stirling Approximation D - Degrees Of Freedom
ρ(p, q) = 1
N !h3N
e−βH (Density Function)
ln (N !) ≈ N ln (N ) − N γ - Power Dependance from Hamiltonian 1
β=
´
KT
Zn = ρ(p, q)d3N p d3N q (Partition Function)
This is very important when doing ensemble For example
1
P3N 2 F = −KT ln (Zn )
calculations, keep it in this form to make the calcu- H = 2m pi
U = F + TS
lations easier later
Having this Hamiltonian shows us that D = 3N
and γ = 2
, Classical Grand Canonical Ensemble
-N is Not Known and Not Constant
-T is Known and Fixed
-µ is Known and Fixed
———————————————————————
∂F
µ= ∂N (µ is Chemical Potential)
Now we have introduced chemical potentials we
have to augment the thermodynamic potential
equations
dU = T dS − P dV + µdN
dG = −SdT + V dP + µdN
dF = −SdT − P dV + µdN
z = eβµ (Z is Fugacity)
Z = z N ZN (Grand Partition)
P
´
ZN = N !h13N d3N p d3N q e−βH (Classical Partition)
PV
ln (Z) = KT
∂
< N >= z ∂z ln (Z)
z - fugacity
Z - Grand Partition Function
ZN - Classical Canonical Partition Function
,Microcanonical Ensemble
26 June 2023 09:24
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