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Statistical Physics Formulas

Complete collection of statistical physics formulas with detailed explanations.

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Statistical Distributions

Maxwell-Boltzmann Distribution

f(v)=4πv2(m2πkT)3/2e−mv22kTf(v) = 4\pi v^2\left(\frac{m}{2\pi kT}\right)^{3/2}e^{-\frac{mv^2}{2kT}}

Probability distribution of molecular speeds in an ideal gas.

Notation:

f(v):Speed distribution function
v:Molecular speed
m:Molecular mass
k:Boltzmann constant
T:Temperature

Units:

f(v):s/m
v:m/s
m:kg
k:1.381 × 10⁻²³ J/K
T:K

Applications:

  • •Gas kinetics
  • •Thermal physics
  • •Molecular dynamics

Limitations:

Ideal gas, equilibrium

Bose-Einstein Statistics

ni=1eεikT−1n_i = \frac{1}{e^{\frac{\varepsilon_i}{kT}} - 1}

Average occupation number for bosons in energy level i.

Notation:

n_i:Average occupation number
ε_i:Energy of level i
k:Boltzmann constant
T:Temperature

Units:

n_i:dimensionless
ε_i:J
k:J/K
T:K

Applications:

  • •Photon statistics
  • •Bose-Einstein condensates
  • •Quantum gases

Limitations:

Bosonic particles

Fermi-Dirac Statistics

ni=1eεi−μkT+1n_i = \frac{1}{e^{\frac{\varepsilon_i - \mu}{kT}} + 1}

Average occupation number for fermions in energy level i.

Notation:

n_i:Average occupation number
ε_i:Energy of level i
μ:Chemical potential
k:Boltzmann constant
T:Temperature

Units:

n_i:dimensionless
ε_i, μ:J
k:J/K
T:K

Applications:

  • •Electron gas
  • •Semiconductor physics
  • •White dwarf stars

Limitations:

Fermionic particles

Boltzmann Distribution

P(E)=1Ze−EkTP(E) = \frac{1}{Z}e^{-\frac{E}{kT}}

Probability of finding a system in energy state E.

Notation:

P(E):Probability
E:Energy
Z:Partition function
k:Boltzmann constant
T:Temperature

Units:

P(E):dimensionless
E:J
Z:dimensionless
k:J/K
T:K

Applications:

  • •Statistical mechanics
  • •Thermal equilibrium
  • •Energy level populations

Limitations:

Canonical ensemble

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Partition Functions

Canonical Partition Function

Z=∑ie−εikTZ = \sum_i e^{-\frac{\varepsilon_i}{kT}}

Canonical partition function equals sum over all states of exponential of negative energy divided by kT.

Notation:

Z:Partition function
ε_i:Energy of state i
k:Boltzmann constant
T:Temperature

Units:

Z:dimensionless
ε_i:J
k:J/K
T:K

Applications:

  • •Thermodynamic properties
  • •Statistical mechanics
  • •Ensemble theory

Limitations:

Canonical ensemble

Grand Canonical Partition Function

Ξ=∑N∑ie−εi−μNkT\Xi = \sum_N \sum_i e^{-\frac{\varepsilon_i - \mu N}{kT}}

Grand canonical partition function equals sum over particle numbers and states.

Notation:

Ξ:Grand canonical partition function
ε_i:Energy of state i
μ:Chemical potential
N:Particle number
k:Boltzmann constant
T:Temperature

Units:

Ξ:dimensionless
ε_i, μ:J
N:dimensionless
k:J/K
T:K

Applications:

  • •Open systems
  • •Chemical reactions
  • •Phase transitions

Limitations:

Grand canonical ensemble

Free Energy

F=−kTln⁡(Z)F = -kT\ln(Z)

Helmholtz free energy equals negative kT times natural logarithm of partition function.

Notation:

F:Helmholtz free energy
k:Boltzmann constant
T:Temperature
Z:Partition function

Units:

F:J
k:J/K
T:K
Z:dimensionless

Applications:

  • •Thermodynamic equilibrium
  • •Phase stability
  • •Statistical mechanics

Limitations:

Canonical ensemble

ℹ️

Entropy and Information

Boltzmann Entropy

S=kln⁡(W)S = k\ln(W)

Entropy equals Boltzmann constant times natural logarithm of number of microstates.

Notation:

S:Entropy
k:Boltzmann constant
W:Number of microstates

Units:

S:J/K
k:J/K
W:dimensionless

Applications:

  • •Statistical mechanics
  • •Thermodynamics
  • •Information theory

Limitations:

Equilibrium systems

Shannon Entropy

H=−∑ipilog⁡(pi)H = -\sum_i p_i\log(p_i)

Shannon entropy equals negative sum of probability times logarithm of probability.

Notation:

H:Shannon entropy
p_i:Probability of state i

Units:

H:bits or nats
p_i:dimensionless

Applications:

  • •Information theory
  • •Data compression
  • •Quantum information

Limitations:

Discrete probability distribution

von Neumann Entropy

S=−Tr(ρlog⁡(ρ))S = -\text{Tr}(\rho\log(\rho))

von Neumann entropy equals negative trace of density matrix times logarithm of density matrix.

Notation:

S:von Neumann entropy
ρ:Density matrix
Tr:Trace

Units:

S:dimensionless
ρ:dimensionless

Applications:

  • •Quantum mechanics
  • •Quantum information
  • •Entanglement measures

Limitations:

Quantum systems

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Phase Transitions

Critical Temperature

Tc=2JkT_c = \frac{2J}{k}

Critical temperature equals twice interaction energy divided by Boltzmann constant.

Notation:

T_c:Critical temperature
J:Interaction energy
k:Boltzmann constant

Units:

T_c:K
J:J
k:J/K

Applications:

  • •Magnetic phase transitions
  • •Ising model
  • •Critical phenomena

Limitations:

Mean field approximation

Order Parameter

M=tanh⁡(βJzM)M = \tanh(\beta J z M)

Order parameter equals hyperbolic tangent of beta times interaction energy times coordination number times order parameter.

Notation:

M:Order parameter
β:1/kT
J:Interaction energy
z:Coordination number

Units:

M:dimensionless
β:J⁻¹
J:J
z:dimensionless

Applications:

  • •Phase transitions
  • •Magnetic systems
  • •Critical phenomena

Limitations:

Mean field theory

Correlation Length

ξ=ξ0∣T−Tc∣−ν\xi = \xi_0|T - T_c|^{-\nu}

Correlation length equals characteristic length times absolute temperature difference to power negative critical exponent.

Notation:

ξ:Correlation length
ξ₀:Characteristic length
T:Temperature
T_c:Critical temperature
ν:Critical exponent

Units:

ξ, ξ₀:m
T, T_c:K
ν:dimensionless

Applications:

  • •Critical phenomena
  • •Phase transitions
  • •Scaling theory

Limitations:

Near critical point