Statistical Physics of Matter

Microstate

A microstate denotes a description of a system where the values of dynamical variables are exactly specified.

Macrostate

A macrostate denotes a collection of system microstates with a common property.

Principle of equal a priori probabilities

An isolated system in equilibrium occupies each accessible microstate with the same probability.

Microcanonical ensemble

An isolated system.

• Every microstate has equal probabilities.

Canonical ensemble

A system that can exchange energy with a large reservoir.

• Every microstate in combined (sys + res) is equally likely.

Grand canonical ensemble

A system that can exchange energy and particles with a large reservoir.

• Every microstate in combined (sys + res) is equally likely.

Zeroth law

If two systems are in thermal equilibrium with a third system, then they are in equilibrium with each other.

First law

The total energy of an isolated system remains constant.

Second law

Heat cannot flow spontaneously from a colder to a warmer body (Clausius statement).

Or: There is an extensive state function, entropy, which for an isolated system either grows or, in ideal reversible transformations, is conserved.

Third law

It is impossible to reduce the absolute temperature to zero in a finite number of steps (Nernst’s unattainability principle).

Or: The entropy of a system at zero temperature is an absolute constant that does not depend on any other thermodynamic variable.

Equipartition of energy

Each term a_jxj² in the energy, or each degree of freedom, contributes a term ½k_BT to the total average canonical energy at a temperature T.

Extensive variable

Scale with the amount of material in the system, such as the energy E, the volume V, and the number of constituents N.

Intensive variable

Remain the same for different amounts of material in the system, such as the temperature T and the pressure p.

Process variables

Heat Q and work W, describe changes to the system but do not specify its state in equilibrium.

Helmholtz free energy

F(T,V,N) = E - TS

Gibbs free energy

G(T,p,N) = E - TS + pV

Enthalpy

H(S,p,N) = E + pV

Grand potential

Φ(T,V,\mu) = E - TS - \muN

Thermodynamic limit

N → $\infty$, where V grows in proportion.

Classical limit

\langle N \rangle « 1, so bosons and fermions become equivalent.

Bose-Einstein condensate

A Bose-Einstein condensate is attained when, for a boson gas at very low temperatures, a finite ratio of the total number of particles (and hence a macroscopic number of particles in the thermodynamic limit) occupy the single-particle ground state.

• This occurs suddenly when the temperature drops below a critical value T_c.

Degenerate Fermi gas

A degenerate Fermi gas occurs when the single-particle energy states are occupied with probability unity up to a maximum single-particle energy ϵ_F, called the ‘Fermi energy’, whilst all single-particle states with energy above the Fermi energy ϵ_F are empty.

• This occurs exactly at zero temperature and approximately when βϵ_F ≫ 1.

Phonons

The collective excitations of bosons in a lattice.

Heat capacities of solids at low temperatures

In the low-T limit of the Debye model, the heat capacity C_V of a solid is proportional to T³

Black body radiation

Electromagnetic radiation in thermal equilibrium at temperature T = 1/(k_Bβ) inside a container of volume V at which there is zero net energy flux between system and reservoir.

Stefan-Boltzmann law

A black body at temperature T irradiates energy with a flux Φ that is proportional to T⁴.

Debye model

For which the vibration of modes have a distribution of frequencies.

• g(\omega) d\omega is the density of states, which is the number of states between frequencies \omega and \omega + d\omega.

Einstein model

For which all vibrational modes have the same frequency, \omega_E.

Wien’s law

For electromagnetic radiation in thermal equilibrium at a temperature T, the maximum energy density occurs at the frequency \omega_\text{max}\approx2.821\frac{k_BT}{\hslash}

Fermi temperature

k_BT_F = ϵ_F.

• The Fermi gas is degenerate if the Fermi energy is much larger than the energy of thermal fluctuations: ϵ_F » k_BT such that T « T_F.