Statistical Physics

1. Microstate: full description of system
   1. Many microstates can correspond to the same macrostate

1. Total entropy is extensive => S_AB = S_A + S_B
   1. => Boltzmann's Entropy: S = k_b ln(omega)
2. Total number of microstates of AB

1. Statistical weight
2. Need to maximise entropy S with constraints: ∑E_j*n_j=U and ∑n_j=N
   1. Lagrange multipliers
      1. Define partition function z = ∑e^(-βE_j)
         1. Specify α from number of particles and partition function
      2. Add dQ and equate to dU to find β= 1/k_B*T
      3. n_j = (N/z)*e^(-E_j/(k_B*T))
   2. Degeneracy g_j:
      1. g_j multiplies Boltzmann factor

Nota:

• Heat baths: heat can leave or enter. e.g. Glass of water, single atom in solid. T is constant

1. Gibbs entropy: S = -k_B*∑p_j*ln(p_j)
2. Can use z to link to thermodynamics
   1. Expressing U in terms of z: U=-N(d(ln(x))/d(β))
      1. 1D SHO
         1. Expressions for U at high and low T regimes
   2. Bridge equation: F = -N*k_B*T*ln(z_1)
      1. Use z to derive thermodynamic properties

1. Maximise Gibbs' entropy with constraints on N, P and U
   1. Grand Partition Function Z: (Ej-uN) instead of Ej
      1. Can write Gibbs' entropy in terms of U, N, T and F
         1. F links to Thermodynamics

1. Density of states
   1. Partition function of classical gas
      1. For indistinguishable particles: Z_n = Z_1^N/N!
         1. Maxwell-Boltzmann distribution describes occupancy: f(E) = A*e^(-E/k_B*T)
            1. Maxwell-Boltzmann distribution of speeds: n° part's with velocity v: n(v)*dv = f(v)*g(v)*dv
   2. Statistical weight of classical gases

      Nota:

      • product((g_j^(n_j))/n_j!)

1. Fermi gas
   1. Statistical weight for Fermi gas

      Nota:

      • omega = product(g_j!/n_j!(g_j-n_j)!)
      1. Maximise at constant U and N to get expression for n_j
         1. Probability distribution is n_j/g_j = FD distribution = 1/(1+e^((E-u)/kT)
   2. Pauli's exclusion principle
   3. Degenerate Fermi gas:
   4. Fermi E: E_F = u at T=0
      1. Fermi T: T_F = E_F/k_B
2. Bose-Einstein gas
   1. Statistical weight for Boson gas

      Nota:

      • omega = product((n_j+g_j-1)!/(n_j!*(g_j-1)!) which is approximately product(n_j+g_j)!/(n_j!*g_j!)
      1. maximise ln(omega) at constant U and N to get
         1. Bose-Einstein distribution: f_BE = 1/((e^((E-u)/kT)-1)
   2. Photon gas: no chemical potential
      1. Energy spectral density: u = E*g(w)*f(w)*dw

         Nota:

         • E = h_bar w. g(w)dw = V/(2pi)^3 * 4*pi*k^2 dw f(w) = 1/(e^(h_bar*omega/kT)-1)
         1. Planck's law of radiation (u(v))
            1. Energy flux: V*integral(u(v)*dv) * c * 1/4

               Nota:

               • Note: integral for u gives pi^4/15
               1. Stefan-Boltzmann law: enery flux = sigma*t^4
   3. Bose-Einstein condensation
      1. At T=0
         1. n_0 is large => u goes to 0
            1. n' is proportional to T^(3/2)

               Nota:

               • And n'/N = (T/T_B)^(3/2), where T_B is Bose Temperature
               1. At T_B all particles are in excited state
                  1. At T_B, average distance between particle is comparable to De Broglie wavelength
                     1. Wavefunctions of atoms overlap => single wavefunction describing the whole system: condensate
3. Both quantum gases reduce to classical gas if very dilute: g_j >> n_j