Statistical Physics

Statistical Physics

1. Definitions

   • Macrostate: description of a thermodynamic system using macroscopic variables
     • Microstate: full description of system
       • Many microstates can correspond to the same macrostate
   • Fundamental postulate of Stat. Physics: every microstate has the same probability
   • If 2 systems A & B are merged:
     • Total entropy is extensive => S_AB = S_A + S_B
       • => Boltzmann's Entropy: S = k_b ln(omega)
     • Total number of microstates of AB

2. Distinguishable particles: solids

   • Isolated: microcanonical ensemble
     • Statistical weight
     • Need to maximise entropy S with constraints: ∑E_j*n_j=U and ∑n_j=N
       • Lagrange multipliers
         • Define partition function z = ∑e^(-βE_j)
           • Specify α from number of particles and partition function
         • Add dQ and equate to dU to find β= 1/k_B*T
         • n_j = (N/z)*e^(-E_j/(k_B*T))
       • Degeneracy g_j:
         • g_j multiplies Boltzmann factor
   • Closed: Canonical ensemble
     • Heat baths: heat can leave or enter. e.g. Glass of water, single atom in solid. T is constant
     • Gibbs entropy: S = -k_B*∑p_j*ln(p_j)
     • Can use z to link to thermodynamics
       • Expressing U in terms of z: U=-N(d(ln(x))/d(β))
         • 1D SHO
           • Expressions for U at high and low T regimes
       • Bridge equation: F = -N*k_B*T*ln(z_1)
         • Use z to derive thermodynamic properties
   • Open: Grand canonical ensemble
     • Maximise Gibbs' entropy with constraints on N, P and U
       • Grand Partition Function Z: (Ej-uN) instead of Ej
         • Can write Gibbs' entropy in terms of U, N, T and F
           • F links to Thermodynamics

3. Indistinguishable particles: gases

   • Classical gases (dilute): g_J >> n_j
     • Density of states
       • Partition function of classical gas
         • For indistinguishable particles: Z_n = Z_1^N/N!
           • Maxwell-Boltzmann distribution describes occupancy: f(E) = A*e^(-E/k_B*T)
             • Maxwell-Boltzmann distribution of speeds: n° part's with velocity v: n(v)*dv = f(v)*g(v)*dv
       • Statistical weight of classical gases
         • product((g_j^(n_j))/n_j!)
   • Quantum gases
     • Fermi gas
       • Statistical weight for Fermi gas
         • omega = product(g_j!/n_j!(g_j-n_j)!)
         • Maximise at constant U and N to get expression for n_j
           • Probability distribution is n_j/g_j = FD distribution = 1/(1+e^((E-u)/kT)
       • Pauli's exclusion principle
       • Degenerate Fermi gas:
       • Fermi E: E_F = u at T=0
         • Fermi T: T_F = E_F/k_B
     • Bose-Einstein gas
       • Statistical weight for Boson gas
         • omega = product((n_j+g_j-1)!/(n_j!*(g_j-1)!) which is approximately product(n_j+g_j)!/(n_j!*g_j!)
         • maximise ln(omega) at constant U and N to get
           • Bose-Einstein distribution: f_BE = 1/((e^((E-u)/kT)-1)
       • Photon gas: no chemical potential
         • Energy spectral density: u = E*g(w)*f(w)*dw
           • E = h_bar w. g(w)dw = V/(2pi)^3 * 4*pi*k^2 dw f(w) = 1/(e^(h_bar*omega/kT)-1)
           • Planck's law of radiation (u(v))
             • Energy flux: V*integral(u(v)*dv) * c * 1/4
               • Note: integral for u gives pi^4/15
               • Stefan-Boltzmann law: enery flux = sigma*t^4
       • Bose-Einstein condensation
         • At T=0
           • n_0 is large => u goes to 0
             • n' is proportional to T^(3/2)
               • And n'/N = (T/T_B)^(3/2), where T_B is Bose Temperature
               • At T_B all particles are in excited state
                 • At T_B, average distance between particle is comparable to De Broglie wavelength
                   • Wavefunctions of atoms overlap => single wavefunction describing the whole system: condensate
     • Both quantum gases reduce to classical gas if very dilute: g_j >> n_j