Statistical Physics

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Graduação Thermodynamics & Stat. Physics Mind Map on Statistical Physics, created by eg612 on 04/08/2014.
eg612
Mind Map by eg612, updated more than 1 year ago
eg612
Created by eg612 over 11 years ago
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Resource summary

Statistical Physics
  1. Definitions
    1. Macrostate: description of a thermodynamic system using macroscopic variables
      1. Microstate: full description of system
        1. Many microstates can correspond to the same macrostate
      2. Fundamental postulate of Stat. Physics: every microstate has the same probability
        1. If 2 systems A & B are merged:
          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
          2. Distinguishable particles: solids
            1. Isolated: microcanonical ensemble
              1. Statistical weight
                1. 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
                        1. n_j = (N/z)*e^(-E_j/(k_B*T))
                        2. Degeneracy g_j:
                          1. g_j multiplies Boltzmann factor
                      3. Closed: Canonical ensemble

                        Annotations:

                        • 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)
                          1. 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
                            2. Open: Grand canonical ensemble
                              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
                            3. Indistinguishable particles: gases
                              1. Classical gases (dilute): g_J >> n_j
                                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

                                      Annotations:

                                      • product((g_j^(n_j))/n_j!)
                                  2. Quantum gases
                                    1. Fermi gas
                                      1. Statistical weight for Fermi gas

                                        Annotations:

                                        • 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
                                          1. Degenerate Fermi gas:
                                            1. 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

                                                Annotations:

                                                • 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

                                                    Annotations:

                                                    • 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

                                                        Annotations:

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

                                                          Annotations:

                                                          • 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
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