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Graduação Thermodynamics & Stat. Physics Mind Map on Statistical Physics, created by eg612 on 04/08/2014.
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Created by eg612
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Statistical Physics
1 Definitions
1.1 Macrostate: description of a thermodynamic
system using macroscopic variables
1.1.1 Microstate: full description of system
1.1.1.1 Many microstates can correspond
to the same macrostate
1.2 Fundamental postulate of Stat. Physics:
every microstate has the same probability
1.3 If 2 systems A & B are merged:
1.3.1 Total entropy is extensive =>
S_AB = S_A + S_B
1.3.1.1 => Boltzmann's Entropy: S = k_b ln(omega)
1.3.2 Total number of microstates of AB
1.3.2.1
2 Distinguishable particles: solids
2.1 Isolated: microcanonical ensemble
2.1.1 Statistical weight
2.1.2 Need to maximise entropy S with
constraints: ∑E_j*n_j=U and ∑n_j=N
2.1.2.1 Lagrange multipliers
2.1.2.1.1 Define partition function z = ∑e^(-βE_j)
2.1.2.1.1.1 Specify α from number of particles and partition function
2.1.2.1.2 Add dQ and equate to dU to find β= 1/k_B*T
2.1.2.1.3 n_j = (N/z)*e^(-E_j/(k_B*T))
2.1.2.2 Degeneracy g_j:
2.1.2.2.1 g_j multiplies Boltzmann factor
2.2 Closed: Canonical ensemble
Annotations:
- Heat baths: heat can leave or enter. e.g. Glass of water, single atom in solid. T is constant
2.2.1 Gibbs entropy: S = -k_B*∑p_j*ln(p_j)
2.2.2 Can use z to link to thermodynamics
2.2.2.1 Expressing U in terms of z: U=-N(d(ln(x))/d(β))
2.2.2.1.1 1D SHO
2.2.2.1.1.1 Expressions for U at high and low T regimes
2.2.2.2 Bridge equation: F = -N*k_B*T*ln(z_1)
2.2.2.2.1 Use z to derive thermodynamic properties
2.3 Open: Grand canonical ensemble
2.3.1 Maximise Gibbs' entropy with constraints on N, P and U
2.3.1.1 Grand Partition Function Z: (Ej-uN) instead of Ej
2.3.1.1.1 Can write Gibbs' entropy in terms of U, N, T and F
2.3.1.1.1.1 F links to Thermodynamics
3 Indistinguishable particles: gases
3.1 Classical gases (dilute): g_J >> n_j
3.1.1 Density of states
3.1.1.1 Partition function of classical gas
3.1.1.1.1 For indistinguishable particles:
Z_n = Z_1^N/N!
3.1.1.1.1.1 Maxwell-Boltzmann distribution describes occupancy:
f(E) = A*e^(-E/k_B*T)
3.1.1.1.1.1.1 Maxwell-Boltzmann distribution of speeds: n° part's with velocity v:
n(v)*dv = f(v)*g(v)*dv
3.1.1.2 Statistical weight of classical gases
Annotations:
- product((g_j^(n_j))/n_j!)
3.2 Quantum gases
3.2.1 Fermi gas
3.2.1.1 Statistical weight for Fermi gas
Annotations:
- omega = product(g_j!/n_j!(g_j-n_j)!)
3.2.1.1.1 Maximise at constant U and N
to get expression for n_j
3.2.1.1.1.1 Probability distribution is n_j/g_j = FD
distribution = 1/(1+e^((E-u)/kT)
3.2.1.2 Pauli's exclusion principle
3.2.1.3 Degenerate Fermi gas:
3.2.1.4 Fermi E: E_F = u at T=0
3.2.1.4.1 Fermi T: T_F = E_F/k_B
3.2.2 Bose-Einstein gas
3.2.2.1 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!)
3.2.2.1.1 maximise ln(omega) at constant U and N to get
3.2.2.1.1.1 Bose-Einstein distribution: f_BE = 1/((e^((E-u)/kT)-1)
3.2.2.2 Photon gas: no chemical potential
3.2.2.2.1 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)
3.2.2.2.1.1 Planck's law of radiation (u(v))
3.2.2.2.1.1.1 Energy flux: V*integral(u(v)*dv) * c * 1/4
Annotations:
- Note: integral for u gives pi^4/15
3.2.2.2.1.1.1.1 Stefan-Boltzmann law: enery flux = sigma*t^4
3.2.2.3 Bose-Einstein condensation
3.2.2.3.1 At T=0
3.2.2.3.1.1 n_0 is large => u goes to 0
3.2.2.3.1.1.1 n' is proportional to T^(3/2)
Annotations:
- And n'/N = (T/T_B)^(3/2), where T_B is Bose Temperature
3.2.2.3.1.1.1.1 At T_B all particles are in excited state
3.2.2.3.1.1.1.1.1 At T_B, average distance between particle is
comparable to De Broglie wavelength
3.2.2.3.1.1.1.1.1.1 Wavefunctions of atoms overlap => single wavefunction
describing the whole system: condensate
3.2.3 Both quantum gases reduce to classical gas if very dilute: g_j >> n_j