Problems in Statistical Mechanics

Exercises in Statistical Mechanics

00-00   Using random variables
01-00   The energy spectrum: N(E) and Z(T)
03-00   (The probability matrix)
 
10-00   Canonical formalism
15-00   Grand canonical formalism
20-00   Application of the canonical formalism
30-00   Bose gas
35-00   Fermi gas
40-00   Chemical equilibrium
50-00   Interacting systems
55-00   Phase transitions
 
60-00   Kinetics: Maxwell-Boltzmann etc
65-00   Boltzmann Equation
70-00   Fluctuations, Theory of response, FD relation, Onsager
80-00   Stochastic processes: random walk, diffusion, Langevin
90-00   Master equations: Fokker-Plank, Pauli, Bloch
 
 
The Hebrew source [HTML] of this database has been badly translated into English by a student, and should be improved. Assembled booklet of all exercises [PDF] is available. You can browse all the files in the pool.  
 

ex_0010[pdf][files/sol][forum]    ex_0010 [H] Average distance between two particles
ex_0020[pdf][files/sol][forum]    ex_0020 [H] Average length of chain molecule
ex_0030[pdf][files/sol][forum]    ex_0030 [H] Fluctuations in the number of particles
ex_0050[pdf][files/sol][forum]    ex_0050 [H] Changing random variables $x=\cos(\theta)$
ex_0060[pdf][files/sol][forum]    ex_0060 [C] Oscillator in a microcanonical state
ex_0070[pdf][files/sol][forum]    ex_0070 [H] The ergodic microcanonical density $\rho(x)$
ex_0080[pdf][files/sol][forum]    ex_0080 [L] [edu] (the spreading of a free particle)
ex_0100[pdf][files/sol][forum]    
ex_0105[pdf][files/sol][forum]    ex_0105 [H] Z(T) for double well
ex_0120[pdf][files/sol][forum]    ex_0120 [e] N(E) for N particles in a box
ex_0122[pdf][files/sol][forum]    ex_0122 [e] N(E) for N harmonic oscillators
ex_0130[pdf][files/sol][forum]    ex_0130 [e] N(E) for general dispersion relation
ex_0140[pdf][files/sol][forum]    ex_0140 [e] N(E) for N spins
ex_0150[pdf][files/sol][forum]    ex_0150 [C] Z(T) for spins
ex_0160[pdf][files/sol][forum]    ex_0160 [H] Z(T) for AB and AA molecules
ex_1000[pdf][files/sol][forum]    ex_1000
 
=== Canonical formalism ===
ex_1010[pdf][files/sol][forum]    ex_1010 [e]
ex_1020[pdf][files/sol][forum]    ex_1020 [e]
ex_1030[pdf][files/sol][forum]    ex_1030 [e]
ex_1040[pdf][files/sol][forum]    ex_1040 [e]
ex_1050[pdf][files/sol][forum]    ex_1050 [e]
ex_1060[pdf][files/sol][forum]    ex_1060 [e]
ex_1510[pdf][files/sol][forum]    ex_1510 [e]
ex_1530[pdf][files/sol][forum]    ex_1530 [e]
ex_1535[pdf][files/sol][forum]    ex_1535 [e] (fluctuations in Grand canonical system)
ex_1536[pdf][files/sol][forum]    ex_1536 [e] (fluctuations in Grand canonical system)
ex_1537[pdf][files/sol][forum]    ex_1537 [e] (fluctuations in Grand canonical system)
ex_1601[pdf][files/sol][forum]    ex_1601 [e]
ex_1602[pdf][files/sol][forum]    ex_1602 [e] Mixing entropy
ex_1604[pdf][files/sol][forum]    ex_1604 [e] Negative temperatures
ex_1627[pdf][files/sol][forum]    ex_1627 [e] Equipartition theorem
ex_1800[pdf][files/sol][forum]    
ex_1808[pdf][files/sol][forum]    ex_1808 [-] Adabatic gas law for generalized dispersion (microcanonical)
ex_1814[pdf][files/sol][forum]    ex_1814 [-] Adiabatic vs. sudden expansion of an ideal gas
ex_1815[pdf][files/sol][forum]    ex_1815 [-] Cooling by demagnetization: heat flow between two subsystems
ex_1816[pdf][files/sol][forum]    ex_1816 [-] Cooling by adiabatic demagnetization
ex_1817[pdf][files/sol][forum]    ex_1817 [C] Adiabatic cooling of spins (A25)
ex_2000[pdf][files/sol][forum]    ex_2000
 
=== Canonical formalism, applications ===
ex_2040[pdf][files/sol][forum]    ex_2040 [H] Gas in gravitation
ex_2042[pdf][files/sol][forum]    ex_2042 [H] Gas in gravitation
ex_2044[pdf][files/sol][forum]    ex_2044 [-] Gas in electric field
ex_2046[pdf][files/sol][forum]    ex_2046 [-] Gas in rotating box
ex_2050[pdf][files/sol][forum]    ex_2050 [H] Pressure by particle in a spring-box system
ex_2051[pdf][files/sol][forum]    ex_2051 [L] Gas in V(x,y,z) box potential
ex_2052[pdf][files/sol][forum]    ex_2052 [L] Pressure in V(x) potenial
ex_2060[pdf][files/sol][forum]    ex_2060 [H] Particle on a ring ("rotor") with electric field
ex_2070[pdf][files/sol][forum]    ex_2070 [H] Polarization of two-spheres system inside a tube
ex_2073[pdf][files/sol][forum]    ex_2073 [-] Polarization of molecules
ex_2080[pdf][files/sol][forum]    ex_2080 [-] Magnetization of Spin 1/2
ex_2082[pdf][files/sol][forum]    ex_2082 [-] Magnetization of Spin 1
ex_2090[pdf][files/sol][forum]    ex_2090 [L] Gas in magnetic fiels, Landau levels
ex_2100[pdf][files/sol][forum]    
ex_2110[pdf][files/sol][forum]    ex_2110 [-] Harmonic Osc
ex_2115[pdf][files/sol][forum]    ex_2115 [-] Harmonic Osc, Debay
ex_2130[pdf][files/sol][forum]    ex_2130 [-] Harmonic Osc, Photons
ex_2200[pdf][files/sol][forum]    
ex_2210[pdf][files/sol][forum]    ex_2210 [-] Density of a gas inside a rotating box
ex_2220[pdf][files/sol][forum]    ex_2220 [-] Classical gas with general dispersion relation
ex_2300[pdf][files/sol][forum]    
ex_2311[pdf][files/sol][forum]    ex_2311 [C] Defects in lattice (A13) (Similar to 2310)
ex_2320[pdf][files/sol][forum]    ex_2320 [-] Tension of rotating device
ex_2340[pdf][files/sol][forum]    ex_2340 [-] Tension of a chain molecule
ex_2345[pdf][files/sol][forum]    ex_2345 [-] One dimensional XY model, exact solution
ex_2351[pdf][files/sol][forum]    ex_2351 [C] Length of a chain (A12) (Similar to 2350)
ex_2353[pdf][files/sol][forum]    ex_2353 [-] Tension of a rubber band, instability criterion
ex_2360[pdf][files/sol][forum]    ex_2360 [C] DNA molecule (A20) (Similar to 2340)
ex_3000[pdf][files/sol][forum]    ex_3000
 
=== Quantum gases ===
ex_3010[pdf][files/sol][forum]    ex_3010 [H] Heat capacity of ideal Bose gas (see also 3011)
ex_3020[pdf][files/sol][forum]    ex_3020 [-] Bose gas in strong magnetic field
ex_3021[pdf][files/sol][forum]    ex_3021 [C] Bosons with Spin in magnetic field (B06) (Similar to 3020)
ex_3030[pdf][files/sol][forum]    ex_3030 [H] Charged Bose gas in box with potential difference
ex_3040[pdf][files/sol][forum]    ex_3040 [H] Quantum Bose Gas with an oscillating piston (Exam2002 Q2)
ex_3042[pdf][files/sol][forum]    ex_3042 [-] Quantum Bose Gases, oscillations of a piston (Exam2004)
ex_3050[pdf][files/sol][forum]    ex_3050
ex_3230[pdf][files/sol][forum]    ex_3230 [-] Heat Capcity of a gapped system (He)
ex_3240[pdf][files/sol][forum]    ex_3240
ex_3336[pdf][files/sol][forum]    ex_3336 [C] Condesation for general dispersion (B03)
ex_3338[pdf][files/sol][forum]    ex_3338 [-] Bose in graviation
ex_3341[pdf][files/sol][forum]    ex_3341 [-] Bose in 2D harmonic trap
ex_3342[pdf][files/sol][forum]    ex_3342 [-] BlackBody
ex_3344[pdf][files/sol][forum]    ex_3344 [e] BEC in harmonic potential
ex_3500[pdf][files/sol][forum]    
ex_3510[pdf][files/sol][forum]    ex_3510 [e] State equations for ideal Fermi gas
ex_3515[pdf][files/sol][forum]    ex_3515 [-] Ideal Fermi gas in 1D space
ex_3520[pdf][files/sol][forum]    ex_3520 [H] Ideal Fermi gas in 2D space
ex_3530[pdf][files/sol][forum]    ex_3530 [-] Ideal Fermi gas in 2D space
ex_3540[pdf][files/sol][forum]    ex_3540 [H] Ideal Fermi gas in semiconductor
ex_3550[pdf][files/sol][forum]    ex_3550 [C] Fermions in magnetic field - Pauli
ex_3555[pdf][files/sol][forum]    ex_3555 [-] Fermions in magnetic field - Landau
ex_3570[pdf][files/sol][forum]    ex_3570 [H] Fermi gas in 2D+3D connected boxes with gravitation
ex_3710[pdf][files/sol][forum]    ex_3710 [-] Fermions in gravitational field, star.
ex_3711[pdf][files/sol][forum]    ex_3711 [-] Fermions in gravitational field, star (duplication)
ex_3745[pdf][files/sol][forum]    ex_3745 [C] Fermions in gravitational field. (B12)
ex_4000[pdf][files/sol][forum]    ex_4000
 
=== Chemical equilibrium ===
ex_4001[pdf][files/sol][forum]    ex_4001
ex_4012[pdf][files/sol][forum]    ex_4012
ex_4013[pdf][files/sol][forum]    ex_4013
ex_4014[pdf][files/sol][forum]    ex_4014
ex_4015[pdf][files/sol][forum]    ex_4015
ex_4016[pdf][files/sol][forum]    ex_4016 [C] adsorbtion of polar molecules to surface (A43) (Similar to 4015)
ex_4017[pdf][files/sol][forum]    ex_4017 [e] Adsorbsion and fractal dimension
ex_4018[pdf][files/sol][forum]    ex_4018 [-] Adsorbsion of molecules with internal degress of freedom
ex_4019[pdf][files/sol][forum]    ex_4019
ex_4200[pdf][files/sol][forum]    
ex_4211[pdf][files/sol][forum]    ex_4211 [C] equilibrium (H2+D2=2HD) (Similar to 4210) (A52)
ex_4213[pdf][files/sol][forum]    ex_4213 [H] equilibrium (A=A+e)
ex_4220[pdf][files/sol][forum]    ex_4220
ex_4410[pdf][files/sol][forum]    ex_4410
ex_4420[pdf][files/sol][forum]    ex_4420
ex_4430[pdf][files/sol][forum]    ex_4430
ex_4440[pdf][files/sol][forum]    ex_4440
ex_4441[pdf][files/sol][forum]    ex_4441 [H] equilibrium (e+,e-=2gamma) (B22)
ex_4445[pdf][files/sol][forum]    ex_4445 [-] [remove?] Adsorbtion and decomposition
ex_4446[pdf][files/sol][forum]    ex_4446 [-] [remove?] Relative abundance of isotopes
ex_4450[pdf][files/sol][forum]    ex_4450
ex_4540[pdf][files/sol][forum]    ex_4540 [-] BEC in mechanical equilibrium with ideal gas
ex_4546[pdf][files/sol][forum]    ex_4546 [-] Quantum statistics of reactions
ex_4549[pdf][files/sol][forum]    ex_4549 [-] Consevation law for the number of nucleons
ex_4550[pdf][files/sol][forum]    ex_4550 [H] Equilib of Fermions and condensed Boson F-->A+B (B25)
ex_4551[pdf][files/sol][forum]    ex_4551 [-] Velocity of sound of a degenerate Fermi gas
ex_4552[pdf][files/sol][forum]    ex_4552 [-] Cooling by adiabatic pressurization
ex_4553[pdf][files/sol][forum]    ex_4553 [-] Beta decay
ex_4554[pdf][files/sol][forum]    ex_4554 [-] Fermions in magnetic field, quantum phase transition
ex_5000[pdf][files/sol][forum]    ex_5000
 
=== Interacting systems, Phase transitions ===
ex_5010[pdf][files/sol][forum]    ex_5010 [H] One dimensional hard sphere gas
ex_5011[pdf][files/sol][forum]    ex_5011
ex_5012[pdf][files/sol][forum]    ex_5012
ex_5015[pdf][files/sol][forum]    ex_5015 [-] 2D coulomb gas
ex_5021[pdf][files/sol][forum]    ex_5021 [e] (virial/equi theorems)
ex_5022[pdf][files/sol][forum]    ex_5022 [-] pressure via the virial theorem
ex_5023[pdf][files/sol][forum]    ex_5023 [-] pressure of hard spheres
ex_5024[pdf][files/sol][forum]    ex_5024 [C] Pressure of Lenard-Jones
ex_5030[pdf][files/sol][forum]    ex_5030 [-] Virial coeff - standard examples
ex_5040[pdf][files/sol][forum]    ex_5040 [-] Virial coeff - ideal Bose/Fermi
ex_5600[pdf][files/sol][forum]    
ex_5641[pdf][files/sol][forum]    ex_5641 [H] Ising with long range interaction
ex_5650[pdf][files/sol][forum]    ex_5650
ex_5651[pdf][files/sol][forum]    ex_5651 [C] Ising in interaction with lattice gas (C24) (duplicate 5650)
ex_5660[pdf][files/sol][forum]    ex_5660 [H] Ising model of adsorption sites
ex_5700[pdf][files/sol][forum]    
ex_5710[pdf][files/sol][forum]    ex_5710
ex_5720[pdf][files/sol][forum]    ex_5720
ex_5721[pdf][files/sol][forum]    ex_5721 [H] Mean Field for Ising antiferromagnet (C03) (same as 5720)
ex_5730[pdf][files/sol][forum]    ex_5730
ex_5740[pdf][files/sol][forum]    ex_5740
ex_5741[pdf][files/sol][forum]    ex_5741 [L] edu (mean field correlation function) same as 5740
ex_5742[pdf][files/sol][forum]    ex_5742 [L] edu (Ising correlation function)
ex_5750[pdf][files/sol][forum]    ex_5750
ex_5800[pdf][files/sol][forum]    
ex_5811[pdf][files/sol][forum]    ex_5811 [C] Symmetry breaking (C05) (Similar to 5810)
ex_5955[pdf][files/sol][forum]    ex_5955 [H] Change of boiling point with altitude. (C01)
ex_5956[pdf][files/sol][forum]    ex_5956 [C] Mean Field for Heisenberg model (C02)
ex_5958[pdf][files/sol][forum]    ex_5958 [H] Ferroelectricity model (C04)
ex_5960[pdf][files/sol][forum]    ex_5960 [-] [remove?] Cubic cystal ferromagnet
ex_5962[pdf][files/sol][forum]    ex_5962 [-] [remove?] Lattice gas with no interation
ex_5963[pdf][files/sol][forum]    ex_5963 [C] Stoner ferromagnetism (C20)
ex_5966[pdf][files/sol][forum]    ex_5966 [C] Ising model 1D, domain walls (C23)
ex_5968[pdf][files/sol][forum]    ex_5968 [e] Disorder averaging
ex_5969[pdf][files/sol][forum]    ex_5969 [H] 2D Coulomb gas (Similar to 5015) (C26)
ex_5970[pdf][files/sol][forum]    ex_5970 [e] Scaling form for free energy
ex_5971[pdf][files/sol][forum]    ex_5971 [H] Potts model (C28)
ex_6000[pdf][files/sol][forum]    ex_6000
 
=== Kinetics ===
ex_6010[pdf][files/sol][forum]    ex_6010
ex_6020[pdf][files/sol][forum]    ex_6020
ex_6030[pdf][files/sol][forum]    ex_6030 [C] Thermionic emission of electrons from a metal
ex_6040[pdf][files/sol][forum]    ex_6040 [C] Effusion of electrons from a box in magnetic field
ex_6050[pdf][files/sol][forum]    ex_6050 [H] Radiation from 1D balckbody fiber
ex_6070[pdf][files/sol][forum]    ex_6070 [H] Landauer formula for 1D conductance.
ex_6071[pdf][files/sol][forum]    ex_6071 [-] Incident flux formula for 2D
ex_6080[pdf][files/sol][forum]    ex_6080 [L] [edu]
ex_6110[pdf][files/sol][forum]    ex_6110
ex_6700[pdf][files/sol][forum]    
ex_6772[pdf][files/sol][forum]    ex_6772 [-] Boltzmann equation: distribution function
ex_6773[pdf][files/sol][forum]    ex_6773 [-] Dissipation phase space volume and entropy
ex_6774[pdf][files/sol][forum]    ex_6774 [-] Boltzmann equation: Conductivity
ex_6775[pdf][files/sol][forum]    ex_6775 [e] Coarse grained entropy
ex_6776[pdf][files/sol][forum]    ex_6776 [e] Boltzmann equation: Emission
ex_6777[pdf][files/sol][forum]    ex_6777 [-] Phase space evolution of confined particle
ex_6778[pdf][files/sol][forum]    ex_6778 [-] Boltzmann equation: particles between two plates
ex_7000[pdf][files/sol][forum]    ex_7000
 
=== Fluctuations and the FD realation ===
ex_7001[pdf][files/sol][forum]    ex_7001 [L] [edu] (power spectrum)
ex_7005[pdf][files/sol][forum]    ex_7005 [-] Shot noise
ex_7020[pdf][files/sol][forum]    ex_7020 [H] FDT for Osc
ex_7040[pdf][files/sol][forum]    ex_7040 [C] FDT for RLC circuit
ex_7041[pdf][files/sol][forum]    ex_7041 [-] FDT for RLC circuit
ex_7050[pdf][files/sol][forum]    ex_7050 [L] The Drude formula
ex_7481[pdf][files/sol][forum]    ex_7481 [e] FDT for velocities
ex_7486[pdf][files/sol][forum]    ex_7486 [e] Linear response and Kubo
ex_7487[pdf][files/sol][forum]    ex_7487 [-] Velocity-velocity correlation and diffusion
ex_7489[pdf][files/sol][forum]    ex_7489 [e] Kubo formula
ex_7491[pdf][files/sol][forum]    ex_7491 [-] [remove?] Onsager
ex_7492[pdf][files/sol][forum]    ex_7492 [-] [remove?] Onsager
ex_8000[pdf][files/sol][forum]    ex_8000
 
=== Stochastic picture, Langevin ===
ex_8001[pdf][files/sol][forum]    ex_8001 [L] Random walk with correlations
ex_8020[pdf][files/sol][forum]    ex_8020 [L] Correlation functions from Langevin dynamics
ex_8025[pdf][files/sol][forum]    ex_8025 [-] Thermal flow via a Brownian particle (Exam 2011)
ex_8030[pdf][files/sol][forum]    ex_8030 [L] Diffusion of Brownian particle from Langevin
ex_8032[pdf][files/sol][forum]    ex_8032 [H] Sub diffusion of Brownian particle
ex_8481[pdf][files/sol][forum]    ex_8481 [C] Mass on a spring (D22)
ex_8483[pdf][files/sol][forum]    ex_8483 [C] Millikan (D24)
ex_8484[pdf][files/sol][forum]    ex_8484 [H] galvanometer (D25)
ex_8490[pdf][files/sol][forum]    ex_8490 [H] Rate Equation with noise (D31)
ex_9000[pdf][files/sol][forum]    ex_9000
 
=== System-Bath ===
ex_9010[pdf][files/sol][forum]    ex_9010 [e] Spin resonance
ex_9012[pdf][files/sol][forum]    ex_9012 [e] Equilibrium of a two level system