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GSM:010117Six cells 3A+3B, K=#A in left three: counts[1,9,9,1]/20, mean 3/2, var 9/20, segregation 1/10, DeltaS/kB=ln20, identical control 0

Gsm1 Hypergeometric Left Block CountGsm1 Gibbs Mixing Identical Control

Ensembles, entropy, and the thermodynamic limitGraduate Statistical Mechanics I · Mixed review

GSM:010118Three-level maxent E=0,eps,2eps, mean 4eps/7: q=1/2 from 10q^2+3q-4=0, p=(4/7,2/7,1/7), beta eps=ln2, H=ln7-(10/7)ln2

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Gsm1 Three Level Boltzmann MaxentGsm1 Maxent Entropy Strict Concavity

Ensembles, entropy, and the thermodynamic limitGraduate Statistical Mechanics I · Mixed review

GSM:010219Debye low-T asymptotes with zero-point excluded: U~(3pi^4/5)NkT(T/thetaD)^3, C~(12pi^4/5)Nk(T/thetaD)^3, CT/U tends to 4

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Gsm1 Debye Thermal Energy IntegralGsm1 Debye Heat Capacity Ratio

Quantum ensembles and ideal Bose and Fermi gasesGraduate Statistical Mechanics I · Mixed review

GSM:010220T=0 spin-1/2 Fermi sphere n=1/(3pi^2 a^3): kF=1/a, epsF=hbar^2/(2m a^2), U/N=3epsF/5, P=2n epsF/5, <k^2>=3kF^2/5, |k|<=kF/2 is 1/8, kappaT=3/(2n epsF)

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Gsm1 Fermi Sphere Density WavevectorGsm1 Fermi T0 Pressure Compressibility

Quantum ensembles and ideal Bose and Fermi gasesGraduate Statistical Mechanics I · Mixed review

GSM:010221Two spinless particles on {0,eps,2eps}, q=1/2: ZB=35/16, ZF=7/8, UB/eps=34/35, UF/eps=11/7, VarB=1434/1225, VarF=26/49, PB(ground)=16/35, PF(lowest)=4/7

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Gsm1 Two Particle Bose Fermi PartitionGsm1 Two Particle Energy Moments

Quantum ensembles and ideal Bose and Fermi gasesGraduate Statistical Mechanics I · Mixed review

GSM:010322Tonks hard rods N=4 on L=10a: free length 6a, Qconf=54 a^4, beta P a=2/3, n=2/(5a), Z=5/3, kappa_T=9a/(10 k_B T)

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Gsm1 Tonks Hard Rod Free VolumeGsm1 Tonks Pressure Compressibility

Virial and cluster expansionsGraduate Statistical Mechanics I · Mixed review

GSM:010323Coupled pair V=k(x1^2+x2^2)/2+g(x1-x2)^2/2 at g=k: Zconf=2pi/(beta k sqrt3), <q+^2>=1/(beta k), rho=1/2, energies 1/(3beta)+2/(3beta)=1/beta

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Gsm1 Coupled Pair Normal ModesGsm1 Gaussian Quadratic Averages

Virial and cluster expansionsGraduate Statistical Mechanics I · Mixed review

GSM:010324Square well on ring L=4a, beta eps=ln3: Zrel=6a, labelled 24 a^2 vs ideal 16 a^2 ratio 3/2, bound 1/2, <u>=-eps/2, C/kB=(ln3)^2/4

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Gsm1 Square Well Ring ConfigurationalGsm1 Two Value Energy Fluctuation

Virial and cluster expansionsGraduate Statistical Mechanics I · Mixed review

GSM:010425Curie-Weiss phi at K=ln3: stationary 0 and +/-1/2, phi''(0)=1-ln3<0, phi''(+/-)=4/3-ln3>0, barrier ln2-(5/8)ln3, chi=3/(4-3ln3)

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Gsm1 Curie Weiss Stationary PointsGsm1 Mean Field Barrier Susceptibility

Phase transitions and mean-field theoryGraduate Statistical Mechanics I · Mixed review

GSM:010426Degree-3 Bethe bond percolation: u=1-p+p u^2, pc=1/2, at p=2/3 physical u=1/2 and Pinf=7/8, below pc S=1+3p/(1-2p), p=1/3 gives S=4

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Gsm1 Bethe Bond Self ConsistencyGsm1 Bethe Percolation Cluster Size

Phase transitions and mean-field theoryGraduate Statistical Mechanics I · Mixed review

GSM:010427Finite two-state rounding X=±1, weights exp(±At/2): Zred=2cosh(At/2), mean=tanh(At/2), var=sech²; A=8 gives t=0 mean0 var1 slope4 and mean=1/2 at t=ln3/8 with p+=3/4

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Gsm1 Two State Boltzmann WeightsGsm1 Finite Amplitude Tanh Rounding

Phase transitions and mean-field theoryGraduate Statistical Mechanics I · Mixed review

GSM:010528Yukawa OZ kernel C=A e^{-r/ξ}/r in R³, A=1/(4πξ²): S(k)=1/(1+k²ξ²); S=1,1/2,1/5 at k=0,1/ξ,2/ξ; ∫C=1, ∫r²C=6ξ², second-moment length ξ

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Gsm1 Yukawa Three Dimensional FourierGsm1 OZ Second Moment Length

Correlations, fluctuations, and responseGraduate Statistical Mechanics I · Mixed review

GSM:010529N=4 independent circle angles, nonzero integer m, ρ=∑e^{imθ}: Eρ=0, E|ρ|²=4, E|ρ|⁴=28, Var|ρ|²=12; S=|ρ|²/N has mean 1 and variance 3/4

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Gsm1 Circular Density Mode MomentsGsm1 Intensive Structure Factor Variance

Correlations, fluctuations, and responseGraduate Statistical Mechanics I · Mixed review

GSM:010630Hard-core lattice gas on C5, independent-set counts (1,5,5), z=2: Ξ=31; P(N=0,1,2)=(1,10,20)/31; ⟨N⟩=50/31, ⟨N²⟩=90/31, Var=290/961, density=10/31

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Gsm1 C5 Independent Set EnumerationGsm1 Hard Core Grand Moments

Lattice models and transfer-matrix methodsGraduate Statistical Mechanics I · Mixed review

GSM:010631q=3 Potts triangle, e^{βJ}=2: classes (all-same 3, B=3), (two-same 18, B=1), (all-different 6, B=0); Z=66, probs 4/11,6/11,1/11; ⟨B⟩=18/11, ⟨B²⟩=42/11, var=138/121, U=-18J/11, C/kB=(138/121)(ln2)²=0.5479546770306759

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Gsm1 Potts Triangle Class EnumerationGsm1 Potts Energy Heat Capacity

Lattice models and transfer-matrix methodsGraduate Statistical Mechanics I · Mixed review

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GSM:010117Six cells 3A+3B, K=#A in left three: counts[1,9,9,1]/20, mean 3/2, var 9/20, segregation 1/10, DeltaS/kB=ln20, identical control 0

GSM:010118Three-level maxent E=0,eps,2eps, mean 4eps/7: q=1/2 from 10q^2+3q-4=0, p=(4/7,2/7,1/7), beta eps=ln2, H=ln7-(10/7)ln2

GSM:010219Debye low-T asymptotes with zero-point excluded: U~(3pi^4/5)NkT(T/thetaD)^3, C~(12pi^4/5)Nk(T/thetaD)^3, CT/U tends to 4

GSM:010220T=0 spin-1/2 Fermi sphere n=1/(3pi^2 a^3): kF=1/a, epsF=hbar^2/(2m a^2), U/N=3epsF/5, P=2n epsF/5, <k^2>=3kF^2/5, |k|<=kF/2 is 1/8, kappaT=3/(2n epsF)

GSM:010221Two spinless particles on {0,eps,2eps}, q=1/2: ZB=35/16, ZF=7/8, UB/eps=34/35, UF/eps=11/7, VarB=1434/1225, VarF=26/49, PB(ground)=16/35, PF(lowest)=4/7

GSM:010322Tonks hard rods N=4 on L=10a: free length 6a, Qconf=54 a^4, beta P a=2/3, n=2/(5a), Z=5/3, kappa_T=9a/(10 k_B T)

GSM:010323Coupled pair V=k(x1^2+x2^2)/2+g(x1-x2)^2/2 at g=k: Zconf=2pi/(beta k sqrt3), <q+^2>=1/(beta k), rho=1/2, energies 1/(3beta)+2/(3beta)=1/beta

GSM:010324Square well on ring L=4a, beta eps=ln3: Zrel=6a, labelled 24 a^2 vs ideal 16 a^2 ratio 3/2, bound 1/2, <u>=-eps/2, C/kB=(ln3)^2/4

GSM:010425Curie-Weiss phi at K=ln3: stationary 0 and +/-1/2, phi''(0)=1-ln3<0, phi''(+/-)=4/3-ln3>0, barrier ln2-(5/8)ln3, chi=3/(4-3ln3)

GSM:010426Degree-3 Bethe bond percolation: u=1-p+p u^2, pc=1/2, at p=2/3 physical u=1/2 and Pinf=7/8, below pc S=1+3p/(1-2p), p=1/3 gives S=4

GSM:010427Finite two-state rounding X=±1, weights exp(±At/2): Zred=2cosh(At/2), mean=tanh(At/2), var=sech²; A=8 gives t=0 mean0 var1 slope4 and mean=1/2 at t=ln3/8 with p+=3/4

GSM:010528Yukawa OZ kernel C=A e^{-r/ξ}/r in R³, A=1/(4πξ²): S(k)=1/(1+k²ξ²); S=1,1/2,1/5 at k=0,1/ξ,2/ξ; ∫C=1, ∫r²C=6ξ², second-moment length ξ

GSM:010529N=4 independent circle angles, nonzero integer m, ρ=∑e^{imθ}: Eρ=0, E|ρ|²=4, E|ρ|⁴=28, Var|ρ|²=12; S=|ρ|²/N has mean 1 and variance 3/4

GSM:010630Hard-core lattice gas on C5, independent-set counts (1,5,5), z=2: Ξ=31; P(N=0,1,2)=(1,10,20)/31; ⟨N⟩=50/31, ⟨N²⟩=90/31, Var=290/961, density=10/31

GSM:010631q=3 Potts triangle, e^{βJ}=2: classes (all-same 3, B=3), (two-same 18, B=1), (all-different 6, B=0); Z=66, probs 4/11,6/11,1/11; ⟨B⟩=18/11, ⟨B²⟩=42/11, var=138/121, U=-18J/11, C/kB=(138/121)(ln2)²=0.5479546770306759

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Problem results

GSM:010117Six cells 3A+3B, K=#A in left three: counts[1,9,9,1]/20, mean 3/2, var 9/20, segregation 1/10, DeltaS/kB=ln20, identical control 0

GSM:010118Three-level maxent E=0,eps,2eps, mean 4eps/7: q=1/2 from 10q^2+3q-4=0, p=(4/7,2/7,1/7), beta eps=ln2, H=ln7-(10/7)ln2

GSM:010219Debye low-T asymptotes with zero-point excluded: U~(3pi^4/5)NkT(T/thetaD)^3, C~(12pi^4/5)Nk(T/thetaD)^3, CT/U tends to 4

GSM:010220T=0 spin-1/2 Fermi sphere n=1/(3pi^2 a^3): kF=1/a, epsF=hbar^2/(2m a^2), U/N=3epsF/5, P=2n epsF/5, <k^2>=3kF^2/5, |k|<=kF/2 is 1/8, kappaT=3/(2n epsF)

GSM:010221Two spinless particles on {0,eps,2eps}, q=1/2: ZB=35/16, ZF=7/8, UB/eps=34/35, UF/eps=11/7, VarB=1434/1225, VarF=26/49, PB(ground)=16/35, PF(lowest)=4/7

GSM:010322Tonks hard rods N=4 on L=10a: free length 6a, Qconf=54 a^4, beta P a=2/3, n=2/(5a), Z=5/3, kappa_T=9a/(10 k_B T)

GSM:010323Coupled pair V=k(x1^2+x2^2)/2+g(x1-x2)^2/2 at g=k: Zconf=2pi/(beta k sqrt3), <q+^2>=1/(beta k), rho=1/2, energies 1/(3beta)+2/(3beta)=1/beta

GSM:010324Square well on ring L=4a, beta eps=ln3: Zrel=6a, labelled 24 a^2 vs ideal 16 a^2 ratio 3/2, bound 1/2, <u>=-eps/2, C/kB=(ln3)^2/4

GSM:010425Curie-Weiss phi at K=ln3: stationary 0 and +/-1/2, phi''(0)=1-ln3<0, phi''(+/-)=4/3-ln3>0, barrier ln2-(5/8)ln3, chi=3/(4-3ln3)

GSM:010426Degree-3 Bethe bond percolation: u=1-p+p u^2, pc=1/2, at p=2/3 physical u=1/2 and Pinf=7/8, below pc S=1+3p/(1-2p), p=1/3 gives S=4

GSM:010427Finite two-state rounding X=±1, weights exp(±At/2): Zred=2cosh(At/2), mean=tanh(At/2), var=sech²; A=8 gives t=0 mean0 var1 slope4 and mean=1/2 at t=ln3/8 with p+=3/4

GSM:010528Yukawa OZ kernel C=A e^{-r/ξ}/r in R³, A=1/(4πξ²): S(k)=1/(1+k²ξ²); S=1,1/2,1/5 at k=0,1/ξ,2/ξ; ∫C=1, ∫r²C=6ξ², second-moment length ξ

GSM:010529N=4 independent circle angles, nonzero integer m, ρ=∑e^{imθ}: Eρ=0, E|ρ|²=4, E|ρ|⁴=28, Var|ρ|²=12; S=|ρ|²/N has mean 1 and variance 3/4

GSM:010630Hard-core lattice gas on C5, independent-set counts (1,5,5), z=2: Ξ=31; P(N=0,1,2)=(1,10,20)/31; ⟨N⟩=50/31, ⟨N²⟩=90/31, Var=290/961, density=10/31

GSM:010631q=3 Potts triangle, e^{βJ}=2: classes (all-same 3, B=3), (two-same 18, B=1), (all-different 6, B=0); Z=66, probs 4/11,6/11,1/11; ⟨B⟩=18/11, ⟨B²⟩=42/11, var=138/121, U=-18J/11, C/kB=(138/121)(ln2)²=0.5479546770306759

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GSM:010117Six cells 3A+3B, K=#A in left three: counts[1,9,9,1]/20, mean 3/2, var 9/20, segregation 1/10, DeltaS/kB=ln20, identical control 0View problem statement

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GSM:010117Six cells 3A+3B, K=#A in left three: counts[1,9,9,1]/20, mean 3/2, var 9/20, segregation 1/10, DeltaS/kB=ln20, identical control 0View problem statement

GSM:010117Six cells 3A+3B, K=#A in left three: counts[1,9,9,1]/20, mean 3/2, var 9/20, segregation 1/10, DeltaS/kB=ln20, identical control 0

GSM:010117Six cells 3A+3B, K=#A in left three: counts[1,9,9,1]/20, mean 3/2, var 9/20, segregation 1/10, DeltaS/kB=ln20, identical control 0