2. 1D Ising model Consider the Ising model in 1D with zero external field. The Ising Hamiltonian in this case is N-1 #=-JΣ σίσί+1, i=1 where σ = ±1, J > 0. In this problem I am asking to plot some graphs. You can use any graphics software, e.g. gnuplot, excel, etc. (a) (b) (c) (d) Calculate the ground-state energy. Calculate the canonical partition function. Calculate the probability that all spins are aligned. Consider the limit N → ∞. Plot this probability versus kBT/J for few values of N, say N = 103 and N = 108. Argue that your result indicates the absence of ordered state at T > 0. Calculate the Helmholtz free energy A, average energy E, heat capacity C, and the entropy. Plot graphs for A/(JN), E/(JN) and C/N vs kBT/J. In few words discuss your results.

2. 1D Ising model Consider the Ising model in 1D with zero external field. The Ising Hamiltonian in this case is N-1 #=-JΣ σίσί+1, i=1 where σ = ±1, J > 0. In this problem I am asking to plot some graphs. You can use any graphics software, e.g. gnuplot, excel, etc. (a) (b) (c) (d) Calculate the ground-state energy. Calculate the canonical partition function. Calculate the probability that all spins are aligned. Consider the limit N → ∞. Plot this probability versus kBT/J for few values of N, say N = 103 and N = 108. Argue that your result indicates the absence of ordered state at T > 0. Calculate the Helmholtz free energy A, average energy E, heat capacity C, and the entropy. Plot graphs for A/(JN), E/(JN) and C/N vs kBT/J. In few words discuss your results.

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