Consider a 2- dimensional square lattice. The total number of square cells is V. The occupation number of i-th cell, ni,
Posted: Wed Jul 06, 2022 11:41 am
Consider a 2- dimensional square lattice. The total number ofsquare cells is V. The occupation number of i-th cell, ni, is 0when the cell is empty and 1 when occupied. (see figure.) The totalnumber of particles is N = ∑i ni . When twoneighboring cells are occupied , the pair of particles acquireenergy -ε. Thus the Hamiltonian is given by
H = -ε ∑(i, j) ninj
Where ∑(i, j) indicates the sum over all nearestneighbor pairs. We assume that the interaction is attractive andthus ε > 0. The grand partition function is given by
Q = ∑{n1 = (0, 1)} …………∑{nv = (0, 1)}eβ(μ∑i ni + ε∑(I,j)ni nj ) .
(a) Show that this grand canonical ensemble problem ismathematically equivalent to the canonical ensemble of the 2 –dimensional Ising model with a uniform magnetic field B. [HINT:Hamiltonian of the Ising model is H = - µB B∑i si - J∑(i, j)sisj
where si ={-1, 1} . [Hint: consider transformation si= 2ni -1. Note that μ and µB are two differentquantities .]
(b) When B = 0 in the Ising model, the mean magnetization, m=<∑i si > /N, undergoes a continuousphase transition. Similarly when μ = 2ε in the lattice gas model,the average density of particles , n = <∑ini > /V, also shows a phase transition. Find theaverage density n above the critical temperature based on thecorrespondence obtained in part (a) without actually computing it.[You can use the mean magnetization obtained from the Ising modelwithout derivation.]
(c) Assume again that μ = 2ε. Find the average density atT=0.
Could you please answer these questions as soon as possible?thank you!
H = -ε ∑(i, j) ninj
Where ∑(i, j) indicates the sum over all nearestneighbor pairs. We assume that the interaction is attractive andthus ε > 0. The grand partition function is given by
Q = ∑{n1 = (0, 1)} …………∑{nv = (0, 1)}eβ(μ∑i ni + ε∑(I,j)ni nj ) .
(a) Show that this grand canonical ensemble problem ismathematically equivalent to the canonical ensemble of the 2 –dimensional Ising model with a uniform magnetic field B. [HINT:Hamiltonian of the Ising model is H = - µB B∑i si - J∑(i, j)sisj
where si ={-1, 1} . [Hint: consider transformation si= 2ni -1. Note that μ and µB are two differentquantities .]
(b) When B = 0 in the Ising model, the mean magnetization, m=<∑i si > /N, undergoes a continuousphase transition. Similarly when μ = 2ε in the lattice gas model,the average density of particles , n = <∑ini > /V, also shows a phase transition. Find theaverage density n above the critical temperature based on thecorrespondence obtained in part (a) without actually computing it.[You can use the mean magnetization obtained from the Ising modelwithout derivation.]
(c) Assume again that μ = 2ε. Find the average density atT=0.
Could you please answer these questions as soon as possible?thank you!