Lebwohl-Lasher model - simulations

NEMATICS 2D

2D square lattice with complex numbers z, |z| = 1.
If \vec{n} = [cos(t1), sin(t1)], then z = exp(2 i t1).

L = np.empty((x, y), dtype=np.complex128)

L[i, j] = random_complex()   # random orientation
L[i, j] = complex(1, 0)   # ordered system

def random_complex():   # abs(z)=1
    return cmath.rect(1.0, 2 * cmath.pi * random.random())

@njit('float64(complex128)')
def interaction(spin):   # spin = z2/z1
    # H12 = P2(cos(t2-t1)) = (3/4) * Re{z2/z1} + (1/4)
    return 0.75 * spin.real + 0.25

@njit('complex128(complex128[:,:])')
def calc_magnetization(L):
    return L.sum()

@njit('float64(complex128[:,:])')
def calc_energy(L): ...

@njit('Tuple((float64,float64,float64,float64))(complex128[:,:],float64)')
def mc_simulation(L, t): ...

40 times faster with numba.

NEMATICS 3D

3D cubic lattice with quaternions.

L = np.empty((x, y, z), dtype=object)   # Quat is not a standard type

L[i, j, k] = random_quat_uniax()   # random orientation
L[i, j, k] = Quat(1)   # ordered system

def interaction(unit_quat): ... # unit_quat = ~spin * neighbor

def calc_energy(L): ...

Changes for numba.

L = np.empty((x, y, z, 4), dtype=np.float64)

L[i, j, k, :] = random_quat_biax().q   # random orientation
L[i, j, k, :] = Quat(1).q   # ordered system

@njit('float64(float64[:],float64[:])')
def interaction(arr1, arr2): ...

@njit('float64(float64[:,:,:,:])')
def calc_energy(L): ...

18 times faster with numba.