q1range = [-3.1*σ1, 3.1*σ1] q2range = [-3.1*σ2, 3.1*σ2] p1range = [-3.1*m1, 3.1*m1] p2range = [-3.1*m2, 3.1*m2] ε = 0.01 ω_plus = numpy.complex(0.0, numpy.sqrt(α*(1.0 + γ))) ω_minus = numpy.complex(0.0, numpy.sqrt(α*(1.0 - γ))) t_plus = 2.0*numpy.pi / numpy.abs(ω_plus) t_minus = 2.0*numpy.pi / numpy.abs(ω_minus) tmax = 2.0*t_minus/3.0 nsample = 10000 U = hmc.bivariate_normal_U(γ, σ1, σ2) K = hmc.bivariate_normal_K(m1, m2) dUdq = hmc.bivariate_normal_dUdq(γ, σ1, σ2) dKdp = hmc.bivariate_normal_dKdp(m1, m2) momentum_generator = hmc.bivariate_normal_momentum_generator(m1, m2) potential_pdf = hmc.potential_distribution(U) momentum_pdf = hmc.momentum_distribution(K) file_prefix = "hmc-bivariate-normal-γ-0.0" # %% H, p, q, accepted = hmc.HMC(q0, U, K, dUdq, dKdp, hmc.momentum_verlet_integrator, momentum_generator, nsample, tmax, ε) # %%
return U_t + K_t
# %%
# Configuration
γ = 0.9
α = 1 / (1.0 - γ**2)
ω_plus = numpy.complex(0.0, numpy.sqrt(α*(1.0 + γ)))
ω_minus = numpy.complex(0.0, numpy.sqrt(α*(1.0 - γ)))
t_plus = 2.0*numpy.pi / numpy.abs(ω_plus)
t_minus = 2.0*numpy.pi / numpy.abs(ω_minus)
nsteps = 500
U = hmc.bivariate_normal_U(γ, 1.0, 1.0)
K = hmc.bivariate_normal_K(1.0, 1.0)
# %%
# Compute solutions using eigenvalues and eigenvectots computed from numercically diagonalizing Hamiltonian Matrix
PQ0 = numpy.matrix([[1.0], [1.0], [1.0], [1.0]])
time = numpy.linspace(0.0, 2.0*t_minus, nsteps)
# Diagonalize Hamiltonian Matrix
H = hamiltonian_matrix(γ, α)
λ, E = linalg.eig(H)
PQ = coordinate_time_series(E, PQ0, λ, time)
q = numpy.array([[numpy.real(PQ[i,2,0]), numpy.real(PQ[i,3,0])] for i in range(nsteps)])
p = numpy.array([[numpy.real(PQ[i,0,0]), numpy.real(PQ[i,1,0])] for i in range(nsteps)])
title = f"Numerical Evaluation: γ={γ}, " + r"$t_{+}=$" + f"{format(t_plus, '2.4f')}, " + r"$t_{-}=$" + f"{format(t_minus, '2.4f')}"