Applications to simulations: Monte-Carlo A.C. Maggs ESPCI, Paris June 2016
Summary Algorithms Faster/simpler codes Thermodynamics of Electric fields Partition function of electric field Thermal Casimir/Lifshitz in general geometries Fluctuations in dielectrics- numerical methods Historical remarks Origins of electromagnetism, FitzGerald
Local Algorithm The charges q live on the vertices D ij field on the links of a cubic lattice. dd When a particle, moves from 1 to 2 then D 12 D 12 q Metropolis rule U = 1 2 <ij> D 2 ij Never calculate the potential
Constrained Monte-Carlo: Particle Motion Flux interpretation of Gauss law D ds = q 1/4 1/4 1/12 1/4 1/12 1/4 1/12 1/4 1/4 3/4 1/12 1/12 1/4 1/12 Gauss constraint is satisfied dynamically (conserved)
Constrained Minimization of Energy U = 1 2ɛ 0 with div D = ρ Lagrange multiplier to minimize: D 2 d 3 r A = 1 2ɛ D2 φ(div D ρ) d 3 r leading to δe : E = φ p D = ɛ 0 E 2 φ p = ρ/ɛ 0
Constrained Statistical Mechanics Define: Z(r) = ( δ(div D ρ) DD exp β 1 2ɛ 0 ) D 2 d 3 r Gaussian, but constrained, sample with two MC moves. 3 E 3,2 2 θ E E 4,3 1,2 4 1 E 4,1
Proof Helmholtz decomposition: with div D = 0, D = ɛ φ Z = Z c D = D + D div (D)e β D 2 d 3 r/2ɛ
Full Algorithm randomly chose a Plaquette or Particle if (Particle) displace, updating field on ONE link if (Plaquette) update 4 fields apply Metropolis Analogies: Hydrodynamics with lattice Boltzmann Dirac treatment of electromagnetism where div D ρ is weakly imposed
Why not the potential There is a well known variational principle in terms of φ: {ɛ0 } U φ = 2 ( φ)2 iρφ d 3 r Require the use of complex weights- awkward for numerical work
Electric diffusion equation Algorithm dynamics and convergence: ɛ 0 D t = J + curl H ξh = ɛ 0 curl E Giving electric diffusion equation Interesting case J = σe E t = ( ɛ 0 2 E grad ρ ) /ξ J/ɛ 0 iωe t = (σ + q 2 )E t Gap in spectrum for transverse modes
Off-lattice Interpolate continuum position to lattice using splines/ Gaussians Monte-Carlo or molecular dynamics algorithms with arbitrary potentials Suitable for atomistic modeling LAAMPS, replacing Fourier methods
Molecular Dynamics We need a dynamic process which conserves Gauss law: Maxwell s equations! Ḃ = c curl E, v i = e i E(r i )/m i Ė = c curl B J, ṙ i = v i Dynamics conserves the energy U m = v 2 i /2 + dr { B 2 /2 + E 2 /2 } However note that there is no Lorentz force. Inspired by the MC algorithm we now violate the constraint div B = 0 by coupling the magnetic degrees of freedom to a thermostat v i = q i E(r i ) γ 1 v i + ξ 1 Ḃ = c curl E γ 2 B + ξ 2, Can prove convergence to correct distribution. (An excercise in Fokker-Planck equations) Note that the noise kills off lots of conservation laws in Maxwell s equations. Converges independent of the speed of light.
works with dielectrics Trivially parallel Born energy for free
Other constrained models Bosonic Hubbard model Continuum limit of dimer models Z(r) = ( δ(div E) exp β ɛ 0 2 E=integer ) E 2 d 3 r Two phases- confining with line tension/ Coulomb-superconducting Use of quantum worm algorithms to replace plaquette updates Echo of phase transition in dynamics- string tension
Other models with arrows - Ice Long range correlations coming from constraints Critical state Macroscopic entropy of water
Spin ice Magnetic monopoles in condensed matter
Inhomogeneous Media Dielectrics The basic trick generalizes using the electric displacement div D = ρ U = D 2 2ɛ(r) d3 r Minimize:- generalized Poisson equation is div (ɛ(r) φ) = ρ Statistical mechanics when ρ = 0 ( Z fluct = DD δ(div D) exp β D 2 ) 2ɛ(r) d3 r
Fluctuation potentials A pair of dipoles p V fluct p4 ktr 6 NOT quantum: Keesom/Debye potential Algorithm sums these potentials, Lifshitz/ Thermal Casimir
Mechanical models of electromagnetism 3 E 3,2 2 θ E 4,3 E1,2 4 E 1 4,1 θ
Conclusions O(N) Monte-Carlo algorithm for Coulomb interactions Inhomogeneous ɛ(r) Automatically adds in thermal Casimir interactions Similar in philosophy to lattice Boltzmann for hydrodynamics Dirac like procedure for imposing Gauss law