Computation of Long Range Interactions
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(2) Efficient Algorithms for Long Range Interactions (... All I say will in principle also be valid for dipolar interactions). 2.
(3) Electrostatics under periodic boundary Electrostatics under pbcconditions • Periodic boundary conditions (pbc) eliminate boundary effects in bulk simulations • Minimum image convention for short ranged potentials • Coulomb potential 1/r is long ranged, many images contribute significantly • Sum is only conditionally convergent • For fully periodic boundary conditions (pbc) many e⇥cient methods exist: Ewald (N 3/2), P3M (N log N ), FMM (N ) • Simulation of surface effects: both periodic and nonperiodic coordinates (2d+h / 1d+2h geometries) 3.
(4) Conditional convergence: Why the summation order does matter. Conditionally Convergence Example: The alternating harmonic series: ⇤ ( 1)k+1 =1 k. k=1. 1 1 + 2 3. 1 1 + 4 5. · · · = ln 2. but look at this... (1. 1 ) 2. 1 1 +( 4 3 =. 1 ) 6 1 2. 1 = 1 2. 1 1 +( 8 5 1 1 + 4 6 1 1 + 2 3. 1 ) 10 1 1 + 8 10 1 1 + 4 5. 1 1 +( 12 7 1 14. 1 ) 14. 1 16. .... ⇥. 1 . . . = ln 2 2 4. ....
(5) Conditional convergence: Why the summation order does matter. Conditionally Convergence Example: The alternating harmonic series: ⇤ ( 1)k+1 =1 k. k=1. 1 1 + 2 3. 1 1 + 4 5. · · · = ln 2. but look at this... (1. 1 ) 2. 1 1 +( 4 3 =. 1 ) 6 1 2. 1 = 1 2. 1 1 +( 8 5 1 1 + 4 6 1 1 + 2 3. 1 ) 10 1 1 + 8 10 1 1 + 4 5. 1 1 +( 12 7 1 14. 1 ) 14. 1 16. .... ⇥. 1 . . . = ln 2 2 5. ....
(6) Conditional convergence: Why the summation order does matter. Conditionally Convergence Example: The alternating harmonic series: ⇤ ( 1)k+1 =1 k. k=1. 1 1 + 2 3. 1 1 + 4 5. · · · = ln 2. but look at this... (1. 1 ) 2. 1 1 +( 4 3 =. 1 ) 6 1 2. 1 = 1 2. 1 1 +( 8 5 1 1 + 4 6 1 1 + 2 3. 1 ) 10 1 1 + 8 10 1 1 + 4 5. 1 1 +( 12 7 1 14. 1 ) 14. 1 16. .... ⇥. 1 . . . = ln 2 2 6. ....
(7) Conditional convergence: Why the summation order does matter. Conditionally Convergence Example: The alternating harmonic series: ⇤ ( 1)k+1 =1 k. k=1. 1 1 + 2 3. 1 1 + 4 5. · · · = ln 2. but look at this... (1. 1 ) 2. 1 1 +( 4 3 =. 1 ) 6 1 2. 1 = 1 2. 1 1 +( 8 5 1 1 + 4 6 1 1 + 2 3. 1 ) 10 1 1 + 8 10 1 1 + 4 5. 1 1 +( 12 7 1 14. 1 ) 14. 1 16. .... ⇥. 1 . . . = ln 2 2 7. ....
(8) Electrostatics Electrostatics Coulomb energy. http://www.icp.uni-stuttgart.de. Pair energy summation N lB X0 qi qj U= 2 |rij | i,j=1. Bjerrum length e2 lB = 4⇡✏0 ✏r kB T. summing up 1/r Coulomb pair potential. electrostatic prefactor / inverse temperature. Bjerrum length lB. for two unit charges:. 1kBT 1lB. A. Arnold. Coulomb interactions. 8. 2/26.
(9) Electrostatics Electrostatics. Coulomb energy. http://www.icp.uni-stuttgart.de. Pair energy summation N X 0 qi qj lB U= 2 |rij | i,j=1. summing up 1/r Coulomb pair potential Bjerrum length lB. Potential summation N. 1X U= qi (ri ) 2 i=1. potential from solving Poisson’s equation r2 (r) =. equivalent approaches A. Arnold. Coulomb interactions. 4⇡lB. N X. qj (rj. r). j=1. 9. 2/26.
(10) Electrostatics in Periodic BC Electrostatics in periodic boundary conditions Coulomb energy. http://www.icp.uni-stuttgart.de. Pair energy summation. Potential summation. N 1 qi qj lB X X X 0 U= 2 |rij + mL| 2 S=0 m =S i,j=1. conditionally convergent — summation order important numerically difficult. U not periodic in coordinates ri. N. 1X U= qi 2. per (ri ). i=1. solve Poisson’s equation imposing periodic boundaries U is periodic in coordinates ri. these two calculate something different! A. Arnold. Coulomb interactions. 10. 3/26.
(11) The Dipole Term. Where the difference comes from: the dipole term. assume summation in periodic shells surrounded by polarizable material of dielectric constant ✏1 ✏ = ✏1. http://www.icp.uni-stuttgart.de. 3. 3 2. 1. 2. 1. 0. 1. 2. 1. 2. ✏=1. 3. Pair energy summation vacuum around: ✏1 = 1. 3. ✏ = ✏1. Potential summation periodic: ✏1 = 1. difference to periodic solution is nonperiodic dipole term ✓X ◆2 2⇡ U (d) = qi ri 3 (1 + 2✏1 )L i. metallic boundary conditions ✏1 = 1 always safe never use ✏1 < 1 for conducting systems. A. Arnold. Coulomb interactions. 4/26 11.
(12) The Ewald Method P. P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale, Ann. Phys. 369(3):253, 1921. http://www.icp.uni-stuttgart.de. The Ewald method. P. P. Ewald, 1888 — 1985 Coulomb potential has 2 problems 1. singular at each particle position 2. very slowly decaying Idea: separate the two problems! one smooth potential — Fourier space one short-ranged potential — real space A. Arnold. Coulomb interactions. 12. 7/26.
(13) Splitting the Potential Ewald: splitting the potential. charge distribution ⇢=. N X X. qi (r. ri. n). http://www.icp.uni-stuttgart.de. n2LZ3 i=1. = replace. +. by Gaussians of width ↵. 1:. p ⇢Gauss (r) = ↵/ ⇡ (r) = ⇢Gauss (r) + [ (r) A. Arnold. 3. e. ↵2 r 2. ⇢Gauss (r)]. Coulomb interactions. 13. 8/26.
(14) The Standard Ewald Sum The Ewald formula. U = U (r ) + U (k ) + U (s) with U. (r ). erfc(↵|rij + mL|) lB X X 0 = qi qj 2 |rij + mL| 3 i,j. http://www.icp.uni-stuttgart.de. m2Z. U. (k ). lB X 4⇡ = 3 e 2 2L k. k 2 /4↵2. k6=0. U. (s). =. real space correction. |b ⇢(k)|2. ↵lB X 2 p qi ⇡. Gaussians in k -space Gaussian self interaction. i. forces from differentiation @ U @ri ... coming soon to ESPResSo (on GPU) Fi =. A. Arnold. Coulomb interactions. 14. 9/26.
(15) ...The%Bible... written by the evangelists R.W.2Hockney,2J.W.2Eastwood,21988 R. W. Hockney J. W. Eastwood replace k -space Fourier sum by discrete FFT • Near field:2Standard2Ewald • Far FT field:2replace bycutoff the discrete discrete is exact —Fourier2space constant real sum space FFT2on2a2regular mesh computational order O(N log N) • Computational order O(N log N ) most frequently used methods: 3 • 3 P M2(Hockney,2Eastwood,21973) P M: optimal method •PME The2wheel got reinvented:2 PME2(Darden2et2al.21993) SPME SPME2(Essmann et2al.,21995) A. Arnold. R. W. Hockney and J. W. Eastwood,. http://www.icp.uni-stuttgart.de. Particle Mesh Ewald Methods. Coulomb interactions. Particle)Particle)Particle)Mesh. 10. Simply the best..... 15.
(16) 3 Steps Steps of of P Mthe P M 3. http://www.icp.uni-stuttgart.de. 1. {ri , qi } ! ⇢(r): interpolate charges onto a grid (window functions: cardinal B-splines) 2. ⇢(r) ! ⇢b(k):. Fourier transform charge distribution. 4. ik ˆ(k) ! Ê(k):. obtain field by Fourier space differentiation. 3. ˆ(k) = Ĝ(k)ˆ ⇢(k): solve Poisson’s equation by multiplication with optimal influence function Ĝ(k) (in continuum: product of Green’s function 4⇡ and k2 2 2 Fourier transform of Gaussians e k /4↵ ) b 4. E(k) ! E(r):. Fourier transform field back. 5. E(r) ! {ri , Fi }: interpolate field at position of charges to obtain forces Fi = qi Ei Instead of ik,differentiation.(4.).I.can also.use finite.difference or differenciate the pullback function of r ..This.saves two FFT‘s.
(17) Charge Assignment Function Charge assignment q7. q8. 1. 1. 2. 0.8. q5. q6. 3. M (P ) (x). 4 5. 0.6. 6. 7. 0.4. q3. http://www.icp.uni-stuttgart.de. q1. a. q4 q2. 0.2 0. 0. 1. 2. 3. 4. 5. 6. 7. x. interpolate charges onto h-spaced grid N X 1 ⇢M (rp ) = 3 qi W (p) (rp h. ri ). i=1. W (p) (r) cardinal B-splines in P3 M / SPME A. Arnold. Coulomb interactions. 17. 12/26.
(18) Optimal Influence Function Optimal influence function minimize the rms error functional Z Z ⇥ Q[G] := dr1 dr (G; r, r1 ) h3. V. (r). ttp://www.icp.uni-stuttgart.de. with the analytic reference potential ˆ(k) = 4⇡ e k2. k 2 /4↵2. leads to Ĝopt (k) = with m = L/h. P. 'ˆ2 (k + ml)Ĝ(k + ml) ⇥P ⇤2 2 ˆ (k + ml) l2Z3 '. l2Z3. 18. ⇤2.
(19) Why Control Errors ? Why to control errors rms force error. F =. q⌦. (Fexact. 10. ↵. FEwald )2 =. rmax=1, kmax=10 rmax=2, kmax=10 rmax=1, kmax=20. 1. s. 1 N. N P. i=1. ttp://www.icp.uni-stuttgart.de. ΔF. 0.1 0.01 0.001 0.0001 1e-05 0. 1. 2. 3. 4. 5. α. optimal ↵ brings orders of magnitude of accuracy at given required accuracy, find fastest cutoffs compare algorithms at the same accuracy 19. Fi2.
(20) How toerror Control Errors How to: estimates 10. total error real space estimate k-space estimate. 1. ΔF. 0.1 0.01 0.001 0.0001. ttp://www.icp.uni-stuttgart.de. 0. 1. 2. 3. 4. 5. α. Kolafa and Perram: Freal. P 2 ⇣ ⌘ qi 2 2 ⇡ p p exp ↵2 rmax N rmax L3. Hockney and Eastwood:. FFourier. P 2 qi ⇡ p N. s. Q[Ĝopt (k)] L3. 20.
(21) Interlaced P3M:. Improving the performance: interlacing A..Neelov,.C..H.,.J.Chem..Phys. 132,%234103 (2010). p://www.icp.uni-stuttgart.de. (int) Fi. 1 = Fi + Fpi 2. force Fi is average of P3 M force and P3 M force with grid shifted by p = (h/2)e doubles the effort, gains order of magnitude in accuracy effort can be reduced by using complex-to-complex FFT. Interlacing plus.analytical differentiation seems to be the fastest.method currently ! 21.
(22) S CA FAC O S ScaFaCoS Library Scalable Fast Coulomb Solver http://www.scafacos.de Library of different Coulomb solvers. http://www.icp.uni-stuttgart.de. “Highly scalable”, MPI-parallelized Common interface for all methods Developed by groups from Jülich, Wuppertal, Chemnitz, Bonn and Stuttgart Project by the German Research Ministry (BMBF), officially ended 2011 Open source: Source code on github Olaf Lenz. http://www.scafacos.de/. Scalable Fast Coulomb Solvers. 6/24.
(23) Methods Implemented Algorithms. http://www.icp.uni-stuttgart.de. S CA FAC O S currently provides 11 method implementations: DIRECT, EWALD, P3M, P2NFFT, VMG, PP3MG, PEPC, FMM, MEMD, MMM1D, MMM2D Features: For reference purposes (not competitive): DIRECT, EWALD 3d-periodic boundaries: P3M, P2NFFT, VMG, PP3MG, PEPC, FMM(, EWALD) Open boundaries: P2NFFT, FMM, PEPC(, DIRECT) Partially periodic boundaries: P2NFFT, FMM, PEPC, MMM*D General triclinic boundaries: P3M, P2NFFT. Distinguish Splitting Methods, Hierarchical Methods and Local Methods (i.e. MEMD). A.#Arnold,#F.#Fahrenberger,#C.#Holm,#O.#Lenz,#M.#Bolten,#H.#Dachsel,#R.#Halver,#I.# Kabadshow,#F.#Gähler,#F.#Heber,#J.#Iseringhausen,#M.#Hofmann,#M.#Pippig,#D.#Potts,#G.# Sutmann,##Comparison of scalable fast0methods for long4range0interactions,#Phys.#Rev.#E# 88,#063308#(2013).# Olaf Lenz Scalable Fast Coulomb Solvers 7/2.
(24) P2NFFT Particle-Particle Nonequidistant FFT ! Simplified P3M is a special case of P2NFFT ! Same structure = same performance ! Minor mathematical details, simpler approach ! Advantage: can be formulated in 3D, 2D,1D, 0D !. 24.
(25) Splitting Methods: Multigrid Multigrid. http://www.icp.uni-stuttgart.de. Solve Poisson equation in far field with multigrid PDE solver use different levels of successively coarser meshes solve poisson equation on these meshes by recursively improving the solution of the coarser mesh. Complexity O(N). l l l l. =4 =3 =2 =1 Restriction. Can be extended to handle periodic BC. Prolongation Smoothing/Solving. In S CA FAC O S: PP3MG (Wuppertal), VMG (Bonn). Olaf Lenz. Scalable Fast Coulomb Solvers. 25. 10/2.
(26) Hierarchical Methods: Barnes-Hut Barnes-Hut Tree code Treecode Hierarchically cluster charges. http://www.icp.uni-stuttgart.de. Multipole expand these clusters Compute interaction with far away cluster multipole moments instead of single particle charges Complexity O(N log N). Can be extended to handle periodic BC In S CA FAC O S: PEPC (Jülich). Olaf Lenz. Scalable Fast Coulomb Solvers. 26. 11/2.
(27) Fast Multipole Hierarchical Methods:Method Fast Multipole Method Expand Treecode: let clusters interact with each other http://www.icp.uni-stuttgart.de. Put everything on a grid Complexity O(N). Can be extended to handle periodic BC In S CA FAC O S: FMM (Jülich). Olaf Lenz. Scalable Fast Coulomb Solvers. 27. 12/2.
(28) MEMD for Molecular Dynamics. F.#Fahrenberger,#CH,#Phys.#Rev.#E#90,#063304#(2014). 28.
(29) Benchmarks Complexity Complexity. http://www.icp.uni-stuttgart.de. MEMD and Multigrid ⇡ ⇥10 slower All algorithms show (close-to-)linear behavior. log N-term of P2NFFT and P3 M is invisible No cross-over with FMM. Time t/#Charges [s]. P2NFFT, P3 M and FMM are fastest. 10. 3. 10. 4. 10. 5. 10. 6. MEMD P2NFFT P3M VMG FMM PP3MG. 104. 105. Silica melt, "pot. 106 107 #Charges 10 3 , P =. 108. 1 (JUROPA). 29. 109.
(30) Conclusions ScaFaCoS Conclusions. Performance depends heavily on architecture, compiler and implementation. http://www.icp.uni-stuttgart.de. . . . and tuning! ⇥2 differences between algorithms are “normal”. Within these limits, FMM, P3 M and P2NFFT perform equally good MEMD slightly worse (⇡ ⇥4), but performs better with larger systems Multigrid methods seem to be worse (⇡ ⇥10). . . . apparently due to large variation in the potential. Olaf Lenz. Scalable Fast Coulomb Solvers. 24/24 30.
(31) Some Coulomb Solvers N"="Number"of"charges Fast Methods for Coulomb Interactions. http://www.icp.uni-stuttgart.de. Year 1820 1921 1974 1977 1986 1987 2002. Method Direct Summation Ewald Summation Particle-Particle Particle-Mesh Multigrid Summation Barnes-Hut Treecode Fast Multipole Method Maxwell Equation Molecular Dynamics. Complexity N2 3 N2 N log N. Reference Laplace Ewald Hockney, Eastwood. N N log N N N. Brandt Barnes, Hut Greengard, Rokhlin Maggs, Rossetto. These methods often are complex.. • Methods"often"complex do not parallelize easily. • They They"do"not"parallelize"easily and parallel scaling are highly platform and • Prefactors Prefactors and"scaling"are"highly"platform"and" implementation dependent. implementation"dependent 315/24 • Olaf MC"and"MD"performance"differs,"check"accuracy Lenz Scalable Fast Coulomb Solvers.
(32) Dielectric contrasts Partially Periodic Systems. riodic systems. z. water. elec. q∆t ∆b εt q ∆t. membrane. x lz. q. http://www.icp.uni-stuttgart.de. q ∆b. water. εm. wate. εb. q ∆b∆t. elec. • 2D$periodic:$slablike films,$ typical twosystems,$surfaces,$thin dimensional systems: thin films, slit po . for slablikemembranes,$air:water systems (surfaces, thin films) interfaces material boundaries =) dielectric contrast • 1D$periodic:$Needles,$rods,$nanopores, rical systems (rods, nanopores) take into account polarization by image charges • 0D$periodic boundaries =$open$systems asts at interfaces. can be handled by MMM2D e employed straightforwardly. • P3M$cannot be employed straightforwardly • P2NFFT$can be formulated to work A. Arnold. 32 Coulomb interactions.
(33) MMM2D and MMM1D Another approach: MMM2D far formula (r ) =. X. p. p. e. (x+k)2 +(y+l)2 +z 2. (x + k)2 + (y + l)2 + z 2 0 1 ✓ ◆ q q 2 X @X 2 + p 2 (y + l)2 + z 2 A eipx = K0 L 2⇡ k,l2LZ. http://www.icp.uni-stuttgart.de. p2 L Z. 2⇡ = 2 L. =. l2LZ. X. p,q2 2⇡ Z L. 0. 2⇡ @ X L2 2 2. e p. p +q >0. p. 2. 2 +p 2 +q 2 |z|. + p2 + q 2. efpq |z| fpq. eipx eiqy 1. eipx eiqy + |z|A +. ⇡ L2. 1. +O. !0 (. ). screened Coulomb interaction in limit of screening length 1 6>*0) • Convergence*factor*based*summation*(limit*! other for z ⇡ 0 • formula Near*formula*for*z*≈0. optimal computation time O(N 5/3 ), comparable to Ewald • Optimal*computation*time*comparable*to*Ewald O(N 5/3 ) analogously for 1d, but then O(N 2 ) • Analogous*formulae*for*1D,*but*then*T CPU ≈ O(N 2 ). A. Arnold A. Arnold,*C.*H.,*Chem.*Phys.*Lett. Coulomb interactions 354,%324 (2002)'% A.*Arnold,*C.H.,*J.*Chem.*Phys. 123,*144103 (2005). 18/26. 33.
(34) 2D PBC approx. with 3D PBC TheThe$Yeh method and$Berkowitz$correction of Yeh+Berkowitz replicated slab system slab system h Lz. http://www.icp.uni-stuttgart.de. replicated slab system L. potential of a charge and its periodic images similar to plate plates cancel due to charge neutrality 2⇡qi. N X j=1. j (|zji. + mLz | + |zji. mLz |) = 4⇡qi nLz. N X. Coulomb interactions. =0. j=1. leave a gap and hope artificial replicas cancel ✓ ◆2 P requires changed dipole term U (d) = 2⇡ i qi zi L3. A. Arnold. j. 34. 21/26.
(35) ⇡ Ulc = 2 L. X. http://www.icp.uni-stuttgart.de. http://www.icp.uni-stuttgart.de. The ELC Method. 2 k2 2⇡ Z L 2 k >0. N X. i,j=1. N ij |k|z Xe |k|z X e|k|z ⇡ ij + i(kx e xij + ijky+ yije ) qUi lcqj= 2 f L eqi qj fpq (efpq Lz fpqL(e pq2⇡z 2 i,j=1 1) k2 L Z k2 >0. |k|zij. 1). ei(. error not known a priori — required gap siz error not known a priori — required gap size? • Calculate)contribution)of)image)layers)exactly. calculate contribution of image layers •calculate Subtract)numerically)=>)smaller)gap)size contribution of image layers subtract numerically ) needs smaller gaps •subtract Change)summation)order)with)dipole)term numerically ) needs smaller gaps 2-4x faster than plain Yeh+Berkowitz •2-4x 2>4x)faster)than)plain)Yeh>Berkowitz)plus)full) faster than plain Yeh+Berkowitz error)control A. Arnold Coulomb interactions. A. Arnold. Coulomb interactions. A.)Arnold,)J.)de)Joannis,)C.H.,)J.)Chem.)Phys.)117,)2496)(2002) 35.
(36) Dipolar Interactions There is also a dipolar P3M available in ESPResSo ! Also a 2D+h Version as DLC !. J.#J.#Cerdà ,V.#Ballenegger,#O.#Lenz,#Ch.#Holm,#P3M$ algorithm$for$dipolar$interactions,#J.#Chem.#Phys,#129,# 234104#(2008)#. 36.
(37) Include Dielectric Interfaces. Where does this matter?. 37.
(38) What does Dielectric Contrast do?. !. Water is more polarizable than the solid wall material, leading to an induced charge of the same sign. !. Force pushing the charge away from the wall. !. For air-water interfaces it is the opposite situation. !. How to compute the electrostatic force efficiently?. !. Use either Image charges or induced charges…..
(39) Planar Dielectric Interfaces ! ="2. Can"be"handled"by • ICMMM2D • ELCIC • S."Tyagi,"A."Arnold,"C."H.,"ICMMM2D:"An"accurate"method"to"include"planar"dielectric" interfaces"via"image"charge"summation,"J."Chem."Phys.127,"154723"(2007)" • S."Tyagi,"A.""Arnold"C."H.,"Electrostatic+layer+correction+with+image+charges:+A+linear+ scaling+method+to+treat+slab+2D+++h+systems+with+dielectric+interfaces,"J."Chem." Phys.129,&204102"(2008)" 39.
(40) rt.de. Arbitrarily shaped dielectric surfac What%about%arbitrarily%curved%interfaces%like%a%nanopore?. 40.
(41) ICC* Algorithm !. Boundary condition for the normal component of the electric field: !. !. can be fulfilled by introducing a charge density:. Discretization of the surface to boundary elements. electric' Field normal'vector. 41.
(42) ICC* Algorithm !. Boundary condition for the normal component of the electric field: !. !. can be fulfilled by introducing a charge density:. Discretization of the surface to boundary elements lead to a set of equations. ‣ Solved'by'an'iterative'scheme • can'be'obtained'by'any'Coulomb'solver ‣ Periodic'boundary'conditions'automatically'fulfilled 42.
(43) ICC* Algorithm !. Boundary condition for the normal component of the electric field: !. !. can be fulfilled by introducing a charge density:. Discretization of the surface to boundary elements lead to a set of equations ICC*$(Induced Charge$Computation)$Algorithm:$$$ S.$Tyagi,$M.$Süzen,$M.$Sega,$M.$Barbosa,$S.$Kantorovich,$and C.$Holm,$Journal$of Chemical$Physics 132,$154112$(2010)$ 43.
(44) Salt can Reduce the Dielectric !r 80 +. −. Na Cl εE = 72/(1+0.278c) + − Na Cl scaled + − K I. 70. εE. 60 50 40 30. 0. 1. 2 3 concentration (M). 4. AllCAtom$MD,$SPC/E$explicit$water$ B.$Hess,$C.$Holm,$N.$van$der$Vegt:$PRL$96,$147801$(2006). 44.
(45) Inhomogeneous Dielectrics. Permittivity)can)be)reduced) inhomogeneously by) presence)of)ions)around) charged)objects!.
(46) onstantVariation speed of the fields and ithe motion of the particles d vative. This method of numerical relaxation is not very % % correct Then the propagation of the 2 fields can be describe Maxwell-like Equations ε0 c but has toartificial be done only once. 2 3 dynamics, in − a Car-Parrinello (CPMD) d r + man A (∇ × A) 2 law, only Fig.from 1. ng this solution of Gauss’ updates of Analog to Pasichnyk and Dünweg, the most general c A.*C.*Maggs and*V.*Rosseto,*PRL*88,*196402*(2002). ⇢ 2system by the I.*Pasichnyk and*B.*Dünweg,*J.*Phys.*Cond.Mat.*16,*3999*(2004).* for the r =a isconstraint tric field following have to applieda The equations of motion for be the particles " eld on be derived from Eq. (17) and the Lagrangia we can assume that the time scales of the propagation ˙ he vector field Ḋ + j − ∇ × ! = 0, r"r = ⇢ use motion of variational calculus. The resulting equ the fields(10) and the of the particles decouple. ) r ·the Dfor = ⇢ displacement (Gauss law) with electric field ε E, the the particles and fields fromDthe=unconstra e propagation ofdensity the fields can be described by an current j , and an arbitrary vector field ! are = 0 r ⇥ D cle •(CPMD) Varying*permittivity dynamics, in a Car-Parrinello manner [35] additional degree of freedom. From this, the Lagrang # • potentials*to*fields ∂U α o∂A Pasichnyk and Dünweg, the most general constraint " r̈ = − − q E + qi v"i × m i i i • most*general*form ! mi 2 fmass ∂ r i ˙ 2 3 1 −ystem , β is General*constraint L= vi − U + ε(r)! d r − ∂ri 2 2 2 ε i 1 ˙ ˙ = 0, B = !, Ḋ + "j − ∇ × ! (5) 2 (11) + A( Ḋ − ∇ × ! ˙ + j )d 3 r c 2 treatment*leads*to*equations*of*motion* e electricLagrangian displacement field D Ḋ==ε cE, electric ∇ the ×B− j, electromagnetic for*the*particles*and*fields. the Lagrange multiplier A46isas usedan to density jis, obtained, and anwhere arbitrary vector field !.
(47) Maggswellian Dynamics with !r(r) • Naturally)formulated)on)a)lattice)(=>fast)and)local) • Changing)speed)of)light)(CPMD))(=>)tricky) • Implemented)in)ESPResSo as)MEMD$(=>)useful) Leads)naturally)to)MaxwellFlike)equations. ṙi. =. ṗi. =. Ȧ. =. Ḋ. =. A.)C.)Maggs and)V.)Rosseto,)PRL)88,)196402)(2002). J.)Rottler and)A.)C.)Maggs,)PRL)93,170201)(2004).. pi mi @U ei + D(ri ) @ri " I.)Pasichnyk and)B.)Dünweg,)J.)Phys.)Cond.Mat.) 16,)3999)(2004).) D F.)Fahrenberger,)C.)Holm,)Phys.)Rev.)E)90,$063304) (2014) " j 2 c r ⇥ (r ⇥ A) 47 ".
(48) F. FAHRENBERGER C. HOLM MEMD withAND Variable Dielectric j. Bx. Dz. By Dx. 2. e p e. 1. Dy Bz. (a) Discretization. d2. d1. q. a. (b) ε interpolation. FIG. 2. (Color online) (a) Discretization of the currents, fields, • permittivity*as*a*vector*(differential15form),*taken*as*the* and permittivities onto a lattice cell. (b) Interpolation of dielectric difference*between*adjacent*lattice*points*(harmonic* permittivity values on the lattice. ε(r) has a position and a direction average) (blue arrow). The values for ε1 and ε2 are determined and the48value. w h s s c p c p.
(49) Wrap up Lecture: •Basic'simulation methods to describe charged systems •Various Advanced Methods for periodic and partially periodic systems •ScaFaCoS •Including dielectric interfaces and heterogeneities. Any$Questions?? 49.
(50) Acknowledgements M."Deserno,"A."Arnold,"F."Weik,"F."Fahrenberger,"Z."Xu,"J."de"Joanis," O."Lenz,"S."Tyagi,"and"many"more….. www.espressomd.org. €€:"EXC"SimTech,"DFG,"Volkswagen"Foundation.
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Interaction potentials at constant temperature Tc- T = 220 mK and different lateral positions D.x = x - xo relative to the substrate (with xo a reference position at the
Abstract—The efficient computation of interactions in charged particle systems is possible based on the well known Ewald summation formulas and the fast Fourier transform for
In the case of 3d-periodic boundary conditions the nonequispaced fast Fourier trans- form (NFFT) 30 can be directly applied to the Fourier space sum in order to achieve a
Abstract—Ewald summation has established as basic element of fast algorithms evaluating the Coulomb interaction energy of charged particle systems in three dimensions subject
