Mesoscale Simulation Methods Ronojoy Adhikari The Institute of Mathematical Sciences Chennai

Outline • What is mesoscale ? • Mesoscale statics and dynamics through coarse-graining. • Coarse-grained equations for a binary fluid-fluid mixture. • Numerical methods of solution. • Example simulations.

Numbers and methods 1 2 3 4 5 6 7 0 Ab initio methods Atomistic methods Continuum methods DFT CPMD MD LD BD DPD CFD LB Number of atoms (log) Simulation method Example of method

How lengths scale with numbers !! Try this for water Molar mass = 18 gm Molar volume = 18 mL N = 1023 L 107˚ A L N 1 3 N = 1 L 1˚ A !! Are other scalings possible ? How does length grow with size for a polymer ? Throughout, (!!) indicate exercises/derivations/points to ponder.

Length and time scales pico milli micro nano nano micro milli L T H = E m ¨ R = ⇥U ˙ R = U ˙ c = D 2c !! Are there systems with millimeter L but picosecond T ?

How times scale with masses and lengths m¨ x + kx = 0 2 0 = k m 0 m k ¨ u = c2 2u 0 = ±cq ⇥0 c ˙ u = D 2u i 0 = Dq2 ⇥0 2 D Harmonic Oscillator Wave Equation Diffusion Equation

Thermal fluctuations P(x |x) = 1 ⇤ 2 D⇥ exp (x x)2 2D⇥ ⇥ Brownian motion : Einstein (1905) D = kBT 6⇥ a temperature size Stokes-Einstein-Sutherland Relation Q. Why does a smaller particle have a higher diffusion coefficient ? A. The ‘root N’ effect! The central limit theorem helps us understand this. !! Is it (central limit) theorem or central (limit theorem) ?

Langevin theory inertia damping reversible force fluctuation deterministic stochastic mean behaviour ‘root N’ fluctuations regression to the mean equilibrium fluctuations v2⇥ = kBT F = 0 Free Brownian particle m˙ v + v F(x) = ⇥(t)

Mesoscale regime 1 2 3 4 5 6 7 0 Ab initio Atomistic Continuum Number of atoms Simulation Coarse grained length scales. Coarse grained time scales. Retain thermal fluctuations. Mesoscale methods } Examples Brownian dynamics. Dissipative particle dynamics. Time-dependent Ginzburg-Landau.

Coarse-graining in degrees of freedom What is the idea ? x1 x2 } P(x1, x2 ) RV y = x1 + x2 “Coarse-grained” sum P(y) P(y) = dx1dx2 (y x1 x2 )P(x1, x2 ) What is the distribution ? contains less information about the system than P(y) P(x1, x2 ) Adequate if we are only interested in the sum variable.

General ‘coarse-graining’ formula P(x1, x2, . . . , xN ) y = f(x1, x2, . . . , xN ) Microstate probability Mesoscale variable/order parameter P(y) = ⇥ N i=1 dxi [y f(x1, . . . , xN )]P(x1, . . . , xN ) Mesostate probability

Coarse-grained Landau-Ginzburg functional Microscopic Hamiltonian Gibbs distribution P(q) = exp[ H(q)] Z H(q) = f(q) Order parameter Definition of Landau functional P(⇤) = ⇥ N i=1 dqi⇥[⇤ f(q)] exp[ H(q) Z ⇥ exp[ L] Z P(⇥) ⇥ exp[ L] Z

Explicit DOF coarse-graining ... H(s1, s2, . . . , sN ) = J ij⇥ sisj Order parameter. Block spin. Coarse-grained magnetization. = 1 N i si

Explicit DOF coarse-graining ... H(s1, s2, . . . , sN ) = J ij⇥ sisj Order parameter. Block spin. Coarse-grained magnetization. = 1 N i si P(⇥ ) ⇥ exp[ L] Z⇥ =? K. Binder, Z. Phys B, 43, 119 (1981)

Explicit DOF coarse-graining ... H(s1, s2, . . . , sN ) = J ij⇥ sisj Order parameter. Block spin. Coarse-grained magnetization. = 1 N i si P(⇥ ) ⇥ exp[ L] Z⇥ =? K. Binder, Z. Phys B, 43, 119 (1981)

Explicit DOF coarse-graining ... H(s1, s2, . . . , sN ) = J ij⇥ sisj Order parameter. Block spin. Coarse-grained magnetization. = 1 N i si P(⇥ ) ⇥ exp[ L] Z⇥ =? K. Binder, Z. Phys B, 43, 119 (1981) Explicit DOF needs this!

Or ansatz based on symmetry. L = A 2 2 + B 4 4 + K 2 (⇥ )2 P(⇥) ⇥ exp[ (A 2 ⇥2 + B 4 ⇥4)] P(s) ⇥ a+ (s 1) + a (s + 1) -J +J local part non-local part K( )2 All equal-time equilibrium properties are determined by this functional. !! Why is this called a “free energy” ? Is it a thermodynamic free energy ?

What about dynamics ? Temporal coarse-graining D = kBT 6⇥ a Why is this independent of particle mass ? ⇥d = m Inertial time scale. Time the velocity ‘remembers’ its initial condition. d ˙ x = v ˙ x = ⇥(t) Overdamped equations of motion. Valid when d m˙ v + v = ⇤(t) = 6⌅⇥a

Overdamped stochastic equations damping reversible force fluctuation deterministic stochastic mean behaviour ‘root N’ fluctuations ˙ x = ⇥U + ⇥(t)

Overdamped coarse-grained equations of motion ˙ ⇤(r) = L ⇤ + ⇥(r, t) ˙ ⇥(r) = [A⇥ + B⇥3 K⇥2⇥] + (r, t) damping reversible force fluctuation Model A equation ⇥⇥(r, t)⇥(r , t )⇤ = 2kBT (r r ) (t t ) Fluctuation Dissipation Relation. !! Zero-mean Gaussian noise (r, t)⇥ = 0 “Stochastic partial differential equations with additive Gaussian noise”

Conserved order parameters Ising spin Lattice gas Ising spins are usually non-conserved d dt (r, t) = 0 Lattice gas models are conserved d dt (r, t) = 0

Conserved coarse-grained dynamics damping reversible force fluctuation Model B equation Fluctuation Dissipation Relation. !! Zero-mean vector Gaussian noise (r, t)⇥ = 0 ˙ ⇤(r) = ⇤2 L ⇤ + ⇤ · ⇥(r, t) ˙ ⇥(r) = ⇤2[A⇥ + B⇥3 K⇤2⇥] + ⇤ · (r, t) !! Derive this. Use the local conservation law ˙ + ⇥ · j = 0 ⇥⇥i (r, t)⇥j (r , t )⇤ = 2kBT ij (r r ) (t t )

Numerical solution Stochastic PDE Stochastic ODE Stochastic realisation Spatial discretisation Temporal integration ⇤t⇥ = D⇤2 x ⇥ + ⇥x = (x + h) (x) h ⇤t⇥(i) = DL2 ij ⇥(j) + (i) ⇥(i, t + t) ⇥(i, t) = t+ t DL2 ij ⇥(j) + (i) !! Obtain the explicit form of the L matrix for a central difference Laplacian

Summary of TDGL mesoscale methods Microscopic Hamiltonian Ginzburg-Landau Functional Langevin equations Explicit Coarse Graining Overamped approximation Symmetry + gradient expansion Numerical solution

Conclusion • Mesoscale methods are appropriate at length scales intermediate between the molecular and continuum scales. • The continuum description must be supplemented by fluctuation terms. At the mesoscale, the number of particles is not so large that fluctuations can be neglected. • Lengths and times must be coarse-grained. Intelligent coarse-graining improves computational efficiency, often by orders of magnitude, compared to direct MD. • Mesoscale methods can be particle-based, for instance dissipative particle dynamics and Brownian dynamics, or field-based like time-dependent Ginzburg-Landau theory and fluctuating hydrodynamics. • TDGL equations can be solved very efficiently on computers using matrix formulations.

Further reading • Stochastic processes in Physics and Chemistry : van Kampen • Handbook of stochastic processes : Gardiner • Principles of Condensed Matter Physics : Chaikin and Lubensky • Modern Theory of Critical Phenomena : Ma • Reviews of Modern Physics article : Halperin and Hohenberg • Numerical Solutions of Stochastic Differential Equations : Kloeden and Platen.

Thank you for your attention.