Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Symplectic integrators for classical spin systems
Klas Modin
Joint work with
Robert McLachlan
Massey University
Olivier Verdier
Western Norway University
Outline
- Symplectic integrators on R2n
- Slightly dissipative systems
- Spherical midpoint method
Illustration
- Non-linear pendulum H(q,p)=p2/2−cos(q)
q˙=p,p˙=−sin(q)
\dot q = p, \quad \dot p = - \sin(q)
q
q
p
p
Illustration
- Non-linear pendulum H(q,p)=p2/2−cos(q)
q˙=p,p˙=−sin(q)
\dot q = p, \quad \dot p = - \sin(q)
Illustration
- Non-linear pendulum H(q,p)=p2/2−cos(q)
q˙=p,p˙=−sin(q)
\dot q = p, \quad \dot p = - \sin(q)
Why?
Hamiltonian dynamics on co-tangent bundles
Phase space T∗Q
Simplest case T∗Rn≃R2n
q˙i=∂pi∂H(q1,…,qn,p1,…,pn)
\displaystyle\dot q^i = \frac{\partial H}{\partial p_i}(q^1,\ldots,q^n,p_1,\ldots,p_n)
p˙i=−∂qi∂H(q1,…,qn,p1,…,pn)
\displaystyle\dot p_i = -\frac{\partial H}{\partial q^i}(q^1,\ldots,q^n,p_1,\ldots,p_n)
Typical examples: celestial mechanics, molecular dynamics
\mathbf{z}=(q,p) \;\Rightarrow\; \dot \mathbf{z} = X_H(\mathbf{z})
Symplectic property of the flow
Symplectic 2-form:
Ω=dqi∧dpi
\Omega = dq^i\wedge dp_i
On R2n:
Ωz(U,V)=U⊤(0In×n−In×n0)V
\displaystyle\Omega_z(U,V) = U^\top \begin{pmatrix} 0 & -I_{n\times n} \\ I_{n\times n} & 0 \end{pmatrix} V
Hamiltonian vector fields fulfill (defining property)
Ωz(XH,⋅)=dH
\Omega_z(X_H,\cdot) = d H
Ωz(XH,XG)={H,G}(z)
\Omega_z(X_H,X_G) = \{H,G \}(z)
Flow φt preserves the symplectic form:
Ωφt(z)(Dφt(z)⋅U,Dφt(z)⋅V)=Ωz(U,V)
\Omega_{\varphi_t(z)}(D\varphi_t(z)\cdot U,D\varphi_t(z)\cdot V) = \Omega_z(U,V)
Geometric interpretation: the "symplectic area" of 2D surfaces in phase space are preserved
Consequences of symplecticness
This explains why symplectic integrators are superior!
- "Almost" conservation of tori for Hamiltonian perturbations of integrable systems (KAM theory)
- Convergence towards correct macroscopic equilibrium (e.g. temperature in MD)
Common symplectic integrators on R2n
- Verlet scheme (many names: Leap-frog, Störmer-Verlet, symmetric splitting, Strang splitting, etc) for H(q,p)=T(p)+V(q)
q(k+1)=q(k)+h∇T(p(k+1/2))
\displaystyle q^{(k+1)} = q^{(k)} + h \nabla T(p^{(k+1/2)})
p(k+1/2)=p(k)−2h∇V(q(k))
\displaystyle p^{(k+1/2)} = p^{(k)} - \frac{h}{2} \nabla V(q^{(k)})
p(k+1)=p(k+1/2)−2h∇V(q(k+1))
\displaystyle p^{(k+1)} = p^{(k+1/2)} - \frac{h}{2} \nabla V(q^{(k+1)})
- Midpoint method
z(k+1)=z(k)+hXH(2z(k+1)+z(k))
\displaystyle z^{(k+1)} = z^{(k)} + h X_H\left(\frac{z^{(k+1)}+z^{(k)}}{2} \right)
Slightly dissipative systems
- Elastic 3D pendulum
q˙=mp,p˙=−k(1−∣q∣ℓ)q+mg
\displaystyle\dot q = \frac{p}{m}, \quad \dot p = -k\left(1-\frac{\ell}{|q|} \right)q + mg
−ε∣q∣2qq⊤p
\displaystyle -\;\varepsilon \frac{qq^\top p}{|q|^2}
Slightly dissipative systems
- Elastic 3D pendulum
q˙=mp,p˙=−k(1−∣q∣ℓ)q+mg
\displaystyle\dot q = \frac{p}{m}, \quad \dot p = -k\left(1-\frac{\ell}{|q|} \right)q + mg
−ε∣q∣2qq⊤p
\displaystyle -\;\varepsilon \frac{qq^\top p}{|q|^2}
non-symplectic
symplectic
Slightly dissipative systems
- Elastic 3D pendulum
q˙=mp,p˙=−k(1−∣q∣ℓ)q+mg
\displaystyle\dot q = \frac{p}{m}, \quad \dot p = -k\left(1-\frac{\ell}{|q|} \right)q + mg
−ε∣q∣2qq⊤p
\displaystyle -\;\varepsilon \frac{qq^\top p}{|q|^2}
Energy behaviour
Classical spin systems
w˙i=wi×∂wi∂H
\displaystyle \dot w_i = w_i\times \frac{\partial H}{\partial w_i}
Phase space (S2)n
Symplectic structure on S2 Ωw(u,v)=det(w,u,v)
+⋯+
+\cdots +
Motivation for spin systems
-
Abundance in physics
(e.g. atomistic spin dynamics)
-
Simplest Kähler manifold
-
Simplest coadjoint orbit
-
Simplest inexact symplectic form
Some example spin systems
Free rigid body w˙=w×I−1w
Heisenberg spin chain
(discrete Landau-Lifshitz) w˙i=wi×(wi−1+wi+1)
Fluid particle tracking on sphere w˙=ξ(t,w),ξ(t,⋅)∈Xμ(S2)
Point vortex dynamics on spheres
(Jupiter's great red spot)
Known symplectic integrators
- Splitting methods
- S2 coadjoint orbit of reduced system on T∗SU(2)
RATTLE ⇒ 9 variables
Variational Lie ⇒ 8 variables
- Collective symplectic integrator
⇒ 4 variables
T∗SU(2)
T^*SU(2)
su(2)∗
\mathfrak{su}(2)^*
Drawbacks: many auxiliary variables, complicated, large error constants
Minimum variables (=3)
and symplectic?
- Classical midpoint method on R3
- Riemannian midpoint method on S2
⇒ not symplectic
⇒ not symplectic
Candidates:
Spherical midpoint method
hWi−wi=∣Wi+wi∣Wi+wi×∂wi∂H(∣W1+w1∣W1+w1,…,∣Wn+wn∣Wn+wn)
\displaystyle\frac{W_i - w_i}{h} = \frac{W_i+w_i}{|W_i+w_i|}\times \frac{\partial H}{\partial w_i}\left(\frac{W_1+w_1}{|W_1+w_1|},\ldots,\frac{W_n+w_n}{|W_n+w_n|} \right)
Main result
- Second order
- Equivariant w.r.t. SO(3)n
- Symplectic
- Preserves single-spin quadratic invariants
- Self-adjoint
S3≃
S^3 \simeq
Hopf fibration
S2
S^2
π
\pi
Extended Hopf fibration
R∗3
\mathbb{R}^3_*
π
\pi
T∗R∗2
T^*\mathbb{R}^2_*
π
\pi
Classical
midpoint
Riemannian
midpoint
Simple interpretation
hW−w=∣W+w∣W+w×∂w∂H(∣W+w∣W+w)
\displaystyle\frac{W - w}{h} = \frac{W+w}{|W+w|}\times \frac{\partial H}{\partial w}\left(\frac{W+w}{|W+w|} \right)
XH(w)=w×∂w∂H(w)
\displaystyle X_H(w) = w\times \frac{\partial H}{\partial w}(w)
⇒W−w=hXH(∣W+w∣W+w)
\displaystyle \Rightarrow W-w = h X_H\left(\frac{W+w}{|W+w|}\right)
XˉH(w)=XH(∣w∣w)
\displaystyle \bar X_H(w) = X_H\left(\frac{w}{|w|} \right)
⇒W−w=hXˉH(2W+w)
\displaystyle \Rightarrow W-w = h \bar X_H\left(\frac{W+w}{2}\right)
Classical midpoint method applied to XˉH
Example 1: free rigid body
w˙=w×I−1w
\dot w = w \times I^{-1}w
Example 2: irreversible rigid body
H(w)=21w⋅I(w)−1w
H(w)= \frac{1}{2} w\cdot I(w)^{-1}w
Example 3: development of chaos
H(w)=21w⋅I−1w+εsin(t)w1
H(w)= \frac{1}{2} w\cdot I^{-1}w + \varepsilon \sin(t)w^1
Example 4: Landau-Lifshitz PDE
w˙=w×w′′,w:R×S1→S2
\dot w = w\times w'', \qquad w\colon \mathbf{R}\times S^1 \to S^2
Example 5: atomistic spin dynamics ???
THANKS!
References:
Symplectic integrators for spin systems, Phys. Rev. E, 89:061301, 2014
A minimal-variable symplectic integrator on spheres, Math. Comp., 86, 2325-2344, 2016
Geometry of discrete-time spin systems, J. Nonlin. Sci., 26(5):1507-1523, 2016
*slides at slides.com/kmodin
Symplectic integrators for classical spin systems Klas Modin
Symplectic integrators for classical spin systems
By Klas Modin
Symplectic integrators for classical spin systems
Presentation given 2017-12 to physicists in Uppsala.
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