invariance and the structure of...maxwell-boltzmann equation methods of structure determination for...

Thanks: Kaushik Dayal, Stefan Müller, Traian Dumitrica

Invarianceand the structure of

matterRichard JamesUniversity of [email protected]

October 24, 2015 Ericksen@90

mailto:[email protected]

Three examples


An invariant manifold of molecular dynamics

Maxwell-Boltzmann equation

Methods of structure determination for non-crystalline structures (Maxwell’s equations)

…of invariance under discrete groups, and implications for thestructure of matter

Isometry groups

Ericksen@90

This product rule implies

Identity

The main subject of The International Tables of Crystallographyare isometry groups containing three linearly independent translations

October 24, 2015

Examples of isometry groups Translation group

Theorem: If a discrete group of isometries does not contain a translation and does not consist entirely of rotations, it is expressible in one of the forms

where

Full list: Dayal, Elliott, James

BAB

DC

A DC

Ericksen@90October 24, 2015

Three examples





…of invariance under discrete groups and implications for thestructure of matter

A time-dependent invariant manifold of the equations of molecular dynamics

simulated atoms

a discrete group of isometries(N can be infinite)

all of the atoms

The elements can depend on t>0, but this time dependence must beconsistent with

,t


Atomic forces

Frame-indifference

Permutation invariance

The force satisfies

where Π is a permutation that preserves species.

(These conditions satisfied, e.g., by the Hellmann-Feynmanforce based on Born-Oppenheimer quantum mechanics)


Potential energy

These conditions can be found by formally differentiating the frame-indifference and permutation invariance of the potential energy,

( but of course this calculation would not make sense when N = ∞ )


TheoremAssume the restrictions on the potential energy above and let be a time-dependent discrete group of isometries satisfying the restriction on the time-dependence given above. If satisfy the equations of molecular dynamics, i.e.,

then also satisfy the equations of molecular dynamics:


Proof


Allowed time dependence of the group elements

The permitted time-dependence,

that is,

This is satisfied (in the absence of excessive assumptions on the solution)if and only if


This is an invariant manifold of the equations of molecular dynamics

October 24, 2015

p = {p1,…, pN}

(p0, q0)

q = {q1,…,qN}

t

Dayal, James, JMPS (2010)

In the case of the translation group…

The form of the manifold is independent of the material

Ericksen@90

Simplest case - translation group

simulated atoms

discrete translation group

all of the atoms

permitted time-dependence

affine in t


All of continuum mechanics inherits this (these) invariant manifold(s)

Lagrangian Eulerian

because

Ericksen@90

If thermodynamics is included, the energy equation becomes an ODE for thetemperature

The solution for the temperature is not universal.October 24, 2015


Other groups besides the translation group

B. de St. Venant, Memoire sur la torsion des prismes, Mem. Des Savants Etrangers 14 (1855), 233-560

J. L. Ericksen, Adv. Appl. Mech. (ed. C-S Yih) 17 (1977), 189-244:

Three examples






Maxwell-Boltzmann equationmolecular densityfunction

Ericksen@90


October 24, 2015

f

f log f

1/e

Solutions on the invariant manifold have their own “statistics”

Use translation group (i.e., gases fill volumes)

0

Ericksen@90

y

October 24, 2015

This yields an exact reduction of the Maxwell-Boltzmann equation

Includes many (all?) known exact solutions of the equations of the moments for special force laws

Does not include the Bobylev-Krook-Wu solution

(with Stefan Müller)

Ericksen@90

satisfies

October 24, 2015

Further simplification for inverse 5th powermolecules

Ericksen@90

Assume

Choose ξ(t) and η(t) to remove the time dependence. Get

removes the time dependence (In each case get explicit formulas for ξ(t) and η(t))

Return to the reduced Maxwell-Boltzmann equation:

October 24, 2015

Example 1


a

n

t

H(t)

H0

!

1.

Example 2

Plate moving atconstant velocity a Stationary

platea

n

Ericksen@90

!t

H(t)

H0

2.

October 24, 2015

Remarks

g(t,w)

w

Caveat: we have not (yet) given an existence theorem for the initial datumG(w), i.e., for

Consistency of the behavior of H and its definition?

Both the solutions of Boltzmann and the numerical results on pulling carbon nanotubes at constant strain rate suggest that there is a statistical mechanics for the invariant manifold. If so, it cannot be based on the invariant measure of ordinary statistical mechanics (Gibbs measure) because

Hamiltonian constant Ericksen@90

f

f log f

1/e

October 24, 2015

Three examples






X-ray diffraction


Coherent diffractive imaging requires a coherent source of hard x-rays, , i.e., one of these:


LINAC at SLAC: 2 mile longlinear accelerator,longest in the world

$ 400,000,000.

Some of the most interesting structures today are not crystals


Buckyball (Smalley, Kurl, Kroto, Nobel Prize 1996) Carbon nanotubes

Graphene (Geim and Novoselov, Nobel Prize 2010)

Amyloid protein (causes Alzheimer's, Parkinson's and

Creutzfeldt-Jakob disease)H1N1 bird flu virus

Ebola virus

Myovirusbacteriophage

Forthcoming work on the crystallizationproblem for these kinds of structures: B. Schmidt and P. Dondl

“Objective structures”

Our method Focus on structures generated by isometry groups (objective

structures) Exploit the mathematical analog

Design radiation for the inverse problem


eigenfunction of the Laplacian(i.e., solves the time-harmonic wave equation)

eigenfunction of the translation group:

Jüstel, Friesecke, James


eigenfunction of the isometry group* that generates the objective structure

Key point:isometries are in invariance groupof the Helmholtzequation. Hence,structural invariancecoincides with invariance of the PDE

Design criterion



eigenfunction of the isometry group* that generates the objective structure

Remarks The solution is vector-valued, and the required action is

The design criterion is

is a character of G. We do not know thestructure ahead of time, so, which group G?

Explicit form of the design criterion for the simplest helical group


We design in a super group G’

G’ is isomorphic to

This implies the character The explicit form of the design criterion is

Problem: find functions of this form satisfying Helmholtz, i.e., of the form

G that generates the structure is a discrete subgroup of G’

October 24, 2015 Ericksen@90Ebola virus

Ebola Virus

Konstantinov et al., Visual Science nsf.gov

Solution: twisted waves



and the magnetic field is given by

Important parameters:α, β

Twisted waves interact with helicalstructures in the same way as

plane waves do for crystals

October 24, 2015 Ericksen@90 Jüstel, Friesecke, James

Slightly off resonance: destructive interference (A. Weil’s theorem)

On resonance: constructive interference

How do you build the machine?


(…for under $ 400,000,000.)

Theoretical concept:

Example: reconstruction of Pf1


formula for far-fieldscattered radiation

unbiased: random initialphases, no structuralinformation used

reconstruction algorithm(JFJ + Elser)


invariance and the structure of...maxwell-boltzmann equation methods of structure determination for...

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