Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for...
Transcript of Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for...
![Page 1: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/1.jpg)
Thanks: Kaushik Dayal, Stefan Müller, Traian Dumitrica
Invarianceand the structure of
matterRichard JamesUniversity of [email protected]
October 24, 2015 Ericksen@90
![Page 2: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/2.jpg)
October 24, 2015 Ericksen@90
![Page 3: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/3.jpg)
October 24, 2015 Ericksen@90
![Page 4: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/4.jpg)
Three examples
October 24, 2015 Ericksen@90
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
![Page 5: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/5.jpg)
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
![Page 6: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/6.jpg)
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
![Page 7: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/7.jpg)
Three examples
October 24, 2015 Ericksen@90
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
![Page 8: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/8.jpg)
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
Ericksen@90October 24, 2015
![Page 9: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/9.jpg)
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)
Ericksen@90October 24, 2015
![Page 10: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/10.jpg)
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 = ∞ )
Ericksen@90October 24, 2015
![Page 11: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/11.jpg)
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:
Ericksen@90October 24, 2015
![Page 12: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/12.jpg)
Proof
Ericksen@90October 24, 2015
![Page 13: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/13.jpg)
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
Ericksen@90October 24, 2015
![Page 14: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/14.jpg)
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
![Page 15: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/15.jpg)
Simplest case - translation group
simulated atoms
discrete translation group
all of the atoms
permitted time-dependence
affine in t
Ericksen@90October 24, 2015
![Page 16: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/16.jpg)
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
![Page 17: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/17.jpg)
October 24, 2015 Ericksen@90
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:
![Page 18: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/18.jpg)
Three examples
October 24, 2015 Ericksen@90
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
![Page 19: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/19.jpg)
Maxwell-Boltzmann equationmolecular densityfunction
Ericksen@90
Maxwell-Boltzmann equation
October 24, 2015
f
f log f
1/e
![Page 20: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/20.jpg)
Solutions on the invariant manifold have their own “statistics”
Use translation group (i.e., gases fill volumes)
0
Ericksen@90
y
October 24, 2015
![Page 21: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/21.jpg)
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
![Page 22: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/22.jpg)
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
![Page 23: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/23.jpg)
Example 1
October 24, 2015 Ericksen@90
a
n
t
H(t)
H0
!
1.
![Page 24: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/24.jpg)
Example 2
Plate moving atconstant velocity a Stationary
platea
n
Ericksen@90
!t
H(t)
H0
2.
October 24, 2015
![Page 25: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/25.jpg)
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
![Page 26: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/26.jpg)
Three examples
October 24, 2015 Ericksen@90
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
![Page 27: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/27.jpg)
X-ray diffraction
October 24, 2015 Ericksen@90
![Page 28: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/28.jpg)
Coherent diffractive imaging requires a coherent source of hard x-rays, , i.e., one of these:
October 24, 2015 Ericksen@90
LINAC at SLAC: 2 mile longlinear accelerator,longest in the world
$ 400,000,000.
![Page 29: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/29.jpg)
Some of the most interesting structures today are not crystals
October 24, 2015 Ericksen@90
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”
![Page 30: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/30.jpg)
Our method Focus on structures generated by isometry groups (objective
structures) Exploit the mathematical analog
Design radiation for the inverse problem
October 24, 2015 Ericksen@90
eigenfunction of the Laplacian(i.e., solves the time-harmonic wave equation)
eigenfunction of the translation group:
Jüstel, Friesecke, James
eigenfunction of the Laplacian(i.e., solves the time-harmonic wave equation)
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
![Page 31: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/31.jpg)
Design criterion
October 24, 2015 Ericksen@90
eigenfunction of the Laplacian(i.e., solves the time-harmonic wave equation)
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?
![Page 32: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/32.jpg)
Explicit form of the design criterion for the simplest helical group
October 24, 2015 Ericksen@90
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’
![Page 33: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/33.jpg)
October 24, 2015 Ericksen@90Ebola virus
Ebola Virus
Konstantinov et al., Visual Science nsf.gov
![Page 34: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/34.jpg)
Solution: twisted waves
October 24, 2015 Ericksen@90
Jüstel, Friesecke, James
and the magnetic field is given by
Important parameters:α, β
![Page 35: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/35.jpg)
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
![Page 36: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/36.jpg)
How do you build the machine?
October 24, 2015 Ericksen@90
(…for under $ 400,000,000.)
Theoretical concept:
![Page 37: Invariance and the structure of...Maxwell-Boltzmann equation Methods of structure determination for non- crystalline structures (Maxwell’s equations) …of invariance under discrete](https://reader035.fdocuments.in/reader035/viewer/2022071106/5fe0b23d9a5947318c1e83d2/html5/thumbnails/37.jpg)
Example: reconstruction of Pf1
October 24, 2015 Ericksen@90
formula for far-fieldscattered radiation
unbiased: random initialphases, no structuralinformation used
reconstruction algorithm(JFJ + Elser)
Jüstel, Friesecke, James