Useful math

This page is about useful math in the wide context of atomically precise manufacturing.

Specific application areas include:

• friction and dissipation
• quantum chemistry
• molecular modelling
• thermally driven self assembly
• 3d modelling
• differential geometry for larger scale gears
• ...

Thermodynamics and statistical physics

Summing up over all the possible microstate configurations of a system.
Thereby deriving a partitioning function – (some exotic math involved in there)
From this partitioning function then thermodynamic laws can be re-derived and explained.
These thermodynamic laws can be (and historically have been) formerly phemomenologically derived.
Meaning derived from their effects not their causes.

Related:

• Thermodynamic potentials and associated statistical ensembles
• Transformation between the potentials – Legendre Transformation
• Conjugated pairs of valuables (extrinsic and intrinsic) – a pairs product always gives the physical unit of energy

General note on solid state physics

Prevalent are long chains of simplifications by approximations that pile up and up and up.
Changing the application area of the models hugely may requires reevaluation of all these approximation steps.
Given that the chains of approximation are not formalized on computers (state 2021) this is difficult error prone and tedious.

Also: Following all the derivations from the lowermost assumptions
it becomes very evident that energy is a relative concept. (Not talking about relativity theory here).

Math for modelling with atomistic detail

From first principles

The exact solutions of the Schrödinger equation for the hydrogen problem.
Using the property of it being a "separable partial differential equation"

• Laguerre polynomials for the radial part
• Spherical harmonics for the angular parts

The major reason why exact solutions are way off for other elements than hydrogen
(and the less relevant highly charged one electron ions) is the shielding effect of the inner electrons.
To get good approximations for orbitals it is necessary to do iterative self-consistent-field methods.
The exact hydrogen solutions can serve as a good initial guess starting point.

Also Useful in getting good starting points:

• the Grahm Schmidt orthogonalization method
• composing Gaussian distributions as base functions for orbitals
• the Hartree-Fock method – helps filling up states consistent with pauli exclusion rules – antideterminant for fermionic states

Related: Density functional theory.

Phenomenological models

• Lennard Jones potential – and similar ones – good for molecular dynamics simulations
• Hund's rule of maximum multiplicity – not particularly useful in the context of chemically bond atoms

Misc

Derivation of London dispersion forces from first principles by
integrating over virtual electron states (related: virtual particles, feynman graphs) ...
Related: Born–Oppenheimer approximation – and its deceiving pseudo convergence (to check)

Generally useful math tools from Analysis & co

• eigenvectors (linear algebra)
• vector spaces with functions as base vectors (aka Hilbert spaces)
• "integral kernels" – just a fancy word for projections in vector spaces with functions as base vectors – "overlap integrals"
• (The crazy math symbol of an integral with a sum drawn over for quantum systems that contain both continuous band and discrete energy states)
• commutators and anti-commutators
• Creation and annihilation operators
• all sorts of tricks an hackery with matrix math – selfadjungatedness & co
• distributions aka generalized functions – including dirac deltas and Heaviside steps – quite a bit of math rules to memorize there
• support function (in the limit a dirac delta) => unusual math
• Liouville's theorem (complex analysis)
• Cauchy–Riemann equations – complex differentiability (aka holomorphic function) – Cauchy's integral theorem
• Fourier transformations – folds
• Einstein notation
• Complete set of commuting observables
• Clebsch–Gordan coefficients
• bra-ket formalism – abstracting math from positional 3D space – treating positional space and impulse equally

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• Lagrangian and Hamiltonian mechanics – principle of least action – variational calculus
• Nöther theorem – linking conserved quantities to invariance under transformations (aka symmetries) – related: generating functions => unusual math
• Liouville's theorem (Hamiltonian)
• (de) Betragsquadrat ~ (en) Absolute square

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• Finding zeros: – Newton method – Regula falsi
• Integrating differential equations: – Runge Kutta methods – Leap frog methods
• Implicit differentiation
• Finding extrema with side conditions: "Lagrange multipliers"
• (Reversely calculated) gradient descent in multi-dimensional scalar fields: ...

Useful algorithms in computer graphics

• GJK algorithm (collision detection)
• ...

Notes

• Not to confuse "Holomorphic function" and "Holonomic constraints"