Useful math
This page is about useful math in the wide context of atomically precise manufacturing.
Specific application areas include:
- friction and dissipation
- thermally driven self assembly
- quantum chemistry
- molecular modelling
- 3d modelling
- differential geometry for larger scale gears
- ...
Contents
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 – e.g. for quantum chemistry
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 – e.g. for molecular modelling
- 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" – Integral transform
- Fourier transformations – easy to do folds they become multiplications
- (Laplace transformations)
- "overlap integrals" – e.g. Orbital overlap – projections in vector spaces with functions as base vectors
- Approximations: Slater type orbital and Gaussian_orbital
- (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 – Commutator ~> Ring theory
- Creation and annihilation operators – (Coherent state)
- all sorts of tricks an hackery with matrix math – selfadjungatedness & co
- Distributions (one class of generalized functions) – including Dirac deltas and Heaviside steps – quite a bit of math rules to memorize there
- support function (de)
- Support functions => Test functions => Bump_function – (in the limit a Dirac delta) ~ unusual math
- Green's function
- Liouville's theorem (complex analysis) – incompessibility of phase space
- Cauchy–Riemann equations – complex differentiability; holomorphic; analytic; ...
- Cauchy's integral theorem
- Einstein notation
- Clebsch–Gordan coefficients – for coupling angular momenta
– a good table and a good video explanation how to use it - Bra-ket notation – abstracting math from positional 3D space – treating positional space and impulse equally
- Density matrix
- Complete set of commuting observables – "the measurement of one observable has no effect on the result of measuring another observable in the set"
- Nöther's theorem – linking conserved quantities to invariance under transformations (aka symmetries) – related: generating functions => unusual math
- Lagrangian and Hamiltonian mechanics – principle of least action – variational principle (and calculus)
- Liouville's theorem (Hamiltonian) – Canonical transformations
- Canonical coordinates – (in Hamiltonian mechanics)
- Generalized coordinates – (in Lagrangian mechanics)
- Absolute square
- Finding zeros: – Newton's method – Regula falsi
- Integrating differential equations: – Runge Kutta methods – Leapfrog integration
- Implicit differentiation
- Lagrange multipliers – finding extrema under geometric side constraints
- Gradient descent
- (Reversely calculated) gradient descent in multi-dimensional scalar fields: Conjugate gradient method
- Equipartition theorem
- Thermodynamic beta (kBT) in the Boltzmann factor in the Boltzmann distribution
- For fermions like electrons: Fermi–Dirac statistics
- For bosons like phonons (and photons): Bose–Einstein statistics
- Maxwell–Boltzmann statistics & Maxwell–Boltzmann distribution
- Lennard-Jones potential
- Arrhenius equation – "a formula for the temperature dependence of reaction rates"
- Onsager reciprocal relations – modelling transport phenomena – statistical physics
- Fluctuation-dissipation theorem – links drag to Brownian motion – friction
– The paper "Evaluating the Friction of Rotary Joints in Molecular Machines (paper)" uses a simplified result from this. - Langevin equation – for modelling brownian motion – statistical physics
– Einstein relation (kinetic theory) – diffusion coefficient from microscopic mobility
Thermodynamics
Statistical ensemble (mathematical physics) (overcounting) (list of ensembles):
- Canonical ensemble – NVE – heat bath
- [1] – NVT – isolated
Most fundamental concepts
- causation vs correlation
- necessity vs sufficiency (if and only if aka iff)
- convergence ...
Useful algorithms in computer graphics
- GJK algorithm (collision detection)
- ...
Notes
- Not to confuse "Holomorphic function" and "Holonomic constraints"
