Difference between revisions of "Useful math"

Latest revision as of 12:27, 19 September 2023

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
• ...

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

Hamiltonian mechanics finds heavy use in in quantummechaincs.
Interestingly in gem-gum systems at slightly larger scales things behave very classically.
Lagrangian mechanics might be useful there.

• Hamiltonian mechanics
• Lagrangian mechanics

Related: principle of least action and variational principle (and calculus)

Basic math for physics

• Finding zeros: – Newton's method – Regula falsi
• Integrating differential equations: – Runge Kutta methods – Leapfrog integration

• Eigenvalues and eigenvectors – (linear algebra)
• Vector spaces with functions as base vectors – (Hilbert spaces)
• "Integral kernels" – Integral transform
• Fourier transformations – (convolution becomes multiplication)
• (Laplace transformations – more used in electrical system engineering)
• Convolution (aka folding)
• Linear functional – Adjoint_functors
• https://en.wikipedia.org/wiki/Self-adjoint_operator Self-adjoint operator]
• Orbital overlap / Overlap integral

• All sorts of tricks an hackery with matrix math – selfadjunctness & co – (Category:Matrix_theory)

• List of equations in quantum mechanics
• (Heisenberg picture and Matrix mechanics)

Category:Perturbation_theory – particularly:

• Perturbation_theory_(quantum_mechanics) which employs the
• Gram–Schmidt_process in Hilbert space (? IIRC)

Category:Condensed_matter_physics

Useful for 3D modelling and robotics

• screws (math object), wrenches (math objects), dual numbers – useful for robotics (forwards & backwards kinematic) and differential geometry for gear flanks
• math tools for surface based commuter graphics (manly dealing with triangulations)
• math tools for volume based computer graphics:
• functional-representation (F-Rep), implicit surfaces, algebraic varieties, distance fields (magnitude of gradient is 1 for these)
• ray-marching algorithms

Some multi-purpouse misc math

• Gradient descent
• (Reversely calculated) gradient descent in multi-dimensional scalar fields: Conjugate gradient method
• Lagrange multipliers – finding extrema under geometric side constraints
• Implicit differentiation

Generalized functions (1D):

• Dirac delta function … derivative of Heaviside function
• Heaviside step function … antiderivative of Dirac delta function

Basic higher dimensional higher order derivatives

• Gradient … First order derivative of a scalar field
• Directional derivative … gradient projected into direction vector at specific point
• Jacobian matrix and determinant
• Hessian matrix

For a taylor series of a scalar field:

• First derivative: Gadient
• Second derivative: Hessian Matrix
• Third derivative: unnamed three index tensor

For a taylor series of a vector field:

• First derivative of a scalar field: Jacobian matrix
• Second derivative: unnamed three index tensor

For a taylor series of a tensor fields of arbitrary rank (all derivatives):

• See: Beautiful differentiation (Conal Elliott)

Other uses of the Jacobian matrix:

• For a transformation of a coordinate system into an other coordinate system.
  For locally linear transformations – unless generalized derivatives are allowed (like multidimensional analogs to Dirac-deltas)
• For propagation of uncertainty in scientific measurements near sigma limit

Useful for analysis of selfassembly and dissipation

• 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

Important for non-qunatum mechanical molecular dynamics simulations

• Molecular dynamics
• Lennard-Jones potential

Tools to set up the right initial distribution of particle motions:

• Equipartition theorem
• Thermodynamic beta (k_BT) 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

Thermodynamics

Statistical ensemble (mathematical physics) (overcounting) (list of ensembles):

• Canonical ensemble – NVE – heat bath
• Microcanonical ensemble – NVT – isolated

Changes in the rate of a chemical reaction against temperature. (chemical kinetics)

• Eyring equation – from first principles – (Transition state theory)
• Arrhenius equation – empirical / phenomenological

For more precise quantum mechanical calculations

• Absolute square – to the the density from the wave function
• Bra-ket notation – abstracting math from positional 3D space – treating positional space and impulse equally

• Schrödinger equation – and exact exact solutions – and iterative methods
• (Helium atom as the simplemost three body case and first case where there is electron shielding)
• Approximations: Slater type orbital and Gaussian_orbital
• "overlap integrals" – e.g. Orbital overlap – projections in vector spaces with functions as base vectors
• (The crazy math symbol of an integral with a sum drawn over for quantum systems that contain both continuous band and discrete energy states)
• Gram–Schmidt process – for getting a reasonable orthonormal basis as a starting point

• Hartree–Fock method
• Density functional theory

• Complete set of commuting observables – "the measurement of one observable has no effect on the result of measuring another observable in the set"
• commutators and anti-commutators – Commutator ~> Ring theory

• Clebsch–Gordan coefficients – for coupling angular momenta
  – a good table and a good video explanation how to use it

• Density matrix

Maybe more relevant for high energy free particle physics

• Green's function – needed for scattering problems
• Liouville's theorem (Hamiltonian) – on incompessibility of phase space
  – Liouville's theorem puts limits on focusing particle beams after they left solid state
  – To cheat and reduce the phase space of a free floating particle beam at least some indirect interaction with solid state matter (for removal of excess phase space) is necessary (Capillary guiding)
• Canonical transformations
• Canonical coordinates – (in Hamiltonian mechanics)
• Generalized coordinates – (in Lagrangian mechanics)

• 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

• Liouville's theorem (complex analysis)
• Cauchy–Riemann equations – complex differentiability; holomorphic; analytic; ...
• Cauchy's integral theorem

• operator Ladder_operator
• Creation and annihilation operators – (Coherent state)

• Einstein notation

• Nöther's theorem – linking conserved quantities to invariance under transformations (aka symmetries) – related: generating functions => unusual math

Most fundamental concepts

• causation vs correlation
• necessity vs sufficiency (if and only if aka iff)
• convergence ... (one or two closely associated topics are missing here ... which ones? ...)

Useful algorithms in computer graphics

• GJK algorithm (collision detection)
• Raymarching and more general "Walk on Spheres"
  (also useful.for physical simulations)
• Octree and more advanced subdivision algorithms
• Marching cubes (for ugly triangulations – somehow this is everyone's favorite though)

Useful math for larger scale gear-train design

Differential geometry for generation of conjugate profiles in generalized gear-sets
(only solved for general axial alignments cycloid gear profiles as of yet 2021)
Associated math includes:

• Plücker coordinates
• screw theory – screw axis
• dual numbers

Potentially extremely useful computer science

Automatic differentiation ...

• generalized to arbitrary dimensionality (Jaconian matrix is first deerivative of vector field, Hessian matrix is second derivative of a scalar field, following are higher tensors)
• generalized to arbitrary degree (Basically a taylor series)

See: "Beautiful differentiation" by Conal Elliott (March 2009) Appeared in ICFP 2009 (link)

Generalized code interpretation for vastly more resuability ...

• generalizing lambda calculus to a category theoretic interpretation that allows for reuse of exactly the same code in vastly different compilation targets

See: "Compiling to categories" by Conal Elliott (February 2017) Appeared at ICFP 2017 (link)

The revolution to "content addressed" systems.
Maybe the most powerful weapon against dependency hell and all its various workaround hacks.

Notes / Misc

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

• Wikipedia [Functional derivative]

Related

• Atomic orbitals
• Hartree-Fock method

• Pages with math

• Exotic math