Difference between revisions of "Useful math"
From apm
(→Generally useful math tools from Analysis & co: added a few links) |
(→Generally useful math tools from Analysis & co: added quite a lot of links) |
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* Creation and annihilation operators | * Creation and annihilation operators | ||
* all sorts of tricks an hackery with matrix math – selfadjungatedness & co | * all sorts of tricks an hackery with matrix math – selfadjungatedness & co | ||
| − | * distributions aka generalized functions – including | + | * distributions aka generalized functions – including [https://en.wikipedia.org/wiki/Dirac_delta_function Dirac deltas] and [https://en.wikipedia.org/wiki/Heaviside_step_function Heaviside steps] – quite a bit of math rules to memorize there |
| − | * support function (in the limit a | + | * [https://en.wikipedia.org/wiki/Support_(mathematics)#Compact_support support function] [https://de.wikipedia.org/wiki/Stetige_Funktion_mit_kompaktem_Tr%C3%A4ger (de)] (in the limit a Dirac delta) => [[unusual math]] |
| − | * Liouville's theorem (complex analysis) | + | * [https://en.wikipedia.org/wiki/Green%27s_function Green's function] |
| + | * [https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis) Liouville's theorem (complex analysis)] – incompessibility of phase space | ||
* Cauchy–Riemann equations – complex differentiability (aka holomorphic function) – Cauchy's integral theorem | * Cauchy–Riemann equations – complex differentiability (aka holomorphic function) – Cauchy's integral theorem | ||
* Fourier transformations – folds | * Fourier transformations – folds | ||
| − | * Einstein notation | + | * [https://en.wikipedia.org/wiki/Einstein_notation Einstein notation] |
* [https://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients Clebsch–Gordan coefficients] – for coupling angular momenta<br> – [https://pdg.lbl.gov/2019/reviews/rpp2019-rev-clebsch-gordan-coefs.pdf a good table] and [https://youtu.be/UPyf9ntr-B8 a good video explanation how to use it] | * [https://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients Clebsch–Gordan coefficients] – for coupling angular momenta<br> – [https://pdg.lbl.gov/2019/reviews/rpp2019-rev-clebsch-gordan-coefs.pdf a good table] and [https://youtu.be/UPyf9ntr-B8 a good video explanation how to use it] | ||
| − | * | + | * [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation Bra-ket notation] – abstracting math from positional 3D space – treating positional space and impulse equally |
---- | ---- | ||
* [https://en.wikipedia.org/wiki/Complete_set_of_commuting_observables Complete set of commuting observables] – "the measurement of one observable has no effect on the result of measuring another observable in the set" | * [https://en.wikipedia.org/wiki/Complete_set_of_commuting_observables Complete set of commuting observables] – "the measurement of one observable has no effect on the result of measuring another observable in the set" | ||
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* [https://en.wikipedia.org/wiki/Square_(algebra)#Absolute_square Absolute square] | * [https://en.wikipedia.org/wiki/Square_(algebra)#Absolute_square Absolute square] | ||
---- | ---- | ||
| − | * Finding zeros: – Newton method – Regula falsi | + | * Finding zeros: – [https://en.wikipedia.org/wiki/Newton%27s_method Newton's method] – [https://en.wikipedia.org/wiki/Regula_falsi Regula falsi] |
| − | * Integrating differential equations: – Runge Kutta methods – | + | * Integrating differential equations: – [https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods Runge Kutta methods] – [https://en.wikipedia.org/wiki/Leapfrog_integration Leapfrog integration] |
| − | * Implicit differentiation | + | * [https://en.wikipedia.org/wiki/Implicit_function#Implicit_differentiation Implicit differentiation] |
| − | * | + | * [https://en.wikipedia.org/wiki/Lagrange_multiplier Lagrange multipliers] – finding extrema under geometric side constraints |
| − | * (Reversely calculated) gradient descent in multi-dimensional scalar fields: .. | + | * [https://en.wikipedia.org/wiki/Gradient_descent Gradient descent] |
| + | * (Reversely calculated) gradient descent in multi-dimensional scalar fields: [https://en.wikipedia.org/wiki/Conjugate_gradient_method Conjugate gradient method] | ||
---- | ---- | ||
* [https://en.wikipedia.org/wiki/Arrhenius_equation Arrgenius equation] – "a formula for the temperature dependence of reaction rates" | * [https://en.wikipedia.org/wiki/Arrhenius_equation Arrgenius equation] – "a formula for the temperature dependence of reaction rates" | ||
Revision as of 15:40, 2 June 2021
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" – 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 (de) (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 (aka holomorphic function) – Cauchy's integral theorem
- Fourier transformations – folds
- 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
- 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
- Arrgenius equation – "a formula for the temperature dependence of reaction rates"
- Onsager reciprocal relations – modelling transport phenomena – statistical physics – 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
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"
