View source for Useful math


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


Specific application areas include:
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* friction and dissipation
* thermally driven self assembly
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* quantum chemistry
* molecular modelling
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* 3d modelling
* differential geometry for larger scale gears
* ...

== Thermodynamics and statistical physics ==

Summing up over all the possible microstate configurations of a system. <br>
Thereby deriving a partitioning function – (some [[exotic math]] involved in there) <br>
From this partitioning function then [[thermodynamic laws]] can be re-derived and explained. <br>
These [[thermodynamic laws]] can be (and historically have been) formerly phemomenologically derived. <br>
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. <br>
Changing the application area of the models hugely may requires reevaluation of all these approximation steps. <bR>
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 <br>
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. <br>
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 <br>
(and the less relevant highly charged one electron ions) is '''the shielding effect of the inner electrons'''. <br>
To get good approximations for orbitals it is necessary to do iterative self-consistent-field methods. <br>
The exact hydrogen solutions can serve as a good initial guess starting point. <br>

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 <br>
integrating over virtual electron states (related: virtual particles, feynman graphs) ... <br>
Related: Born–Oppenheimer approximation – and its deceiving pseudo convergence (to check)

== Generally useful math tools ==

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

* [https://en.wikipedia.org/wiki/Hamiltonian_mechanics Hamiltonian mechanics]
* [https://en.wikipedia.org/wiki/Lagrangian_mechanics Lagrangian mechanics]
Related: [https://en.wikipedia.org/wiki/Stationary_Action_Principle principle of least action] and [https://en.wikipedia.org/wiki/Variational_principle variational principle (and calculus)]

=== Basic math for physics ===

* 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: – [https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods Runge Kutta methods] – [https://en.wikipedia.org/wiki/Leapfrog_integration Leapfrog integration]
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* [https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors Eigenvalues and eigenvectors] – (linear algebra)
* Vector spaces with functions as base vectors – ([https://en.wikipedia.org/wiki/Hilbert_space Hilbert spaces]) 
* "Integral kernels" – [https://en.wikipedia.org/wiki/Integral_transform Integral transform]
* [https://en.wikipedia.org/wiki/Fourier_transform Fourier transformations] – (convolution becomes multiplication)
* ([https://en.wikipedia.org/wiki/Laplace_transform Laplace transformations] – more used in electrical system engineering) 
* [https://en.wikipedia.org/wiki/Convolution Convolution (aka folding)]
* [https://en.wikipedia.org/wiki/Linear_form Linear functional] – [https://en.wikipedia.org/wiki/Adjoint_functors Adjoint_functors]
* https://en.wikipedia.org/wiki/Self-adjoint_operator Self-adjoint operator]
* [https://en.wikipedia.org/wiki/Orbital_overlap Orbital overlap / Overlap integral]
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* All sorts of tricks an hackery with matrix math – selfadjunctness & co – [https://en.wikipedia.org/wiki/Category:Matrix_theory (Category:Matrix_theory)]
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* '''[https://en.wikipedia.org/wiki/List_of_equations_in_quantum_mechanics List of equations in quantum mechanics]'''
* <small>([https://en.wikipedia.org/wiki/Heisenberg_picture Heisenberg picture] and [https://en.wikipedia.org/wiki/Matrix_mechanics Matrix mechanics])</small>
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[https://en.wikipedia.org/wiki/Category:Perturbation_theory Category:Perturbation_theory] – particularly:
* [https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics) Perturbation_theory_(quantum_mechanics)] which employs the
* [https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process Gram–Schmidt_process] in Hilbert space (? IIRC)

=== 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 ===

* [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/Lagrange_multiplier Lagrange multipliers] – finding extrema under geometric side constraints
* [https://en.wikipedia.org/wiki/Implicit_function#Implicit_differentiation Implicit differentiation]

=== Useful for analysis of selfassembly and dissipation ===

* [https://en.wikipedia.org/wiki/Arrhenius_equation Arrhenius equation] – "a formula for the temperature dependence of reaction rates"
* [https://en.wikipedia.org/wiki/Onsager_reciprocal_relations Onsager reciprocal relations] – modelling transport phenomena – [[statistical physics]]
* '''[https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem Fluctuation-dissipation theorem]''' – links drag to Brownian motion – [[friction]] <br>– The paper "[[Evaluating the Friction of Rotary Joints in Molecular Machines (paper)]]" uses a simplified result from this.
* [https://en.wikipedia.org/wiki/Langevin_equation Langevin equation] – for modelling brownian motion – [[statistical physics]] <br>– [https://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory) Einstein relation (kinetic theory)] – diffusion coefficient from microscopic mobility

=== Important for non-qunatum mechanical molecular dynamics simulations ===

* [https://en.wikipedia.org/wiki/Molecular_dynamics Molecular dynamics]
* [https://en.wikipedia.org/wiki/Lennard-Jones_potential Lennard-Jones potential]
Tools to set up the right initial distribution of particle motions:
* '''[https://en.wikipedia.org/wiki/Equipartition_theorem Equipartition theorem]'''
* [https://en.wikipedia.org/wiki/Thermodynamic_beta Thermodynamic beta (k<sub>B</sub>T)] in the Boltzmann factor in the [https://en.wikipedia.org/wiki/Boltzmann_distribution Boltzmann distribution]
* For fermions like electrons: [https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics Fermi–Dirac statistics]
* For bosons like phonons (and photons): [https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics Bose–Einstein statistics]
* [https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_statistics Maxwell–Boltzmann statistics] & [https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution Maxwell–Boltzmann distribution]

=== Thermodynamics ===

[https://en.wikipedia.org/wiki/Statistical_ensemble_(mathematical_physics) Statistical ensemble (mathematical physics)]
[https://en.wikipedia.org/wiki/Statistical_ensemble_(mathematical_physics)#Correcting_overcounting_in_phase_space (overcounting)]
[https://en.wikipedia.org/wiki/Category:Statistical_ensembles (list of ensembles)]:
* [https://en.wikipedia.org/wiki/Canonical_ensemble Canonical ensemble] – NVE – heat bath
* [https://en.wikipedia.org/wiki/Microcanonical_ensemble Microcanonical ensemble] – NVT – isolated
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Changes in the rate of a chemical reaction against temperature. (chemical kinetics)
* [https://en.wikipedia.org/wiki/Eyring_equation Eyring equation] – from first principles – ([https://en.wikipedia.org/wiki/Transition_state_theory Transition state theory])
* [https://en.wikipedia.org/wiki/Arrhenius_equation Arrhenius equation] – empirical / phenomenological

=== For more precise quantum mechanical calculations ===

* [https://en.wikipedia.org/wiki/Square_(algebra)#Absolute_square Absolute square] – to the the density from the wave function
* [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
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* [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation] – and exact exact solutions – and iterative methods
* ([https://en.wikipedia.org/wiki/Helium_atom Helium atom] as the simplemost three body case and first case where there is electron shielding)
* Approximations: [https://en.wikipedia.org/wiki/Slater-type_orbital Slater type orbital] and [https://en.wikipedia.org/wiki/Gaussian_orbital Gaussian_orbital]
* "overlap integrals" – e.g. [https://en.wikipedia.org/wiki/Orbital_overlap Orbital overlap] – projections in vector spaces with functions as base vectors
* <small>(The crazy math symbol of an integral with a sum drawn over for quantum systems that contain both continuous band and discrete energy states)</small> 
* [https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process Gram–Schmidt process] – for getting a reasonable orthonormal basis as a starting point
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* '''[https://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method Hartree–Fock method]'''
* '''[https://en.wikipedia.org/wiki/Density_functional_theory Density functional theory]'''
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* [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"
* commutators and anti-commutators – [https://en.wikipedia.org/wiki/Commutator#Ring_theory Commutator ~> Ring theory]
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* [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]
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* [https://en.wikipedia.org/wiki/Density_matrix Density matrix]

=== Maybe more relevant for high energy free particle physics ===

* [https://en.wikipedia.org/wiki/Green%27s_function Green's function] – needed for scattering problems
* [https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian) Liouville's theorem (Hamiltonian)] – on incompessibility of phase space <br>– Liouville's theorem puts limits on focusing particle beams after they left solid state <br>– 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 [http://www.iap.tuwien.ac.at/www/atomic/surface/capillaries (Capillary guiding)]
* Canonical transformations
* [https://en.wikipedia.org/wiki/Canonical_coordinates Canonical coordinates] – (in [https://en.wikipedia.org/wiki/Hamiltonian_mechanics Hamiltonian mechanics])
* [https://en.wikipedia.org/wiki/Generalized_coordinates Generalized coordinates] – (in [https://en.wikipedia.org/wiki/Lagrangian_mechanics Lagrangian mechanics])
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* [https://en.wikipedia.org/wiki/Distribution_(mathematics) Distributions] (one class of [https://en.wikipedia.org/wiki/Generalized_function 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
* [https://en.wikipedia.org/wiki/Support_(mathematics)#Compact_support support function] [https://de.wikipedia.org/wiki/Stetige_Funktion_mit_kompaktem_Tr%C3%A4ger (de)]
* Support functions => Test functions => [https://en.wikipedia.org/wiki/Bump_function Bump_function] – (in the limit a Dirac delta) ~ [[unusual math]] 
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* [https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis) Liouville's theorem (complex analysis)]
* [https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations Cauchy–Riemann equations] – complex differentiability; holomorphic; analytic; ... 
* Cauchy's integral theorem
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* [https://en.wikipedia.org/wiki/Ladder operator Ladder_operator]
* [https://en.wikipedia.org/wiki/Creation_and_annihilation_operators Creation and annihilation operators] – ([https://en.wikipedia.org/wiki/Coherent_state Coherent state])
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* [https://en.wikipedia.org/wiki/Einstein_notation Einstein notation]
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* '''[https://en.wikipedia.org/wiki/Noether%27s_theorem 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 ...

== Useful algorithms in computer graphics ==

* GJK algorithm (collision detection)
* Raymarching
* Octree and more advanced subdivision algorithms
* Marching cubes (for ugly triangulations – somehow this is everyone's favorite though)

== 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 [http://conal.net/papers/beautiful-differentiation/ (link)]

Compiling to categories
* 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 [http://conal.net/papers/compiling-to-categories/ (link)]

== Notes ==

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

== Related ==

* [[Atomic orbitals]]
* [[Hartree-Fock method]]