PlanetPhysics/Category of Molecular Sets 3

Molecular sets and representations of chemical reactions

A uni-molecular chemical reaction is represented by the [[../VariableCategory2/|natural transformations]] $\eta :h^A⟶h^B$, through the following [[../Commutativity/|commutative diagram]]:

Failed to parse (unknown function "\def"): {\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d]_{h^A(t)} & h^B (A) = Hom(B,A)\ar[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}} }

with the states of molecular sets $A_u=a_1,\dots ,a_n$ and $B_u=b_1,\dots b_n$ being represented by certain endomorphisms in $Hom(A,A)$ and $Hom(B,B)$, respectively. In general, molecular sets $M_S$ are defined as finite sets whose elements are molecules defined in terms of their molecular [[../QuantumSpinNetworkFunctor2/|observables]] that are specified next. One need to define first the [[../PreciseIdea/|concept]] of a molecular class variable. A molecular class variable , or $m.c.v$ is defined as a family of molecular sets $[M_S{]}_{i\in I},$ with $I$ being an indexing set, or class, defining the molecular range of variation of the $m.c.v$. Most applications in Physics, Chemistry or Biochemistry require that $I$ is a finite set, (that is, without any sub-classes). A homomorphism of molecular sets $M_t:M_S\to M_S$, with $t\in T$ being real time values, is defined as a time-dependent mapping or [[../Bijective/|function]] $M_S(t)$ also called a $M_t$ molecular transformation .

An $m.c.v.$ observable of $B$, characterizing the products of chemical [[../Bijective/|type]] "B" of a chemical reaction is defined as a [[../TrivialGroupoid/|morphism]]:

$\gamma :Hom(B,B)⟶\mathrm{\Re },$ where $\mathrm{\Re }$ is the set or [[../CosmologicalConstant/|field]] of real numbers. This mcv-observable is subject to the following [[../TrivialGroupoid/|commutativity]] conditions:

Failed to parse (unknown function "\def"): {\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{Hom(A,A) \ar[r]^{f} \ar[d]_{e} & Hom(B,B)\ar[d]^{\gamma} \\ {Hom(A,A)} \ar[r]_{\delta} & {R},} }

~

with $c:A_u^{*}⟶B_u^{*}$, and $A_u^{*}$, $B_u^{*}$ being, respectively, specially prepared fields of states of the molecular sets $A_u$, and $B_u$ within a measurement uncertainty range, $\mathrm{\Delta }$, which is determined by Heisenberg's uncertainty [[../Bijective/|relation]], or the [[../Commutator/|commutator]] of the observable [[../QuantumOperatorAlgebra4/|operators]] involved, such as $[A^{*},B^{*}]$, associated with the observable $A$ of molecular set $A_u$, and respectively, with the obssevable $B$ of molecular set $B_u$, in the case of a molecular set $A_u$ interacting with molecular set $B_u$.

With these concepts and preliminary data one can now define the [[../ChemicalTransformations/|category of molecular sets]] and their transformations as follows.

Category of molecular sets and their transformations

The category of molecular sets is defined as the [[../Cod/|category]] $C_M$ whose [[../TrivialGroupoid/|objects]] are molecular sets $M_S$ and whose morphisms are molecular transformations $M_t$.

This is a mathematical [[../CategoricalGroupRepresentation/|representation]] of chemical reaction [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] in terms of molecular sets that vary with time (or $msv$'s), and their transformations as a result of diffusion, [[../Collision/|collisions]], and chemical reactions.