Occupancy and Occurrence
Possible structure, realized count, and realized state
Multiplicity tells us what can be realized.
It does not tell us what is realized.
R2D therefore separates three different kinds of count:
\[ \boxed{ W_{n,i} = \text{possible count} } \]
\[ \boxed{ \nu_{n,i} = \text{realized occurrence count} } \]
\[ \boxed{ c_{n,i} = \text{realized occupancy} } \]
These quantities are related.
They are not interchangeable.
This distinction is essential because scientific descriptions often compress possible structure, realized events, and realized state into one statistical language.
R2D keeps them separate.
Multiplicity Comes First
At boundary \(B_n\), macrostate \(i\) has multiplicity
\[ W_{n,i} = \left| \pi_n^{-1}(i) \right|. \]
This is the number of compatible formal microstates classified as \(i\).
It defines possible structure.
The corresponding logarithmic coordinate is
\[ S_{n,i} = \ln W_{n,i}. \]
Nothing in this definition requires any particular microstate actually to be realized.
Multiplicity therefore answers:
How many compatible realizations are possible?
Occurrence and occupancy answer different questions.
Occurrence Is Realized Count
A formal microstate
\[ \mu_n \in \mathfrak{M}_n \]
is mathematically possible.
When one such microstate is actually realized within a specified finite count domain, it becomes a microstate occurrence.
Write the \(r\)-th realized microstate as
\[ \mu_n^{(r)} \in \mathfrak{M}_n. \]
If
\[ \pi_n \left( \mu_n^{(r)} \right) = i, \]
then that realization contributes one occurrence associated with macrostate \(i\).
Let
\[ \nu_{n,i} \]
be the number of realized microstate occurrences classified as macrostate (i) within the specified count domain.
Thus:
\[ \boxed{ \nu_{n,i} = \text{number of realized occurrences associated with } i } \]
Occurrence answers:
How many realizations occurred?
Occurrence Does Not Yet Mean Time
This point is important.
The finite domain over which
\[ \nu_{n,i} \]
is counted does not have to be a physical time interval.
It is simply a specified domain of realized count.
For positive occurrence count, R2D defines
\[ \tau_{n,i} = \ln \nu_{n,i}. \]
The symbol
\[ \tau_{n,i} \]
is therefore a logarithmic occurrence coordinate.
At this stage it is not physical time.
Likewise,
\[ \Delta \tau \]
is not automatically a rate.
Physical time, frequency, and recurrence are later physical readings of different relations.
Occupancy Is Realized Structure
Occurrence counts events.
Occupancy describes the realized state structure produced through those events.
Let
\[ c_{n,i} \]
denote the realized occupancy of macrostate \(i\).
For positive occupancy,
\[ G_{n,i} = \ln c_{n,i}. \]
The occupancy landscape is the collection
\[ \mathcal{G}_n = \{ G_{n,i} \}_{i\in\mathcal{M}_n}. \]
Occupancy answers:
How is realized structure distributed across the available macrostates?
This is different from asking how many realizations occurred.
A Simple Two-State Example
Suppose a system has two readable macrostates:
\[ 1 \qquad\text{and}\qquad 2. \]
Their possible multiplicities might be
\[ W_{n,1}=100, \qquad W_{n,2}=100. \]
That tells us only that the two macrostate classes contain equal numbers of compatible formal microstates.
Now suppose a finite realization domain contains
\[ \nu_{n,1}=1000 \]
occurrences associated with state 1 and
\[ \nu_{n,2}=1000 \]
occurrences associated with state 2.
That describes realized occurrence count.
But the realized occupancy at one particular configuration might still be
\[ c_{n,1}=80, \qquad c_{n,2}=20. \]
These are three different descriptions:
- multiplicity describes the available state structure;
- occurrence describes realized events;
- occupancy describes the realized state distribution.
They must not be collapsed into one quantity.
Occupancy Is Not Multiplicity
In general,
\[ \boxed{ c_{n,i} \neq W_{n,i} } \]
Multiplicity tells us how many compatible realizations belong to macrostate \(i\).
Occupancy tells us how much realized structure is currently expressed at \(i\).
So it is misleading to say that occupancy is simply “realized multiplicity.”
A more precise statement is:
Occupancy is realized structure within a multiplicity-defined landscape.
The multiplicity landscape defines the possible structure.
The occupancy landscape tells us what structure is presently expressed.
Occurrence Is Not Occupancy
Likewise,
\[ \boxed{ \nu_{n,i} \neq c_{n,i} } \]
in general.
Many occurrences can contribute to one occupancy structure.
And repeated occurrences can occur without changing the structural macrostate.
For example, a molecular ensemble may undergo many subordinate transitions while remaining in the same larger-scale occupancy mode.
Thus:
Occurrence counts realizations.
Occupancy records realized structure.
The Three Landscapes
R2D therefore distinguishes three mathematical landscapes over the same macrostate domain.
Multiplicity landscape
\[ \mathcal{W}_n = \{ W_{n,i} \}_{i\in\mathcal{M}_n} \]
Possible structure.
Occurrence landscape
\[ \mathcal{T}_n = \{ \tau_{n,i} \} \]
where
\[ \tau_{n,i} = \ln \nu_{n,i}. \]
Realized occurrence structure.
Occupancy landscape
\[ \mathcal{G}_n = \{ G_{n,i} \} \]
where
\[ G_{n,i} = \ln c_{n,i}. \]
Realized state structure.
The three landscapes answer three different questions:
\[ \boxed{ W_{n,i}: \text{How many compatible realizations are possible?} } \]
\[ \boxed{ \nu_{n,i}: \text{How many realizations occur?} } \]
\[ \boxed{ c_{n,i}: \text{What realized structure is expressed?} } \]
Logarithms Do Not Change the Count Type
Each count can be represented logarithmically:
\[ S_{n,i} = \ln W_{n,i}, \]
\[ \tau_{n,i} = \ln \nu_{n,i}, \]
\[ G_{n,i} = \ln c_{n,i}. \]
But taking a logarithm does not turn these quantities into the same thing.
They remain:
\[ S = \text{log possible count}, \]
\[ \tau = \text{log occurrence count}, \]
\[ G = \text{log occupancy}. \]
Their mathematical similarity should not erase their conceptual distinction.
Macrostate Differences
For a readable transition
\[ i \rightarrow k, \]
the multiplicity difference is
\[ \Delta S_{n,i\rightarrow k} = \ln \left( \frac{W_{n,k}}{W_{n,i}} \right). \]
The occurrence-count difference is
\[ \Delta \tau_{n,i\rightarrow k} = \ln \left( \frac{\nu_{n,k}}{\nu_{n,i}} \right). \]
The occupancy difference is
\[ \Delta G_{n,i\rightarrow k} = \ln \left( \frac{c_{n,k}}{c_{n,i}} \right). \]
These have the same logarithmic form.
But they compare different count structures.
That distinction becomes important when R2D later asks how possible structure becomes realized occurrence and how occurrence becomes expressed as occupancy.
Multiplicity Structures Occurrence
R2D proposes that, within a fixed realization relation and a sufficiently resolved realization domain, relative occurrence count proportionally realizes relative multiplicity.
For two macrostates \(i\) and \(k\),
\[ \frac{\nu_{n,k}}{\nu_{n,i}} \approx \frac{W_{n,k}}{W_{n,i}}. \]
Equivalently,
\[ \boxed{ \Delta \tau_{n,i\rightarrow k} \approx \Delta S_{n,i\rightarrow k} } \]
This is the proportional occurrence-realization relation.
It does not state
\[ \nu_{n,i} = W_{n,i}. \]
Multiplicity and occurrence remain different count types.
The relation concerns their relative structures:
relative possible count is realized as relative occurrence count.
This Still Does Not Define Direction
Even if occurrence proportionally realizes multiplicity, this does not yet tell us how a particular realized system becomes directionally biased.
The native multiplicity landscape may remain fixed while enclosing conditions change.
Those enclosing conditions can alter the realized occupancy.
This is why R2D separates two relations:
\[ \text{multiplicity} \rightarrow \text{occurrence} \]
and
\[ \text{enclosing asymmetry} \rightarrow \text{occupancy bias}. \]
Direction requires more than multiplicity alone.
It requires the across-boundary relation developed through possibility and retention.
Asymmetry Acts on Occupancy
For a readable transition,
\[ i \rightarrow k, \]
R2D defines local asymmetry through
\[ \Delta A_{n,i\rightarrow k} = \Delta P_{n,i\rightarrow k} - R_{n,i\rightarrow k}. \]
The occupancy relation is then expressed schematically as
\[ \boxed{ \Delta G = \Delta \tau + \Delta A } \]
The terms play different roles.
The occurrence relation reflects realized multiplicity structure.
The asymmetry term changes how that realized occurrence is expressed as occupancy.
Thus:
Multiplicity structures occurrence.
Asymmetry biases occupancy.
Structural Macrostate
Because occupancy is a realized distribution, it can possess a mode.
If the occupancy landscape has a unique maximum, define the structural macrostate by
\[ i_n^* = \operatorname*{arg\,max}_{i\in\mathcal{M}_n} c_{n,i}. \]
Equivalently, because the logarithm is monotonic,
\[ i_n^* = \operatorname*{arg\,max}_{i\in\mathcal{M}_n} G_{n,i}. \]
A change
\[ i_n^* \rightarrow k_n^* \]
is a structural macrostate transition.
This does not identify one unique subordinate trajectory.
Many different microstate occurrences can realize the same change in occupancy mode.
That is precisely why R2D separates the structural state from the subordinate mechanism that realizes it.
Muscle Makes the Distinction Concrete
Muscle provides a useful physical example.
Individual myosin molecules undergo many state transitions.
Those transitions are occurrences.
At the ensemble level, however, experiments can read the distribution of molecular states.
That distribution is occupancy.
The available combinatorial state structure of the switch ensemble is the multiplicity landscape.
So:
\[ \text{switch multiplicity} \neq \text{switch transitions} \neq \text{ensemble occupancy}. \]
All three are physically related.
They are not the same mathematical object.
This separation was one of the empirical routes by which R2D emerged.
Why the Distinction Matters for Causality
If occurrence, occupancy, and multiplicity are conflated, a subordinate event can appear to create the larger state structure merely because the event is necessary for its realization.
R2D instead gives them different causal roles.
Multiplicity defines possible structure.
Occurrence realizes possibilities.
Occupancy records the resulting realized structure.
Schematically:
\[ \boxed{ \text{multiplicity} \rightarrow \text{occurrence} \rightarrow \text{occupancy} } \]
But this sequence is incomplete.
It does not yet explain direction.
The enclosing relation supplies the asymmetry that determines how occurrence is expressed as occupancy.
The fuller architecture is therefore
\[ \boxed{ \text{multiplicity} \rightarrow \text{occurrence} } \]
together with
\[ \boxed{ \text{enclosing possibility and constraint} \rightarrow \text{occupancy bias}. } \]
The subordinate mechanism realizes the change.
The enclosing relation organizes which realization is favored.
Possible Structure and Realized Structure
We can now sharpen a distinction introduced earlier.
Possible structure
Defined by
\[ W_{n,i}. \]
This belongs to the boundary-defined classification.
Realized event structure
Defined by
\[ \nu_{n,i}. \]
This counts occurrences within a specified finite count domain.
Realized occupancy structure
Defined by
\[ c_{n,i}. \]
This describes how realized structure is distributed across macrostates.
These distinctions let R2D ask separate questions that are often compressed together:
What can happen?
What happens?
What state structure is realized?
Why is one realized structure favored over another?
The first three are now defined.
The fourth requires possibility and constraint.
Core Equations
Possible count
\[ \boxed{ W_{n,i} = \left| \pi_n^{-1}(i) \right| } \]
Occurrence count
\[ \boxed{ \nu_{n,i} = \text{realized occurrences classified as } i } \]
Occupancy
\[ \boxed{ c_{n,i} = \text{realized occupancy of macrostate } i } \]
Logarithmic coordinates
\[ \boxed{ S_{n,i} = \ln W_{n,i} } \]
\[ \boxed{ \tau_{n,i} = \ln \nu_{n,i} } \]
\[ \boxed{ G_{n,i} = \ln c_{n,i} } \]
Relative occurrence realization
\[ \boxed{ \Delta\tau \approx \Delta S } \]
Occupancy bias
\[ \boxed{ \Delta G = \Delta\tau+\Delta A } \]
The Key Distinction
The simplest summary is:
Multiplicity is what can be realized.
Occurrence is what happens.
Occupancy is the structure that is realized.
Keeping those three counts separate allows R2D to ask the next question cleanly:
What makes one realized structure directionally favored over another?
That requires an enclosing relation.