Macrostate Probability Distributions (MPDs)

• Macrostate Probability Distributions are coarse-grained probability laws defined on reduced descriptors that capture the essential macroscopic behavior of complex systems.
• They enable efficient computation of moments, large-deviation analysis, and Bayesian estimation, bypassing the need for full microscopic simulations.
• Applications range from statistical mechanics and turbulent dynamics to molecular simulations and multicomponent adsorption process optimization using dynamic reweighting techniques.

Macrostate Probability Distributions (MPDs) are probability laws defined on coarse-grained descriptors of a system rather than on its full microscopic state. In the cited literature, a macrostate may be an occupancy-count vector in a closed cohort Markov model, a small interval of values of a macroscopic observable, an adsorbate occupancy vector in the grand-canonical ensemble, a discrete large-scale-circulation state in turbulent convection, a fuzzy membership function on microstate space, or a partition-derived count vector induced by a uniformly random permutation (Iskandar et al., 2022, Mori, 2016, Yoon et al., 17 Aug 2025, Maity et al., 2021, Sikorski et al., 2024, Sills, 2019). Taken together, these works use MPDs for several related but non-identical constructions; the common structure is the projection of microstate-level randomness onto a lower-dimensional macro-description whose law, moments, large-deviation properties, or transition statistics can be analyzed directly.

1. Formalizations of macrostates

A macrostate is defined differently across domains, but always as a reduced descriptor that suppresses microscopic detail. In a closed cohort of size $N$ with $S$ mutually exclusive states $S_1,\dots,S_S$, the macrostate is the count vector

$N(t)=[N_1(t),\dots,N_S(t)]',$

with $N_s(t)\in\{0,1,\dots,N\}$ and $\sum_{s=1}^S N_s(t)=N$. In statistical mechanics, a macrostate may instead be the event that a macro-observable $X$ lies in a small interval $[x,x+dx)$, with probability

$$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$

For multicomponent adsorption, the macrostate is the occupancy vector $\vec n=(n_1,\dots,n_k)$ of the components in a pore of fixed volume $S$0 and temperature $S$1. In fuzzy coarse-graining for Markov systems, the macrostates themselves are membership functions $S$2 satisfying $S$3. In combinatorial models derived from random permutations, the macrostate is either a padded part vector $S$4 or a multiplicity vector $S$5 associated with a partition $S$6 (Iskandar et al., 2022, Mori, 2016, Yoon et al., 17 Aug 2025, Sikorski et al., 2024, Sills, 2019).

Setting	Macrostate	Probability object
Cohort Markov model	$S$7	multinomial PMF
Ensemble large deviations	$S$8	$S$9
Grand-canonical adsorption	$S_1,\dots,S_S$0	$S_1,\dots,S_S$1
Fuzzy Markov coarse-graining	$S_1,\dots,S_S$2 or $S_1,\dots,S_S$3	$S_1,\dots,S_S$4, $S_1,\dots,S_S$5, holding-time laws
Random-permutation partitions	$S_1,\dots,S_S$6 or $S_1,\dots,S_S$7	partition-weighted PMFs

This diversity matters conceptually. It shows that MPDs are not tied to a single ontology of macrostates: they may be discrete counts, continuous macrovariables, distributions over partitions, or even fuzzy observables. A plausible implication is that the phrase is best understood as denoting a class of coarse-grained probability constructions rather than a unique formal object.

2. Multinomial occupancy laws and exact moment structure

For cohort state-transition models, the MPD is derived by first representing each individual $S_1,\dots,S_S$8 at time $S_1,\dots,S_S$9 by a one-hot occupancy vector

$N(t)=[N_1(t),\dots,N_S(t)]',$0

with

$N(t)=[N_1(t),\dots,N_S(t)]',$1

This is a Multinoulli law with one trial. Under independence across individuals and identical law, aggregation yields the cohort macrostate distribution

$N(t)=[N_1(t),\dots,N_S(t)]',$2

which is the standard $N(t)=[N_1(t),\dots,N_S(t)]',$3-category multinomial PMF. The occupancy-probability vector evolves by the Chapman-Kolmogorov update

$N(t)=[N_1(t),\dots,N_S(t)]',$4

where $N(t)=[N_1(t),\dots,N_S(t)]',$5 is the $N(t)=[N_1(t),\dots,N_S(t)]',$6 transition matrix with entries $N(t)=[N_1(t),\dots,N_S(t)]',$7. The first and second moments follow exactly:

$N(t)=[N_1(t),\dots,N_S(t)]',$8

$N(t)=[N_1(t),\dots,N_S(t)]',$9

$N_s(t)\in\{0,1,\dots,N\}$0

These formulas provide an exact moment representation at each time step once $N_s(t)\in\{0,1,\dots,N\}$1 is known (Iskandar et al., 2022).

The computational consequences are explicit. Updating $N_s(t)\in\{0,1,\dots,N\}$2 costs one matrix-vector multiplication, $N_s(t)\in\{0,1,\dots,N\}$3 per step, and the moments cost $N_s(t)\in\{0,1,\dots,N\}$4 work. By contrast, a cohort microsimulation with $N_s(t)\in\{0,1,\dots,N\}$5 replications of $N_s(t)\in\{0,1,\dots,N\}$6 individuals over $N_s(t)\in\{0,1,\dots,N\}$7 steps costs $N_s(t)\in\{0,1,\dots,N\}$8 random draws plus bookkeeping, while master-equation approaches involve solving a system of $N_s(t)\in\{0,1,\dots,N\}$9 coupled ODEs. The multinomial representation is exact up to floating-point arithmetic, with no sampling error, whereas Monte Carlo error scales as $\sum_{s=1}^S N_s(t)=N$0 (Iskandar et al., 2022).

The same multinomial structure also supports Bayesian estimation. If one views transitions out of state $\sum_{s=1}^S N_s(t)=N$1 as a multinomial likelihood and places independent Dirichlet priors on the rows,

$\sum_{s=1}^S N_s(t)=N$2

then the posterior is conjugate:

$\sum_{s=1}^S N_s(t)=N$3

When only macrostate counts $\sum_{s=1}^S N_s(t)=N$4 are observed, the paper notes either an approximation based on $\sum_{s=1}^S N_s(t)=N$5 or latent-data MCMC, but the central point remains that the cohort likelihood is built from multinomials (Iskandar et al., 2022).

An illustrative four-state example uses $\sum_{s=1}^S N_s(t)=N$6, $\sum_{s=1}^S N_s(t)=N$7-year cycles, and $\sum_{s=1}^S N_s(t)=N$8 microsimulations. The empirical mean $\sum_{s=1}^S N_s(t)=N$9 SD of $X$0 from microsimulation coincides exactly with the multinomial mean $X$1 SD curves over $X$2, with no appreciable difference in either mean or variance trajectory. In this setting, the MPD is not an approximation to the count law; it is the count law.

3. Large deviations, rate functions, and ensemble equivalence

In statistical mechanics, MPDs are often studied through large deviations rather than finite-state multinomial formulas. For a macro-observable $X$3, one considers

$X$4

in the sense that

$X$5

with $X$6 taken infinitesimal after $X$7. The nonnegative function $X$8 is the large-deviation rate function, and the typical values are the points where $X$9. For two ensembles $[x,x+dx)$0 and $[x,x+dx)$1, one defines $[x,x+dx)$2 and $[x,x+dx)$3 analogously and says that the ensembles are macrostate equivalent if

$[x,x+dx)$4

Partial equivalence is the weaker inclusion relation $[x,x+dx)$5 or vice versa (Mori, 2016).

Mori’s central result relates these rate functions to specific relative Rényi entropies. For $[x,x+dx)$6, with

$[x,x+dx)$7

and

$[x,x+dx)$8

the inequality

$[x,x+dx)$9

holds for any $$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$0. In the limit $$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$1, this yields the von Neumann entropy version

$$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$2

where

$$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$3

If measure equivalence holds, meaning $$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$4 and $$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$5, then measure equivalence implies macrostate equivalence (Mori, 2016).

The importance of this framework is that the MPD becomes an asymptotic fluctuation object. It is not merely a histogram over coarse states; it encodes the exponential cost of macroscopic deviations. Mori uses this to formulate a sufficient condition for thermalization in isolated quantum systems via the diagonal ensemble $$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$6 and the microcanonical ensemble $$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$7, to bound the timescale on which macrovariables can drift under unitary dynamics when the specific relative entropy stays near zero, and to sharpen the second law in a quantum quench by showing that macrostate nonequivalence implies an extensive relative entropy and hence a thermodynamic entropy jump of order $$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$8 (Mori, 2016).

This perspective also clarifies a frequent source of confusion. In this large-deviation usage, an MPD need not be a probability mass function on a finite state space. It may instead be a family of probabilities whose principal content is summarized by a rate function and its zero set.

4. Dynamic macrostates and Markovian transition laws

MPDs also arise in explicitly dynamical settings, where the coarse-grained object of interest is a transition matrix, a stationary distribution over macrostates, or a holding-time law. Maity, Koltai, and Schumacher study turbulent Rayleigh-Bénard convection in a cubic cell at $$P_N^{(\rho)}(X\in[x,x+dx))=\mathrm{Tr}\!\left[\rho_N \Pi_{X\in[x,x+dx)}\right].$$9 and define a macrostate by coarse-graining the large-scale circulation (LSC) using the orientation angle $\vec n=(n_1,\dots,n_k)$0 of the dominant Fourier mode of the vertical velocity and the coherence indicator

$\vec n=(n_1,\dots,n_k)$1

They identify six macrostates: four long-lived LSC states with diagonal orientations and $\vec n=(n_1,\dots,n_k)$2, one short-lived edge-aligned LSC state, and one Null state with $\vec n=(n_1,\dots,n_k)$3. For some analyses the four long-lived states are merged into a single macrostate $\vec n=(n_1,\dots,n_k)$4, producing a three-state model $\vec n=(n_1,\dots,n_k)$5 (Maity et al., 2021).

For lag time $\vec n=(n_1,\dots,n_k)$6, the transition matrix is

$\vec n=(n_1,\dots,n_k)$7

The stationary distribution satisfies

$\vec n=(n_1,\dots,n_k)$8

For the three-state aggregation at $\vec n=(n_1,\dots,n_k)$9, the stationary distributions are reported as approximately $S$00 for $S$01, $S$02 for $S$03, and $S$04 for $S$05. The subdominant eigenvalues of the six-state matrices satisfy

$S$06

for $S$07 up to about $S$08, which is taken as strong evidence for approximate Markovianity up to that horizon. Persistence-time distributions for the long-lived and short-lived LSC states have exponential tails,

$S$09

$S$10

corresponding to mean persistence times of about $S$11 and $S$12, respectively. The joint distribution of successive persistence times nearly factorizes, indicating that the previous holding time provides no significant extra information about the next one beyond the current macrostate. At $S$13, no transition occurred in $S$14 (Maity et al., 2021).

A distinct but related dynamical construction appears in coarse-graining Markov systems through fuzzy membership functions. Instead of a crisp partition of microstate space, one uses smooth functions $S$15 with $S$16. In the two-state case, the macrostate is represented by a single function $S$17, with $S$18. The holding-time law is postulated as

$S$19

so that

$S$20

The stationary weight is

$S$21

and the lag-$S$22 transition probabilities are

$S$23

where $S$24 is the Koopman transfer operator. The ISOKANN method learns the dominant nonconstant membership function by fitting the Koopman relation

$S$25

with

$S$26

In the $S$27-opioid-receptor example, the data consist of $S$28 independent MD trajectories, each $S$29 long with $S$30 frames, using all C$S$31-distances $S$32 after $S$33-scoring. A neural network with hidden-layer widths $S$34, sigmoid activations, one linear output, $S$35 regularization $S$36, ADAM, and learning rate $S$37 is trained. After about $S$38 steps, the MSE residual is about $S$39, and the extracted exit rate is $S$40. The resulting fuzzy macrostate acts as a one-dimensional reaction coordinate, and a shortest-path construction with $S$41 yields about $S$42 frames for a maximal-likelihood reactive path from inactive to active conformations (Sikorski et al., 2024).

These two dynamical lines of work show that MPDs can encode not only static occupancy statistics but also memory structure, holding times, stationary weights, and transition probabilities. They also show that the macrostates themselves need not be crisp sets.

5. Algebraic and combinatorial constructions

In the semantics of probability and non-determinism, MPDs arise from a distributive law of the multiset monad over the distribution monad. If $S$43 denotes finite multisets and $S$44 denotes discrete distributions, there is a natural transformation

$S$45

which turns a multiset of micro-distributions into a distribution over count-multisets. Concretely, if

$S$46

then for each macrostate $S$47, with $S$48,

$S$49

where

$S$50

The same law is given in the paper in three additional equivalent forms: as shuffle or string-interleaving semantics on $S$51, as a Kleisli coequaliser induced by the accumulation map $S$52, and as the induced Eilenberg-Moore algebra on the free commutative monoid $S$53. A parallel hypergeometric law $S$54 governs sampling without replacement and commutes with multivariate hypergeometric structure. The paper states that the usual identities of sequential sampling, parallel sampling, and commutation with conditioning and updating fall out of this distributive-law framework (Jacobs, 2021).

A concrete example uses $S$55 with

$S$56

and

$S$57

The resulting distribution over three-element multisets is

$S$58

The same framework recovers the ordinary multinomial law for Maxwell-Boltzmann occupancy statistics via $S$59 and the multivariate hypergeometric law for without-replacement sampling from multiple sub-populations (Jacobs, 2021).

A different combinatorial MPD appears in Sills’s study of random permutations. Choosing a permutation uniformly from the symmetric group $S$60 induces a random partition $S$61, hence either a multiplicity vector

$S$62

or a part vector

$S$63

The multiplicity-vector PMF is

$S$64

and the part-vector PMF has the same denominator when expressed through the multiplicities of equal parts. For $S$65, the exact mean vector is

$S$66

and the covariance matrix has entries

$S$67

For $S$68, exact closed forms are not proved except in trivial cases, but Sills reports conjectured formulas for the scaled means $S$69, including degree-$S$70 polynomial behavior for $S$71 (Sills, 2019).

These algebraic and combinatorial constructions broaden the notion of MPD beyond stochastic-process modeling. They show that the same basic passage from micro-configurations to occupancy counts can be formalized either categorically, as a distributive law, or combinatorially, as a weighted distribution on partitions.

6. Grand-canonical MPDs in multicomponent adsorption

For multicomponent adsorption in porous materials, the MPD is the grand-canonical probability of an adsorbate occupancy vector $S$72:

$S$73

where

$S$74

$S$75

and

$S$76

Introducing a density-of-states or weight function $S$77, one writes

$S$78

with $S$79 estimated up to a multiplicative constant by flat-histogram Monte Carlo. In a biased simulation at reference state $S$80, if $S$81 is the bias and $S$82 the visit histogram, then

$S$83

The unbiased reference MPD is recovered as

$S$84

followed by normalization (Yoon et al., 17 Aug 2025).

The key computational feature is reweighting. For any new state $S$85,

$S$86

again normalized so that $S$87. A first-order Taylor expansion in $S$88 may be written using the macrostate-average potential energy $S$89. Observable loadings are then

$S$90

with conversion to $S$91 through the crystal cell mass (Yoon et al., 17 Aug 2025).

This MPD is integrated directly into cyclic PSA/VSA modeling. For each process condition $S$92, one sets

$S$93

reweights $S$94 to $S$95, computes $S$96, converts to loadings $S$97, and inserts these into the mass-, energy-, and momentum-balance PDEs through a linear driving-force kinetic model. The PDEs are discretized by finite-volume/WENO methods and marched to cyclic steady state. Multi-objective optimization uses NSGA-II to construct the CH$S$98 purity/recovery Pareto front by varying pressures, step times, reflux ratios, and flow rates. The paper states that no new Monte Carlo simulations are needed once the reference MPD is built; all subsequent equilibrium data are generated on-the-fly by simple reweighting (Yoon et al., 17 Aug 2025).

Within this framework, the MPD is both a statistical-mechanical object and a process-model input. It is not only descriptive of equilibrium occupancy fluctuations; it is also operational in optimization workflows.

7. Scope, interpretation, and recurring distinctions

The surveyed literature indicates that MPD is a family resemblance term. Across the cited works, macrostates may be occupancy counts, macro-observable intervals, coarse-grained circulation states, fuzzy memberships, or partition-derived vectors. Consequently, the associated probability objects range from exact finite-$S$99 multinomial PMFs and grand-canonical occupancy laws to large-deviation rate functions and lag-time transition matrices (Iskandar et al., 2022, Mori, 2016, Maity et al., 2021, Sikorski et al., 2024, Yoon et al., 17 Aug 2025).

Several distinctions recur. First, exactness is domain-dependent. In the cohort Markov model, the multinomial representation is exact up to floating-point arithmetic and has no sampling error. In ensemble-equivalence theory, the central statements concern thermodynamic-limit rate functions. In the Rayleigh-Bénard study, Markovianity is supported empirically by eigenvalue decay, persistence-time distributions, and factorization of successive holding-time distributions, but the data are accompanied by considerable noise and the initial exponential decay is interpreted as evidence only up to about $S_1,\dots,S_S$00. In adsorption, the MPD is derived from grand-canonical statistical mechanics and then estimated and transported computationally through flat-histogram sampling and reweighting (Iskandar et al., 2022, Mori, 2016, Maity et al., 2021, Yoon et al., 17 Aug 2025).

Second, macrostates need not be crisp partitions. The membership-function formulation explicitly replaces disjoint sets by smooth functions $S_1,\dots,S_S$01 and treats these functions as the macrostates themselves. This directly counters the common assumption that coarse-graining must be set-valued (Sikorski et al., 2024).

Third, MPDs are not exclusively static equilibrium objects. They may encode transition probabilities $S_1,\dots,S_S$02, stationary weights $S_1,\dots,S_S$03, exponential holding-time laws, or process-condition-dependent equilibrium loadings produced on demand by reweighting. A plausible implication is that the central unifying idea is not equilibrium per se, but tractable probabilistic structure after coarse-graining (Maity et al., 2021, Sikorski et al., 2024, Yoon et al., 17 Aug 2025).

Across these formulations, the function of an MPD is stable even when its mathematical form changes: it renders macroscopic behavior computable without retaining the full microstate description. Depending on context, the payoff is exact moment computation, a criterion for ensemble equivalence, a reduced Markov model of turbulent or molecular dynamics, a categorical account of sampling semantics, or a direct bridge from molecular simulation to adsorption-process optimization.