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A reaction's stoichiometric vector outer product multiplied by its reaction propensity function is its contribution to the infinitesimal state-increment covariance. Summing over channels gives $B$. Its normalized form $D$ divides by the two mean copy numbers. The Fokker-Planck equation writes the associated second-derivative term with $B/2$; the covariance evolution and normalized stationary Lyapunov equation instead use $B$ and $D$ themselves. A simultaneous unit loss of two species contributes positive cross noise even when their long-time stationary covariance is negative.

At chemical equilibrium, a reversible chemical reaction has equal forward and reverse reaction rates, so its macroscopic composition stays constant. Individual molecules can keep changing identity: the reaction has not stopped. This dynamic balance is the definition described by the ^[ref]. Equal reaction rates also need not mean equal concentrations. A small population converting rapidly can balance a large population converting slowly.

This distinction gives two useful questions. Which composition is favored at the specified temperature and other constraints? Thermodynamics answers through chemical potentials and an equilibrium constant. How quickly does a prepared mixture approach that composition? Chemical kinetics answers through reaction rates. Knowing the destination does not determine the journey's duration. A composition that barely changes during observation may reflect extremely slow chemical kinetics, so constancy alone does not establish chemical equilibrium.

Consider the idealized elementary chemical reactionAssume a well-mixed closed system at fixed temperature and volume, with first-order law of mass action rates in both directions and no competing chemical reactions. Write $a=[\mathrm{A}]$ and $b=[\mathrm{B}]$ for concentrations in $\mathrm{M}=\mathrm{mol}/\mathrm{L}$. The total $a+b=C=1\mathrm{M}$ is conserved because each conversion replaces one molecule by one molecule. This conservation is special to the chosen stoichiometry.

The forward and reverse reaction rates, measured as conversions per unit volume per unit time, are $J_f=k_{f}a$ and $J_r=k_{r}b$. HenceSetting this derivative to zero givesWith $a(0)=1\mathrm{M}$ and $b(0)=0$, solving the ordinary differential equation yieldsAt $t=5\mathrm{s}$, these are approximately $0.3515\mathrm{M}$ and $0.6485\mathrm{M}$. Initially $J_f=0.3M/s$ and $J_r=0$. At chemical equilibrium, both are $0.075M/s$, while the net reaction rate $J_f-J_r$ is zero. The persistent exchange is visible in the lower panel even after the upper panel has nearly flattened.

Why does chemical equilibrium also involve a free energy minimum? For a closed system at fixed temperature and pressure, without imposed non-expansion work, stable equilibrium minimizes Gibbs free energy over compositions allowed by conserved elemental inventories. See W. C. Carter, MIT thermodynamics notes, sections 2-1.14 and 2-1.19. This is constrained Gibbs minimization for chemical equilibrium: freely minimizing over unrelated species amounts would change the inventory. At fixed temperature and volume, the appropriate potential is Helmholtz free energy. In either setting, varying an allowed extent of reaction gives the corresponding free energy slope ${\sum }_i{\nu }_i{\mu }_i$, with molar chemical potentials and signed stoichiometric vector entries. An interior equilibrium has zero slope; stability requires a minimum. For A converting to B, this says ${\mu }_{\mathrm{B}}={\mu }_{\mathrm{A}}$.

The beauty of this connection is that molecular exchange, a macroscopic composition, and a geometric minimum describe the same balance. In the ideal concentration model, let $x=b/C$ and choose the standard molar chemical potential difference consistent with $K=3$. Up to a composition-independent constant, the fixed-volume Helmholtz free energy then obeyswhere $n$ is the conserved total amount of substance and $R$ the molar gas constant. Its derivative is $\ln [x/(1-x)]-\ln 3$, vanishing at $x=3/4$. Its second derivative is $1/x+1/(1-x)>0$ inside $0<x<1$, so this balance is the unique minimum. The mixing entropy terms explain why the favored composition still contains A.

For reactions with stoichiometric vectors ${\nu }_r$ and reaction propensity functions $a_r$, the probability mass function $p(x,t)$ obeysIt is the forward operator of a Markov jump process equation for the copy-number continuous-time Markov chain.

Consider two internally equilibrated reservoirs A and B at the same imposed temperature and pressure. They may have different compositions or represent different phases. Transfer a small positive amount of substance $\delta n$ of the same chemical species from A to B, holding the other inventories fixed. To first order, their total Gibbs free energy changes byHence exchange toward lower chemical potential lowers Gibbs free energy. If transfer is allowed in both directions and the two reservoirs contain the exchangeable component, thermodynamic equilibrium requires equal chemical potentials for that component. See MIT 5.60, Lecture 14, pages 1–2.

An impermeable partition can maintain unequal chemical potentials; equality is a condition on an allowed exchange. Two pure samples of the same ideal gas already have equal chemical potentials at the same temperature and pressure. Different compositions at a common pressure can instead give different chemical potentials for one component through its thermodynamic activity.

Equality concerns the same exchangeable component in different places, not all different chemical species. For a permitted chemical reaction, the corresponding interior chemical equilibrium criterion is ${\sum }_i{\nu }_i{\mu }_{\mathrm{m},i}=0$, with signed stoichiometric vector entries ${\nu }_i$. The existing chemical equilibrium and constrained Gibbs minimization for chemical equilibrium sections develop those composition constraints.

A chemical reaction network specifies chemical species, reactions between them, and their stoichiometric vectors. It can be modelled deterministically by reaction-rate equations or stochastically by molecule-copy-number jumps.

At fixed temperature and pressure, a closed reacting system reaches thermochemical equilibrium by minimizing Gibbs free energy over nonnegative species amounts subject to conserved elemental abundances. For an allowed stoichiometric vector $\nu$, an interior minimum has ${\sum }_i{\nu }_i{\mu }_i=0$. More generally the chemical potential vector lies in the row span of the elemental conservation matrix at positive concentrations, with the appropriate one-sided inequalities for absent species. The first law of thermodynamics gives the thermodynamic differential, while the Second law of thermodynamics supplies the minimization direction.

For a specified chemical reaction and standard state convention, the dimensionless equilibrium constant is the equilibrium value of the reaction quotient $Q={\prod }_i{a}_i^{{\nu }_i}$, where $a_i$ denotes thermodynamic activity and ${\nu }_i$ a signed stoichiometric vector entry. Thermodynamic activities incorporate the reference state and departures from ideal behavior. See IUPAC Gold Book, equilibrium constant.

In the ideal concentration model for A converting to B, use $a_{\mathrm{A}}=[\mathrm{A}]/c^{\circ }$ and $a_{\mathrm{B}}=[\mathrm{B}]/c^{\circ }$ with the same reference $c^{\circ }=1\mathrm{M}$. Their ratio gives $Q=[\mathrm{B}]/[\mathrm{A}]$ and $K=0.75/0.25=3$. Equality $K=k_f/k_r$ applies to this elementary first-order pair; it is not a universal ratio of two measured rate constants for an arbitrary multistep chemical reaction.

At a specified constant temperature, with consistent standard states, the standard molar reaction Gibbs free energy isHere ${\Delta }_{r}G$ is the local slope per mole of extent of reaction at fixed temperature and pressure, rather than the finite total Gibbs free energy change between two mixtures. At equilibrium ${\Delta }_{r}G=0$, while ${\Delta }_r{G}^{\circ }$ generally remains nonzero. For $K=3$ at $298\mathrm{K}$, it is about $-2.72kJ/mol$. For positive activities, $Q<K$ favors forward conversion and $Q>K$ favors reverse conversion; neither sign supplies a reaction rate. See MIT 5.111, Lecture 18, reaction free energy and equilibrium.

For a closed system with one specified chemical reaction, the extent of reaction $\xi$ records progress from $\xi =0$ through $n_i=n_{i,0}+{\nu }_i\xi$. Here $n_i$ is the amount of substance of species $i$, and the signed stoichiometric vector has negative entries for reactants and positive entries for reaction products. For the water chemical equation in the order ${\mathrm{H}}_2,{\mathrm{O}}_2,{\mathrm{H}}_2\mathrm{O}$, it is $(-2,-1,2)$. The normalization of $\xi$ depends on the chosen coefficients. See IUPAC Gold Book, extent of reaction.

With particle numbers instead of molar amounts, a molecular extent ${\xi }_N$ obeys $d{N}_i={\nu }_{i}d{\xi }_N$, and the chemical potentials are measured per particle. At fixed temperature and volume, the Helmholtz free energy then satisfies $dF=({\sum }_i{\nu }_i{\mu }_i)d{\xi }_N$. An interior chemical equilibrium requires ${\sum }_i{\nu }_i{\mu }_i=0$; the molar convention gives the same relation with chemical potentials per mole.

Keep the stoichiometry attached to the enthalpy. Use the same abstract $A,B,C$ and suppose each of two entries fixes an amount of substance for every species:Here each unit coefficient represents one mole on the common basis. To obtain $A\to C$, halve $R_1$ and reverse $R_2$. Adding those transformed chemical equations givesThe equal amounts of $B$ on opposite sides cancel, leaving $A\to C$. Doubling this final chemical equation would give $2A\to 2C$ and $\Delta H=-14\mathrm{kJ}$; its change per mole of $A$ would remain $-7\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$.

More generally, write each chemical equation as a stoichiometric vector ${\mathbf{\nu }}_j$, with negative entries for reactants and positive entries for reaction products. Finding coefficients $a_j$ with ${\mathbf{\nu }}_{\mathrm{target}}={\sum }_j{a}_j{\mathbf{\nu }}_j$ also gives $\Delta H_{\mathrm{target}}={\sum }_j{a}_j\Delta H_j$, using compatible state and amount conventions. An identical chemical species cancels only with the same thermodynamic phase and composition conditions. A leftover water or dilution step is a real part of the balance.

Let $\mu =⟨X⟩$. The Markov jump-process generator applied to $X$ gives the exact moment equation $\dot{\mu }=3\lambda -\beta ⟨\sqrt{X}⟩$. With the prescribed moment closure, $⟨\sqrt{X}⟩\approx \sqrt{\mu }$, stationarity givesThe production flux counts molecules, not reaction events: it is $v^{+}=3\lambda$, while the removal flux is $v^{-}=\beta \sqrt{\mu }=3\lambda$. The reaction-rate elasticity of removal relative to constant production isThe mean molecular lifetime is population divided by its stationary molecular removal flux. ThereforeThe noise-relevant average chemical event size is the molecular-flux-weighted reaction event size. For the stoichiometric vectors $+3$ and $-1$ it isThis weights each absolute jump by the molecular turnover it contributes. Equivalently, the balanced production and removal fluxes give equal weight to their sizes $3$ and $1$, yielding $(3+1)/2$. An event-count-weighted average would be $3/2$, since three quarters of firing events are deaths at stationarity; that different average does not determine the diffusion term in the linear noise approximation.

The normalized stationary fluctuation-dissipation relation for a reaction network isHere $M$ is the negative drift Jacobian matrix expressed in normalized population coordinates, and $D$ is the normalized noise matrix. More explicitly, with $S=\mathrm{diag}({\mu }_i)$, drift Jacobian $A$, stoichiometric vectors ${\nu }_r$, and stationary reaction propensity functions $a_r$, these matrices are $M=-S^{-1}AS$ and $D=S^{-1}B{S}^{-1}$, where $B={\sum }_r{a}_r{\nu }_r{\nu }_r^{\mathsf{T}}$. The linear noise approximation gives a covariance evolution $\dot{\eta }=-M\eta -\eta M^{\mathsf{T}}+D$; stationarity is the displayed Continuous Lyapunov equation.

In one dimension, the normalized restoring coefficient is $M=H/\tau =1/(2\tau )$. Squared chemical jumps determine the noise:The factor nine from a birth burst is essential; its three molecules arise in one correlated reaction event. The scalar Continuous Lyapunov equation is $2M\eta =D$, givingEquivalently, the normalized variance is $⟨s⟩/(H\mu )=2/((1/2)\mu )$. The Fano factor is approximately four. This calculation uses the prescribed mean closure and the linear noise approximation, so its small-relative-fluctuation regime is $\mu \gg 1$; it is not an exact assertion for arbitrarily small molecular copy numbers.

For $\beta >0$, symmetry of the unique stationary mean-field state gives $\mu =⟨X_1⟩=⟨X_2⟩$ and the balance $\lambda =\beta \mu +C{\mu }^2$. The fraction of the molecular removal flux using the joint reaction and the mean molecular lifetime are consequentlyIn particular $0\le E<1$, $C\mu =E/\tau$, and $\beta =(1-E)/\tau$.

The total loss flux of species one is $v_1^{-}={\mu }_1(\beta +C{\mu }_2)$, whereas its production flux $v_1^{+}=\lambda$ is constant. The reaction-rate elasticities are therefore $H_{11}=1$ and $H_{12}=C{\mu }_2/(\beta +C{\mu }_2)=E$ at the symmetric state. Species two gives $H_{22}=1$, $H_{21}=E$. All chemical jumps in either component have unit absolute size, so their molecular-flux-weighted reaction event sizes satisfy $⟨r_1⟩=⟨r_2⟩=1$. With $M_{ij}=H_{ij}/{\tau }_i$, this provesThe same matrix follows from $M=-S^{-1}AS$ with $S=\mu I$ and the drift Jacobian matrix in part (b); its positive eigenvalues are the restoring rates.

For the chemical diffusion matrix, each component receives noise from one unit birth, its own unit death, and the joint unit loss. The unnormalized diagonal noise intensity isOnly the joint reaction changes both components in one event. Its stoichiometric vector is $(-1,-1)$, so $B_{12}=B_{21}=(-1)(-1)C{\mu }^2=C{\mu }^2$. The positive sign reflects simultaneous decreases, rather than independent death events. Divide by the product of the means to normalize:Thus the chemical diffusion matrix in the normalized stationary fluctuation-dissipation relation for a reaction network isThe unequal roles of $M$ and $D$ matter: $M$ describes restoration of perturbations, while $D$ records the covariance of the immediate stochastic reaction increments.

Let the closed atmospheric parcel exchange heat and volume work with reservoirs at fixed temperature $T$ and pressure $P$, while reactions rearrange its conserved elemental inventory. For a reversible differential between equilibrium states, the first law of thermodynamics, with chemical work included, isThe Gibbs free energy $G=U+PV-TS$ therefore obeysThe first law of thermodynamics supplies this differential but does not alone choose the direction of a reaction. The Second law of thermodynamics does: for the parcel plus its reservoirs, at fixed $T,P$ and with no non-volume work,Thus spontaneous reactions decrease Gibbs free energy, and stable thermochemical equilibrium is its constrained minimum. For an allowed reaction of extent $d\xi$, let $d{N}_i={\nu }_{i}d\xi$, with positive stoichiometric vector entries for products and negative entries for reactants. At an interior minimum,The condition must hold for every independent allowed reaction, with fixed total atoms of each element and any relevant charge constraint. At a boundary with absent species, only feasible one-sided variations are permitted. These are the hypotheses of constrained Gibbs minimization for chemical equilibrium.

For an ideal gas mixture, ${\mu }_i={\mu }_i^{\circ }(T)+\mathcal{R}T\log (x_{i}P/P^{\circ })$, where $x_i$ is the volume mixing ratio and $\mathcal{R}$ the molar gas constant. ThereforeThis is a dimensionless activity-based equilibrium constant, not a bare product of dimensional pressures.

Write the molecular copy numbers as $\mathbf{x}=(x_1,x_2,x_3,x_4)$ and use the power-law reaction propensity convention of the paper. The seven propensities and stoichiometric vectors areA channel is assigned zero propensity whenever its update would leave the nonnegative integer lattice.

The Gillespie algorithm starts from $\mathbf{x}=(5,0,0,0)$ and $t=0$. At the current state compute $a_0={\sum }_{r=1}^7{a}_r$. If $a_0=0$, terminate the path. Otherwise draw independent random variables $U_1,U_2$ from the uniform distribution on $(0,1)$, set the next waiting time toand choose the least index $r$ satisfyingThen update $t\leftarrow t+\tau$ and $\mathbf{x}\leftarrow \mathbf{x}+{\nu }_r$, and repeat. The minimum of the seven competing reaction clocks has an exponential distribution of rate $a_0$, while reaction $r$ wins with probability $a_r/a_0$.

For a chemical reaction with signed stoichiometric vector entries ${\nu }_i$, positive for products and negative for reactants, the reaction quotient is the product of the species' thermodynamic activities raised to these powers. It equals the equilibrium constant at chemical equilibrium, with all quantities using the same standard state convention.

Stochastic chemical kinetics models molecular copy numbers as a continuous-time Markov chain. Each reaction has a state-dependent reaction propensity function and changes the state by its stoichiometric vector when it fires.

Stoichiometry relates the quantities consumed and formed in a chemical reaction. Its coefficients are ratios of entity counts, or equivalently of amount of substance, rather than ratios of masses. For the water chemical equation, consuming two molecular hydrogen molecules per molecular oxygen molecule gives the same $2:1$ ratio as consuming two moles of molecular hydrogen per mole of molecular oxygen. Multiplying every entity count by the same factor preserves the ratio. See Flowers et al., Chemistry 2e, reaction stoichiometry.

Coefficients and subscripts do different jobs. The coefficient in $2{\mathrm{H}}_2\mathrm{O}$ counts two water molecules; the subscript says that each contains two hydrogen chemical atoms. Changing coefficients scales quantities. Changing a subscript changes the specified chemical substance.

This distinction catches a common counting error: one mole of water molecules contains two moles of hydrogen chemical atoms and one mole of oxygen chemical atoms. Always specify what is counted.

Stoichiometry records reactant and product multiplicities in a chemical reaction or a chemical reaction network. The change caused by one reaction is its stoichiometric vector; multiplying by the reaction rate gives its contribution to the concentration equations. For two identical reactants the rate-constant convention absorbs any combinatorial factor.