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A transformation of phase space is canonical when it preserves the symplectic form, or equivalently all Poisson brackets. For one degree of freedom this isequivalently $\{Q,P{\}}_{q,p}=1$.

Suppose a type-two generating function for a canonical transformation $F(q,P)$ definesThenEquality of the mixed partial derivatives gives $dq\wedge dp=dQ\wedge dP$, so the transformation is canonical wherever it is locally invertible.

Forthe fundamental theorem of calculus and differentiation under the integral sign giveThusand in particular $P=(p^2+q^2)/2$.

For the unit-frequency simple harmonic motion, the transformed Hamiltonian is simply $H=P$. Hamilton's equations becomeThereforeand transformation back gives the familiar solution

Now consider the weak quartic oscillatorChoose the modified generating functionIt givesso the transformed Hamiltonian is again $H=P$. Hence $P=E$ and $Q=t-t_0$. Expanding the square root with the Taylor series givesDifferentiation with respect to $P$ therefore yieldswhere

Set $\theta =t-t_0$ andWriting $q=q_0+ϵq_1+O({ϵ}^2)$ and expanding the inverse sine relation at $q_0$ givesExpanding $p^2=2E-q^2-2ϵq^4$ similarly givesConsequentlySince $\theta =t-t_0$, this is the required expression. The formula is understood away from turning points where $p_0=0$ and the ratio $q/p$ itself is singular.

This is a birth-death process with birth and death ratesIts transition diagram has the two arrowsThe gain into state $n$ comes from a death in state $n+1$ or a birth in state $n-1$, while the loss is the sum of both rates out of state $n$. The birth-death master equation is consequently

Multiplying by $n$, summing over the nonnegative integers, and shifting the summation indices shows that each birth contributes $+1$ and each death contributes $-1$. Thus the first-moment equation of a birth-death process givesAt a stationary state, use the variance identity $⟨n^2⟩=⟨n{⟩}^2+(\Delta n{)}^2$. Solving the resulting quadratic equation givesThe minus root is inadmissible whenever it is negative, namely when $(\Delta n{)}^2<\gamma /\alpha$. Equality would give zero mean, which is also incompatible with a positive immigration rate $\gamma$, so for $\gamma >0$ the minus branch requires $(\Delta n{)}^2>\gamma /\alpha$ even to be a possible mean.

For a continuum approximation, write $b(n)=\gamma +\beta n$ and $d(n)=\alpha n(n-1)$. Applying the Kramers-Moyal expansion to the two gain terms and retaining derivatives through second order gives the Fokker-Planck equationwhere the negative drift and infinitesimal jump variance areThis truncation requires the typical population and the scale on which $P$ varies to be much larger than the unit jump size, with the rates varying smoothly across that scale.

The positive zero of $g$ isso in particular $n_{\ast }\sim \beta /\alpha$ as $\alpha /\beta \to 0$. Put $x=n-n_{\ast }$. The linear noise approximation useswhereand, because $b(n_{\ast })=d(n_{\ast })$,In a stationary state with zero Fokker-Planck probability current,Its normalized solution is the normal distributionIt follows that

These estimates agree with the exact stationary first-moment relation on its plus branch to leading order: inserting $(\Delta n{)}^2\sim \beta /\alpha$ gives $⟨n⟩\sim \beta /\alpha$. Moreover,Thus the stationary mass lies far from the boundary $n=0$, is narrow relative to its mean, yet changes across many lattice sites. These are precisely the large-population and slow-variation conditions needed for the diffusion approximation.

In the nonrelativistic regime,The assumption $E-\mu \gg k_{B}T$ puts the Fermi-Dirac distribution in its Maxwell-Boltzmann limit, soTherefore the nonrelativistic Maxwell--Boltzmann number density is

For $p+e^{-}\leftrightarrow H+\gamma$, chemical equilibrium and the vanishing photon chemical potential implyApplying the number-density formula to each massive species and eliminating the chemical potentials givesUse charge neutrality, $n_p\simeq n_e$, the mass approximation $m_H\simeq m_p$, and the convention in the question that suppresses the order-one internal-degeneracy ratio $g_e{g}_p/g_H$. With the Hydrogen binding energywe obtain the Saha ionization equationThe assumptions are thermal equilibrium and chemical equilibrium before cosmological recombination, a nonrelativistic nondegenerate gas, zero photon chemical potential, charge neutrality, negligible proton-electron mass correction in the translational prefactor, and the stated degeneracy-factor approximation.