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For one molecule whose translational motion separates from its internal degrees of freedom, the classical one-particle partition function iswhere $q_{\mathrm{int}}$ contains rotational, vibrational, and other internal states. For $N$ indistinguishable noninteracting molecules, the classical ideal-gas partition function is $Z_N=z_1^N/N!$. The thermodynamic pressure therefore obeyssoThe internal structure changes the energy, entropy, and heat capacity through $q_{\mathrm{int}}(T)$, but it does not change the ideal gas law while the internal partition function is independent of volume and the molecules have negligible interactions.

For a dilute monatomic gas $q_{\mathrm{int}}=1$. Using the Stirling formula in $Z_N=(V/{\lambda }^3{)}^N/N!$ and differentiating the Helmholtz free energy $F=-k_{B}T\log Z_N$ gives the Sackur-Tetrode equationWith $N$ fixed, an isentropic process therefore satisfies $V{T}^{3/2}=\text{constant}$, or $T{V}^{2/3}=\text{constant}$. Eliminating $T$ with the ideal-gas law yields the monatomic reversible ideal-gas adiabat

For a normalized trial state $\psi$, expand it in an orthonormal basis of energy eigenstates, $\psi ={\sum }_n{c}_n{\psi }_n$. The Rayleigh quotient is thenThis is the quantum variational principle: minimizing the expectation of the Hamiltonian operator over any trial family gives an upper bound on the ground-state energy.

Let $\psi$ be the normalized ground-state wavefunction and form the normalized dilation ${\psi }_{\lambda }(x)={\lambda }^{1/2}\psi (\lambda x)$. Its kinetic and potential expectations areThe true ground state makes the energy stationary at $\lambda =1$, soThus the power-law quantum virial theorem gives

If $n\le -3$ and $A>0$, the two sides have opposite signs, because $T>0$ and $V>0$. If $A<0$, the scaled energy ${\lambda }^{2}T+{\lambda }^{-n}V$ tends to $-\infty$ as $\lambda \to \infty$, since $-n>2$, so the Hamiltonian is not bounded below. If $A=0$, the free particle has no normalizable ground state. Hence there is no normalizable ground state for $n\le -3$.

For $n=1$ take the normalized Gaussian functionThe Gaussian integral givesand thereforeIts unique minimum satisfiesAt this point $⟨V⟩=2⟨T⟩$, consistently with the virial theorem, and the resulting variational estimate is

A map is Devaney-chaotic on $\Lambda$ when it is topologically transitive, it has dense periodic points in $\Lambda$, and it has sensitive dependence on initial conditions. A one-dimensional map is Glendinning-chaotic when some iterate has a horseshoe for an interval map: there are an interval $J$ and two disjoint subintervals $K_0,K_1\subseteq J$ such that $F^n(K_0)=F^n(K_1)=J$.

The map $F(x)=ax(\mathrm{mod}1)$ is the beta transformation with slope $a>1$. On every interval of continuity of $F^n$, it is affine with slope $a^n$. The endpoints of these intervals are iterated preimages of the discontinuity set. Choosing $n$ with $a^n>2$, the standard full-branch lemma for the beta transformation supplies two disjoint continuity intervals $K_0,K_1$ on which $F^n$ maps bijectively onto $[0,1)$. Thus $F^n$ has a horseshoe, proving Glendinning chaos.

For Devaney chaos one may either use the standard theorem that a full-branch expanding interval map is Devaney-chaotic, or verify its ingredients directly. Every nonempty open interval expands by the factor $a$ on each uninterrupted iterate; after subdivision at preimages of discontinuities, one of its images contains a full branch and hence eventually covers $[0,1)$. This proves topological transitivity. Applying an inverse branch of a sufficiently high iterate to a full branch gives a contraction whose unique fixed point is periodic; choosing the branch inside any prescribed open interval proves that periodic points are dense. Finally, two sufficiently close points that follow the same branch separate by the factor $a$ on each iterate until their distance exceeds a fixed positive constant, proving sensitive dependence. Hence $F$ is also Devaney-chaotic for every real $a>1$.