The electric potential difference across the inner mitochondrial membrane, contributing to the proton-motive force. A change in potential can alter both useful phosphorylation and leak behavior.

For $ϵ=km/{\gamma }^2\ll 1$, momentum relaxes in ${\tau }_m=m/\gamma$ while the trap relaxes in ${\tau }_k=\gamma /k$. After the fast transient, the slow amplitude is $x_0+{\tau }_m{v}_0+O(ϵ)$. Neglecting the initial-velocity contribution additionally requires $|{\tau }_m{v}_0|\ll |x_0|$; small Reynolds number alone does not prove this condition.

A maximal independent set is an independent set that cannot be enlarged by adding another vertex. Equivalently, every vertex outside it has a neighbor inside it. This is maximality by inclusion, rather than maximum possible size.

In a three-vertex path, the middle vertex alone is a maximal independent set, as is the set containing both endpoints. Only the latter is a maximum independent set. A greedy procedure that repeatedly chooses a vertex and excludes its neighbors finds a maximal independent set, but can miss a maximum one. The distinction is important in approximation and scheduling algorithms.

The independence polynomial of a finite graph $G$ is $I_G(x)={\sum }_{k\ge 0}{i}_k{x}^k$, where $i_k$ counts independent sets of size $k$, including $i_0=1$. For a three-vertex path it is $1+3x+x^2$.

Choosing whether to include a vertex $v$ gives $I_G(x)=I_{G-v}(x)+x{I}_{G-N[v]}(x)$, where $N[v]$ consists of $v$ and its neighbors. Disjoint unions turn into products, because independent choices in the components combine freely. At positive $x$, the same polynomial is the partition function of the hard-core model, with $x$ weighting occupied vertices.

In a finite impartial game with the normal-play convention and no draws, a losing position is one from which the player whose turn it is cannot force a win against best play. It is also called a P-position, since the previous player wins.

A terminal position is losing. Recursively, a position is winning if it has at least one move to a losing position, and losing if every legal move goes to a winning position. Induction on the finite remaining game tree proves this classification. For a single Nim heap, only the empty heap is losing. Infinite plays and misère play need separate conventions and can change the base case.

An impartial game played with heaps of counters: a move removes a positive number of counters from exactly one heap. Under the normal-play convention, a position is losing exactly when the bitwise exclusive or of its heap sizes is zero.

The value $g(P)=\mathrm{mex}\{g(Q):P\to Q\}$ of a game position, where mex is the smallest nonnegative integer missing from the values of its legal successors. Terminal positions have value zero. The Sprague-Grundy theorem explains why these values combine by exclusive or.

A Euclid game is an impartial game based on subtraction steps of the Euclidean algorithm.^[ref] Here use the following normal-play convention: a position is an ordered pair $0\le a\le b$; if $a>0$, remove a positive multiple $ka\le b$ from $b$ and reorder. A position with $a=0$ has no legal move and is losing. Thus moving to a zero coordinate wins; variants that instead stop at equal positive coordinates require a different endpoint convention.

The sum of the coordinates strictly decreases, so every play terminates. If $a$ divides $b$, subtract $b$ and win immediately. If $a<b<2a$, there is only one move, to $(b-a,a)$, so the position has the opposite winning status from that successor.

If $b=qa+r$ with $q\ge 2$ and $0<r<a$, two legal choices leave $(r,a)$ or $(a,a+r)$. The latter has exactly one move, to the former, so exactly one of these two successors is losing. Choose it. For example, $(2,3)$ is losing because its only move reaches $(1,2)$, where the opponent takes the whole larger coordinate. From $(2,5)$, subtract $2$ to leave that losing position. These cases give a recursive strategy without exploring every subtraction amount.

Fibonacci Nim is a normal play impartial game on one heap. Start with $N\ge 2$ counters. The first move removes between $1$ and $N-1$ counters; thereafter a move removes at least one and at most twice the number just removed by the opponent, without exceeding the remaining heap. Taking the last counter wins.

For a position with $n>0$ counters and permitted maximum $r$, let $z(n)$ be the smallest Fibonacci number appearing in the Zeckendorf representation of $n$, using the sequence $1,2,3,5,8,\dots$. The position is losing exactly when $r<z(n)$. If $r\ge z(n)$, removing $z(n)$ is a winning move. When this does not empty the heap, the next smallest summand exceeds $2z(n)$, so the opponent's new limit is too small. Conversely, the Zeckendorf subtraction lemma gives $z(n-t)\le 2t$ whenever $0<t<z(n)$, so every legal move from a losing position leaves a winning one.^[ref]

Initially $r=N-1$, so the losing initial heap sizes are precisely the Fibonacci numbers. For example, $10=8+2$: take $2$, leaving the opponent $8$ counters and a limit of $4$, a losing position. The move limit is part of the state; heap size alone does not determine later positions.

In a disjunctive sum of games, each move changes exactly one component game. Under normal play, a player loses when no component has a legal move.

A subtraction game has a heap of nonnegative integer size. A legal move subtracts an allowed positive amount without making the size negative; normal play declares an empty move set losing.

Sprague-Grundy theory studies finite acyclic impartial games under the normal-play convention by assigning each position a Grundy number. The assignment recursively takes the minimum excludant of the values of all legal followers. The Sprague-Grundy theorem then reduces a disjunctive sum of games to the bitwise exclusive or of its component values. This framework explains both why a zero value is losing and how independently understood games can be combined without exploring the entire product of their state spaces.

An impartial game played under the normal-play convention.

For noninteracting particles in a three-dimensional isotropic quantum harmonic oscillator, measure the one-particle ground energy as zero. The single-particle partition function is $Z_1=(1-e^{-\beta \hslash \omega }{)}^{-3}$, from the product of three geometric series. In the dilute Maxwell–Boltzmann statistics regime, $\ln \Xi =e^{\beta \mu }{Z}_1$, so the grand potential is the displayed expression and $N=e^{\beta \mu }{Z}_1$. For $k_{B}T\gg \hslash \omega$,The same approximation follows from integrating the density of states $g(E)\simeq E^2/[2(\hslash \omega {)}^3]$ against $e^{-\beta E}$. Extrapolating the classical formula until the fugacity is one gives the scale of Bose condensation in a three-dimensional harmonic trap, but the quantitative transition temperature requires the Bose occupation law rather than Maxwell-Boltzmann statistics.

An ideal gas whose particles move in an area $A$ with two independent translational momentum components. In its dilute classical limit the one-particle translational partition function is $q=A/{\Lambda }^2$, where $\Lambda$ is the thermal de Broglie wavelength, and $Z_N=q^N/N!$. The translational internal energy is $N{k}_{\mathrm{B}}T$, and the surface pressure conjugate to area is $N{k}_{\mathrm{B}}T/A$. A mobile dilute adsorbed gas is an example; a fully occupied array of fixed adsorption sites has different state counting.

The gas constant per unit mass for a specified composition. An ideal gas obeys $p=\rho \mathcal{R}T$. With dimensionless mean molecular weight $\mu$ relative to an atomic mass unit, $\mathcal{R}=k_B/(\mu m_u)$; the common astrophysical notation $R^{\ast }/\mu$ uses a modified mass-based constant rather than the molar gas constant without a mass conversion.

A gas whose constituent molecules have two atoms. In the classical ideal-gas regime with three translational and two rotational degrees of freedom active but vibration frozen, $c_v=5\mathcal{R}/2$, $c_p=7\mathcal{R}/2$ and $\gamma =7/5$. Vibrational excitation, rotational freezing or chemical changes alter this heat-capacity model.

For an ideal chain of $N$ independent statistical segments of length $a$, the end-to-end probability density is proportional to $\exp [-3R^2/(2N{a}^2)]$. Its constrained Helmholtz free energy has extension-dependent part $3k_{B}T{R}^2/(2N{a}^2)$, so the restoring force is $-3k_{B}T\mathbf{R}/(N{a}^2)$. This is an entropic spring because extension removes available chain configurations, and its stiffness grows with temperature. The Gaussian law applies at extensions small compared with the contour length $Na$; near full extension, finite extensibility matters.

A freely rotating chain has fixed bond length and fixed angle between successive forward bond vectors, with independently uniform torsion angles. It neglects interactions between distant chain segments. The fixed angle introduces short-range orientational correlations even though the torsional rotations are unrestricted. Its characteristic ratio of a freely rotating chain converts those correlations into an end-to-end size; its equivalent long-chain link is the Kuhn length of a freely rotating chain.