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Monthly Notices of the Royal Astronomical Society 360, 1013–1021. doi.org/10.1111/j.1365-2966.2005.09090.x

The original analysis tests possible foreground origins of the WMAP–HEAO1 temperature/background cross correlation.

An ion-dipole attraction couples an ion's electric field to a molecule's electric dipole moment. In the elementary point-electric dipole approximation, the interaction potential energy is $U=-\mathbf{p}\cdot \mathbf{E}$. An orientation with the electric dipole moment along the electric field lowers this energy. For water, the oxygen end preferentially faces a cation, and the hydrogen end an anion. At nonzero temperature, fluctuations and competing hydrogen bonds prevent perfect alignment of every molecule.^[ref]

The product of hydrogen-ion and hydroxide-ion activities at a specified temperature. In dilute aqueous solution at 25 degrees Celsius, its usual concentration approximation is ten to the minus fourteen.

An elementary ionization-chamber charge-loss model makes the inference explicit. Let a positively charged electrode and its electrometer have effective capacitance $C$, electric potential difference $U$, and enclosed gas volume $\mathcal{V}$. Assume constant $C$, a short enough interval that collection conditions remain stable, and no multiplication of charge in the gas. If ionization produces $q$ ion pairs per unit volume per unit time and a fraction $\eta$ is collected, thenHere $e$ is the elementary charge and $I_{\mathrm{leak}}$ is the parasitic electric current through imperfect insulation. Each fully collected ion pair transfers one elementary charge through the measuring circuit; counting its two opposite carriers as two transfers would introduce a factor-of-two error. The relation follows from charge conservation, rather than from a model of the incoming particle's identity.

For an invented example, take $C=2.00\mathrm{pF}$, $\mathcal{V}=2.00\mathrm{L}$, $\eta =1$, a measured fall of $6.00\mathrm{V}{\mathrm{h}}^{-1}$, and a separately determined leakage-equivalent fall of $0.60\mathrm{V}{\mathrm{h}}^{-1}$. The leakage-subtracted fall is $5.40\mathrm{V}{\mathrm{h}}^{-1}$, soThis is about $1.87\times 10^4$ collected ion pairs per second throughout the chamber, not that many primary cosmic rays. Unknown collection losses make $\eta <1$ another uncertainty; changes in capacitance would require differentiating $Q=CU$ with both factors variable.

Gas density is a separate issue. Under a fixed radiation field, fixed gas composition and approximately constant energy deposited per unit gas mass, the volume-based ionization rate scales as $q\propto \rho$. For a vented ionization chamber, the ideal gas law gives the illustrative correctionwhere $T$ is absolute temperature. A pressure-only correction presumes equal temperature. A sealed rigid chamber instead fixes gas number density as it rises: its internal pressure may change with temperature without losing gas. This does not make its insulation, electrometer calibration or collection efficiency insensitive to temperature. Nor can dividing by gas density correct arbitrary radiation from the vessel walls or changing particle energies. Those are distinct effects, not a universal normalization recipe.

A neutral-hydrogen column density $N$ traces total hydrogen column $N/x_{\mathrm{HI}}$ only after specifying its neutral fraction $x_{\mathrm{HI}}$. Integrating the latter against the distribution per absorption distance gives the comoving gas mass density in the displayed expression. The factor ${\mu }_H$ includes helium if desired. In photoionization equilibrium for optically thin, almost fully ionized hydrogen, $x_{\mathrm{HI}}\simeq \alpha (T)n_H/\Gamma$, so gas density, temperature, photoionization rate and absorber size are required. An observed distribution alone does not fix the total baryon inventory.

Spatial variation of the ion temperature. The inverse logarithmic scale in the title is positive for an outward-decreasing profile. A nonzero value is a possible free energy source; it does not by itself establish ion-temperature-gradient instability without the appropriate dynamics and threshold.

An isenthalpic process preserves enthalpy for a fixed amount of material, or preserves specific enthalpy between inlet and outlet when comparing a flowing fluid of fixed composition. It need not preserve temperature or entropy. An insulated throttle satisfying the steady-flow energy balance assumptions is an important example.

At fixed atom number, the entropy of the classical ideal gas in a spherical linear trap depends on temperature and strength through $Z_1\propto T^{9/2}{a}^{-3}$. Reversible adiabatic change therefore holds this product constant, giving the displayed relation. Reducing trap strength lowers the temperature while increasing cloud size. Its peak classical phase-space occupation $N/Z_1$ remains constant, so this cooling alone does not make the gas more quantum degenerate. Heat exchange, particle loss or irreversible entropy production would change this relation.

For $J>0$ and $S_i^z=\pm 1/2$, a parallel bond has energy $-J/2$ while an antiparallel bond has $+J/2$. A wall therefore costs J, rather than the $2J$ of a differently normalized $-J\sum s_i{s}_{i+1}$ model. At zero field its density is $[1-\tanh (\beta J/2)]/2$, positive at every nonzero temperature. Walls occur in pairs on a periodic ring, but that global parity restriction does not remove their nonzero density in the thermodynamic limit. The low-temperature correlation length is approximately $a{e}^{\beta J}/2$.

The Ising model is a spin model with an Ising spin ${\sigma }_v\in \{-1,1\}$ at every graph vertex $v$. For a finite graph with vertex set $V$ and edge set $E$, a uniform coupling $J$ and field parameter $h$ give the HamiltonianEach edge is counted once. Here $T>0$ is temperature, $k_B$ is the Boltzmann constant, and $\beta$ is inverse temperature; $J$ and $h$ have units of energy. For magnetic applications, $h$ includes the magnetic moment multiplying the applied magnetic field. The canonical ensemble assigns the Boltzmann distributionThe partition function $Z$ makes the probabilities sum to one. Positive $J$ rewards equal neighboring Ising spins, modeling ferromagnetism; negative $J$ rewards opposite neighbors, giving an antiferromagnetic Ising model. Positive $h$ rewards $+1$, while negative $h$ rewards $-1$.^[ref]

A single local rule can organize a whole graph: rewarding agreement on each edge makes all-equal Ising spin patterns favorable even though no edge directly couples distant vertices. The four-spin Ising ring makes this visible with only sixteen microstates. It also separates three different questions: whether neighbors agree, whether the mean spin magnetization is nonzero, and how large its thermal fluctuations are.

A motivating experiment is the University of Colorado Boulder's Curie-temperature demonstration. An iron rod suspended from a twisting wire on a stand is initially held near a permanent magnet by ferromagnetism. A gas burner supplied from a propane cylinder heats the rod; the readout is its motion away from the magnet, followed by return as it cools. Repeated cooling supplies a reversible comparison on the same sample. The source explicitly calls for heating the rod while keeping the magnet cool, since heating the magnet can reduce its strength. The inference is that warming the sample weakens its magnetic attraction across the Curie temperature. This is a qualitative force readout, with no calibrated temperature or magnetization trace supplied: it motivates competition between alignment and thermal fluctuations, but does not identify $J$, measure individual Ising spins, or realize a four-spin Ising ring.^[ref]

Ernst Ising's 1925 paper examined a linear chain of two-orientation magnetic elements with neighboring interactions. In its calculation for a large chain, the zero-field mean magnetization vanishes and the model has no spontaneous magnetization. The inspected English translation states this immediately after equation (8) in §2. This historical result concerns the chain; it does not establish the behavior of a two-dimensional Ising model.^[ref]

The Ising ring bond correlation is the neighboring two-point correlation function $c_1=⟨{\sigma }_i{\sigma }_{i+1}⟩$ in the four-spin Ising ring. All four edges are equivalent. At zero field each mean Ising spin is zero by spin inversion symmetry, so $c_1$ also equals the connected correlation function. Agreement gives bond product $+1$ and disagreement $-1$; thus $P({\sigma }_i={\sigma }_{i+1})=(1+c_1)/2$.

Differentiating the finite partition function with respect to $K=\beta J$ inserts the sum of four bond products. ThereforeHere $U$ is internal energy. At $K=(\ln 2)/4$, the class-weight calculation gives $c_1=(4-1)/17=3/17$ and $U=-12J/17$. The agreement probability for a selected bond is $10/17$, despite the exactly zero mean spin magnetization.

A second derivative gives the energy variance, and hence the heat capacity $C_H=dU/dT$ at fixed $J$ and $h=0$:To see the variance identity directly, a derivative of an expected value under weights $e^{KS}$ inserts $S-⟨S⟩$, where $S={\sum }_i{\sigma }_i{\sigma }_{i+1}=-H/J$. The factor $K^2$ follows from $dK/dT=-K/T$. At the same exact example, $\mathrm{Var}(H)/J^2=1216/289$. The heat capacity vanishes both as $K\to 0$ and as $K\to \infty$: at high temperature its dimensionless factor $K^2$ vanishes, while at low temperature excited microstates become exponentially rare.

Define the instantaneous spin magnetization $m=\frac{1}{4}{\sum }_i{\sigma }_i$. Global reversal sends $m$ to $-m$ without changing the zero-field energy. Pairing every microstate with its reversed partner proves $⟨m⟩=0$ at every positive temperature, even when almost all probability lies on the two fully aligned patterns. A vanishing mean does not imply weak alignment: the limiting probability distribution is half at $m=+1$ and half at $m=-1$.

The Ising ring magnetization fluctuations measure this distinction through variance. In the $d=0$ class, both patterns have $m^2=1$. Among the $d=2$ patterns, eight have $m^2=1/4$ and four have $m^2=0$; the two $d=4$ patterns also have $m^2=0$. ConsequentlyIt equals $6/17$ at $K=(\ln 2)/4$, approaches $1/4$ for four independent fair Ising spins at $K\to 0$, and approaches $1$ as $K\to \infty$. Unlike the energy variance, the spin magnetization variance remains large in the low-temperature symmetric mixture.

The same thermal fluctuation controls the field response. Allow $h$ to vary before setting it to zero; differentiating the Boltzmann distribution inserts $\beta {\sum }_i{\sigma }_i=4\beta m$, givingThis is a spin-normalized magnetic susceptibility with respect to the energy-valued parameter $h$; conversion to physical magnetization and magnetic field requires the magnetic moment and sample volume factors.

For finite $N$, $Z$ is a finite positive sum of exponential functions. Its free energy $-k_{B}T\log Z$ is a real analytic function for $T>0$, so the smooth heat capacity peak here is not a critical phase transition. Spontaneous magnetization concerns the order of limits: first a thermodynamic limit, then removal of a symmetry-breaking field. The four-spin Ising ring alone establishes neither that limit nor any two-dimensional Ising model result.^[ref]

An ideal gas with constant heat capacity ratio $\gamma$ has $U=n{c}_{V,m}T$ up to a consistent thermal-energy reference, and $c_{V,m}=R/(\gamma -1)$. Combining this with $pV=nRT$ gives the displayed thermal internal-energy density. At fixed pressure it does not grow with temperature: the particle density falls as $1/T$. A fixed-volume space exchanging gas with the atmosphere therefore differs from a sealed fixed-particle system. Composition changes, activated internal modes or a pressure change require a modified comparison.

Consider an ideal-gas cycle with constant heat-capacity ratio $\gamma >1$: expand isobarically from $V_1$ to $r{V}_1$, cool isochorically, then compress reversibly and adiabatically to the initial state. With $P_{\ast }=p_1{V}_1=nR{T}_1$, closure gives $T_2=r{T}_1$ and $T_3=r^{1-\gamma }{T}_1$. The heat input is $Q_h=\gamma P_{\ast }(r-1)/(\gamma -1)$ and the rejected heat magnitude is $Q_c=P_{\ast }(r-r^{1-\gamma })/(\gamma -1)$. Thus $\eta =1-(r-r^{1-\gamma })/[\gamma (r-1)]$. Reservoir temperatures determine feasibility and entropy generation but need not coincide with the gas temperatures at heat-exchange endpoints.

At fixed pressure and fixed composition, density varies as $T^{-1}$. With temperature-independent heating per particle, low-density cooling per particle scales as $T^{-3/2}{e}^{-\theta /T}$ and is restoring for $T<2\theta /3$. For constant heating per volume instead, the cooling scales as $T^{-5/2}{e}^{-\theta /T}$ and the threshold is $2\theta /5$. The heating normalization and perturbation constraint must be specified.

At fixed density below critical density, optically thin two-level cooling scales as $\mathcal{C}\propto T^{-1/2}{e}^{-\theta /T}$, with $\theta =\Delta E/k_B$. Under temperature-independent heating, $d\ln \mathcal{C}/d\ln T=-1/2+\theta /T$ gives restoring thermal balance for $T<2\theta$ and instability for $T>2\theta$. This boundary assumes fixed ionic abundance and does not describe a constant-pressure perturbation.

An isolated system exchanges neither matter nor energy with its surroundings. Its total energy is fixed, while the Second law of thermodynamics requires its total entropy to be nondecreasing. Its internal energy alone is fixed when bulk kinetic energy and external potential energy also remain unchanged. For example, an initially spinning stirrer inside an insulated rigid vessel can lose kinetic energy while the contents gain internal energy, with no external input.

An isolated system need not yet be in thermodynamic equilibrium. Two enclosed bodies at different temperatures can exchange heat internally until they reach thermal equilibrium; their combined energy stays constant. Their internal transfer is absent from the outer boundary ledger. A vacuum-insulated container is only an approximation to isolation: residual thermal conduction and thermal radiation can still carry energy across its boundary.

An isotherm is a curve of equilibrium states at fixed temperature, often drawn in the pressure-volume plane. For an ideal gas of fixed amount it obeys $pV=\mathrm{constant}$. The curve is not itself a specification of the rate, reversibility or experimental mechanism by which the states are visited.

An isothermal atmosphere has constant temperature. Under constant gravity and ideal-gas hydrostatic equilibrium, its pressure and density decay exponentially with altitude.