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True thermal equilibrium requires a thermal ionizing radiation field as well as chemical balance; equality of photoionization and recombination rates alone would require a specified external radiation field. Under the stronger equilibrium assumption, apply the Saha equation to pure hydrogen. The electron spin factor cancels the ground-state electronic degeneracy, givingAt $T=10^4\mathrm{K}$, $k_{B}T=0.862\mathrm{eV}$ and $S=3.38\times 10^{20}{\mathrm{m}}^{-3}$. Charge neutrality gives $n_e=n_p$. Importantly, the known quantity from d is the neutral number density, not the total number density; use Saha neutral fraction from a measured neutral density to obtain $n_p=\sqrt{S{n}_{\mathrm{HI}}}=5.31\times 10^7{\mathrm{m}}^{-3}$. ThereforeThis extraordinarily small neutral fraction is the result of applying thermal equilibrium to dilute intergalactic gas; it is not a realistic inference from photoionization equilibrium with a nonthermal cosmic background. For the axial geometry of d, the equilibrium neutral fraction is smaller by $\sqrt{20}$.

Adopt the stipulated neutral fraction $f_{\mathrm{HI}}=10^{-6}$, independently of the equilibrium estimate in e. For pure hydrogen, $n_e=(1-f_{\mathrm{HI}})n_{\mathrm{HI}}/f_{\mathrm{HI}}\simeq 8.3{\mathrm{m}}^{-3}$. The thermal Sunyaev-Zeldovich effect measures integrated electron pressure:Using the same absorbing line of sight as in d,Thus one such filament is below the stated measurement sensitivity. If the number density inferred transversely is instead used to view the same cylinder end-on, its path is twenty times longer and $y=2.9\times 10^{-10}$, still below the stated sensitivity. Changing the assumed path of the original absorption measurement while keeping its column density fixed does not change its predicted $y$. The quoted neutral-fraction floor is a premise of this estimate, not a universal bound derivable from thermal equilibrium.

The Jeans equations express local momentum conservation for a collisionless stellar system, just as the Euler equations do for a gas. The mean streaming velocity plays the role of fluid flow, the gravitational force is a body force, and random stellar motions supply a pressure tensor through $\rho {\sigma }_{ij}^2$. Taking moments smooths over individual stellar orbits and produces fields of density, mean velocity and random-motion stress; this is the sense of “stellar hydrodynamics”.

Ordinary gas pressure is usually nearly isotropic because collisions relax its local velocity distribution. A collisionless stellar system can retain an anisotropic velocity ellipsoid, so its pressure tensor is not generally a scalar times the identity. There are more unknown second moments than momentum equations, and their evolution introduces higher moments. The Jeans equations therefore require further dynamical assumptions or a specified one-particle distribution function for closure; they do not by themselves establish local thermal equilibrium or a gas equation of state.

Use the energy-density convention of part (a), a normalized common spectral line profile $\int \varphi (\nu )d\nu =1$, and isotropic spontaneous emission. For a narrow spectral line centered at ${\nu }_0$,The factor $1/c$ follows from $u_{\nu }=4\pi J_{\nu }/c$: angle-integrated absorption removes $c{\kappa }_{\nu }{u}_{\nu }$ of energy per volume and time. The emission coefficient is per solid angle, whereas the absorption coefficient has units of inverse length.

The Boltzmann distribution in thermal equilibrium gives $n_2/n_1=(g_2/g_1)e^{-h{\nu }_0/(k_{B}T)}$. Dividing the two expressions and using the relations between Einstein coefficients,Thus the source function equals the Planck function at the transition frequency, and in the usual narrow-line approximationWith an exactly common finite-width $\varphi$ and constant line-center Einstein coefficients, their ratio is constant across the profile; exact equality to a varying $B_{\nu }$ at every frequency is only a narrow-line approximation. Exact thermal radiative transfer requires frequency-resolved detailed balance, which supplies the corresponding small profile corrections. This distinction does not alter the intended equilibrium result.

evaporative cooling removes atoms selectively from the high-energy tail of a trapped gas. The remaining sample has less energy per atom; elastic collisions then redistribute energy and restore a thermal distribution at lower temperature. Random atom loss would generally remove average energy and would not provide the same cooling mechanism. Repeated energy selection and rethermalization, rather than one removal event, is the essential cycle.

Using the evaporation truncation parameter $\eta$, let the escape threshold be ${ϵ}_{\mathrm{cut}}=\eta k_{B}T$. As the gas cools, a fixed threshold is populated by fewer atoms, so evaporation stalls. In forced evaporative cooling the threshold is lowered during the sequence. In a magnetic trap, a radiofrequency evaporation knife flips trapped weak-field-seeking atoms into untrapped spin states where the Zeeman splitting matches the photon energy. Energetic atoms reach that resonance surface; sweeping the radiofrequency lowers the energy knife. In an optical dipole trap, lowering the trap depth offers a different implementation, although it usually also weakens confinement.

A simple evaporative cooling energy balance shows when atom loss is useful. For a classical gas in a three-dimensional harmonic trap of fixed frequencies, equipartition gives $E=3N{k}_{B}T$, counting both kinetic and potential energy. Suppose escaping atoms remove average energy ${ϵ}_{\mathrm{ev}}=(\eta +\kappa )k_{B}T$, with $\kappa$ a distribution-dependent excess above the threshold. For $dN<0$,Thus preferential removal cools when the removed energy exceeds $3k_{B}T$. More stringently, cooling must raise the phase-space density of an atomic gas, not merely lower the temperature:For this harmonic-trap phase-space density, with fixed geometric-mean trap frequency $\bar{\omega }$, $d\ln \mathcal{D}/d\ln N=1-3\alpha$. Therefore phase-space density rises during atom loss if $\alpha >1/3$. The numerical factors here belong to the stated harmonic-trap model, not every evaporative apparatus.

There must be enough elastic collisions to repopulate the tail: ${\Gamma }_{\mathrm{el}}\sim n\sigma \bar{v}$ should exceed damaging loss and heating rates, and the ramp should allow rethermalization. Increasing density can improve elastic collisions but also enhances inelastic and three-body losses. Very rapid removal can leave a nonequilibrium cloud; very slow removal exposes it to background-gas losses. Lowering an optical trap can reduce density and collision rate, so a falling temperature is not by itself evidence of efficient evaporation.

The dilute-gas condensation experiments made this tradeoff concrete. Eric A. Cornell and Carl Wieman's account explains how selective escape and contraction of a trapped cloud raise phase-space density, while the need for rapid two-body equilibration and slow three-body loss restricts usable density. It also describes expansion imaging as an experimental diagnostic.^[ref] Useful controls compare different knife ramps and hold times, measure both atom number and expansion temperature, and check trap-frequency calibration. Pure adiabatic expansion can cool without increasing phase-space density. Successful evaporation pays an atom-number cost to increase the remaining gas's phase-space occupancy.

Dynamic measurements explain which conformations a protein visits, how quickly it exchanges between them, and which transitions control biological activity. A static protein structure identifies possible contacts; protein dynamics can establish whether those contacts form before substrate binding, during enzyme turnover, or when a membrane protein switches between transport states. The useful comparison is therefore among conformational ensembles, transition rates and function under matched conditions.

Consider two interconverting conformations $A$ and $B$. For an ideal two-state system at thermal equilibrium, detailed balance gives $p_A{k}_{AB}=p_B{k}_{BA}$, soHere $G_A,G_B$ are molar Gibbs free energies, $R$ is the molar gas constant, and $T$ is temperature. Measuring only the population ratio cannot identify the exchange timescale: multiplying both rates by the same factor preserves that ratio. Conversely, a measured relaxation rate without populations does not determine the individual forward and reverse rates. Under this model, if $p_B=0.1$ and $k_{\mathrm{ex}}=100{\mathrm{s}}^{-1}$, then $k_{AB}=10{\mathrm{s}}^{-1}$ and $k_{BA}=90{\mathrm{s}}^{-1}$. These illustrative values are a consistency check, not measurements of a particular protein. A driven ATP-consuming cycle need not satisfy detailed balance around the entire cycle.

NMR spectroscopy is particularly powerful for a soluble globular protein. Residue-resolved chemical shifts report local environments; NMR relaxation dispersion measures exchange contributions to line broadening and can detect low-population states exchanging on microsecond-to-millisecond timescales. Different experiments probe faster local fluctuations or slower exchange. Analysis can estimate populations, rates and chemical shift differences, allowing a minor state's environment to be compared with a known ligand-bound state. dihydrofolate reductase provides a concrete example: relaxation-dispersion measurements identified excited conformations resembling states used elsewhere in its catalytic cycle.^[ref] This supports access to functionally relevant conformations before the corresponding chemical step; it does not establish that every detected motion accelerates the bond-making reaction.

This distinction helps test conformational selection against induced fit. If an unbound protein already samples a binding-competent conformation and the ligand stabilizes it, conformational selection contributes. If substantial rearrangement follows initial binding, induced fit contributes. Both can occur in the same mechanism. Measure binding and structural signals over a range of ligand concentrations, include independently measured exchange rates, and fit explicit kinetic models. A matching chemical shift alone does not establish pathway flux, and different models can fit a single relaxation trace. Combining equilibrium populations, rapid-mixing kinetics and site-directed mutagenesis can constrain which route dominates under the tested conditions.

single-molecule Förster resonance energy transfer reveals heterogeneity concealed by ensemble averaging. For a calibrated label pair, the ideal distance dependence iswhere $R_0$ is the Förster distance. Distinct efficiencies and transitions between them can reveal domain closure, opening and dwell times; $r=R_0$ gives $E=1/2$. A distribution of efficiencies is not automatically a distribution of structural states: dye orientation, blinking, label flexibility and photobleaching also contribute. Use donor-only and acceptor-only samples, alternating excitation where appropriate, activity measurements after labeling, and more than one label geometry. For a single memoryless exit with rate $k$, the dwell survival is $e^{-kt}$ and the mean dwell is $1/k$; missed fast events and multiple hidden states violate that simple interpretation.

These measurements connect protein dynamics to allostery. A ligand at one site can change the populations or exchange barriers at a distant functional site, even if the mean protein structure changes little. Measure both sites, compare their thermodynamic coupling and test whether an interface mutation changes coupling while preserving protein folding. hydrogen-deuterium exchange mass spectrometry identifies changes in protection across a protein and is useful for large assemblies. Interpretation needs matched pH and temperature, correction for exchange during analysis, and recognition that protection combines solvent access, hydrogen bonds and structural opening. A protected region can be an interface or a stabilized internal region; it is not direct proof of contact.

For membrane proteins, the environment is part of the experiment. micelles of detergents, lipid bilayer nanodiscs and lipid vesicles can stabilize different conformational ensembles. NMR spectroscopy, electron paramagnetic resonance distance measurements and single-molecule Förster resonance energy transfer can compare cytoplasmic domain movements in these environments. In the beta-2 adrenergic receptor, combined fluorine NMR spectroscopy and double electron-electron resonance showed that agonist occupancy did not fully stabilize one active conformation; interaction with a G protein or a mimetic further favored the active state.^[ref] This explains how ligands with similar binding can differ in signaling efficacy: binding occupancy and population of a productive signaling conformation are distinct quantities.

Ion channels provide an unusually direct connection between molecular switching and function. Patch clamp records individual openings and closures, permitting measurement of conductance, open probability, dwell times and responses to membrane potential or ligand. For $N$ identical independent channels with single-channel current $i$ and open probability $P_o$, the mean current is $Ni{P}_o$. A changed whole-cell current could therefore reflect channel number, conductance or gating; single-channel measurements help separate them. Correlating a conformational probe with electrical activity can assign a structural motion to opening, but a multi-state gating model is usually more realistic than simply equating all structural fluctuations with current-producing transitions. Control membrane integrity, recording bandwidth, label effects and the relevant electrochemical gradient.

For a carrier protein, structural dynamics tests alternating access: the substrate-binding region becomes accessible successively from opposite sides without maintaining an open pore across the lipid bilayer. Measure conformational exchange together with uptake into lipid vesicles, varying substrate, ATP or the driving electrochemical gradient. A conformation trapped by a mutation may be an on-pathway intermediate, an off-pathway dead end or a misfolded state; compare binding, transport and the driving-energy reaction. cryo-electron microscopy classification supplies candidate structural states, but particle-class fractions can be biased by preparation and classification and do not directly provide transition rates. molecular dynamics can suggest connecting routes and transient interactions; predictions must reproduce observables and are limited by sampling and the physical model.

The strongest functional inference comes from perturbing a particular transition and measuring the resulting activity. Preserve abundance, localization, protein folding and binding where possible, then ask whether the measured rate change predicts the functional change. Slow reaction product release may determine enzyme turnover even when chemical conversion is fast; a membrane protein can bind a substrate normally while failing to complete its transport cycle. Biophysical dynamics turns a structural catalogue into a mechanism when state populations, transition rates and activity constrain the same model.

At thermal equilibrium, detailed balance equates the opposing transition fluxes: $k_f{p}_A^{\mathrm{eq}}=k_b{p}_B^{\mathrm{eq}}$. Define the state free energy difference explicitly by $\Delta G_{AB}=G_B-G_A$. The equilibrium Boltzmann distribution then givesAny state degeneracy is included in the state free energies. Detailed balance is stronger than simply requiring a time-independent total occupancy: it equates each pair of equilibrium transition fluxes.

Source illustration TODO: the source's reaction coordinate diagram is missing. The definition above fixes the sign independently; no barrier heights or ordering of drawn energy levels are inferred. Below, a large forward free-energy drop means $G_A-G_B\gg k_{B}T$. If instead $\Delta G_{AB}=G_B-G_A\gg k_{B}T$, forward transitions are unfavorable: for example $\Delta G_{AB}=10k_{B}T$ gives $k_f/k_b=e^{-10}$, so neglecting reverse transitions would be wrong.

Assume thermal equilibrium, nondegenerate carriers, and isotropic parabolic band dispersions, with the supplied effective masses interpreted as density-of-states masses. Take one effective valley with two electron spin states for each band; any additional valley multiplicity must instead be absorbed into the masses or explicit degeneracy factors.

Let the conduction band edge be $E_c$, the valence band edge $E_v$, and $E_g=E_c-E_v$. The spin-summed conduction density of states per volume isUsing the nondegenerate limit of the Fermi-Dirac distribution and the substitution $x=(E-E_c)/(k_{B}T)$ givesThe factor $\sqrt{\pi }/2$ in the supplied integral produces the normalization. Counting missing valence band electrons similarly givesMultiplication cancels the chemical potential, establishing the semiconductor mass-action law:The result does not require an intrinsic semiconductor: semiconductor doping changes $n$ and $p$ separately while preserving this product under the stated equilibrium and nondegeneracy assumptions.

For a monatomic ideal gas, write the thermal energy density as $u=3n{k}_{B}T/2$. At low optical depth, radiative loss uses an astrophysical cooling function, often written $\mathcal{C}=n_e{n}_H\Lambda (T,Z)$, while $\mathcal{H}$ contains heating. At fixed density the net radiative gas cooling time is approximately $u/(\mathcal{C}-\mathcal{H})$ when the denominator is positive; a cooling flow closer to constant pressure must include enthalpy and compressional work. Comparing this time with the free-fall time of a uniform sphere determines whether pressure can be lost during gravitational contraction.

Hot ionized gas cools by bremsstrahlung and recombination; collisional excitation followed by photon escape supplies atomic and metal-line cooling at lower temperatures. molecular line cooling through rotational or vibrational lines and dust thermal emission become important in cold dense gas. Abundances, ionization state, density-dependent collisional de-excitation, and photon trapping all change the cooling function. Optically thick gas requires radiative transfer, not a simple local density-squared loss law.

Heating comes from photoionization, grain photoelectric heating driven by ultraviolet light, cosmic rays, shocks, turbulent dissipation and compression. photoionization heating deposits the photon energy above the ionization threshold as electron kinetic energy. A thermal equilibrium satisfies $\mathcal{H}=\mathcal{C}$; its stability depends on how the net loss changes under the relevant pressure or density constraint, not merely on equality. Different stable balances permit a multiphase interstellar medium. Spectral-line ratios constrain temperature and ionization, but interpreting them requires density diagnostics, interstellar extinction corrections and a radiation-field model.

The official notes, PDF pp. 24–27, handwritten pp. 17–20 obtain the photon number density by integrating the Planck distribution, explain the temperature and dilution scalings, and compare the interaction time with the expansion time for thermal equilibrium. They also correct the missing cube and factor-of-two error in the paper's printed photon number density.

The official notes, PDF p. 27, handwritten p. 20 derive the early-time photon count, its vanishing limit, and the failure of thermal equilibrium when the count is of order one. Two details need supplementing.

First, their constant $N_{\gamma 0}$ is the normalization obtained by extrapolating the early-time particle horizon formula to $z=0$. It is not the count within today's actual particle horizon. Let $N_0=N_{\gamma }(t_0)$ and let $r_{h0}$ be that horizon's physical radius. Using the paper's early-time approximation givesFor a present matter-dominated approximation, $r_{h0}\simeq 2c/H_0$. Alternatively, retaining both components in the flat model exactly, ${\Omega }_R=1/(2+z_{\mathrm{eq}})$, and integrating the particle horizon givesThe second boxed formula uses the exact two-component normalization and the radiation-dominated early-time limit. Its large-$z_{\mathrm{eq}}$ leading term agrees with the first expression and $r_{h0}\simeq 2c/H_0$.

Second, a particle horizon containing only order-one photons cannot support a smooth, locally representative radiation density: discrete-particle fluctuations are of the same order as the mean, and a single photon's direction is strongly anisotropic. Thus the fluid description assuming spatial homogeneity and cosmological isotropy is not self-consistent on that causal scale. This does not by itself disprove an ensemble-averaged cosmological symmetry; it limits the extrapolation of the smooth thermal equilibrium description.

Assume an ideal nondegenerate hydrogen gas in thermal equilibrium and chemical equilibrium, with ground-state atoms and charge neutrality. Let $n_p=n_e=X{n}_B$ and $n_H=(1-X)n_B$. The ground-state Saha equation, with the usual electron/proton spin factors cancelling the atomic ground-state degeneracy, isInvert it and use $n_B=n_{\gamma }/s=(0.24/s)(k_{B}T/(\hslash c){)}^3$. This givessoHere $m_e$ is the electron mass denoted $M_e$ in the source, and $0.24(2\pi {)}^{3/2}=3.78$. At $X=1/2$, the left side is two. At $k_{B}T$ of order $B$, the right side is extremely small because $1/s\sim 10^{-9}$ and $B/(m_e{c}^2)\ll 1$. A large exponential is needed to compensate, requiring $B/(k_{B}T)\gg 1$ rather than a temperature of order the binding energy.

Physically, there are enormously many photons per baryon, so even a small high-energy tail of the thermal photon distribution can maintain ionization. Recombination is delayed until that tail is sufficiently depleted. This explains the low recombination temperature without solving the equilibrium equation; actual cosmological recombination also eventually requires rate equations once equilibrium fails.

A photon travels at $c$, so its typical scattering interval isabout four minutes. The supplied universe age isThusA photon scatters extremely frequently compared with the expansion timescale, of order the cosmic age. The plasma is opaque and its photons are tightly coupled to the electrons rather than freely streaming over a cosmological distance. In rate language ${\Gamma }_{\gamma }=n_e{\sigma }_{e}c\gg H$, where $H$ is the Hubble parameter.^[ref]

This is a statement about transport and momentum coupling. Elastic Thomson scattering alone does not create or destroy photons and hence is not, by itself, proof that an arbitrary photon spectrum reaches complete blackbody thermal equilibrium. The interaction interval calculated here is the interval experienced by a photon; the interval per electron differs because there are many more photons than electrons. Even replacing the inconsistent supplied age by the standard age of order a year at this scale factor would leave scattering much faster than expansion.

Spatial homogeneity makes $\rho$ independent of position and makes the two-particle number density depend only on the separation. Isotropy removes its directional dependence. ConsequentlyHere $g_2$ is a radial distribution function, with the finite-$N$ normalization inherited from a; statistical independence is unnecessary. In particular, the translated pair in c has separation $\mathbf{R}$ for every $s$, so its $s$ integral is simply ${\rho }^2{g}_2(R,t)$.

Isotropy of the random velocity $\mathbf{c}=\mathbf{v}-\mathbf{u}$ makes its second velocity moment proportional to the identity. Taking its trace fixes the proportionality constant:Thus $\theta$ is the one-component random velocity variance; for unit mass in thermal equilibrium, $\theta =k_{B}T$. A Maxwell-Boltzmann velocity distribution is not required for the pressure tensor to be isotropic.

For the interaction contribution, the angular integral is $\int d\Omega {\hat{R}}_i{\hat{R}}_j=(4\pi /3){\delta }_{ij}$, by isotropy and its trace. With $d^{3}R=R^{2}dRd\Omega$, c becomesCombining the two pressure tensors gives the virial pressure from pair correlations:For a repulsive pair potential, the outward radial force is $-{\varphi }^{\prime }(R)\ge 0$. Since $g_2\ge 0$, the second term in $p$ is nonnegative, and positive when occupied separations feel nonzero repulsion. It expresses additional momentum transfer through interactions. The displayed integral assumes a differentiable pair potential and convergence; a hard core requires a contact formula or a limiting argument.

The long-wavelength limit in d is resonant heating. Most electrons screen the slowly varying spatial field. The noise nevertheless excites collective Langmuir waves, whose sharp temporal spectrum lies near $\omega =\pm {\omega }_{pe}$. Electrons with $v\simeq \omega /k$ remain in phase with a wave and exchange energy with it. A stable tail absorbs net wave energy through Landau damping. The heating is initially deposited in narrow tail regions of velocity space, even though the volume-integrated rate equals that in c.

The short-wavelength limit in c is stochastic heating in the terminology intended here. With $ϵ\simeq 1$, the imposed electric field is essentially unscreened. Its temporally uncorrelated kicks cause a random walk of electron velocity, so $⟨(\Delta v{)}^2⟩=2D_0\Delta t$. Electrons throughout the thermal population contribute to the resulting kinetic energy increase; no narrow collective wave peak selects a small tail population.

Both mechanisms are driven by white noise, and the formal condition $\omega =kv$ enters the stationary quasilinear diffusion calculation in both cases. The physical distinction is the sharp collective resonance in d versus the broad direct random forcing in c, rather than the mere presence of randomness. In collisionless theory this heating is an increase of the averaged kinetic energy; subsequent collisions can establish thermal equilibrium.

A quasi-Fermi level is the chemical potential parameter of an approximately thermalized carrier population within one energy band. Rapid intraband scattering can establish a Fermi-Dirac distribution in each energy band before slower interband recombination equilibrates their populations. Thus separate $E_{\mathrm{Fc}}$ and $E_{\mathrm{Fv}}$ describe a pumped population inversion even though the entire semiconductor is far from thermal equilibrium.

At a common positive temperature, the Fermi-Dirac distribution is strictly decreasing in $E-E_F$. ConsequentlyThe joint density of states is nonzero only above the band gap, soIn particular the quasi-Fermi-level splitting must exceed $E_g$. The upper boundary is transparency, where stimulated emission balances absorption; the lower boundary has zero joint density of states in this ideal three-dimensional parabolic model. The strict occupation comparison uses $T>0$; at exactly zero temperature occupations become step functions, and their supports should be checked explicitly, as in the next part.

Use $X_i=n_i/n_B$, so $X_D$ counts deuterium nuclei per baryon, not baryons bound into deuterium. For the radiative capture reaction, chemical equilibrium and the zero photon chemical potential give ${\mu }_D={\mu }_n+{\mu }_p$. Dividing the three nonrelativistic Maxwell--Boltzmann number densities removes all chemical potentials:Here $g_D=3$ and $g_n=g_p=2$. The binding convention in this PDF is a negative mass defect, $B_D=m_D-m_n-m_p<0$. The positive nuclear binding energy is ${\epsilon }_D=-B_D$, so the abundance is enhanced by $e^{-B_D/T}$, not by $e^{B_D/T}$.

Since $X_D/(X_n{X}_p)=n_B{n}_D/(n_n{n}_p)$, insert the baryon-to-photon ratio $n_B=\eta n_{\gamma }$ and the photon number density $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$. This givesAt equal nucleon mass $m_N$ in the prefactors, $m_D\simeq 2m_N$, the deuterium equilibrium abundance simplifies toThe dimensionless prefactor is about $8.14$. These expressions describe thermal equilibrium; residual primordial deuterium after nuclear reactions cease must instead be obtained from the reaction kinetics.

Use radiation-era $H=C_{\ast }{T}^2/m_{\mathrm{pl}}$, with $C_{\ast }=1.66\sqrt{g_{\ast }}$. The stipulated interaction rate is ${\alpha }_Y^4{T}^5/m_Y^4$, not ${\alpha }_Y^2{T}^5/m_Y^4$. Decoupling therefore givesAt the threshold $T_D=m_{Y\mathrm{nr}}$, the assumed $g_{\ast }\simeq 100$ is self-consistent with the stated high-temperature condition. Smaller masses decouple nonrelativistically, with $x_D=m_Y/T_D=(m_{Y\mathrm{nr}}/m_Y{)}^{1/3}\gg 1$.

For an thermal equilibrium species with zero chemical potential, $n_Y=2(m_Y{T}_D/2\pi {)}^{3/2}{e}^{-x_D}$, while $s=2{\pi }^2{g}_{\ast s}{T}_D^3/45$. ThusThe numerical coefficient before rounding is about $2.9\times 10^{-3}$ for $g_Y=2$ and $g_{\ast s}=100$. This is mass-dependent fermion freeze-out. It assumes the rate maintains chemical equilibrium until sudden cosmological particle freeze-out and that entropy is conserved afterwards. Elastic scattering alone would not justify the thermal equilibrium number-density estimate. For decoupling below the high-temperature range, changes in effective degrees of freedom modify the numerical estimate; the requested formula treats them as roughly constant.

Above the transition,for $g_{\ast }\sim 100$. The ratio increases during cooling, rather than at higher temperature. A GUT transition at $T_c\sim 10^{15}$–$10^{16}$ GeV occurs well above this scale, so the interaction is too slow to establish or maintain thermal equilibrium throughout $T>T_c$. In fact its integrated interaction probability from an arbitrarily high temperature down to $T_c$ is at most of order $T_{\mathrm{eq}}/T_c\ll 1$, since $dt=-dT/(HT)$.

An initial Fermi-Dirac distribution with zero chemical potential therefore cannot be assumed from this high-temperature rate alone. If imposed initially and collisionlessly redshifted while the particles are massless, its thermal-looking form can persist, but this is not dynamically maintained thermal equilibrium. Other initial states can remain nonthermal. This is the thermalization temperature for a massless gauge mediator.

The below-transition rate stipulated in the model has $\Gamma /H=(T/T_D{)}^3$. If $T_c\gg T_D$ and it includes efficient number-changing reactions, it can restore chemical equilibrium after the transition and erase the initial-state memory, making the estimates in a and b applicable. If it only equilibrates momenta, or if reheating never populates the relevant range, the thermal relic calculation must be reconsidered.

The baryon asymmetry is an excess of net baryon number over zero, conveniently measured by ${\eta }_B=(n_B-n_{\bar{B}})/n_{\gamma }$. Ordinary matter contains abundant baryons, whereas no comparable primordial antibaryon population is observed. Large nearby domains of matter and antimatter would produce radiation through particle-antiparticle annihilation at their boundaries. Antiparticles produced in energetic collisions are not evidence for a cosmological antimatter domain. The small nonzero baryon-to-photon ratio inferred from Big Bang nucleosynthesis is independently tested by baryon-density consistency between nucleosynthesis and the microwave background.

Starting with zero net baryon number, the Sakharov conditions require baryon-number violation, C and CP violation, and departure from thermal equilibrium. Conservation of baryon number would keep the initial value zero. C symmetry or CP symmetry would pair baryon production with equal antibaryon production; CP rate cancellation in baryogenesis explains why C violation alone does not suffice. In thermal equilibrium with zero imposed charge chemical potentials, CPT symmetry forbids the required net baryon number. An asymmetric rate in one reaction must not be mistaken for net production after all reverse reactions are included.

A simple example is heavy-boson decay baryogenesis. Take equally many heavy $X$ bosons and their antiparticles, with decay channelsThe symbols in parentheses are branching fractions. The two $X$ channels carry baryon numbers $2/3$ and $-1/3$, so both cannot conserve one assigned baryon number for $X$. Their mean baryon number is $r-1/3$, while the mean for $\bar{X}$ is $-\bar{r}+1/3$. Consequently,where $ϵ$ is the asymmetry per decaying particle. CP violation permits $r\ne \bar{r}$ without violating the equality of total decay widths required by CPT symmetry. Microscopically a direct CP asymmetry requires interference between decay amplitudes with both relative weak phases and absorptive phases; specifying complex couplings alone would not establish unequal partial decay widths.

For radiation domination, a sufficient decay condition is ${\Gamma }_X\lesssim H(T=M_X)=1.66\sqrt{g_{\ast }}{M}_X^2/m_P$. The heavy population can then survive as the equilibrium abundance drops, before decaying while inverse production is suppressed. Its actual abundance differs from its thermal equilibrium value. If subsequent baryon-number washout is negligible and cosmological entropy conservation holds, an initial total abundance $Y_X=(n_X+n_{\bar{X}})/s$ producesHere $s$ is the entropy density. Residual baryon-number washout multiplies this expression by an efficiency factor below one. The example realizes all three Sakharov conditions; it illustrates a mechanism, rather than identifying the actual cosmological source of the baryon asymmetry.