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This circular lensing potential has reduced deflection $\alpha (r)={\theta }_{E}r/\sqrt{r^2+{\theta }_c^2}$, mean lensing convergence $\bar{\kappa }={\theta }_E/\sqrt{r^2+{\theta }_c^2}$ and local convergence $\kappa ={\theta }_E({\theta }_c^2+r^2/2)/(r^2+{\theta }_c^2{)}^{3/2}$. Its central convergence is finite for ${\theta }_c>0$. As the core shrinks to zero, the deflection tends to that of a singular isothermal sphere lens. This softening of the projected lensing potential must not be confused with replacing a three-dimensional density by a different cored profile.

A point-mass gravitational lens is the ideal gravitational lens obtained by concentrating a positive mass $M$ at one location. Use geometrical optics, the weak-field approximation, small angles in radians, and a stationary lens whose deflecting region is short compared with the observer-lens and lens-source distances. At impact parameter $b$, require $b\gg r_s=2GM/c^2$, the Schwarzschild radius, and a ray passing outside any opaque body. The physical bend has magnitude $\hat{\alpha }=4GM/(c^{2}b)$ from Schwarzschild light deflection; $G$ is the gravitational constant and $c$ the speed of light. Multiple images can occur within this weak-field approximation: strong gravitational lensing does not require a ray to approach a black hole closely.^[ref]

Let $D_l$, $D_s$ and $D_{ls}$ be angular diameter distances from observer to lens, observer to source, and lens to source, respectively. They refer to the background geometry used in the thin gravitational lens equation. In nearby Euclidean geometry, $D_{ls}=D_s-D_l$; in cosmology, do not make that subtraction. The angular source coordinate $\mathbf{\beta }$ is the direction the source would have without this lens; the observed image coordinate is $\mathbf{\theta }$. Both are measured relative to the observer-lens optical axis.^[ref]

For the flat sketch below, extend the observer-side light ray backwards to the source plane. Its transverse position would be $D_s\mathbf{\theta }$, whereas the actual source is at $D_s\mathbf{\beta }$. Their separation is $D_{ls}\hat{\mathbf{\alpha }}$. Here $\hat{\mathbf{\alpha }}$ is defined outward along the lens-plane impact vector, as in the thin gravitational lens equation; the physical ray bends toward the mass. Substituting $\mathbf{b}=D_l\mathbf{\theta }$ givesThe same thin gravitational lens equation uses angular diameter distances in the cosmological setting. The factor $D_{ls}/D_s$ converts the physical bend into the reduced angular deflection; these are different angles.

A passive astrophysical disk receives most of its heat from external irradiation rather than internal viscous dissipation. For a stellar disk, a rough criterion is $GM\dot{M}/R_{\ast }\ll L_{\ast }$. The actual local requirement compares absorbed stellar flux with viscous flux; incidence angle, shadowing and absorption can invalidate a luminosity comparison alone.

The admissible rates obey ${\varphi }_{,F}:\dot{F}=h^{\prime }(\theta )\dot{\theta }$. A reaction multiplier $q$ in the reversible nominal stress tensor therefore has an accompanying specific entropy contribution:Their reaction contributions cancel in the Clausius-Duhem inequality on admissible rates. Omitting the entropy term would generally violate the constraint's power balance.

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.

A closed system exchanges no matter with its surroundings, but can exchange energy through heat or work. A sealed cylinder with a moving piston is a typical idealized example. “Closed” fixes the material inventory, rather than the volume or the internal energy. An insulated cylinder can receive compression work; a rigid sealed vessel can receive heat. Neither is necessarily an isolated system.

A thermodynamic system is in thermodynamic equilibrium when its macroscopic state has no spontaneous tendency to change through any exchange allowed by its constraints. For an isolated system, stable thermodynamic equilibrium maximizes entropy over the accessible states at fixed total energy and conserved material inventories. A wall can forbid an exchange, so always specify what can cross or move. Thermal, mechanical and chemical conditions answer different questions about that same constrained state. ^[ref]

For example, two compartments separated by a rigid, impermeable wall that transmits heat can reach thermal equilibrium while retaining different pressures and compositions. Allowing an unloaded piston to move adds a mechanical equilibrium condition. Allowing one chemical species to cross adds chemical potential equality for particle exchange; allowing a chemical reaction adds its chemical equilibrium condition. A fixed barrier can therefore support a constrained thermodynamic equilibrium that ceases to be an equilibrium when the barrier is removed.

Uniformity is not the definition of thermodynamic equilibrium. An isothermal atmosphere in hydrostatic equilibrium can have a common temperature while its pressure and mass density vary with height: the pressure gradient balances gravity. In an external field, the exchange condition must include the relevant potential energy; for charged matter this leads to the electrochemical potential. Relativistic gravitational corrections to thermal equilibrium are described separately by the Tolman temperature law.

A steady state only says that the macroscopic description is constant in time. A bar maintained between hot and cold thermal reservoirs has steady heat flux and produces entropy, so it is not in thermodynamic equilibrium. Even matching temperatures at one perfect-contact interface can coexist with heat transfer through the surrounding material; matching those two local values does not establish equilibrium of the whole apparatus. See thermal conduction for that spatial transport problem.

The useful simplification is that equilibrium conditions predict an endpoint without computing every collision or flow during relaxation. The first law of thermodynamics supplies the conserved quantities, and the Second law of thermodynamics supplies the direction of change. For a system held at fixed temperature and volume, the corresponding criterion is minimization of Helmholtz free energy; at fixed temperature and pressure, it is minimization of Gibbs free energy, with the material and exchange constraints still imposed.

A thermodynamic system for which the reversible mechanical work is entirely pressure-volume work. At fixed composition its internal energy is specified by entropy and volume; additional work modes require additional state variables.

A free energy is a thermodynamic potential that compares possible states under specified environmental constraints. For a closed system of fixed composition, treated as a simple compressible system with only pressure-volume work, stable thermodynamic equilibrium minimizes Helmholtz free energy at fixed temperature and volume, or Gibbs free energy at fixed temperature and pressure. The choice follows the experiment: a rigid sealed vessel and a sealed cylinder with a freely moving loaded piston impose different constraints. Both may exchange heat with a thermal reservoir. See W. C. Carter, MIT thermodynamics notes, sections 2-1.14 and 2-1.19.

The useful insight is that internal energy alone cannot rank these alternatives: exchanging heat also changes the surroundings' entropy. Subtracting $TS$ packages that accounting into a property of the chosen system. Adding $pV$ additionally accounts for displacing the surroundings at fixed pressure. A free energy is a state function; supplied heat describes a transfer during a process. Under the stated constraints, supplied heat is $Q=\Delta U$ at fixed volume or $Q=\Delta H$ at fixed pressure, whereas neither $\Delta F$ nor $\Delta G$ generally equals $Q$.

A lower free energy identifies a thermodynamic preference, without specifying a timescale. A barrier can trap a system in a local minimum, producing metastability although a lower global minimum exists. The enzyme example illustrates the corresponding distinction for chemical reactions: changing the route can change the reaction rate while preserving the endpoint Gibbs free energy difference and chemical equilibrium.

A phase fraction is the proportion of a mixture occupying a specified thermodynamic phase. State the basis: by mass, $x_{\beta }=m_{\beta }/{\sum }_{\alpha }{m}_{\alpha }$; by volume, divide the occupied volume by the total volume. These fractions need not agree when the mass densities differ. The latent heat heating curve uses the liquid fraction by mass, whereas liquid fraction in a mushy layer is defined by volume.

A symbolic potential is a real-valued function on a symbolic sequence space, for example $U:{\mathcal{A}}^{\mathbb{N}}\to \mathbb{R}$ on a finite full shift. Its orbit sumweights a length-$n$ orbit segment in a partition function. Here $\sigma$ is the left shift. The potential can encode geometric contraction or a chosen statistical preference; it is not determined by the alphabet alone.

Its variation at depth $r$ isSummable variations mean ${\sum }_{r\ge 1}{\mathrm{var}}_{r}U<\infty$. If $x$ and $y$ agree on their first $n$ symbols, their shifted sequences agree for $n-j$ symbols at step $j$. ThereforeThe right side stays bounded independently of $n$. This uniform orbit-weight distortion is what allows a reference tail to be changed without changing topological pressure.

For a potential depending only on $x_0$, every variation with $r\ge 1$ is zero. The choice $U(x)=\log p_{x_0}$ generates independent-symbol weights, while a geometric potential of an expanding map instead describes inverse-branch contraction.

The sequences $a_n+D$ and $a_n-D$ are respectively subadditive and superadditive. Apply Fekete's lemma to the first and its negative to the second; their normalized difference is $2D/n$, so their limits coincide. For full-shift weighted orbit sums, summable variations of the potential bound the energy change at a concatenation boundary, giving this estimate for the logarithm of the partition function. A fixed initial symbol or fixed remote tail changes only a bounded normalization and not the limit.

For a finite full shift with alphabet $\mathcal{A}$ and a continuous potential $\varphi$ of summable variations, topological pressure is the limiting logarithmic weight per step:Here $w\eta$ concatenates a finite word and a fixed reference tail, and $\sigma$ is the left shift. Summable variations bound the effect of changing that tail by a constant independent of $n$. The almost-additive partition-function limit consequently proves existence and tail independence.

For a fixed potential $U$, the function $\gamma \mapsto P(\gamma U)$ is convex. At finite $n$, differentiating $n^{-1}\log Z_n(\gamma U)$ twice gives the variance of the orbit sum under its normalized weights, divided by $n$, which is nonnegative. Passing to the limit preserves convexity.

If $-B\le U\le -b<0$, then for $\delta >0$ each orbit weight is multiplied by a factor between $e^{-nB\delta }$ and $e^{-nb\delta }$. HencePressure is therefore strictly decreasing and Lipschitz continuous. If the alphabet has $m\ge 2$ symbols, $P(0)=\log m>0$ and $P(\gamma U)\to -\infty$ as $\gamma \to +\infty$. The intermediate value theorem then gives exactly one positive zero. For a one-symbol alphabet the zero is instead at $\gamma =0$; the alphabet-size qualification is essential.

For example, the constant potential $U=-c$, with $c>0$, has $Z_n(\gamma U)=m^n{e}^{-n\gamma c}$ and $P(\gamma U)=\log m-\gamma c$. Its zero is $(\log m)/c$, making the role of alphabet size explicit.

A Gibbs measure gives probabilities controlled by exponential weights from a potential. The precise convention must be specified: finite-volume conditional distributions and symbolic orbit-weight bounds are related formulations with different hypotheses.

For a finite-alphabet full shift, a standard symbolic Gibbs condition for a probability measure $\mu$ is that a constant $C\ge 1$ satisfyfor every length-$n$ word $w$ and every $x$ in its cylinder set $[w]$. Here $S_n\varphi (x)={\sum }_{j=0}^{n-1}\varphi ({\sigma }^{j}x)$, and $P(\varphi )$ is topological pressure. The same constant must work for every word and every length. Summable variations make the orbit-weight comparison within one cylinder uniform. In applications to equilibrium states, shift invariance is also required.

For a one-symbol potential $\varphi (x)=\log p_{x_0}$, where every $p_a>0$ and ${\sum }_a{p}_a=1$, the independent-symbol law hasThe bound is exact with $C=1$. Equal symbol probabilities and unequal symbol probabilities are both Gibbs laws, although their geometric images can be very different: an arbitrary Gibbs measure need not be absolutely continuous.

An arbitrary unnormalized list of one-sided weights need not give consistent conditional probabilities for every finite horizon. Normalization and compatibility are part of constructing the measure, rather than consequences of calling the weights Gibbs.

Rojas Cardenas, Graur, Perrier and Méolans studied thermal transpiration in helium using a glass microtube of diameter $490\pm 1\mu \mathrm{m}$ and length $3.05\pm 0.01\mathrm{cm}$ in their 2012 experiment. They heated one reservoir, leaving the other near room temperature, and measured tube-end temperatures of $300$ and $351\mathrm{K}$. Three contact thermometers and an infrared camera checked the thermal conditions; the hot reservoir was warmer than the adjacent tube end, so heater temperature was not an adequate substitute for tube-end temperature. Two capacitance diaphragm gauges recorded the reservoir pressures; a third gauge supported calibration. ^[ref]

Their earlier 2011 paper explains the measurement sequence. A wide bypass links the reservoirs in parallel with the microtube. After isolating the gas supply and pump, keep the bypass open to equalize pressures with the temperature difference already established. Closing the bypass defines $t=0$. The hot-side pressure rises and the cold-side pressure falls; ultimately their slopes approach zero at unequal pressures. Repeating at different initial pressures changes the degree of rarefaction without changing the microtube. The 2011 study also compared helium and nitrogen and imposed temperature differences of $50$, $60$ and $70\mathrm{K}$. ^[ref]

The quantities to record are both $p_c(t)$ and $p_h(t)$, the reservoir volumes, and the temperatures relevant to their particle inventories. In an idealized reservoir of fixed volume $V_i$ and uniform fixed temperature $T_i$, the ideal gas law gives ${\dot{N}}_i=[V_i/(k_B{T}_i)]{\dot{p}}_i$, so opposite inventory changes test particle conservation. If temperature changes, differentiation instead gives ${\dot{N}}_i=(V_i/k_B)[{\dot{p}}_i/T_i-p_i{\dot{T}}_i/T_i^2]$: a thermal drift can imitate particle transfer. The authors' 2011 presentation explicitly checks this thermal correction when extracting an initial mass flow rate from the pressure slope. ^[ref]

The 2012 paper compares the final pressure ratio across near-free-molecular through slip conditions with a kinetic model and empirical corrections. It does not establish the thin-hole square-root law at every measured pressure. ^[ref] For a replication, useful additional controls follow from the interpretation: remove the temperature difference to test the null signal; reverse it to test the direction; check gauge zero offsets and leaks; and wait for stable temperatures before closing the bypass. These are suggested controls, not claims that the cited paper performed each one. A reproducible pressure split at zero net transfer makes the failure of a single continuum wall model observable, while the Knudsen number organizes which molecular and apparatus scales are competing.

The linear thermal expansion coefficient is the fractional change in a specified length per change in temperature, at fixed external pressure for a sample free to expand. For small changes, $\Delta L/L\simeq {\alpha }_L\Delta T$. An isotropic sample with the same response in all directions has volumetric coefficient of thermal expansion $\alpha =3{\alpha }_L$, since differentiating $V\propto L^3$ gives $dV/V=3dL/L$. Direction-dependent materials require a specified measurement direction. ^[ref].

Consider two rigid bodies forming an isolated system, with fixed composition and constant whole-body heat capacity at constant volume values $C_1=1\mathrm{kJ}{\mathrm{K}}^{-1}$ and $C_2=3\mathrm{kJ}{\mathrm{K}}^{-1}$. Their initial temperatures are $T_{1,0}=300\mathrm{K}$ and $T_{2,0}=400\mathrm{K}$. Assume each body remains internally uniform, their contact stores negligible energy, no phase transition occurs and no work is done or extracted. These are stipulated model values, not material measurements.

Because $d{U}_i=C_{i}d{T}_i$, write $U_i=a_i+C_i{T}_i$ with arbitrary fixed reference constants $a_i$. The conserved relative internal energy isThis is an energy reference convention, not a claim about actual heat capacities down to zero temperature. At thermal equilibrium the two temperatures equal $T_f$, so the first law of thermodynamics givesThe larger heat capacity pulls the common temperature closer to its initial value: body 1 warms by $75\mathrm{K}$, while body 2 cools by $25\mathrm{K}$.

To predict a timescale, stipulate a constant contact conductance $G=1/R_{\mathrm{th}}=0.1\mathrm{kJ}{\mathrm{s}}^{-1}{\mathrm{K}}^{-1}$, the reciprocal of the contact's thermal resistance. The rate of heat transfer into body 1 is $G(T_2-T_1)$, givingAdding proves conservation of $E$. Subtracting gives exponential decay of the temperature difference, with time constantand thereforeExact equality is approached as $t\to \infty$ in this ideal model. Changing $G$ changes the rate but not $T_f$: equilibrium and the speed of reaching it are separate predictions. Real bodies can have internal temperature gradients, changing heat capacities and contact losses, so the plotted $7.5\mathrm{s}$ is an assumed model timescale.

If a species and the remaining bath share a temperature at decoupling, their initial entropy ratio is the ratio of their entropy-weighted degrees of freedom. Subsequent conservation of each sector's comoving entropy keeps that fraction constant, even when the bath heats relative to the relic. For a two-state neutrino species leaving an ideal Standard Model bath, the fraction is $(7/4)/(106.75+7/4)\simeq 1.61\%$.

A finite-wavenumber convection instability with a resonant three-wave triad can yield three coupled amplitude equations. For the gradient nonlinearity $\nabla \cdot (|\nabla \Theta {|}^2\nabla \Theta )$ and weak quadratic asymmetry $-ϵ\gamma \nabla \cdot (\Theta \nabla \Theta )$, projection onto a triad of length squared $K$ gives the stated equation and its cyclic copies. The quadratic coefficient is $+\gamma K$, while both self and cross cubic coefficients are $-3K^2$ for angles of $120$ degrees. The finite critical mode requires positive linear damping in the underlying long-wave equation.

Convection driven primarily by buoyancy generated by heating or cooling, rather than an externally imposed mean flow. In a constant-flux unstable surface layer, the local speed scales as $(B_{0}z{)}^{1/3}$.