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Use the reduced attractive reduced BCS pairing Hamiltonian with volume V, ${\xi }_k={\epsilon }_k-\mu$ and attraction $\lambda >0$:A specified pairing shell or an ultraviolet renormalization is part of the model. The trace is grand canonical. Insert fermionic coherent states into its time-sliced trace. The resulting Grassmann variables are antiperiodic, and with inverse-energy imaginary time $0\le \tau \le \beta$ the dimensionless action isThis boundary condition is essential for Fermi-Dirac statistics; replacing it by a bosonic periodic trace changes the answer. A complex Hubbard–Stratonovich transformation uses the local identitywith time-slice factors understood. The pairing field renders the fermion action quadratic. For a uniform static field the independent Nambu spinor block $({\psi }_{k\uparrow },{\bar{\psi }}_{-k\downarrow }{)}^T$ hasIts determinant is $-({\omega }_n^2+E_k^2)$. The phase-independent determinant sign cancels under consistent reference normalization. Integrating the fermions gives the BCS pairing-field effective action, ${\mathcal{S}}_{\mathrm{eff}}={\int }_0^{\beta }V|\Delta {|}^2/\lambda d\tau -\mathrm{Tr}\ln {\mathcal{G}}^{-1}$, with the normal-ordering/reference term retained. The fermionic pairing-block determinant normalization uses one independent pairing block for every k; a fully doubled spin-particle-hole trace instead needs one half. The subtracted grand potential of the reduced BCS model is particularly unambiguous:Performing the Matsubara frequency sum yieldsup to a common constant. At zero gap this reduces exactly to the two-spin free-gas result. The Matsubara derivation of the BCS saddle follows by differentiating with respect to ${\Delta }^{\ast }$ and using ${\beta }^{-1}{\sum }_n({\omega }_n^2+E^2{)}^{-1}=\tanh (\beta E/2)/(2E)$ givesThe second equation fixes the chemical potential at specified number. This stationary action is BCS mean-field theory, not an exact integration of pairing-field fluctuations. With constant single-spin density of states ${\nu }_F$ per volume and a symmetric energy cutoff $\Lambda$, weak coupling gives $\Delta (0)\simeq 2\Lambda e^{-1/(\lambda {\nu }_F)}$ and $k_B{T}_c\simeq (2e^{{\gamma }_E}/\pi )\Lambda e^{-1/(\lambda {\nu }_F)}$. The zero-temperature condensation grand-potential density is $-{\nu }_F{\Delta }^2/2$ to leading weak-coupling order; using a spin-summed density would change these displayed factors. The BCS coherence factors are $u_k^2=(1+{\xi }_k/E_k)/2$ and $v_k^2=(1-{\xi }_k/E_k)/2$, and the associated BCS ground state is the normalized product of empty and occupied pair states. ^[ref]

Allowing a slowly varying complex pairing field generates phase stiffness and, near the transition, the Ginzburg-Landau theory of superconductivity. A neutral paired system has a gapless phase sound mode; in a charged bulk superconductor electromagnetic coupling produces the Anderson-Higgs mechanism in a superconductor and the Meissner effect. These collective fluctuations are distinct from the gapped fermionic excitations. A microscopic dynamical calculation is needed to fix collective-mode dispersion and damping; the static saddle alone does not determine them. ^[ref]

Use a real, even interaction $V_{-\mathbf{k}}=V_{\mathbf{k}}$. Let $\mathcal{V}=L^3$ and $N_{\mathrm{ex}}={\sum }_{\mathbf{k}\ne 0}{b}_{\mathbf{k}}^{†}{b}_{\mathbf{k}}$. In the Bogoliubov theory expansion, keep terms with zero or two noncondensed operators. The zero-momentum term gives $V_0{N}_0(N_0-1)/(2\mathcal{V})$. The terms with two noncondensed operators giveHere $V_0$ is the direct condensate interaction and $V_{\mathbf{k}}$ is its exchange contribution. At fixed total particle number, $N_0=N-N_{\mathrm{ex}}$, soThe second term cancels the direct normal term. Adding the kinetic energy yields the number-conserving Bogoliubov quadratic HamiltonianThe constant is the condensate mean-field theory interaction energy; the normal term describes moving a particle out of the condensate, while the anomalous term creates or annihilates an opposite-momentum pair by exchanging two particles with the condensate. Total particle number remains fixed in the underlying theory; the condensate amplitudes in this notation are treated as classical amplitudes.

For condensate fluctuation power counting, put $\delta =N_{\mathrm{ex}}/N\ll 1$. Formal interaction-amplitude counting gives $H_2-E_{\mathrm{mf}}\sim E_{\mathrm{mf}}\delta$, cubic terms $\sim E_{\mathrm{mf}}{\delta }^{3/2}$ and quartic terms $\sim E_{\mathrm{mf}}{\delta }^2$; these are not uniform bounds on individual modes. Thus the latter are down by $\sqrt{\delta }$ and $\delta$ relative to the quadratic interaction terms. The $N_0-1$ correction is relative $1/N$. Homogeneity eliminates terms linear in nonzero-momentum operators. Small depletion and weak interactions, not low temperature alone, justify the truncation. ^[ref]

Use SI units, uniform absolute permittivity $\epsilon$ and a neutral bulk reservoir. Local equilibrium of ideal mobile ions gives $n_i=n_{i,\infty }{e}^{-z_{i}e\psi /(k_{B}T)}$. Combining their charge with the Poisson equation yieldsThe reference is $\psi =0$ in the bulk. Boundary data specify fixed charge, fixed potential or a surface reaction/charge regulation law. These are physically different problems. The mean-field theory assumes point ions, thermal equilibrium and a continuum solvent; ion-size packing, dielectric structure and strong electrostatic correlations can invalidate it.

For a symmetric monovalent electrolyte let $n_{\infty }$ count each sign and define $u=e\psi /(k_{B}T)$, ${\kappa }_D^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$. Outside fixed sources,For $|u|\ll 1$, the Debye-Hückel approximation is ${\nabla }^{2}u={\kappa }_D^{2}u$. A planar wall bounding electrolyte in $z>0$ has $\psi ={\psi }_0{e}^{-{\kappa }_{D}z}$; assuming zero field on the other side, its charge is $\sigma =\epsilon {\kappa }_D{\psi }_0$. A sphere of radius $a$ and total charge $Q$ instead givesThe factors arise from geometry and boundary conditions; planar screening cannot be substituted unchanged for a small sphere. The weak-potential condition must hold where the potential is largest, particularly at the surface.

The nonlinear planar Poisson-Boltzmann profile is also exactly integrable. Multiply $u^{\prime \prime }={\kappa }_D^2\sinh u$ by $u^{\prime }$ and use $u,u^{\prime }\to 0$ at infinity:The sign selects a profile decaying to the bulk for either sign of surface potential. Integration givesFor weak $u_0$ this reduces to the linear solution. Strong surface potentials have a bounded far-field amplitude $4\gamma$, while the near-wall ion density still grows strongly. There is no need to permit the negative ion densities produced by an invalid extrapolation of a linearized density formula.

The same first integral yields the Grahame equation and its diffuse-layer differential capacitance per unit area:These describe the diffuse layer. A compact interfacial layer adds a voltage drop and can place its capacitance in series with $C_D$; a measured electrode capacitance need not equal the ideal diffuse contribution.^[ref]

For comparison, a planar counterion-only double layer next to a negative plane has positive counterions and no salt reservoir. Its profile is algebraic:Integrating $en(z)$ over $z>0$ gives $|\sigma |$, checking charge neutrality. This problem cannot be obtained by putting a finite reservoir density into the symmetric-electrolyte answer without changing its boundary conditions.

Applications include screening around synthetic colloids, diffuse-layer electrode capacitance and forces between charged plates. In planar geometry, the bulk-subtracted normal stress is constant:At the midpoint of identical plates, symmetry sets $E=0$, so the remaining excess ionic osmotic pressure determines their repulsion. A comparison with force or capacitance measurements should vary salt and surface potential independently, test charge regulation and adsorption, and distinguish compact-layer effects from failures of diffuse screening. Nonlinear Poisson-Boltzmann theory retains finite-potential Boltzmann populations, but still omits correlations and finite-ion-volume saturation.

Write $\epsilon ={\epsilon }_0{\epsilon }_r$ and take $n_0$ to be the reservoir density of each monovalent ion species. Equal electrochemical potentials in equilibrium give the Boltzmann ion densities in a symmetric electrolyte,Combining this with the electrostatic Poisson equation gives the planar Poisson-Boltzmann equation,The Debye length is ${\kappa }^{-1}$. If the symbol $n_0$ instead denotes the combined density of both species, the factor two is absent; the per-species convention above makes the normalization explicit. At $|e\psi |\ll k_{B}T$, the linearized planar screening solution obeys ${\psi }^{\prime \prime }={\kappa }^2\psi$. With $\psi (0)=\zeta <0$ and $\psi (\infty )=0$,

The positive ions are counterions and the negative ions are co-ions. At weak potential,They approach $n_0$ from above and below respectively, with equal first-order excess and deficit. For stronger negative potential, interpret the source's strong-potential comparison as a statement about $|e\psi |$, since $\psi$ itself is negative. Counterions become strongly concentrated near the wall, while co-ions are exponentially depleted and remain nonnegative. Linearization misses this asymmetry and would eventually predict unphysical negative co-ion density. Point-ion mean-field theory itself eventually needs finite-size and correlation corrections; it provides no steric cap at arbitrarily large counterion concentration.

For steady tangential electro-osmosis, with no imposed pressure gradient and negligible inertia, the Stokes equation is $\eta v^{\prime \prime }+{\rho }_{e}E=0$. Since ${\rho }_e=-\epsilon {\psi }^{\prime \prime }$,The conditions $v(0)=0$ and $v^{\prime }(\infty )=0$ giveThe printed constant speed is the bulk velocity outside the double layer, not the velocity at the no-slip wall. With negative $\zeta$ the positive counterion excess drives fluid along the applied field. The far-field value is the Helmholtz-Smoluchowski electro-osmotic velocity.^[ref]

For the finite channel, first model the added wall as an ideally uncharged insulating boundary with no normal field on its outer side: ${\psi }^{\prime }(d)=0$, while $\psi (0)=\zeta$. The one-charged-wall electro-osmotic channel then hasIntegrating the same force balance and enforcing $v(0)=v(d)=0$ givesFor $U>0$, the maximum occurs atIn that limit $v/U\simeq 1-x/d-e^{-\kappa x}$: the velocity rises rapidly from zero at the charged wall, reaches a maximum a few screening lengths away, then falls approximately linearly to zero at the neutral wall. One charged wall does not give a symmetric plug across a channel with no-slip at the other wall.

An alternative electrode condition $\psi (d)=0$ means a wall held at reservoir potential; it is not the same as zero surface charge. Under that convention,with $\cosh [\kappa (d-x_{\max })]=\sinh (\kappa d)/(\kappa d)$. Both boundary conventions have the same leading thin-double-layer profile above. An externally imposed pressure gradient would add a pressure-driven profile; none is specified here.

For dilute nonadsorbing uncharged polymers in equilibrium or sufficiently weak flow, the entropic polymer depletion near nonadsorbing walls suppresses chain configurations near each wall. A depletion layer of order the coil radius of gyration is expected at both walls, with a bulk-like central region when $d$ is much larger than the coil size. Strong confinement rounds this into a central enrichment. Neutrality removes a direct electrophoretic force, not excluded-volume entropy. Adsorption, concentration and strong-flow migration were not specified, so this is a conditional equilibrium sketch rather than a universal flowing-polymer profile.

The expansion is a local phenomenological description in a small scalar order parameter near a transition. Its coefficients summarize microscopic physics and generally depend smoothly on temperature. At zero field, spin-reversal Z2 symmetry requires even powers of magnetization; the conjugate field supplies the term $-hm$. The additive $F_0$ does not affect minimization with respect to $m$.

For the ordinary continuous-transition quartic model, $b>0$ ensures stability, and the quadratic coefficient changes sign, $a=a_{0}t+O(t^2)$ with $a_0>0$. If $b$ vanishes or is negative, stabilizing higher powers are required and the phase structure can change. Treating one uniform magnetization neglects spatial gradients, fluctuations and domains; the resulting exponents are predictions of mean-field theory.

Mean-field theory replaces fluctuating interactions with the average field produced by the other degrees of freedom. For an interacting spin system, write ${\mathbf{S}}_i=\mathbf{S}+\delta {\mathbf{S}}_i$ and neglect connected products $\delta {\mathbf{S}}_i\cdot \delta {\mathbf{S}}_j$. This reduces the many-body partition function to independent single-site factors coupled through a self-consistency equation for $\mathbf{S}$.

The approximation is most reliable when many effectively independent neighbors contribute, for sufficiently long-range interactions, or above the upper critical dimension away from the fluctuation-dominated critical region. It usually becomes least reliable near a continuous transition in low dimensions, where correlations extend over long distances. It can estimate ordering scales and reveal candidate broken symmetries while giving incorrect critical exponents or even a spurious transition. The two-dimensional isotropic model in the later parts is an especially important example of this limitation.

In mean-field theory all sites have the same normalized single-site distribution $e^{x\cos \theta }/Z_1(x)$. Consequentlywhere $\mathcal{L}$ is the Langevin function. Requiring the average used to construct the field to equal the average in that field givesSetSince $S=\tau x$, the self-consistency equation is exactly $\tau x=\coth x-1/x$. It also follows by stationarity of $F_{\mathrm{MF}}/N=z_{c}J{S}^2/2-k_{B}T\log Z_1(\beta z_{c}JS)$. This independent route checks the constant term and all neighbor-counting factors in the partition function.

Treat $\psi ,{\psi }^{\ast }$ independently in the Legendre transform in mechanics. Their canonical momenta areThe time-derivative terms cancel in ${\pi }_{\psi }\dot{\psi }+{\pi }_{{\psi }^{\ast }}{\dot{\psi }}^{\ast }-\mathcal{L}$, giving the Hamiltonian densityThe first-order time dependence makes the momenta constrained; this direct calculation still gives the canonical density. In physical terms this is the grand-canonical Hamiltonian of a Bose field, $K=H-\mu N$. The physical energy density is obtained by adding $\mu |\psi {|}^2$ back. A stationary field in this chemical-potential frame has the usual factor $e^{-i\mu t/\hslash }$ in the physical frame.

Complete the square:For uniform or periodic boundary conditions, or in the homogeneous bulk, both nonconstant terms are nonnegative and vanish simultaneously atThe ground states form a circle. A particular nonzero ${\psi }_0$ is changed by every nontrivial phase rotation, although the full family is invariant. Selecting one such mean-field theory state therefore exhibits spontaneous symmetry breaking of $U(1)$. This describes phase selection in the thermodynamic limit; an exact finite-system particle-number eigenstate need not itself have a nonzero field expectation.

The mean-field approximation replaces the fluctuating environment of each microscopic variable by a uniform average field. For spins, write $S_i=S+\delta S_i$ and drop products $\delta S_i\delta S_j$ between different sites. The resulting single-site problem can be solved independently, and its average spin must reproduce the assumed order parameter $S$: this is the self-consistency equation.

It is reliable when correlated thermal fluctuations about the mean are small, for example with many interacting neighbours, long-range interactions or high spatial dimension. Near a thermodynamic critical point the quantitative test is the Ginzburg criterion. For short-range scalar ordering the upper critical dimension is four, with logarithmic qualifications at four; a two-dimensional nearest-neighbour lattice is below this and its asymptotic critical region of a phase transition is not described accurately by simple mean-field theory.^[ref]

Order parameter and the free-energy landscape. In Landau theory, take a scalar magnetization component at fixed volume and composition. It is an order parameter: its nonzero spontaneous value distinguishes an ordered phase with ferromagnetism from the symmetric high-temperature phase. One first constructs a constrained order-parameter free energy at specified magnetization by summing over microscopic states compatible with it. An external magnetic field is an independent control, so it can also appear in that constrained function. For a total magnetic moment the conjugate term is $-BM$; if $M$ is a magnetization density, the total-energy term is $-VBM$, or $-BM$ in free-energy density. Minimizing over $M$ then yields the equilibrium thermodynamic potential as a function of the imposed field.

Thus treating both $M$ and $B$ as arguments of a Landau function does not treat them as independent natural variables of the already-minimized equilibrium free energy. The first is the fluctuating or constrained order parameter, and the second its conjugate field.

At zero field, reversal symmetry requires $F(M)=F(-M)$, excluding odd powers. Near a continuous transition, assume the homogeneous constrained free energy is analytic for small $M$, that the quadratic coefficient crosses zero linearly, and that the quartic coefficient is positive. ThenThe temperature $T_c$ is where the zero-field quadratic coefficient vanishes and spontaneous ordering begins in this approximation. Higher powers are negligible sufficiently close to it; spatial fluctuations are neglected in this mean-field treatment.

Stationarity gives $2M[b(T-T_c)+2d{M}^2]=0$. Above $T_c$, the only minimum is at zero. Below it, zero becomes a maximum and two minima appear:The two signs describe spontaneous symmetry breaking: they are the average magnetizations in the symmetry-broken pure phases. A strictly finite system at exactly zero field samples the two wells symmetrically and has ensemble average $⟨M⟩=0$. A spontaneous pure-phase value is defined by selecting a phase, for example taking the thermodynamic limit before removing a small field. This qualification distinguishes a minimum's location from the average over a symmetric two-well distribution.

Minimum free energy and the heat-capacity jump. At a nonzero minimum, $M^2=-b(T-T_c)/(2d)$. Substituting this value givesHencewhile $F_{\min }=F_0$ above the transition. Take the temperature derivatives only after minimizing, so the equilibrium order parameter is allowed to change with temperature. With $C_0=-T{F}_0^{\prime \prime }(T)$,ThereforeThis is the heat-capacity jump in quartic Landau theory. The entropy is continuous at $T_c$, so there is no latent heat. If $F$ denotes free-energy density rather than total free energy, these expressions denote heat capacity per unit volume with the same normalization.

Stabilizing a tricritical expansion. At a tricritical point, the quartic coefficient is tuned to zero at the same point where the quadratic coefficient changes sign. Reversal symmetry still excludes odd powers. The next available stabilizing term is thereforeA negative or vanishing leading sextic coefficient would not stabilize this truncation for large $|M|$; a higher positive term would then be needed and the present tricritical form would be insufficient. The ordinary quartic theory has a continuous transition for $d>0$, while a stabilized theory with $d<0$ can have a first-order transition. Their meeting at zero field is what makes the simultaneous tuning tricritical.

The tricritical minimum. At zero field and $T<T_c$, stationarity givesThe nonzero solutions are minima: their curvature is $2b(T-T_c)+30d^{\prime }{M}^4=8b(T_c-T)>0$. Put $t=T_c-T>0$. Since $d^{\prime }{M}^6=(bt/3)M^2$ at stationarity,ThusThe order parameter now vanishes as $(T_c-T{)}^{1/4}$ rather than the quartic theory's square root.

Heat-capacity singularity. Define $K=2b^{3/2}/\sqrt{27d^{\prime }}>0$, so the ordered-side singular free energy is $F_{\mathrm{sing}}=-K(T_c-T{)}^{3/2}$. Its derivatives areConsequentlyThis is the tricritical heat-capacity amplitude in a sextic Landau expansion. It is positive on the ordered side and diverges as $T\to T_c^{-}$, rather than remaining a finite jump. This value of the heat-capacity critical exponent is the mean-field result along the path keeping the quartic term zero; a detuning to $d>0$ ultimately restores ordinary quartic behavior. Fluctuation corrections to the mean-field theory are not included in the stipulated polynomial minimization.

Giorgio Parisi's 1979 work replaced a single replica order parameter by an infinite hierarchy in the mean-field theory of spin glasses. The claim concerns a thermodynamic model, rather than inequivalence of the labels attached to two finite laboratory copies.^[ref]

For equilibrium states with weights $w_{\alpha }$ and local means $m_i^{\alpha }$, define $Q_{\alpha \gamma }=N^{-1}{\sum }_i{m}_i^{\alpha }{m}_i^{\gamma }$. The distribution $P_J(q)={\sum }_{\alpha ,\gamma }{w}_{\alpha }{w}_{\gamma }\delta (q-Q_{\alpha \gamma })$ records their similarities. This state-based overlap distribution is distinct from the finite instantaneous spin overlap used in the triangle example.^[ref]

The liquid-gas critical point of the Van der Waals equation occurs where its two subcritical stationary points merge into a horizontal inflection point. The conditions are $(\partial p/\partial v{)}_T=({\partial }^{2}p/\partial v^2{)}_T=0$. The short calculation isDividing these relations gives $2/(v_c-b)=3/v_c$, henceSubstitution gives the Van der Waals law of corresponding states and the parameter-independent ratio $p_c{v}_c/(k_B{T}_c)=3/8$. That ratio belongs to this equation of state; reducing a real fluid's variables does not enforce it.^[ref]

An exact rearrangement of the reduced critical Van der Waals isotherm makes its flatness visible:The first nonzero departure near $x=1$ is $-3(x-1{)}^3/2$. On the critical particle volume approached from $t>1$, $(\partial q/\partial x{)}_{x=1}=-6(t-1)$ andThe divergent isothermal compressibility and cubic flattening explain why the critical region is mechanically soft. These are mean-field theory predictions; thermal fluctuations limit their use for real fluids very close to a thermodynamic critical point.

The Van der Waals equation is a simple real gas equation of state that can also describe a liquid-gas critical point. With particle count $N$ fixed, write the volume per particle as $v=V/N$:or equivalently $(p+a{N}^2/V^2)(V-Nb)=N{k}_{B}T$. Here $p$ is pressure, $T$ is absolute temperature, and $k_B$ is the Boltzmann constant. In SI, $v$ and $b$ have units ${\mathrm{m}}^3$ per particle, $a$ has units $\mathrm{Pa}{\mathrm{m}}^6=\mathrm{J}{\mathrm{m}}^3$, and $k_B$ has units $\mathrm{J}{\mathrm{K}}^{-1}$; the particle count is dimensionless. The pole at $v=b$ excludes smaller volume values from this model.

The $b$ correction represents excluded volume; the attraction term lowers pressure at fixed temperature and particle number density $1/v$. This is a mean-field theory, replacing detailed intermolecular structure by two parameters. At $v\gg b$ with $a/v^2\ll k_{B}T/v$, it approaches the ideal gas law $pv=k_{B}T$. It supplies a useful qualitative route from a dilute gas to phase coexistence, without a claim of quantitative accuracy for every fluid.^[ref]

Do not mix this particle convention with the molar volume convention elsewhere in this article. With amount of substance $n_{\mathrm{mol}}=N/N_A$, Avogadro constant $N_A$, and molar gas constant $R=N_A{k}_B$, one has $v_m=N_{A}v$, $b_m=N_{A}b$, and $a_m=N_A^{2}a$, so $p=RT/(v_m-b_m)-a_m/v_m^2$. The molar parameters have units ${\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$ and $\mathrm{Pa}{\mathrm{m}}^6\mathrm{mo}{\mathrm{l}}^{-2}$, respectively.

Johannes Diderik van der Waals introduced this equation of state in 1873, connecting molecular attraction with departures from the dilute ideal gas law. Its historical significance is the unified treatment of gases and liquids, rather than a universally exact fit.^[ref]

Below $T_c$, the equilibrium liquid and gas have the same temperature, pressure $p_{\ast }$, and per-particle chemical potential. For macroscopic bulk thermodynamic phases, neglect interface contributions. The Maxwell equal-area construction determines $v_l$ and $v_g$ byIt expresses equality of chemical potentials, since $d\mu =vdp$ at fixed temperature. Equivalently, the equilibrium Helmholtz free energy per particle follows a common tangent line between the two endpoint states, lying below the homogeneous continuation between them. The construction uses that mathematical continuation to compare endpoints; it is not a physical compression path through negative isothermal compressibility states.^[ref]

Here is a reproducible calculated example at $t=0.9$. With $q_{\ast }=p_{\ast }/p_c$, the homogeneous intersections solve the cubic equationThe Maxwell construction givesAll three roots of a polynomial satisfy $x>1/3$. Only the smallest and largest are the phase coexistence endpoints; the middle one has negative isothermal compressibility. Equal pressures at three admissible values of volume therefore do not imply three stable thermodynamic phases. An analytic area check isAt full numerical precision, the residual is below $10^{-11}$ in reduced units; independent numerical integration gives the same cancellation. At $t=0.9$, the spinodal points are instead $x\simeq 0.718597189$ and $1.528504964$.

The equilibrium plateau changes the phase fraction rather than the endpoint densities. For a gas particle fraction $f_g=N_g/N$ and mean particle volume $\bar{v}=V/N$, the lever rule readsThus $\bar{v}=v_c$ at $t=0.9$ gives $f_g\simeq 0.227219$: this is a particle fraction, not the volume fraction occupied by gas. Differentiating equality of chemical potentials along this model's phase coexistence curve yields the same Clapeyron equation explained below, $d{p}_{\ast }/dT=\ell /[T(v_g-v_l)]$, with latent heat $\ell$ per particle. Both endpoint differences vanish as the Van der Waals critical point is approached; the equation's critical limit must be taken as a ratio.

The Maxwell construction presumes a pure bulk fluid at thermodynamic equilibrium. Its endpoint equality does not specify how rapidly a real compression reaches equilibrium. A homogeneous metastable sample can persist while nucleation is delayed; the spinodal marks loss of local mechanical stability in this mean-field theory, rather than the equilibrium onset of condensation. Surface tension, finite sample size and impurities can affect the observed transition. These qualifications matter when comparing a mathematical Van der Waals isotherm with a visible liquid meniscus.

A hypothetical Van der Waals equation model illustrates positive and negative Joule-Thomson coefficient regions and a finite throttle that first warms, then cools. Use the molar convention already distinguished in Van der Waals equation:Here $v_{\mathrm{m}}$ is molar volume, $R$ the molar gas constant, $u_{\mathrm{m}}$ internal energy per mole, and $C_{V,\mathrm{m}}$ the stipulated constant molar heat capacity at fixed volume. The second equation is an additional caloric assumption: an equation of state alone does not specify the temperature dependence of internal energy. It satisfies the equilibrium identity $(\partial u_{\mathrm{m}}/\partial v_{\mathrm{m}}{)}_T=T(\partial p/\partial T{)}_{v_{\mathrm{m}}}-p=a_{\mathrm{m}}/v_{\mathrm{m}}^2$. The arbitrary energy reference is set to zero.

Define dimensionless variables $r=v_{\mathrm{m}}/b_{\mathrm{m}}$, $\theta =R{b}_{\mathrm{m}}T/a_{\mathrm{m}}$, $\pi =b_{\mathrm{m}}^{2}p/a_{\mathrm{m}}$ and $\eta =b_{\mathrm{m}}{h}_{\mathrm{m}}/a_{\mathrm{m}}$, where $h_{\mathrm{m}}$ is molar enthalpy. These differ from the critical-value scaling in Van der Waals law of corresponding states. The throttle calculation reduces toThus a prescribed inlet fixes $\eta$, and solving the first and last equations at the outlet pressure gives the final temperature. For an inlet $({\theta }_1,{\pi }_1)=(0.8,0.12)$ and outlet ${\pi }_2=0.025$, the model gives ${\theta }_2\simeq 0.752287$, a decrease of $0.047713$. The inlet $(2.2,0.12)$ instead gives ${\theta }_2\simeq 2.205786$, an increase of $0.005786$. The purple inlet $(1.45,0.30)$ initially warms but ends at ${\theta }_2\simeq 1.437585$ for the same outlet pressure. These are reduced toy values, not kelvin readings or experimental evidence. No physical gas parameters were fitted.

For a short inversion derivation, implicit differentiation of the Van der Waals equation at fixed pressure givesInsert this into $T(\partial v_{\mathrm{m}}/\partial T{)}_p=v_{\mathrm{m}}$ and substitute back into the equation of state. The Joule-Thomson inversion curve isPositive inversion pressure requires $r>3/2$. Its maximum pressure is $\pi =1/3$ at $r=3$, while the upper dilute limit is ${\theta }_{\mathrm{inv}}\to 2$. Using Van der Waals critical point values, the latter is $T_{\mathrm{inv},\max }/T_c=27/4$. The inversion locus depends on the equation of state; the chosen molar heat capacity additionally determines how much temperature changes along each finite throttle.^[ref]

The displayed region has $\theta >8/27$, above the model's critical temperature, and uses only $r>1$, $p>0$, $(\partial p/\partial v_{\mathrm{m}}{)}_T<0$ and positive $C_{p,\mathrm{m}}$. Subcritical calculations require attention to phase coexistence, metastability and the Maxwell construction; an unstable homogeneous continuation is not an equilibrium throttle endpoint. Even on a stable branch, this mean-field theory with constant $C_{V,\mathrm{m}}$ is a qualitative model, not an accurate inversion map for a particular real gas. Phase transitions, changing composition, appreciable heat leakage or bulk kinetic energy changes require a different property calculation or steady-flow energy balance. At phase coexistence, the smooth single-thermodynamic phase coefficient formula cannot replace a balance involving the phase fraction.