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Coexistence of multiple stable long-time states at the same parameter values. Bistability is the case of two states; distinct basins of attraction allow a system to remember its initial state or a sufficiently large perturbation.

// Target: mathematical-biology.bigb

Nucleation forms a sufficiently large region of a new phase within an old metastable phase. In classical nucleation theory, positive surface energy competes with bulk free-energy gain, producing a critical nucleus and a barrier to conversion. The Arrhenius nucleation time depends exponentially on this barrier; boundaries and impurities can alter it. This explains how metastability and hysteresis can persist despite a different globally stable phase.

Direction arises from switching between mutually incompatible states, with feedback providing memory and regulated destruction resetting the system. Different transitions use related logic, but a steep response, a bistable switch and an irreversible molecular event are not identical. The cell cycle also requires checkpoints and once-per-cycle licensing so that the ordered biochemical states produce a correct sequence of physical events.

At the G1/S transition, growth signals promote cyclin-dependent kinase activity and relieve RB-mediated restraint on E2F. E2F-dependent expression of additional cell-cycle components can reinforce the transition. More generally, activation of a kinase together with removal of its inhibitor creates positive feedback. In yeast, multisite phosphorylation of Sic1 permits recognition by the Cdc4 ubiquitin-ligase system and degradation of an inhibitor of S-phase kinase. This supplies a threshold that prevents weak or premature signals from launching the next programme.^[ref] The detailed regulators vary across organisms and biological cell types; the conserved point is a coordinated transition from an inhibited state to one that reinforces its own activation.

Mitotic entry gives an experimentally accessible example of bistability. cyclin B binding increases the available CDK1 complex, but Wee1-dependent inhibitory phosphorylation can keep it inactive. Cdc25 removes the inhibitory phosphate. Active CDK1 promotes Cdc25 activity and suppresses Wee1, giving a positive-feedback loop and a double-negative loop. Over a suitable input range, both a low-activity and a high-activity state can be stable. If cyclin input rises from interphase, activation occurs at a higher threshold than the input required to maintain a pre-existing mitotic state. These distinct thresholds constitute hysteresis and prevent small fluctuations from repeatedly reversing a committed transition.

In egg extracts, nondegradable cyclin could be supplied while CDK activity was approached from different initial states. The same cyclin input then supported different persistent activities depending on history, directly distinguishing hysteresis from a single steep but history-independent response.^[ref] These are extract measurements, not universal numerical thresholds for every intact biological cell. A useful conceptual representation iswhere $u$ represents effective activating input. hysteresis supplies memory within this interval; it does not mean that the switch can never be reset.

Reset is achieved by coupling activation to delayed negative feedback and proteolysis. Mitotic kinase activity helps prepare activation of the anaphase-promoting complex with Cdc20, while the spindle checkpoint prevents productive progression until attachment conditions are met. Once this restraint is satisfied, APC/C-dependent ubiquitination promotes destruction of securin and cyclin B. securin loss allows separase to cleave cohesin, making sister-chromatid separation possible. Experiments identifying cohesin cleavage connected regulated proteolysis to a physical transition that simple reversal of a phosphorylation state cannot undo.^[ref] Chromatids cannot be returned to a pre-anaphase chromosome merely by reducing separase activity.

cyclin B destruction lowers CDK1 activity, allowing phosphatases to reverse mitotic phosphorylation and promoting an interphase state. Activation of anaphase-promoting complex with Cdh1 at low kinase activity can help keep cyclins low, while high kinase activity opposes that state. Thus low-CDK and high-CDK programmes are mutually antagonistic. Cyclin synthesis eventually provides a new activating input. Early experiments in extracts showed the importance of cyclin synthesis and destruction, including failure of normal exit with nondegradable cyclin.^[ref] A bistable activation module combined with an appropriate delayed reset can generate an ordered oscillation; a bistable module held at constant input need not oscillate on its own.

replication licensing enforces another essential directionality. At low CDK activity, origin recognition complex with Cdc6 and Cdt1 loads MCM2-7 double hexamers, establishing competence to replicate. In S phase, activating kinases promote origin firing while high CDK activity and additional controls prevent renewed licensing. In animal biological cells, geminin inhibition of Cdt1 and regulated degradation help separate these states. Purified-protein reconstitution demonstrated loading of the replication-helicase precursor as a distinct biochemical event.^[ref] Separating licensing from firing allows many origins to be prepared but prevents fired regions from being copied repeatedly during the same cycle. Lowering mitotic kinase activity must eventually restore licensing competence for the next cycle, rather than permit uncontrolled rereplication during the current S phase.

Checkpoint pathways modify or hold these switches when DNA is unreplicated, damaged or chromosomes are improperly attached; they do not supply the entire oscillator. Energy-consuming phosphorylation cycles, ubiquitination, proteolysis and resynthesis make the system a driven process, while feedback stabilizes distinct states. Thresholds provide commitment, hysteresis provides memory, destruction and cleavage enforce selected transitions, and licensing/checkpoints couple those transitions to one correct round of replication and segregation.

In spectrin unfolding by force spectroscopy, the cantilever measures tension through its calibrated deflection, while displacement of the pulling point increases the separation of the protein attachments. The protein extension is obtained after allowing for cantilever compliance and any linker extension; it is not simply the commanded displacement.

A folded spectrin repeat stores much of its backbone in a compact domain. Between unfolding events, the already released chain is stretched: increasingly straight conformations have fewer accessible configurations, and the resisting entropic force rises nonlinearly. A peak is reached when a still-folded domain crosses an unfolding barrier on the timescale of the imposed loading. Its released backbone increases the available contour length abruptly. At nearly the same end-to-end distance the fractional extension $x/L$ is then smaller, so tension drops. Pulling resumes on the next, longer-chain elastic branch. Repeated release of domains produces the saw-tooth sequence.

The record is therefore evidence for sequential mechanical unfolding of modular structure, not repeated stretching of a fixed-length rod. For the nominal eight-domain construct, successive branches correspond to increasing numbers of unfolded repeats. The order and rupture forces can fluctuate because barrier crossing is thermally activated and depends on loading rate, structural heterogeneity and pulling geometry; equivalent repeats need not rupture at identical force.

The horizontal separation between worm-like-chain fits at the same force estimates the added contour length, whereas the vertical force drop at one extension reflects redistribution of tension. These are distinct measurements. A repeat increment near the specified $31.7\mathrm{nm}$ is consistent with repeated similar structural units. The gain is a net release relative to the folded span and attachment geometry; $31.7/106\mathrm{nm}$ is not automatically the universal contour length of a peptide residue.

During unloading, relaxation and possible refolding can restore compact domains. Hysteresis would indicate dissipative or kinetically delayed processes. Neither the peak force alone nor an arbitrary saw-tooth area directly equals an equilibrium unfolding free energy: one must separate chain elasticity, cantilever storage, irreversible loss and pulling-rate effects. The worm-like-chain branches describe released-chain elasticity, not the molecular transition state responsible for a rupture peak.

On releasing the tensile load, the extended chain first retracts towards less aligned conformations because they have much greater configurational multiplicity. Viscous friction and internal relaxation determine the time course; stored free energy is dissipated as heat. An unconstrained ideal chain does not shrink to a point: its mean vector is zero but its root mean square end-to-end length is $b\sqrt{N}\simeq 2.37\mathrm{nm}$ for the supplied ideal-chain parameters.

A real polypeptide also has sequence-dependent interactions. If the original native structure is the stable state in the prevailing solvent and the molecule remains intact and capable of folding, it can subsequently refold, gaining favourable intramolecular and solvent free energy. The folded state is not selected by chain entropy alone, and formation of native contacts need not be instantaneous. Kinetic traps, proline isomerization, protein aggregation, misfolding, surface attachments or chemical damage may prevent complete recovery or produce hysteresis. Thus reversible refolding is a physically motivated possibility, not guaranteed simply by removal of force. This distinction connects the recoil of a stretched polymer to the thermodynamics discussed in Question 6(c).

iron remains Fe(II), $d^6$, in both states; the change is a spin crossover, rather than a change in oxidation state. At the higher temperature the high-spin complex has $t_{2g}^4{e}_g^2$: four unpaired electrons, $S=2$, andThis matches the observed moment. At low temperature a low-spin complex with $t_{2g}^6{e}_g^0$ has $S=0$ and shows diamagnetism. Cooling switches high-spin $d^6$ to low-spin $d^6$. Emptying the metal–ligand antibonding $e_g$ orbitals strengthens and shortens the Fe–N bonds, accounting for the contraction by roughly $0.2\text{A˚}$.

The octahedral splitting and electron pairing energy must be sufficiently close for both states to be accessible. The low-spin state gains electronic energy, while the high-spin state has greater spin degeneracy and usually greater vibrational entropy because of its softer bonds. The resulting Gibbs free energy balance favors high spin at higher temperature. The two endpoint observations do not by themselves specify hysteresis or transition cooperativity; those require a temperature series. Coupled structural and magnetic measurements distinguish this explanation from a mere bond-length anomaly.

Pendant alcohols can form reversible borate bridges, schematically $\mathrm{P}-\mathrm{O}-\mathrm{B}(\mathrm{OH})-\mathrm{O}-{\mathrm{P}}^{\prime }$ and $\mathrm{B}(\mathrm{O}-\mathrm{P}{)}_3$, where P and P′ belong to different chains. Tetrahedral borate species are also possible depending on pH and water content. A bridge must connect different network strands to act as a cross-link; an intrachain loop does not contribute equivalently to elasticity.

Between cross-links, stretching reduces the number of accessible chain conformations. A simple incompressible rubber elasticity model gives the nominal tensile stresswhere $\lambda$ is the extension ratio and ${\nu }_e$ the density of elastically active strands. The initial response is soft; stronger extension gives increasing stress and ultimately finite-chain extensibility. Reversible borate exchange permits stress relaxation, rate dependence and a loading–unloading hysteresis loop, rather than permanent ideal elasticity at arbitrarily long times.

Frequency-dependent ac susceptibility probes relaxation: a peak in the out-of-phase component occurs approximately when $\omega \tau =1$. Measuring its displacement with frequency determines $\tau (T)$; an activated region may fit $\tau ={\tau }_0\exp (U_{\mathrm{eff}}/k_{B}T)$. A low-temperature departure can reveal tunnelling or other relaxation channels.

Low-temperature, sweep-rate-dependent magnetization hysteresis tests whether reversal is slow on the measurement timescale. A loop opening below the blocking temperature, variation with sweep rate and possible tunnelling steps provide complementary evidence. Isothermal remanent magnetization decay after field removal directly measures the associated relaxation time. A single open loop alone would not distinguish isolated-molecule blocking from collective ordering or impurities.

Room-temperature iron is ferromagnetic: interactions among its electronic moments produce spontaneous local order. The independent-ion Curie law therefore needs an interaction and an ordered-state description. In a Weiss mean-field approximation, replace the applied field by $H_{\mathrm{eff}}=H+\Lambda M$. Above the ordering temperature, linearizing the single-ion response yieldsThis is the Curie–Weiss approximation on the paramagnetic side. Merely substituting room temperature below $T_C$ into that high-temperature formula would produce a negative response and is not a valid model of stable iron.

Below $T_C$, use the full nonlinear moment response to the effective field; it has nonzero zero-field solutions. On a chosen ordered branch the local small-field response has the form $M\simeq M_s(T)+{\chi }_{\mathrm{diff}}(T)H$, rather than $M\propto H/T$ through the origin. Macroscopic magnetic domains and hysteresis further determine the measured net moment. Exchange interactions, arising from electronic quantum statistics and Coulomb correlations, supply the alignment energy; bare dipolar fields alone do not explain ordinary iron's strong ferromagnetism. A localized-ion Weiss model describes the qualitative ordering, while iron's itinerant electronic structure limits a literal independent-fixed-ion interpretation.

In equilibrium without hysteresis, the system selects the lower free-energy minimum. At $T_2$ the order parameter jumps between $0$ and $2b/(3c)$, which identifies a first-order phase transition. Differentiating the minimized free energy does not add a $dP/dT$ contribution because $F^{\prime }=0$. Thus the polarization-dependent entropy is $S=-a{P}^2/2$, givingThe heat absorbed on heating from the ordered to the disordered phase isThe same amount is released on cooling. If $\Delta S$ is defined in the opposite direction, the signed transition heat is correspondingly negative.

The displayed horizontal variable is ${\mu }_{0}H$ in tesla. The source formulas written with a flux quantum actually have the dimensions of magnetic flux density, so define $B_{c1}={\mu }_0{H}_{c1}$ and $B_{c2}={\mu }_0{H}_{c2}$ when applying them:From the onset of curvature away from the low-field line in panel A, $B_{c1}$ is roughly $0.06\mathrm{T}$ with uncertainty of order $0.01\mathrm{T}$. Panel B's branches approach the normal background over approximately $2$–$3\mathrm{T}$; use $B_{c2}\simeq 2.5\mathrm{T}$ as a coarse central estimate. Resolving irreversible loop closure from the true thermodynamic upper field, and correcting demagnetization, would require better data and geometry. These readings support scales, not precise material constants.

The upper field givesThe leading-log critical-field inversion starts by dividing the two field formulas to eliminate $\xi$:The large-$\kappa$ solution is $\kappa \simeq 6.15$, henceThe penetration-depth range is an indicative graph/model scale, not a propagated confidence interval. There is a mathematical ambiguity worth making explicit: $f(\kappa )=\ln \kappa /(2{\kappa }^2)$ rises from zero at one, peaks at $\sqrt{e}$, then falls. The same central ratio also gives $\kappa \simeq 1.055$ and $\lambda \simeq 12.1\mathrm{nm}$, so the printed restriction $\kappa >1$ alone does not select a unique solution. The leading-log lower-field formula is intended for appreciably large $\kappa$, which motivates the $\kappa \simeq 6$ branch; near one it needs the omitted core-energy correction. Even the larger-branch estimate has systematic uncertainty from that approximation.

$\lambda$ is the London penetration depth, the magnetic screening scale, while $\xi$ is the superconducting coherence length, the recovery scale of the superconducting order parameter after spatial suppression. In the mixed state the complex order parameter vanishes in vortex cores of radius of order $\xi$, recovers between separated cores and winds in phase by $2\pi$ around each singly quantized vortex. Currents and magnetic field spread over scale $\lambda$; vortex density grows with the mean internal field according to $n_v{\Phi }_0=\overline{B}$. Near $H_{c2}$ cores overlap and the order-parameter amplitude becomes small throughout the sample. A schematic core profile appears in solution.

The hysteresis is not the reversible equilibrium magnetization: defects pin vortices, so entry and exit leave different spatial flux distributions. Macroscopic currents oppose changes in flux, trapped vortices give remanence, and depinning and creep depend on history and temperature. Surface barriers can also contribute. The loop width probes an irreversible current scale, but its conversion to current density additionally needs sample geometry and an appropriate critical-state model.

For a long cylindrical wire carrying axial current $I$, Ampère's law gives $B_s={\mu }_{0}I/(2\pi a)$ at its surface. With $a\gg \lambda$, the current resides mainly in a shell of thickness $\lambda$. If “critical” denotes the first equilibrium entry of vortices in an otherwise Meissner wire, set $B_s=B_{c1}$:The dependence on $a$ is necessary because no numerical radius is supplied. The local surface current density at this entry scale is different:Its conversion to the cross-section average is $\overline{J}\simeq (2\lambda /a)j_s$. Treating the shell current density as uniform over the entire thick wire would overestimate its total current by $a/(2\lambda )$.

Meaning of a maximum: in a type-II material, first vortex entry need not be the measured dissipative transport limit. Pinned vortices can sustain greater current, whereas freely moving vortices generate voltage. The hysteresis loop indicates pinning, but no geometry-dependent pinning force or transport criterion is given, so a unique practical maximum cannot be deduced solely from $\lambda ,\xi$ and $a$.

For comparison, the intrinsic homogeneous Ginzburg–Landau depairing scale follows from $j(q)={\Phi }_{0}q(1-{\xi }^2{q}^2)/(2\pi {\mu }_0{\lambda }^2)$. Maximizing at $q=1/(\sqrt{3}\xi )$ givesThis is a local, approximately uniform-flow theoretical scale, not the average current density throughout a thick self-field wire. Surface barriers, nonlinear screening, vortex motion and temperature determine how that scale relates to its actual critical current. Thus the entry estimate, local shell density and intrinsic depairing limit answer distinct physical criteria; the source's word “maximum” does not supply the missing pinning information.

Using the constants as printed, with volume flux measured in Sverdrups, givesIn particular the two small expansion coefficients should not be silently replaced by different oceanographic values.

For a positive steady volume flux, the equation in 2(a) is $q^2-Aq+BE=0$. For a negative steady volume flux, it is $q^2-Aq-BE=0$, with only its negative root admissible. Therefore for $E>0$ the branches areThe upper positive branch $q_{+}$ is stable, the lower positive branch $q_{-}$ is unstable, and the reversed branch $q_{\mathrm{rev}}$ is stable under the usual salt budget dynamics with fixed $\Delta T$. Thus there is bistability below the positive-branch fold. At the fold the two positive equilibria merge with a zero restoring rate; the point is not an ordinary asymptotically stable interior point of either branch. The negative algebraic root of the positive-$q$ quadratic, or positive root of the negative-$q$ quadratic, must never be added as extra solutions.

At $E=0.3\mathrm{Sv}$, the three steady volume fluxes areThe present thermally dominated, northward-upper-flow Atlantic meridional overturning circulation corresponds to $q_{+}\simeq 17.7\mathrm{Sv}$. These algebraic data alone also allow a stable reversed state, so selecting the present branch uses the stated circulation direction. On the large positive branch, the predicted salinity contrast is $\Delta S=E{S}_0/q_{+}\simeq 0.594\mathrm{g}\mathrm{k}{\mathrm{g}}^{-1}$, consistent with the reference-salinity approximation. The weaker branches give larger contrasts, approximately $3.16$ and $4.20\mathrm{g}\mathrm{k}{\mathrm{g}}^{-1}$, so the approximation is less accurate there.

The saddle-node bifurcation occurs atFor $E>E_c$, only the reversed steady volume flux remains. At $E=E_c$ there are still two distinct values, the double positive root and the negative root; “single-valued” holds strictly above this threshold.

Slowly increasing $E$ from the present value causes the stable positive volume flux to weaken. Its salinity contrast then increases because salt exchange is weaker, reducing the cold-box density advantage further: this is the salt-advection feedback. At the fold the positive steady state disappears, producing an abrupt transition to the reversed branch in the quasistatic limit. At the same $E_c$ that branch hasFor a finite slow ramp there can be a small dynamical delay near the fold. In this model subsequently reducing $E$ while keeping it positive follows the stable reversed branch instead of automatically restoring the positive branch, an example of hysteresis. The two small branches approach $q=0$ as $E\to 0^{+}$. At $E=0$ this is an equal-density, no-exchange equilibrium with $\Delta S=\alpha \Delta T/\beta =3.75\mathrm{g}\mathrm{k}{\mathrm{g}}^{-1}$, whereas the positive branch tends to $q=A$ with $\Delta S=0$. One must use the original density relation at this endpoint, rather than divide by $|q|=0$.

The reversal is thermally opposed but salinity driven: freshwater makes the cold box sufficiently fresh that the warm, salty box becomes denser. A real weakening would reduce the northward ocean heat flux and alter regional climate and ocean ventilation. The fold and full reversal here describe the ideal two-box overturning model; its numerical threshold is not a forecast for the real ocean, whose other controls appear in 2(c).

For a rubber-like solid, a spring is a first model of recoverable elasticity, while a viscous element can represent rate-dependent loss and delayed recovery. An immediate extension followed by bounded delayed material creep is represented by series spring and Voigt-branch creep, a standard linear solid model:The first term is immediate and recoverable; the second is delayed and also recoverable. For a load held until $t=t_h$, recovery after a finite-duration stress load predicts an immediate drop of ${\sigma }_0/E_0$ on unloading, followed byThis supplies a direct comparison with a loading-and-unloading observation: the model returns completely to its original length, with separate immediate and delayed recovery. A single Kelvin-Voigt model misses the immediate compliance, while a free series dashpot would predict unbounded flow that an intact crosslinked rubber need not show. A spring alone suffices over a timescale on which the measured response is effectively immediate and elastic.

These are local small-strain analogues. Large-strain rubber elasticity is nonlinear, nearly incompressible and sensitive to temperature, loading history and internal damage. A small set of linear springs and dashpots cannot reproduce every amplitude, timescale or hysteresis loop.

No numerical rubber trace or fitted constants are supplied, so these are explicit qualitative model predictions for comparison with the practical observations.

The sensor must survive the stress range and repeated loading without fracture, fatigue or excessive mechanical compliance. The tensor character of the piezoelectric coefficient makes crystal orientation important; a ceramic's poling quality and domain stability also matter. Compare linearity, hysteresis, reproducibility, aging and the usable frequency bandwidth, including resonance and dynamic response.

Electrical design is equally important: leakage resistance and readout input impedance determine low-frequency loss, and cable/input capacitance loads a voltage output. A charge amplifier may favour a different material ranking from a passive open-circuit voltage measurement. temperature changes can alter the coefficients, produce a pyroelectric signal in a polar material and depole a ferroelectric near its transition. Humidity, electrode bonding, electrical breakdown, manufacturability, cost and environmental constraints such as lead content also affect a practical choice. Calibration should separate applied stress from temperature drift and readout loading.

For $\mu >0$, the cubic equation has the same sign as $a$ everywhere away from zero: small positive or negative disturbances grow without saturation, and the cubic truncation even predicts finite-time blow-up. An amplitude of convection cannot grow indefinitely under that local approximation. The conclusion is that additional nonlinear physics and higher-order terms are needed, not that the small-amplitude reduction is valid at arbitrarily large amplitude.

For the modified equation put $\eta =C_1\mu$ and $C=C_2>0$. It can be writtenThe quintic term points inward for sufficiently large $|a|$, so all forward solutions are bounded and approach equilibria in this scalar problem. The quadratic term breaks $a\mapsto -a$ symmetry. Besides $a=0$, the equilibrium diagram isThis stability criterion follows from $f^{\prime }(a)=-a{g}^{\prime }(a)$ on the equilibrium curve, where $g^{\prime }=1-2Ca+4a^3$. Near zero the new branch is $a=\eta +O({\eta }^2)$, crossing the trivial branch transcritically. It is unstable for small negative $\eta$ and stable for small positive $\eta$; the trivial branch has the opposite stability.

The negative-amplitude part of $g$ has exactly one minimum. Above its negative fold value but below zero, a stable finite-amplitude negative state coexists with the stable origin, separated by an unstable negative equilibrium. This gives hysteresis and finite-amplitude triggering of convection. For positive $\eta$ the origin is unstable and stable nonzero states provide saturation. Additional positive-amplitude folds depend on $C$: $g^{\prime }$ has two positive roots for $C>3/2$, a double root at $C=3/2$ and none for $C<3/2$, so further multistability can occur. The right-hand figure illustrates $C=1$, without those extra positive folds; it does not assume that every possible $C_2$ gives that same detailed diagram. These are the qualitative changes in quintic saturation of a symmetry-broken pitchfork.

To determine the nonlinear pitchfork bifurcation, solve the four slaved steady equations exactly. SetThe nonzero equilibrium points satisfyThusIf $k\le 1$, the coefficient of $u$ is positive for every physical $q\ge 0$: there is no change of pitchfork direction at a nonnegative rotation parameter. If $k>1$, the change occurs atBelow $q_D$ the nonzero branch lies on $r>r_P$; above $q_D$ it initially lies on $r<r_P$. At $q_D$, $r-r_D=k{u}^2+O(u^3)$, so $|a|$ grows like $(r-r_D{)}^{1/4}$ rather than the ordinary square-root scaling.

The rational equilibrium curve at a degenerate pitchfork also gives the nearby saddle-node bifurcations of the two reflection-related nonzero branches. For $q>q_D$,andThe two saddle-node bifurcations occur on one parameter curve by reflection symmetry; its separation from the pitchfork bifurcation curve is quadratic in $q-q_D$.

The stability statement needs a further condition. At $r_P$, the other coupled eigenvalues have polynomial $s^2+(1+2\sigma )s+\sigma [(1+\sigma )-(1-\sigma )q]$. They are stable only for $q<q_{TB}$. If $q_D<q_{TB}$, a one-dimensional centre manifold gives the local form $a^{\prime }=\chi a[\Delta -[1-q(k-1)]a^2-k{a}^4+\cdots ]$, where $\Delta =r-r_P$ and $\chi =\sigma /[(1+\sigma )-(1-\sigma )q_D]>0$. The quintic term stabilizes the large branch. The interval $r_{SN}<r<r_P$ then has a stable conductive state and two stable finite-amplitude states, with the two small unstable branches separating their basins of attraction. This is the usual subcritical convection hysteresis. If $q_D>q_{TB}$, the branch still reverses its direction, but the trivial state already has an additional unstable mode; this equilibrium-curve calculation alone does not imply a stable bistable wedge. At equality a two-dimensional centre manifold is essential.

The Maxwell construction replaces a nonconvex homogeneous free-energy density by its coexistence envelope. If $U_0(M)$ excludes the source, the coexisting values $M_1,M_2$ satisfyThus one straight line is tangent at both endpoints. Equivalently,the equal-area form of the common-tangent construction. At fixed average $\overline{M}$ between the endpoints, the lever rule gives $\overline{M}=p{M}_1+(1-p)M_2$. Such a mixture has the tangent-line bulk energy, below the energy of an unstable homogeneous state.

The two phases occupy spatial domains separated by domain walls. The gradient term gives a positive wall energy, but a macroscopic interface has area of order $L^{D-1}$ rather than bulk volume $L^D$, so its cost per volume vanishes in the thermodynamic limit. For the symmetric quartic example a scalar quartic domain wall obeys $\kappa {\varphi }^{\prime \prime }=r\varphi +u{\varphi }^3$ and interpolates between $\pm M_0$ as $\varphi =M_0\tanh (x/\ell )$, with $M_0^2=|r|/u$ and $\ell =\sqrt{2\kappa /|r|}$. Multiplication of the differential equation by ${\varphi }^{\prime }$ gives the first integral $\kappa ({\varphi }^{\prime }{)}^2/2=u({\varphi }^2-M_0^2{)}^2/4$, explaining the finite positive interface cost. A uniform unconstrained system need not form both domains: phase separation is particularly relevant when the average order parameter is fixed or boundaries enforce competing phases.

Hysteresis occurs when the system follows a metastable local minimum after another minimum has become the equilibrium one. For $r<0,u>0$, the equation of state is $h=rM+u{M}^3$. The hysteresis spinodals of a scalar quartic potential follow from $r+3u{M}^2=0$:Following local minima until these limits gives different switching fields on increasing and decreasing $h$. Equilibrium instead switches at $h=0$. Actual switching can occur before a mean-field spinodal through nucleation, and depends on time scales; hysteresis is not a property of a globally minimized equilibrium potential. The second panel below distinguishes the local branches and their limiting switches.

Along the in-plane hard axis, the ideal uniform-magnetization model gives reversible rotation from the easy direction toward the applied field. With uniaxial anisotropy energy density $K_u{\sin }^2\theta$, minimization givesfollowed by saturation at $\pm M_s$. The right panel of the preceding figure therefore has a nearly linear central section, essentially zero remanence and ideally no hysteresis. Actual domains, misalignment and defects can produce rounding or a small loop. This is an in-plane hard axis; the much larger out-of-plane demagnetizing field is a different effect.

Only the magnetic-response physics is addressed here; biological and clinical application is user-deferred.

A single-domain particle has an approximately coherent magnetic dipole moment, but single-domain does not by itself imply rapid thermal reversal. If moments equilibrate within the measurement time, a dilute, noninteracting ensemble is superparamagnetic: its quasistatic magnetization is reversible, odd in field and approaches saturation with negligible remanence or magnetic coercivity.

For freely orientable classical moments,At small field, $\mathcal{L}(z)\simeq z/3$; at large field moments align. This Langevin form is an ideal limit, not a universal exact curve for anisotropic fixed-axis particles.

The missing timescale/magnetic anisotropy information must be stated: blocked single-domain particles can already show hysteresis. Also, a reversible quasistatic curve does not preclude a finite-frequency loss loop when relaxation lags an alternating field. The original curves illustrate an equilibrated dilute limit and possible blocked/interacting alternatives, not measured treatment performance.