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Below the critical temperature an isotherm of the Van der Waals equation oscillates as shown. Image:MaxwellEqArea.svg in the phase transition zone by defining it as a certain isobar in that zone. The above isotherm is for a reduced temperature of $T\_R=0.9$. The Maxwell correction is at a vapor pressure of $P\_V\backslash approx\; 0.64700$ between the reduced volume of the pure liquid $V\_L\backslash approx\; 0.60340$ and the pure gas $V\_G\backslash approx\; 2.3488$ at the vapor pressure.]] Along the red portion of the isotherm $\backslash scriptstyle\; \backslash left(\{\{\backslash partial\; P\}/\{\backslash partial\; V\}\}\backslash right)\_\{T,N\}>0$ which is unstable; the Van der Waals equation fails to describe real substances in this region because the equation always assumes that the fluid is uniform while between a and c on the isotherm it becomes more stable to be a coexistence of two different phases, a denser phase which we normally call liquid and a sparser phase which we normally call gas. To fix this problem James Clerk Maxwell (1875) replaced the isotherm between a and c with a horizontal line positioned so that the areas of the two hatched regions are equal. The flat line portion of the isotherm now corresponds to a liquid-vapor equilibrium in which part of the system is in a liquid phase and part is in a gas phase. The portions a–d and c–e are interpreted as metastable states of super-heated liquid and super-cooled vapor respectively.Maxwell,J.C. The scientific papers of James Clerk Maxwell Dover 1965(c1890) p424 The equal area rule can be expressed as:

:$P\_V(V\_G-V\_L)=\backslash int\_\{V\_L\}^\{V\_G\}P\backslash ,dV$

where $P\_V$ is the vapor pressure (flat portion of the curve), $V\_L$ is the volume of the pure liquid phase at point a on the diagram, and $V\_G$ is the volume of the pure gas phase at point c on the diagram). The sum of these two volumes will equal the total volume V.

Maxwell justified the rule by saying that work done on the system in going from c to b should equal work released on going from a to b. (The area on a PV diagram corresponds to mechanical work). That’s because the change in the free energy function A(T,V) equals the work done during a reversible process the free energy function being a state variable should take on a unique value regardless of path. In particular, the value of A at point b should calculate the same regardless of whether the path came from left or right, or went straight across the horizontal isotherm or around the original Van der Waals isotherm.

Maxwell’s argument is not totally convincing since it requires a reversible path through a region of thermodynamic instability. Nevertheless, more subtle arguments based on modern theories of phase equilibrium seem to confirm the Maxwell Equal Area construction and it remains a valid modification of the Van der Waals equation of state. Cross, Michael First Order Phase Transitions,http://www.pma.caltech.edu/~mcc/Ph127/b/Lecture3.pdf

The Maxwell equal area rule can also be derived from an assumption of equal chemical potential $\backslash mu$ of coexisting liquid and vapour phases.A E Elhassan, R J B Craven, K M de Reuck, The Area Method for pure fluids and an analysis of the two-phase region, Fluid Phase Equilibria 130 (1997) 167-187. On the isotherm shown in the above plot, points a and c are the only pair of points (at the given temperature) which fulfill the equilibrium condition of having equal pressure, temperature and chemical potential. Define $P\_v,\backslash ,T\_v$, and $\backslash mu\_v\backslash ,$ as the fixed pressure, temperature and chemical potential for a two-phase system. The independent variables V and T suggest the use of the Helmholtz free energy thermodynamic potential (A), for which the pressure is given by

:$P=-\backslash left(\backslash frac\{\backslash partial\; A\}\{\backslash partial\; V\}\backslash right)\_\{T,N\}$

The integral of the pressure with respect to volume of the Van der Waals equation between the pure liquid phase and the pure gas phase is:

:$\backslash int\_\{V\_L\}^\{V\_G\}\; P(V,T,N)\backslash ,dV$

= -\int_{V_L}^{V_G} \left(\frac{\partial A}{\partial V}\right)_{T,N} dV

= A(V_L,T,N)-A(V_G,T,N)

since $A=U-TS=-PV+\backslash mu\; N\backslash ,$ and since the chemical potential is the same at both limits, it follows that:

:$\backslash int\_\{V\_L\}^\{V\_G\}\; P(V,T,N)\backslash ,dV\; =\; (P\_v\; V\_G-\backslash mu\_v\; N)-(P\_v\; V\_L-\backslash mu\_v\; N)=P\_v(V\_G-V\_L)$

which is just Maxwell's equal area rule. The Van der Waals equation may be solved for $V\_G$ and $V\_L$ as functions of the temperature and the vapor pressure $P\_V$. The vapor pressure may be expressed as:

:$P\_V=\backslash frac\{A(V\_G,T,N)-A(V\_L,T,N)\}\{V\_G-V\_L\}=\backslash frac\{kNT\}\{V\_G-V\_L\}\backslash ln\backslash left(\backslash frac\{V\_G-Nb\}\{V\_L-Nb\}\backslash right)-\backslash frac\{aN^2\}\{V\_G\; V\_L\}$

Since the gas and liquid volumes are functions of $P\_V$ and T only, this equation is then solved numerically to obtain $P\_V$ as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes.

A Maxwell equal-area construction can be made for any pair of conjugate variables, while holding two other non-conjugate variables constant. For example, on a plot of temperature versus entropy, on a curve of constant pressure, the proper thermodynamic potential to be used is ???. For the two-phase system, the integral of temperature between the pure liquid entropy $S\_L$ and the pure gas entropy $S\_G$ is given by

:$\backslash int\_\{S\_L\}^\{S\_G\}\; T(S,V,N)\backslash ,dS$

= -\int_{S_L}^{S_G} \left(\frac{\partial U(S,V,N)}{\partial S}\right)_{V,N} dS

= U(S_G,V,N)-U(S_L,V,N)

since $U=TS-PV+\backslash mu\; N\backslash ,$ and since the temperature and chemical potential is the same at both limits, it follows that:

:$\backslash int\_\{S\_L\}^\{S\_G\}\; T(S,V,N)\backslash ,dS\; =\; (T\_v\; S\_G-P\_v\; V\; +\backslash mu\_v\; N)-(T\_v\; S\_L-P\_v\; V\; +\backslash mu\_v\; N)=T\_v(S\_G-S\_L)$

which is Maxwell's equal area rule for the temperature-entropy conjugate pair at constant volume.

{{Reflist}}

For many distributions, the standard deviation is not a particularly natural way of quantifying the structure. For example, uncertainty relations in which one of the observables is an angle has little physical meaning for fluctuations larger than one period.{{Citation |first=D. |last=Judge |title=On the uncertainty relation for angle variables|journal=Il Nuovo Cimento|year=1964|volume=31|issue=2|pages=332–340|doi=10.1007/BF02733639}}{{Citation |title=On an uncertainty relation for angle variables|journal=Il Nuovo Cimento|date=1965 |year=1965|volume=37|issue=3|pages=1119–1125|doi=10.1007/BF02773197 |last1=Bouten |first1=M. |last2=Maene |first2=N. |last3=Leuven |first3=P. }}{{Citation |first=W. H. |last=Louisell|title=Amplitude and phase uncertainty relations|journal=Physics Letters|year=1963|volume=7|issue=1|pages=60–61|doi=10.1016/0031-9163(63)90442-6|bibcode = 1963PhL.....7...60L }} Other examples include highly bimodal distributions, or unimodal distributions with divergent variance.

A solution that overcomes these issues is an uncertainty based on entropic uncertainty instead of the product of variances. While formulating the many-worlds interpretation of quantum mechanics in 1957, Hugh Everett III conjectured a stronger extension of the uncertainty principle based on entropic certainty.{{Citation |last1=DeWitt |first1=B. S. |last2=Graham |first2=N. |year=1973 |title=The Many-Worlds Interpretation of Quantum Mechanics |location=Princeton |publisher=Princeton University Press |pages=52–53 |isbn=0-691-08126-3 }} This conjecture, also studied by Hirschman{{Citation |first=I. I., Jr. |last=Hirschman |title=A note on entropy |journal=American Journal of Mathematics |year=1957 |volume=79 |issue=1 |pages=152–156 |doi=10.2307/2372390 |postscript=. |jstor=2372390 }} and proven in 1975 by Beckner{{Citation |first=W. |last=Beckner |title=Inequalities in Fourier analysis |journal=Annals of Mathematics |volume=102 |issue=6 |year=1975 |pages=159–182 |doi=10.2307/1970980 |postscript=. |jstor=1970980 }} and by Iwo Bialynicki-Birula and Jerzy Mycielski{{Citation |first1=I. |last1=Bialynicki-Birula|last2= Mycielski|first2=J.|title=Uncertainty Relations for Information Entropy in Wave Mechanics|journal=Communications in Mathematical Physics |volume=44 |year=1975 |pages=129 |doi=10.1007/BF01608825 |issue=2|bibcode = 1975CMaPh..44..129B }} is that, for two normalized, dimensionless Fourier transform pairs f(a) and g(b) where

:$f(a)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; g(b)\backslash \; e^\{2\backslash pi\; i\; a\; b\}\backslash ,dx$    and     $\backslash ,\backslash ,\backslash ,g(b)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f(a)\backslash \; e^\{-\; 2\backslash pi\; i\; a\; b\}\backslash ,dx$

the Shannon information entropies

:$H\_a\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f(a)\; \backslash log(f(a))\backslash ,dx,$

and

:$H\_b\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; g(b)\; \backslash log(g(b))\backslash ,dy$

are subject to the following constraint:

{{Equation box 1

|indent =:

|equation =$H\_a\; +\; H\_b\; \backslash ge\; \backslash log\; (e/2)$

|cellpadding= 6

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

where the logarithms may be in any base. The probability distribution functions associated with the position wave function ψ(x) and the momentum wave function φ(x) have dimensions of inverse length and momentum respectively, but the entropies may be rendered dimensionless by:

:$H\_x\; =\; -\; \backslash int\; |\backslash psi(x)|^2\; \backslash ln\; (x\_0\backslash ,|\backslash psi(x)|^2\; )\; \backslash ,dx\; =-\backslash left\backslash langle\; \backslash ln\; (x\_0\backslash ,|\backslash psi(x)|^2\; )\; \backslash right\backslash rangle$

:$H\_p\; =\; -\; \backslash int\; |\backslash phi(p)|^2\; \backslash ln\; (p\_0\backslash ,|\backslash phi(p)|^2)\; \backslash ,dp\; =-\backslash left\backslash langle\; \backslash ln\; (p\_0\backslash ,|\backslash phi(p)|^2\; )\; \backslash right\backslash rangle$

where x₀ and p₀ are some arbitrarily chosen length and momentum respectively, which render the arguments of the logarithms dimensionless. Note that the entropies will be functions of these chosen parameters. Due to the Fourier transform relation between the position wave function ψ(x) and the momentum wavefuction φ(p), the above constraint can be written for the corresponding entropies as:

{{Equation box 1

|indent =:

|equation = $H\_x\; +\; H\_p\; \backslash ge\; \backslash log\; \backslash left(\backslash frac\{e\backslash ,h\}\{2\backslash ,x\_0\backslash ,p\_0\}\backslash right)$

|cellpadding= 6

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

where h is Planck's constant. Depending on one's choice of the x₀ p₀ product, the expression may be written in many ways. If x₀ p₀ is chosen to be h'', then:

:$H\_x\; +\; H\_p\; \backslash ge\; \backslash log\; \backslash left(\backslash frac\{e\}\{2\}\backslash right)$

If x₀ p₀ is chosen to be ћ'', then:

:$H\_x\; +\; H\_p\; \backslash ge\; \backslash log\; (e\backslash ,\backslash pi)$

If x₀ and p₀ are chosen to be unity in whatever system of units are being used, then

:$H\_x\; +\; H\_p\; \backslash ge\; \backslash log\; \backslash left(\backslash frac\{e\backslash ,h\; \}\{2\}\backslash right)$

where h is interpreted as a dimensionless number equal to the value of Planck's constant in the chosen system of units.

The quantum entropic uncertainty principle is more restrictive than the Heisenberg uncertainty principle. From the inverse logarithmic Sobolev inequalities{{citation |first=D. |last=Chafaï |title=Séminaire de Probabilités XXXVI |chapter=Gaussian maximum of entropy and reversed log-Sobolev inequality|series=Lecture Notes in Mathematics |arxiv=math/0102227 |doi=10.1007/978-3-540-36107-7_5 |year=2003 |volume=1801 |pages=194–200 |isbn=978-3-540-00072-3 }}

:$H\_x\; \backslash le\; \backslash frac\{1\}\{2\}\; \backslash ln\; (\; 2e\backslash pi\; \backslash sigma\_x^2\; /\; x\_0^2\; )~,$

:$H\_p\; \backslash le\; \backslash frac\{1\}\{2\}\; \backslash ln\; (\; 2e\backslash pi\; \backslash sigma\_p^2\; /p\_0^2\; )~,$

(equivalently, from the fact that normal distributions maximize the entropy of all such with a given variance),

it readily follows that this entropic uncertainty principle is stronger than the one based on standard deviations, because

:$\backslash sigma\_x\; \backslash sigma\_p\; \backslash ge\; \backslash frac\{\backslash hbar\}\{2\}\; \backslash exp\backslash left(H\_x\; +\; H\_p\; -\; \backslash log\; \backslash left(\backslash frac\{e\backslash ,h\}\{2\backslash ,x\_0\backslash ,p\_0\}\backslash right)\backslash right)\; \backslash ge\; \backslash frac\{\backslash hbar\}\{2\}~.$

A few remarks on these inequalities. First, the choice of base e is a matter of popular convention in physics. The logarithm can alternatively be in any base, provided that it be consistent on both sides of the inequality. Second, the normal distribution saturates the inequality, and it is the only distribution with this property, because it is the maximum entropy probability distribution among those with fixed variance (cf. here for proof).

class="toccolours collapsible collapsed" width="70%" style="text-align:left" !Entropic uncertainty of the normal distribution

We demonstrate this method on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. The length scale can be set to whatever is convenient, so we assign :$\backslash ell=\backslash sqrt\{\backslash frac\{\backslash hbar\}\{2m\backslash omega\}\}$ : &= \left(\frac{1}{2\pi \ell^2}\right)^{1/4} \exp{\left( -\frac{x^2}{4\ell^2}\right)} \end{align} The probability distribution is the normal distribution :$|\backslash psi(x)|^2\; =\; \backslash frac\{1\}\{\backslash ell\; \backslash sqrt\{2\backslash pi\}\}\; \backslash exp\{\backslash left(\; -\backslash frac\{x^2\}\{2\backslash ell^2\}\backslash right)\}$ with Shannon entropy : &= -\frac{1}{\ell \sqrt{2\pi}} \int_{-\infty}^{\infty} \exp{\left( -\frac{x^2}{2\ell^2}\right)} \ln \left[\frac{1}{\sqrt{2\pi}} \exp{\left( -\frac{x^2}{2\ell^2}\right)}\right] \, dx \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \exp{\left( -\frac{u^2}{2}\right)} \left[\ln(\sqrt{2\pi}) + \frac{u^2}{2}\right] \, du\\ &= \ln(\sqrt{2\pi}) + \frac{1}{2}.\end{align} A completely analogous calculation proceeds for the momentum distribution. Choosing a standard momentum of $\backslash hbar/\backslash ell$: :$\backslash phi(p)\; =\; \backslash left(\backslash frac\{2\; \backslash ell^2\}\{\backslash pi\; \backslash hbar^2\}\backslash right)^\{1/4\}\; \backslash exp\{\backslash left(\; -\backslash frac\{\backslash ell^2\; p^2\}\{\backslash hbar^2\}\backslash right)\}$ :$|\backslash phi(p)|^2\; =\; \backslash sqrt\{\backslash frac\{2\; \backslash ell^2\}\{\backslash pi\; \backslash hbar^2\}\}\; \backslash exp\{\backslash left(\; -\backslash frac\{2\backslash ell^2\; p^2\}\{\backslash hbar^2\}\backslash right)\}$ : &= -\sqrt{\frac{2 \ell^2}{\pi \hbar^2}} \int_{-\infty}^{\infty} \exp{\left( -\frac{2\ell^2 p^2}{\hbar^2}\right)} \ln \left[\sqrt{\frac{2}{\pi}} \exp{\left( -\frac{2\ell^2 p^2}{\hbar^2}\right)}\right] \, dp \\ &= \sqrt{\frac{2}{\pi}} \int_{-\infty}^{\infty} \exp{\left( -2v^2\right)} \left[\ln\left(\sqrt{\frac{\pi}{2}}\right) + 2v^2 \right] \, dv \\ &= \ln\left(\sqrt{\frac{\pi}{2}}\right) + \frac{1}{2}.\end{align} The entropic uncertainty is therefore the limiting value : &= 1 + \ln \pi = \ln(e\pi).\end{align}