Bose?Einstein phase transition

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1: 25.17 Physical Applications

… ►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …

2: 10.19 Asymptotic Expansions for Large Order

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§10.19(iii) Transition Region

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$$J_{\nu }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\sim \frac{2^{\frac{1}{3}}}{{\nu }^{\frac{1}{3}}}\mathrm{Ai}\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{2}{3}}}{\nu }{\mathrm{Ai}}^{\prime }\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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$$Y_{\nu }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\sim -\frac{2^{\frac{1}{3}}}{{\nu }^{\frac{1}{3}}}\mathrm{Bi}\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}-\frac{2^{\frac{2}{3}}}{\nu }{\mathrm{Bi}}^{\prime }\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$.

… ►with sectors of validity $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}\nu \le \frac{3}{2}\pi -\delta$. … ►with sectors of validity $-\frac{1}{2}\pi +\delta \le \mathrm{ph}\nu \le \frac{3}{2}\pi -\delta$ and $-\frac{3}{2}\pi +\delta \le \mathrm{ph}\nu \le \frac{1}{2}\pi -\delta$, respectively. …

3: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). Such a solution is given in terms of a Riemann theta function with two phases. …

4: 33.25 Approximations

§33.25 Approximations

►Cody and Hillstrom (1970) provides rational approximations of the phase shift ${\sigma }_0\left(\eta \right)=\mathrm{ph}\mathrm{\Gamma }\left(1+\mathrm{i}\eta \right)$ (see (33.2.10)) for the ranges $0\le \eta \le 2$, $2\le \eta \le 4$, and $4\le \eta \le \mathrm{\infty }$. …

5: 2.11 Remainder Terms; Stokes Phenomenon

… ►In $ℂ$ both the modulus and phase of the asymptotic variable $z$ need to be taken into account. …Then numerical accuracy will disintegrate as the boundary rays $\mathrm{ph}z=\alpha$, $\mathrm{ph}z=\beta$ are approached. … ►In the transition through $\theta =\pi$, $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case $\rho =100$. … ►uniformly with respect to $\mathrm{ph}z$ in each case. ►The relevant Stokes lines are $\mathrm{ph}z=\pm \pi$ for $w_1(z)$, and $\mathrm{ph}z=0,2\pi$ for $w_2(z)$. …

6: Bibliography O

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Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.

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Notes:

See also Olver (1997b)

External Links:

Document, MathReview (L. Berg), ZentralBlatt

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§2.3(iv).

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BibTeX

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Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

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External Links:

FullText, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v).

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BibTeX

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7: Bibliography T

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Special functions in phase space: Mathieu functions. J. Phys. A 31 (31), pp. 6725–6739.

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External Links:

ISSN 0305-4470, Document, MathReview (Bing Hong Wang), ZentralBlatt

Referenced by:

8th item.

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BibTeX

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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.1, Notations E, Notations E.

Encodings:

BibTeX

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8: 36.7 Zeros

… ►The zeros in Table 36.7.1 are points in the $𝐱=(x,y)$ plane, where $\mathrm{ph}{\mathrm{\Psi }}_2\left(𝐱\right)$ is undetermined. … ►The zeros are lines in $𝐱=(x,y,z)$ space where $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$ is undetermined. …, $y=0$), the number of rings in the $m$th row, measured from the origin and before the transition to hairpins, is given by …

9: 36.3 Visualizations of Canonical Integrals

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§36.3(ii) Canonical Integrals: Phase

►In Figure 36.3.13(a) points of confluence of phase contours are zeros of ${\mathrm{\Psi }}_2\left(x,y\right)$; similarly for other contour plots in this subsection. … ►

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Figure 36.3.13: Phase of Pearcey integral $\mathrm{ph}{\mathrm{\Psi }}_2\left(x,y\right)$.

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Figure 36.3.14: Density plots of phase of swallowtail canonical integrals.

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Figure 36.3.21: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,3\right)$. …

10: 9.16 Physical Applications

… ►These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability. The investigation of the transition between subsonic and supersonic of a two-dimensional gas flow leads to the Euler–Tricomi equation (Landau and Lifshitz (1987)). …