Phase diagram of a polydisperse soft-spheres model for liquids and colloids

The phase diagram of soft spheres with size dispersion has been studied by means of an optimized Monte Carlo algorithm which allows to equilibrate below the kinetic glass transition for all sizes distribution. The system ubiquitously undergoes a first order freezing transition. While for small size dispersion the frozen phase has a crystalline structure, large density inhomogeneities appear in the highly disperse systems. Studying the interplay between the equilibrium phase diagram and the kinetic glass transition, we argue that the experimentally found terminal polydispersity of colloids is a purely kinetic phenomenon.Comment: Version to be published in Physical Review Letter

Global regularity in ultradifferentiable classes

Se estudia la w-regularidad de soluciones de ciertos operadores que son globalmente hipoelípticos en el toro N-dimensional. Se aplican estos resultados para probar la w-regularidad global de ciertas clases de sublaplacianos. En este sentido, se extiende trabajo previo en el contexto de la clases analíticas y de Gevrey. Se dan varios ejemplos de w-hipoelipticidad local y global.The research of the authors was partially supported by MEC and FEDER Project MTM2010-15200.Albanese, AA.; Jornet Casanova, D. (2014). Global regularity in ultradifferentiable classes. Annali di Matematica Pura ed Applicata. 193(2):369-387. https://doi.org/10.1007/s10231-012-0279-5S3693871932Albanese A.A., Jornet D., Oliaro A.: Quasianalytic wave front sets for solutions of linear partial differential operators. Integr. Equ. Oper. Theory 66, 153–181 (2010)Albanese A.A., Jornet D., Oliaro A.: Wave front sets for ultradistribution solutions of linear pertial differential operators with coefficients in non-quasianalytic classes. Math. Nachr. 285, 411–425 (2012)Albanese A.A., Zanghirati L.: Global hypoellipticity and global solvability in Gevrey classes on the n–dimensional torus. J. Differ. Equ. 199, 256–268 (2004)Albanese A.A., Popivanov P.: Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions. Ann. Mat. Pura e Appl. 185, 395–409 (2006)Albanese A.A., Popivanov P.: Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators. Ann. Univ. Ferrara 52, 65–81 (2006)Baouendi M.S., Goulaouic C.: Nonanalytic–hypoellipticity for some degenerate elliptic operators. Bull. Am. Math. Soc. 78, 483–486 (1972)Bergamasco A.P.: Remarks about global analytic hypoellipticity. Trans. Am. Math. Soc. 351, 4113–4126 (1999)Bonet J., Meise R., Melikhov S.N.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14, 425–444 (2007)Braun R.W., Meise R., Taylor B.A.: Ultradifferentiable functions and Fourier analysis. Results Math. 17, 206–237 (1990)Chen W., Chi M.Y.: Hypoelliptic vector fields and almost periodic motions on the torus $${\mathbb{T}^n}$$ . Commun. Part. Differ. Equ. 25, 337–354 (2000)Christ M.: Certain sums of squares of vector fields fail to be analytic hypoelliptic. Commun. Part. Differ. Equ. 16, 1695–1707 (1991)Christ M.: A class of hypoelliptic PDE admitting non-analytic solutions. Contemp. Math. Symp. Complex Anal. 137, 155–168 (1992)Christ M.: Intermediate optimal Gevrey exponents occur. Commun. Part. Differ. Equ. 22, 359–379 (1997)Cordaro P.D., Himonas A.A.: Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus. Math. Res. Lett. 1, 501–510 (1994)Cordaro P.D., Himonas A.A.: Global analytic regularity for sums of squares of vector fields. Trans. Am. Math. Soc. 350, 4993–5001 (1998)Dickinson D., Gramchev T., Yoshino M.: Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena. Proc. Edinb. Math. Soc. 45, 731–759 (2002)Ferńandez C., Galbis A., Jornet D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Appl. Math. 340, 1153–1170 (2008)Gramchev T., Popivanov P., Yoshino M.: Some note on Gevrey hypoellipticity and solvability on torus. J. Math. Soc. Jpn. 43, 501–514 (1991)Gramchev T., Popivanov P., Yoshino M.: Some examples of global Gevrey hypoellipticity and solvability. Proc. Jpn. Acad. 69, 395–398 (1993)Gramchev T., Popivanov P., Yoshino M.: Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients. Rend. Sem. Univ. Pol. Torino 51, 145–172 (1993)Greenfield S., Wallach N.: Global hypoellipticity and Liouville numbers. Proc. Am. Math. Soc. 31, 112–114 (1972)Greenfield S.: Hypoelliptic vector fields and continued fractions. Proc. Am. Math. Soc. 31, 115–118 (1972)Hanges N., Himonas A.A.: Singular solutions for sums of squares of vector fields. Commun. Part. Differ. Equ. 16, 1503–1511 (1991)Hanges N., Himonas A.A.: Analytic hypoellipticity for generalized Baouendi–Goulaouic operators. J. Funct. Anal. 125, 309–325 (1994)Helfer B.: Conditions nécessaires d’hypoanalyticité puor des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué. J. Differ. Equ. 44, 460–481 (1982)Himonas A.A.: On degenerate elliptic operators of infinite type. Math. Z. 220, 449–460 (1996)Himonas A.A.: Global analytic and Gevrey hypoellipticity of sublaplacians under diophantine conditions. Proc. Am. Math. Soc. 129, 2001–2007 (2000)Himonas A.A., Petronilho G.: Global hypoellipticity and simultaneous approximability. J. Funct. Anal. 170, 356–365 (2000)Himonas A.A., Petronilho G.: Propagation of regularity and global hypoellipticity. Mich. Math. J. 50, 471–481 (2002)Himonas A.A., Petronilho G.: On Gevrey regularity of globally C ∞ hypoelliptic operators. J. Differ. Equ. 207, 267–284 (2004)Himonas A.A., Petronilho G.: On C ∞ and Gevrey regularity of sublaplacians. Trans. Am. Math. Soc. 358, 4809–4820 (2006)Himonas A.A., Petronilho G., dos Santos L.A.C.: Regularity of a class of subLaplacians on the 3–dimensional torus. J. Funct. Anal. 240, 568–591 (2006)Hörmander L.: Hypoelliptic second order differential equations. Acta Mat. 119, 147–171 (1967)Langenbruch M.: Ultradifferentiable functions on compact intervals. Math. Nachr. 140, 109–126 (1989)Meise R.: Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals. J. Reine Angew. Math. 363, 59–95 (1985)The Lai P., Robert D.: Sur un probléme aus valeurs propres non linéaire. Israel J. Math. 36, 169–186 (1980)Petronilho G.: On Gevrey solvability and regularity. Math. Nachr. 282, 470–481 (2009)Petzsche H.-J.: Die Nuklearität der Ultradistributionsräume und der Satz von Kern I. Manuscripta Math. 24, 133–171 (1978)Tartakoff D.: Global (and local) analyticity for second order operators constructed from rigid vector fields on product of tori. Trans. Am. Math. Soc. 348, 2577–2583 (1996