Mathematical Physics and Stochastic Analysis Session-Mathematical Congress Capricornio-COMCA 2019

DMATULS PROCEEDINGS No 1, 2019

XXVIII COMCA Mathematical Physics and Stochastic Analysis Session Universidad de La Serena. July 31- August 2, 2019

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XXVIII COMCA Mathematical Physics and Stochastic Analysis Session Universidad de La Serena-Chile, July 30-August 2, 2019

Coordination Soledad Torres Díaz & Marco Corgini Videla

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PREFACE The international Capricorn Mathematical Congress (COMCA) is an initiative annually organized by the Mathematical Departments of the chilean northern zone universities: Universidad de Tarapacรก (UTA)- Arica, Universidad Arturo Prat (UAP) )- Iquique, University of Antofagasta (UA) -Antofagasta, Universidad Catรณlica del Norte (UCN)- Antofagasta, Universidad de Atacama (UDA)- Copiapรณ, Universidad de La Serena (ULS)- La Serena.

The organization of the twenty-eight version (XXVIII COMCA) has corresponded to the Department of Mathematics of the Universidad de La Serena.

This issue of DMATULS-Proceedings contains the summaries of the works presented in the Mathematical Physics and Stochastic Analysis Session of this congress ( fteen contributions in applied mathematics, mathematical physics and stochastic analysis).

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Contents

1. Transformada de Wigner Descomponible y

Cuantización de Constantes de Movimiento F. Belmonte

REFERENCES

2. Anomalous averages, Bose-Einstein

condensation and spontaneous symmetry breaking of continuous symmetries, revisited M. Corgini

REFERENCES

3. Cuantización de las constantes de movimiento

del oscilador armónico S. Cuéllar

REFERENCES

4. Técnica de integración multivariable:

Formalización del Método de Brackets I. González

REFERENCES

5. Non-Markovian random walks with memory

lapses

M. González REFERENCES

6. On nonparametric depth based classi cation

of continuous stochastic processes P. Ilmonen

7. M2-branes on a constant ux background C. las Heras

1 2

3 4

6 7

8 9

11 14

15 17

XXVIII COMCA, Math.Phys. & Stoch. An. Session

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REFERENCES

8. Existencia de Energía Libre y Unicidad de

Medidas de Gibbs en modelos de spins con interacciones aleatorias J. Littin

REFERENCES

9. The supermembrane with central charges

and the twisted torus P. León

REFERENCES

10. Solución numérica de EDEAR dirigidas por

movimiento Browniano H. Mardones

REFERENCES

11. Sobre la ecuación de Schrödinger tiempo-

espacio fraccionario no lineal tipo Hartree F. Ramírez

REFERENCES

12. Relativistic kinetic theory with applications

in astrophysics P. Rioseco

13. Función de probabilidad compuesta aplicada

a sistemas físicos fuera del equilibrio E. Sánchez

REFERENCES

18

20 21

23 24

26 28

30 30

31

32 33

14. Transformaciones de Bäcklund e integrabilidad A. Sotomayor 34 REFERENCES 34

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15. Note on a Non Linear Perturbation of the

Ideal Bose Gas R. Tabilo

REFERENCES

35 35

REFERENCES

37 37

16. Delone sets and (bi-)Lipschitz equivalence R. Viera

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1. Transformada de Wigner Descomponible y

Cuantización de Constantes de Movimiento F. Belmonte

Es sabido que en general una cuantización canónica no necesariamente intercambia el corchete de Poisson con el conmutador de operadores (en el caso de la cuantización de Weyl, tal resultado se conoce como el teorema de Groenewold - van Hove). Esto sugiere que deberían haber Hamiltonianos para los cuales tal cuantización no preserva constantes de movimiento.

En relación a lo anterior, en esta charla abordaremos los siguientes dos problemas: a) Dados ciertos Hamiltonianos, ¾es posible construir una cuantización que preserve las correspondientes constantes de movimiento? Bajo ciertas condiciones, mostraremos que es posible y para ello introduciremos el concepto de transformada de Wigner descomponible. La construcción de la cuantización requerida se basa en constatar la analogía entre el proceso de reducción simpléctica y la diagonalización de operadores autoadjuntos sobre espacios de Hilbert. b) ¾Podemos determinar cuales son los Hamiltonianos para los cuales una cuantización predeterminada preserva constantes de movimiento? Daremos una condición geométrica sobre un tal Hamiltoniano en terminos de la transformada de Wigner, que no solo garantiza que la cuantización dada preserve constantes de movimiento, sino

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que además garantiza que se preserve el cálculo funcional y que el espectro cuántico sea absolutamente continuo y que éste esencialmente coincida con el espectro clásico.

REFERENCES Quantization of Constants of Motion , arXiv: 1505.07250.

G. B. Folland, Harmonic Analysis in Phase Space , Annals of Mathematics Stud-

[1] F. Belmonte, [2]

ies,

122.

Princeton University Press, Princeton, NJ, 1989.

Fabián Belmonte

Departamento de Matemáticas Universidada Católica del Norte, Antofagasta, Chile fbelmonte@ucn.cl

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2. Anomalous averages, Bose-Einstein condensation

and spontaneous symmetry breaking of continuous symmetries, revisited M. Corgini

The study of the so-called spontaneous symmetry breaking (SSB) of continuous symmetry is a fundamental notion in quantum statistical physics [1]-[7], speci cally in the phase transitions theory (PTT).

At nite volume, the breaking of a continuous symmetry (U(1) symmetry group) is associated with many in nitely degenerated ground states connected between them by unitary transformations. In this sense, these states are physically equivalent, having the same energy, and being the ground state of the system understood as a superposition of them. However, in the thermodynamic limit, these connections vanish, and an in nite collection of inequivalent ground states, orthogonal to each other, arise.

On the other hand symmetries can be broken by small disturbances. Mathematically speaking, the disturbance once chosen and provided that the parameters on which it depends are xed, selects a unique ground state for the system (vacuum).

In this sense, in the framework of the study of the free Bose gas and in the case of a super uid model N. N. Bogoliubov [12] eliminates the above mentioned degeneracy by introducing a small term on the original energy operator, preserving the self-adjointness but suppressing the symmetry corresponding to the total number conservation law. In this context the

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limit thermal averages de ned by using such kind of perturbations of the original energy operators have been denominated Bogoliubov quasiaverages (QA) or anomalous averages (AA).

It must be taken into account that both, Bose condensation understood as macroscopic occupation of the ground state, as SSB of the continuous symmetry occur only in the thermodynamic limit which in real physical systems is never reached.

In this sense, the introduction of an external eld does not explain by itself the broken of symmetry. Moreover an underlying question is whether there is only one restricted class of perturbations, constituted by operators associated to the same sort of particles in the condensate, compatible with the chosen order parameter and with the existence of pure states and ODLRO (o diagonal long range order) [7].

The scenario becomes even more complicated considering that all current experiments are carried out on nite atom systems (trapped Bose gases for which the total number conservation law is preserved).

In this work, the aforementioned problems, their consequences on the fundamental principles of the quantum PTT and the viability of experimentally testing the QA approach shall be discussed.

REFERENCES [1] Y. Nambu, Phys.Rev. 117, 648 (1960); Phys. Rev. Lett., 4, 380 (1960).

[2] ] J. Goldstone, Nuovo Cim. 19, 154-164 (1961) [3] J. Goldstone, Phys.Rev. 127, 965-970 (1962) [4] P. Higgs, Phys. Rev., 145, 1156 (1966).

[5] S. Weinberg, Phys.Rev.Lett., 29, 1698 (1972).

[6] N.N. Bogoliubov, Lectures on Quantum Statistics: Quasiaverages. Vol.2 (Gordon and Beach, New York, 1970

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[7] W. F. Wreszinski, V. A. Zagrebnov, On ergodic states, spontaneous symmetry breaking and the Bogoliubov quasiaverages. HAL Id:

hal.archives-ouvertes.fr/hal-01342904

hal-01342904

https://

(2014)

Marco Corgini Videla

Departamento de Matemรกticas Universidad de La Serena, Chile mcorgini@userena.cl

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3. Cuantización de las constantes de movimiento del

oscilador armónico S. Cuéllar

Si f es un observable clásico, denotaremos por Op(f ) el operador asociado a f a través de la cuantización de Weyl de nida en [4]. Durante esta charla mostraremos que el cálculo de Weyl preserva las constantes de movimiento del oscilador armónico y describiremos algunas consecuencias de este resultado. Más explícitamente mostramos que eitH Op(f ) = Op(f )eitH para toda constante de movimiento f de h, donde h es el oscilador armónico clásico y H el oscilador armónico cuántico. Una consecuencia directa de lo anterior es que Op(f ) admite una descomposición a través de la diagonalización espectral canónica de H . También, usando la transformada de Wigner W en conjunto con el teorema de N representación, mostramos que toda constante de movimiento claásica es de P la forma |α|=|β| cα,β W (φβ , φα ), en donde φγ es una función de Hermite n dimensional y γ es un mutiíndice. Se muestran nuevas condiciones que garantizan que: i Op(f ) sea acotado ii que Op(f ) vaya de S(Rn ) en S(Rn ), donde S(Rn ) es la clase de Schwartz.

Finalmente se comenta el comportamiento del producto de Moyal sobre el álgebra de constantes de movimiento.

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Acknowledgements. Financiado por Beca CONICYT Doc-

torado Nacional 2017

REFERENCES [1] Agorram F, Benkhadra A., El Hamyani, Ghanmi A. (Julio 2015)

mite functions as Fourier-Wigner transform.

Complex Her-

Integral Transforms and Special

Functions 27(2). DOI: 10.1080/10652469.2015.1095742 [2] Estrada

R.

Gracia

Bondia

J.

pansions of twisted products. .

y

Várilly

Journal

of

J.

(1989),

Mathematical

On asymptotic exPhysics

30,

2789.

https://doi.org/10.1063/1.528514 [3] Federer, Herbert (1969),

Geometric measure theory, . Die Grundlehren der math-

ematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc [4] Folland, G. (1989)

Harmonic Analysis in Phase Space. (AM-122) Princeton Uni-

versity Press. Obtenido de http://www.jstor.org/stable/j.ctt1b9rzs2.8

Sebastián Cuéllar Carrillo

Departamento de Matemáticas Universidad Católica del Norte, Chile sebastian.cuellar01@ucn.cl Trabajo en conjunto con Fabián Belmonte

Departamento de Matemáticas Universidada Católica del Norte, Antofagasta, Chile fbelmonte@ucn.cl

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4. Técnica de integración multivariable:

Formalización del Método de Brackets I. González

En estas presentación formalizaremos un nuevo y poderoso método de integración, el cual hasta ahora ha sido presentado como un método heurístico. Dicha técnica está destinada a la evaluaci ón de integrales multivariables cuyo intervalo de integración está de nido por [0, ∞[ . Este método se denomina Método de Brackets (MoB, su sigla en inglés) [1,2,3], el cual tiene su origen en los formalismos matemáticos desarrollados en la teoría cuántica de campos para la evaluación de las integrales asociadas a los diagramas de Feynman. En este trabajo mostramos las reglas (teoremas) que conforman MoB y su aplicación a diversos problemas de integración, tanto de una variable como multivariable.

Es importante mencionar que desde el punto de vista de la evaluación analítica de integrales, el uso de paquetes computacionales, tales como Mathematica o MAPLE, no resultan ser su cientes para los problemas multivariables que aparecen en la matemática aplicada, bajo esta perspectiva es que MoB resulta ser una poderosa herramienta de cálculo analítico para estas integrales. Desde el punto de vista de lo procedural, la tarea fundamental de MoB es convertir la integral en una "serie" muy particular, denominada "serie de brackets", a partir de esta serie y a través de la aplicación de un conjunto de procedimientos ad-hoc se obtiene la solución de la integral al resolver un sistema de ecuaciones lineales. A la pregunta: ¾qué

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es un bracket?, podemos por ahora indicar que es una estructura matemática que representa a una integral divergente, la cual por de nición está representada por la siguiente ecuación: Z

∞

xa1 +...+an −1 dx = ha1 + ... + an i

0

siendo

{ai } (i = 1, ...n) índices arbitrarios y el símbolo h·i , un

bracket. Desde el punto de vista de lo operacional, el comportamiento del bracket es similar al de una delta de Kronecker, cosa que mostraremos en la presentación.

Las ideas relevantes que podemos indicar sobre esta técnica de integración (MoB), es que posee varias ventajas respecto a otras técnicas avanzadas de integración: • MoB resuelve N integrales múltiples de manera simultánea.

Convencionalmente se evaluan N integrales iteradamente.

• MoB no requiere herramientas de cálculo avanzado, solo elementos básicos de álgebra lineal • MoB se basa en reglas y procedimientos sistemáticos, un algoritmo. Estas características hacen que esta técnica sea altamente automatizable.

REFERENCES [1]

[2]

I. González and V. Moll, De nite integrals by method of brackets. Part 1, Advances in Applied Mathematics, Vol. 45, Issue 1, 50-73 (2010).

I. González, V. Moll and A. Straub, The method of brackets. Part 2: examples and applications, Contemporary Mathematics, Gems in Experimental Mathematics, Volume 517, 157-171 (2010).

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I. Gonzalez , K. Kohl , L. Jiu , V. H. Moll, An extension of the Method of Brackets. Part 1. Open Mathematics, Volume 15 (2017), Issue 1, 1181 1211. (arXiv:1707.08942).

Iván P. González G.

Departamento de Física y Astronomía Universidad de Valparaíso, Chile ivan.gonzalez@uv.cl

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5. Non-Markovian random walks with memory lapses M. GonzĂĄlez

In Gonzålez-Navarrete and Lambert [2, 3, 4] it was characterized a family of dependent processes with the property of memory lapses. The motivation coming from the random walks (RW) proposed in [5], in such RW each step at n ≼ 1 depends on the whole history of the process. Recently [1] showed a relation with the classical Polya-urn model.

In this sense, the paper [4] proposes a two colors urn model evolving as follows. Imagine the urn starts with R0 red and B0 blue balls. The composition of the urn at time n ∈ N is given by (Rn , Bn ), where Rn means the number of red and Bn the number of blue balls. At each time the urn is reinforced with new balls using the Bagchi-Pal mechanism, given by the so-called replacement matrix (1)

M=

a b

c d

! ~ = (a, b)T ; B ~ = (c, d)T . ; R

~ -column determines the balls to add if the In this case, R chosen color is red. That means we put a red and b blue balls. ~ -column.

If a blue-colored ball is obtained we proceed to the B It indicates that we add c red and d blue balls into the urn. As usual, we suppose that the urn is balanced, which means that a + b = c + d = K . Therefore, at each step we add a xed number K of balls. For simplicity assume all entries positive.

In [4] it was supposed that there exist two players reinforcing the urn and follow di erent strategies. One of them looks to the urn composition, and the other does not. Nonetheless,

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both players employ the replacement matrix (1), each of them following a distinct rule.

Player A follows a (generalized) Pólya-type regime. That is, she draws a ball uniformly at random, observe its color and put it back to the urn. Then she chooses with probability p the same color and with probability 1 − p the opposite color.

Finally, she uses the replacement matrix (1).

Player B behaves simpler. He chooses a color, say blue or red, with probabilities p and 1 − p, respectively. Then, he puts the balls into the urn following (1). We remark that player B behaves independently of the current urn composition.

The player choice is made as follows. At each step, a coin is ipped, and the result says who will add the next balls into the urn. Particularly, player A is chosen with probability θ.

At this time, we remark that a memory lapse is a period in which the decisions are done without taking into account the history of the process. That is, an interval of time in which player B is chosen to assume the game.

Particular cases for (1) are illustrated in Figure 1, that is, ~ = (2, 1)T , the state-spaces of the RW (Rn , Bn ) in the cases R ~ = (1, 2)T and K = 3, also the situation a = 3, c = 1 and B K = 3. On the other hand, the RW studied in [1, 5] can be recovered if a = K = 1 and θ = 1.

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State-space of two RW's with K = 3, starting at (R0, B0) = (1, 1). The gray dashed lines represent the deterministic cases, that is, θ = 0, and p ∈ {0, 1}.

Figure 1.

Particular cases for (1) as the state-spaces of the RW (Rn , Bn ) ~ = (2, 1)T , B ~ = (1, 2)T and K = 3, the situation a = 3, for R c = 1 and K = 3 will be discussed. Moreover, the RW studied in [1, 5] can be recovered if a = K = 1 and θ = 1.

In [4] we explicitly determine the long-term behavior of this family of RW with memory lapses, particularly it was proved a strong law of large numbers, stating that for all 0 ≤ θ, p ≤ 1, (2)

Rn Bn , Tn Tn

a.s

−−→

pc + (1 − p)a K − c − (a − c)(θ(2p − 1) + (1 − p)) , K − θ(2p − 1)(a − c) K − θ(2p − 1)(a − c)

as

n → ∞.

As a consequence, we remark the case p = 1/2, then the RW

a+c converges almost surely to a+c , 1 − 2K 2K , as n diverges. This means that, these RW are symmetric around the state-space's diagonal.

Moreover, we proved a functional limit theorem whenever K 1 1 p ≤ pc := a−c 4θ + 2 . Essentially, under suitable time-scale, the proportion of balls centered on the right hand side value in (2), converges as n → ∞ to a continuous bivariate Gaussian

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process. Similar results have been proved for other models in the works [2, 3]. Note that there are some cases of (1) for which pc < 1, then there is a phase transition on the parameters space. The parameters space is splitted in one region where the displacement of the walker is di usive and another for which is superdi usive.

REFERENCES Elephant random walks and their connection to PĂłlya-type urns. Phys. Rev. E 94, 052134.

GonzĂĄlez-Navarrete, M. and Lambert, R. Non-Markovian random walks with memory lapses. J. Math. Phys. 59, 113301 (2018).

GonzĂĄlez-Navarrete, M. and Lambert, R. (2018) The di usion of opposite opinions in a random-trend environment . Preprint arXiv:1811.12070.

GonzĂĄlez-Navarrete, M. and Lambert, R. (2019) Urn models with two types of strategies. Preprint arXiv:1708.06430.

[1] Baur, E. and Bertoin, J. (2016).

[2]

[3]

[4]

[5] SchĂźtz, G. and Trimper, S. (2004) Elephants can always remember: Exact long-range memory e ects in a non-Markovian random walk.

Phys. Rev. E

70,

045101.

Manuel GonzĂĄlez-Navarrete

Departamento de EstadĂ­stica Universidad del BĂ­o-BĂ­o ConcepciĂłn, Chile magonzalez@ubiobio.cl

Joint work with Rodrigo Lambert

Faculdade de Matemåtica Universidade Federal de Uberlândia Uberlândia, Brazil. rodrigolambert@ufu.br

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6. On nonparametric depth based classi cation of

continuous stochastic processes P. Ilmonen

Consider two sets of observed continuous stochastic processes (functional observations). The goal is to classify a new incoming observation to the correct set.

The concept of statistical depth was originally introduced as a way to provide a nonparametric center-outward ordering from a depth-based multivariate median. Several di erent depth functions for functional observations have been presented in the literature. Most of these approaches, however, are based on assessing the location of the function as a measure of typicality. As a result, they are missing some important features inherent to functional data such as variation in shape.

Another problem in assessing typicality of functional observations and in classifying functional observations is that we often observe only part of the function. One may overcome this problem by extrapolating and interpolating i.e. by adding the missing parts. However, doing that, at least implicitly, requires model assumptions.

We discuss assessing typicality of functional observations.

Moreover, we provide a new classi cation method that is based on j-th order k-th moment integrated depths. For j=1 and k=1 this is equal to applying the mean halfspace depth of a functional value with respect to the corresponding univariate marginal distribution. When j is larger than 1, the method is not based on comparing location only but considers shape of the function as well. Moreover, the method can be applied to

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partially observed functions without extrapolation or interpolation. Theoretical properties of the new approach are explored and several real data examples are presented to demonstrate its excellent classi cation performance.

Pauliina Ilmonem

Aalto University Finland pauliina.ilmonen@aalto.fi

Joint work with: S. Helander S. Nagy, G. Van Beve, L. Viitasaari

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7. M2-branes on a constant ux background C. las Heras

We describe a Supermembrane, or M2-brane, in a compacti ed target space M9 Ă&#x2014; T 2 with 2-form uxes that are turned on in the internal manifold. These uxes are generated by constant three-forms. We compare this theory with the one describing a 11D M2-brane formulated on M9 Ă&#x2014; T 2 target space subject to an irreducible wrapping condition [1]. We show that the ux generated by the bosonic 3-form under consideration is in one to one correspondence to the irreducible wrapping condition [2]. After a canonical transformation both Hamiltonians are exactly the same up to a constant shift in one particular case. Consequently both of them, share the same spectral properties [3]. We conclude that the Hamiltonian of the M2-brane with 2-form target space uxes on a torus has a purely discrete spectrum with eigenvalues of nite multiplicity and it can be considered to describe a new sector of the microscopic degrees of freedom of M-theory [4]. We also show that the total membrane momentum in the direction associated to the ux condition adquires a quantized contribution in correspondence to the ux units that have been turned on.

Acknowledgements. Partially funded by CONICYT-PFCHA/ Doctorado Nacional/2019-21190263.

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REFERENCES [1] I. Martin, A. Restuccia and R. S. Torrealba,

On the stability of compacti ed D

= 11 supermembranes. Nucl. Phys. B. 521, 117 (1998).[hep-th/9706090]

[2] M. P. García Del Moral, C. Las Heras, P. Leon, J. M. Pena and A. Restuccia,

Supermembrane theory on a curved constant background

(2018) arXiv:1811.11231

[hep-th].

Discreteness of the spectrum of the compacti ed D = 11 supermembrane with nontrivial winding . Nucl. Phys.

[3] L. Boulton, M. P. García del Moral and A. Restuccia,

B.

671,

343 (2003) [hep-th/0211047].

[4] M. P. García Del Moral, C. Las Heras, P. León, J. M. Pena and A. Restuccia,

Mass operator of the M2-brane on a background with constant three-form,

arXiv:1905.08376 [hep-th].

Camilo las Heras

Departamento de Física Universidad de Antofagasta Chile camilo.lasheras@ua.cl Joint work with: Maria Pilar García del Moral

Departamento de Física Universidad de Antofagasta Chile maria.garciadelmoral@uantof.cl Pablo León

Departamento de Física Universidad de Antofagasta Chile pablo.leon@ua.cl

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Joselen Pena

Departamento de Física Universidad de Antofagasta Chile joselen.pena@uantof.cl Álvaro Restuccia

Departamento de Física Universidad de Antofagasta Chile alvaro.restuccia@uantof.cl

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8. Existencia de EnergĂ­a Libre y Unicidad de

Medidas de Gibbs en modelos de spins con interacciones aleatorias J. Littin

Uno de los objetivos principales de la mecĂĄnica estadĂ­stica es la deducciĂłn de propiedades macroscĂłpicas de un sistema basĂĄndose en el estudio de propiedades microscĂłpicas de ĂŠste.

En este trabajo consideramos un modelo de spins uni-dimensional, con un Hamiltoniano aleatorio de nido sobre un volumen nito V

1 X X Jx,y Ď&#x192;x Ď&#x192;y HV [Ď&#x192;] = 2 |x â&#x2C6;&#x2019; y|axy xâ&#x2C6;&#x2C6;V yâ&#x2C6;&#x2C6;V x6=y

AcĂĄ Jx,y es una colecciĂłn de variables aleatorias independientes sub-Gaussianas y distribuciĂłn invariante por traslaciones, esto es â&#x20AC;˘ Para todo x â&#x2C6;&#x2C6; V , y â&#x2C6;&#x2C6; V se tiene E[Jx,y ] = 0

2 2 Dx,y = E[Jx,y ].

â&#x20AC;˘ Para todo Îť â&#x2030;Ľ 0 Îť2 2 D E[exp {ÎťJx,y }] â&#x2030;¤ exp 2 x,y

â&#x20AC;˘ Para todo x, y â&#x2C6;&#x2C6; Z Ă&#x2014; Z las variables aleatorias axy

se asumen independientes e idĂŠnticamente distribuidas, con valores no negativos.

Dada cualquier realizaciĂłn Ď&#x2030; de las variables aleatorias, la energĂ­a libre y la funciĂłn de particiĂłn en el Volumen V al inverso de la temperatura β vienen dadas por las expresiones.

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FV (ω) =

1 ln Zβ,V , |V |

ZV,β =

X

21

exp{−βH[σ]}

σ∈SV

donde SV = {−1, +1}V es el conjunto de todas las posibles con guraciones en el volumen V , de cardinalidad |V | .

Este trabajo generaliza los artíulos publicados originalmente por Khanin [1] y [2], los resultados más importantes que se discutirán son la existencia de una energía libre no aleatoria y la unicidad de las medidas de Gibbs para el modelo propuesto en el límite termodinámico. Se discuten además algunas de las posibles extensiones y aplicaciones para una expansión en clusters.

REFERENCES [1] K. M. Khanin and Ya. G. Sinai. Existence of free energy for models with longrange random hamiltonians. Journal of Statistical Physics, 20(6):573-584, Jun 1979.

[2] K. M. Khanin. Absence of phase transitions in one-dimensional long-range spin systems with random hamiltonian. Theoretical and Mathematical Physics, 43(2):445-449, May 1980.

[3] David Ruelle. Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics. Cambridge Mathematical Library. Cambridge University Press, 2 edition, 2004.

[4] Sacha Friedli and Yvan Velenik. Statistical Mechanics of Lattice Systems: Concrete Mathematical Introduction. Cambridge University Press, 2017.

Jorge Andres Littin

Universidad Católica del Norte Antofagasta, Chile jlittin@ucn.cl

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Joint work with: César Maldonado

Instituto Potosino de Ciencia y Tecnología San Luis de Potosí, México cesar.maldonado.ahumada@gmail.com

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9. The supermembrane with central charges and the

twisted torus P. LeĂłn

We will discuss about the relation between the supermembrane with central charges proposed in [1], with the geometrical structure better known as the twisted torus. The twisted 3-torus was initially studied in the context of string theory compacti cations by the authors in [2] as a non trivial T 2 â&#x2C6;&#x2019; bunlde over a circle. This idea was generalized by [3], in which the twisted n-torus is formulated as a compact group manifold in which the set of one-forms well-de ned over the twisted torus must satisfy the Maurer-Cartan equation. Now, the main feature of the formulation the supermembrane with central charges is the existence of a well de ned 2-form curvature on the world volume (a Riemann surface ÎŁ) , whose integral must be an integer di erent from zero. This condition can be interpreted in a more geometrical way, as the existence of a non trivial U (1) bundle over the world-volume of the supermembrane. Furthermore, it can be show that the embedding maps of the M2-brane ful ll the Maurer-Cartan equation.

In this sense it seem that there exists a relation between the supermembrane with central charges and the twisted torus.

This relation can be very useful to make a link between the central charge condition with other topological invariants, like the monodromy which is closed related with the compacti cations in the twisted torus [4].

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Acknowledgements. Partially funded by CONICYT PFCHA/

DOCTORADO BECAS CHILE/2019 -2119051

REFERENCES [1] I. Martin, A. Restuccia, and Rafael S. Torrealba. On the stability of compacti ed

Nucl. Phys., B521:117 128, 1998. hep-th/9706090.

N. Kaloper, R.C. Myers The Odd story of massive supergravity. JHEP, 05:010, D = 11 supermembranes.

[2]

1999. hep-th/9901045.

[3] A. Connes, M.R. Douglas, and A. S. Schwarz. Noncommutative geometry and matrix theory: Compacti cation on tori.

[4] M.Trigiante,

Gauged

JHEP, 02:003, 1998. hep-th/9711162.

Supergravities.

Phys.

Rept.

680,

1,

2017.doi:10.1016/j.physrep.2017.03.001 hep-th 1609.09745.

Pablo León

Departamento de Física Universidad de Antofagasta Chile pablo.leon@ua.cl

Joint work with: M.P García del Moral

Departamento de Física Universidad de Antofagasta Chile e-mail: maria.garciadelmoral@uantof.cl C. Las Heras

Departamento de Física Universidad de Antofagasta Chile camilo.lasheras@ua.cl

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J.M Pena

Departamento de Física Universidad Central de Venezuela Venezuela e-mail: joselen@yahoo.com A. Restuccia

Departamento de Física Universidad de Antofagasta Chile alvaro.restuccia@uantof.cl

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10. Solución numérica de EDEAR dirigidas por

movimiento Browniano H. Mardones

Las ecuaciones diferenciales estocásticas hacia adelante y reversivas (EDEAR) ayudan a describir procesos de difusión relacionados a fenómenos que involucran perturbaciones ambientales. Los términos de tendencia constituyen la parte descriptiva de un ambiente sin aleatoriedad, mientras que los procesos de Wiener describen las perturbaciones aleatorias involucradas en su dinámica mediante coe cientes de difusión. Dado T > 0, para un tiempo particular t ∈ [0, T ] y una posición especí ca x ∈ Rd interesa estudiar las EDEAR sobre [t, T ] de la forma (1)

 X = x + R s b (r, X , Y ) dr + R s σ (r, X , Y ) dW , r r r s r r t t RT RT Ys = g (XT ) + h (r, Xr , Yr ) dr − Zr dWr . s

s

Aquí b : [0, T ] × Rd × Rn → Rd , σ : [0, T ] × Rd × Rn → Rd×m , g : Rd → Rn , h : [0, T ] × Rd × Rn → Rn son funciones > dadas y W = W 1 , . . . , W m es un proceso de Wiener mdimensional de nido sobre el espacio de probabilidad com pleto Ω, F, (Ft )t≥0 , P . Una solución adaptada del sistema de EDEAR se de ne mediante los procesos (X, Y, Z) ∈ L2F Ω; C [t, T ] ; Rd ×L2F

× L2F (Ω; C ([t, T ] ; Rn ))

Ω; C [t, T ] ; Rn×m

que satisfacen (1) P-casi seguramente.

Los sistemas de EDEAR dirigidas por movimiento Browniano están relacionados con ecuaciones diferenciales parciales (EDP) no lineales a través de la ecuación de Feynman-Kac.

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Por ejemplo, la ecuación d-dimensional de Burgers (2)

  ∂u + (u · ∇) u + ν 4u + f = 0 ; 0 ≤ t < T, ∂t 2  u (T, ·) = g,

donde f : [0, T ] × Rd → Rd , g : Rd → Rd y viscosidad cinemática ν > 0 , está asociada al sistema acoplado de EDEAR (3)

 X = x + R s Y dr + R s √νdW , r s t t r RT √ RT Ys = g (XT ) + f (r, Xr ) dr − νZr dWr . s

s

Por lo tanto, la fórmula de Feynman-Kac resulta Yt = u (t, x) ;

∀ (t, x) ∈ [0, T ] × Rd .

Observe que la solución determinista u (t, x) de la ecuación de Burgers (2) se puede obtener a través de la simulación del sistema de EDEAR (3) mediante el proceso estocástico Yt . Dado que el proceso Y es no anticipativo, un estrategia es calcular su esperanza condicional con respecto al evento Xt = x, con lo cual se evita su dependencia con respecto al proceso Z .

Durante la exposición, se aborda la simulación numérica de un sistema de partículas estocásticas de acuerdo a un novedoso sistema de EDEAR asociado a las ecuaciones de NavierStokes incompresibles en dimensión d = 2, 3. Para ello, se discretizan las ecuaciones localmente en el tiempo y se consideran esquemas de integración de tipo Euler junto con la cuantización de los incrementos de Wiener involucrados. Así, se aproximan las esperanzas condicionales sobre cada nodo de un dominio computacional espacio-temporal uniforme usando parámetros de discretización en tiempo h > 0 y espacio δ > 0

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tales que h, δ â&#x2C6;&#x2C6; (0, 1) y h > δ . Resultados numĂŠricos son presentados para soluciones analĂ­ticas espacialmente periĂłdicas, en particular un vĂłrtice de Taylor-Green bidimensional y ujos de Beltrami tridimensionales, por ejemplo un ujo de Arnold-Beltrami-Childress. Los algoritmos de simulaciĂłn son completamente probabilĂ­sticos.

REFERENCES [1] C. Navier, MĂŠmoire sur les lois du mouvement des uides, Mem. Acad. Sci. Inst.

France 6 (1822) 389 440.

[2] G. G. Stokes, On the theories of the internal friction of uids in motion, and of the equilibrium and motion of elastic solids, Trans. Camb. Phil. Soc. 8 (1849) 287 319.

[3] R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. 20 (1948) 367 387.

[4] M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc. 65 (1) (1949) 1 13.

[5] Ă&#x2030;. Pardoux, S. G. Peng, Adapted solution of a backward stochastic di erential equation, Syst. Control Lett. 14 (1990) 55 61.

[6] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948) 171 199.

[7] F. Delbaen, J. Qiu, S. Tang, Forward-backward stochastic di erential systems associated to Navier-Stokes equations in the whole space, Stoch. Proc. Appl. 125 (2015) 2516 2561.

[8] F. Delarue, S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab. 16 (2006) 140 184.

[9] F. Antonelli, Backward-forward stochastic di erential equations, Ann. Appl.

Probab. 3 (1993) 777 793.

[10] A. J. Majda, A. L. Bertozzi, Vorticity and incompressible ow, Cambridge University Press, Cambridge, 2002.

[11] H. A. Mardones GonzĂĄlez, Numerical solution of stochastic di erential equations with multiplicative noise, Ph.D. thesis, Universidad de ConcepciĂłn (2017).

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[12] S. Corlay, G. Pagès, J. Printems, The optimal quantization website (2005).

Hernán Mardones González

Departamento de Matemáticas Universidad de La Serena, Chile hmardones@userena.cl

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11. Sobre la ecuación de Schrödinger tiempo-espacio

fraccionario no lineal tipo Hartree F. Ramírez

El objetivo de la charla es mostrar existencia, unicidad y propiedades de regularidad de la solución (débil) de una clase de ecuaciones de Schrödinger fraccionaria no lineal tipo Hartree, de nida por una derivada de orden fraccionaria.

REFERENCES The time fractional Schrödinger equation on Hilbert space. Integral Equations Operator Theory. 87 (2017), 1-14.

Prado, H and RamÃrez, J. The fractional in time Schrödinger equation with a Hartree perturbation. Submitted (2019).

Tao, T. Nonlinear Dispersive Equations: local and Global Analysis 106 (Rhode

[1] Górka, P. Prado, H. and Trujillo, J.

[2]

[3]

Island: Amer. Math. Soc., 2006).

José Ramírez Molina

Departamento de Matemática y C.C. Universidad de Santiago de Chile, Chile jose.ramirezm@usach.cl

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12. Relativistic kinetic theory with applications in

astrophysics

P. Rioseco

I will review relativistic kinetic theory and apply it to two phenomena. The rst phenomenon is accretion of matter around the Schwarzschild black hole, where we calculated its accretion rate. The second phenomenon concerns the dynamics of the kinetic gas for a thin disk in the equatorial plane of the Kerr black hole. In this case, the so-called mixing phenomenon appears, which causes that the gas con guration relaxes to stationary axisymmetric state.

Paola Rioseco

Departamento de FĂ­sica y AstronomĂ­a Universidad de La Serena, Chile pcriosec@gmail.coml

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13. Función de probabilidad compuesta aplicada a

sistemas físicos fuera del equilibrio E. Sánchez

Las distribuciones de energía de sistemas físicos fuera del equilibrio, que experimentan estados estacionarios, se han abordado con éxito a través del modelo Superestadístico (modelo BCS) presentado por Beck C. y Cohen E.G.D. en [1] el año 2003. Dicho modelo está basado en la superposición de factores de Boltzmann (cada factor caracteriza a un sistema en equilibrio) cada uno a una cierta temperatura. En términos matemáticos el modelo está dado por Z B(E) =

f (E|β)g(β)dβ.

donde se han encontrado tres clases relevantes, [2] a saber, χ2 , χ2 inversa y lognormal superestadística, donde en cada caso se tiene que

n/2 nβ 1 n g(β) = β n/2−1 e− 2β0 , Γ(n/2) 2β0

y

n/2 nβ0 β0 nβ0 g(β) = β −n/2−2 e− 2β , Γ(n/2) 2 β/µ)2 1 − (ln 2s 2 g(β) = √ e .

2πsβ

Siendo β el inverso de la temperatura (un parámetro intensivo positivo).

Esto, en términos estadísticos puede verse como el resultado de la composición entre la distribución exponencial de la variable aleatoria E y la función de peso del parámetro β del

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cual depende. Lo anterior se analiza con más detalle, desde el punto de vista estadístico de funciones de distribución compuestas, que han tenido una fuerte presencia en la descripción de algunos fenómenos en ciencias aplicadas, [3], [4] y podría permitirnos el desarrollo de otros modelos alternativos al modelo BCS aplicables a sistemas complejos y fuera del equilibrio.

Pero la compatibilidad de la expresión matemática considerada, con algunos requisitos físicos fundamentales, como el principio de máxima entropía, no siempre es posible, por lo que pareciera que solo cierto modelos obtenidos por medio de esta via podrían ser físicamente aceptables. Esto último es lo que se discutirá a través de la consideración de algunas distribuciones ampliamente conocidas en el campo de la física.

REFERENCES Superstatistics. Physica A 322, 267-275 (2003).

Beck C., Recent developments in superstatistics . Brazilian Journal of Physics, 39,

[1] Beck C., Cohen E.G.D., [2]

357 (2009).

A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families . Biometrika,

[3] Marshall A.W. & Olkin I.,

84, 641-652 (1997).

[4] Charytoniuk W. & Nazarko J.,

tions to electric load modeling ,

An application of compound probability distribuStochastic Analysis and Applications, 12, 31-40

(1994).

Ewin Sánchez Casanga

ewinsanchez.mail@gmail.com

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14. Transformaciones de Bäcklund e integrabilidad A. Sotomayor

En este trabajo se presentan transformaciones tipo Bäcklund, en el sentido de Wahlquist-Estabrook [1] , tanto para una extensión de la ecuación Korteweg-de Vries a valores en un álgebra de Cayley-Dickson así como para una extensión supersimétrica de la misma ecuación. Asimismo, se relacionan las transformaciones con la integrabilidad de tales sistemas, en el sentido de poseer in nitas cantidades conservadas.

REFERENCES [1] H. D. Wahlquist and F. B. Estabrook, Phys. Rev. Lett. 31, 1387 (1973).

[2] A. Restuccia, A. Sotomayor y J. P. Veiro, J. Phys. A: Math. Theor. 51,345203 (2018).

[3] A. Restuccia y A. Sotomayor, J. Phys. Conf. Ser. 738(1):012039 (2016).

Adrián Sotomayor

Departamento de Matemáticas Universidad de Antofagasta, Chile adrian.sotomayor@uantof.cl Alvaro Restuccia

Departamento de Física Universidad de Antofagasta, Chile alvaro.restuccia@uantof.cl Joint work with Jean Pierre Veiro

Departamento de Matemáticas Universidad Simón Bolívar Venezuela. jpveiro@usb.ve

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15. Note on a Non Linear Perturbation of the Ideal

Bose Gas

R. Tabilo

In this work we show that the introduction of a U (1) symmetry breaking eld in the energy operator of the boson-free gas, is equivalent, in the thermodynamic limit, to the inclusion, in the Hamiltonian of the ideal gas, of a non-linear function of the number operator associated with the zero mode. In other words, the limit pressures coincide. Moreover, both models undergo non conventional Bose-Einstein condensation (BEC).

REFERENCES Girardeau. Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension . Jour. Math. Phys. 1, 516-523, 1960.

S. R. S. Varadhan. Asymptotic Probabilities and Di erential Equations , Comm.

[1] M.

[2]

Pure Appl. Math.

[3] W. J. Mullin. ature

106,

19,

261-286 (1966).

Bose-Einstein condensation in harmonic potential . J. Low temper-

615-642 (1997).

[4] Alexander L. Gaunt, Tobias F. Schmidutz, Igor Gotlibovych, Robert P. Smith,

Bose-Einstein Condensation of Atoms in a Uniform Potential. Phys. Rev. Lett. 110, 200406 (2013).

Nir Navon, Alexander L. Gaunt, Robert P. Smith, Zoran Hadzibabic. Critical dynamics of spontaneous symmetry breaking in a homogeneous Bose gas . Science, and Zoran Hadzibabic.

[5]

Vol. 347, Issue 6218, pp. 167-170

On the theory of superfuidity . J. Phys. (USSR) 11, 23 (1947) On the asymptotic exactness of the Bogoliubov approximation for many boson systems, Commun. Math. Phys. 8, 26 (1968).

E. H. Lieb, R. Seiringer, and J. Yngvason, Justi cation of c-Number Substitutions in Bosonic Hamiltonians Phys. Rev. Lett. 94, 1-4, 2005 M. Corgini and R. Tabilo. Non Linear Perturbation of the Ideal Bose Gas in: Bose Gases: Beyond the In nitely extended systems I , DMATULS Essays No

[6] N.N. Bogolubov, [7] J. Ginibre,

[8]

[9]

1, chapter 22, pp. 92-102, 2018 (Ed. M. Corgini). DMATULS Digital Editions.

https://arxiv.org/abs/1812.06380

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36

Model of Interacting Spin One Bosons . U. Jour. of

Phys. an Appl., 8, 42-47- 2014 [11] A. Süt®,

Equivalence of Bose-Einstein condensation and symmetry breaking .

Phys. Rev. Lett. 94, 080402 (2005) [12] N.N. Bogoliubov. Quasi-Averages in Problems of Statistical Mechanics, in N.N. Bogoliubov, Jr., Quantum Mechanics: selected Works of N. N. Bogoliubov. World Scienti c Publishing, 2015.

[13] R. B. Gri ths.

Phys.

5

A Proof that de free energy of a spin system is extensive . J. Math.

(1964) 1215-1222.

Rosanna Tabilo Segovia

Departamento de Matemáticas Universidad de La Serena, Chile rtabilo@userena.cl Joint work with: Marco Corgini Videla

Departamento de Matemáticas Universidad de La Serena, Chile mcorgini@userena.cl

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16. Delone sets and (bi-)Lipschitz equivalence R. Viera

Motivated by problems in many branches in mathematics (e.g. discrete geometry, geometric group theory, information theory, mathematical physics of quasi-crystals), over the last years there has been a lot of activity on Lipschitz equivalence of discrete sets. The goal of this talk is to deal with the (bi)Lipschitz equivalence of Delone sets in Rd .

Acknowledgements. Partially funded by Fondecyt 1160541 REFERENCES [1] D. Burago & B. Kleiner.

[2]

Separated nets in Euclidean space and Jacobians of

bi-Lipschitz maps. Geom. Funct. Anal. 8 (1998), 273-282.

D. Burago & B. Kleiner. Rectifying separated nets . Geom.

Funct. Anal.

12(1)

(2002), 80-92.

[3] M.I. Cortez & A. Navas.

Some examples of repetitive, non-recti able Delone sets.

Geom. & Top. (2016).

[4] M. Dymond, V. Kaluºa & E. Kopecká.

Mapping n grid points onto a square forces

an arbitrarily large Lipschitz constant Geom. Funt. Anal., 28(3):589â ¿ 644, 2018.

Equivalence classes of codimension one cut-and-project nets. Ergod.

[5] Haynes, A.

Th. & Dynam. Sys. (2016),

36,

816-831.

[6] Haynes, A., Kelly, M., Weiss, B.

Equivalence relations on separated nets arising

from linear toral ows. Proc. London Math. Soc. 109 (2014), 1203-1228.

Lipschitz maps and nets in Euclidean space . Geom. Funct.

[7] C. McMullen.

8

Anal.

(1998), 304-314.

Densities non-realizable as the Jacobian of a 2-dimensional bi-Lipschitz map are generic.. J. of Top. and Anal., 10(04), 933-940 (2018).

[8] R. Viera.

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Rodolfo Viera

Departamento de Matemรกticas y C.C. Universidad de Santiago Santiago, Chile rodolfo.viera@usach.c

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DMATULS PROCEEDINGS Departamento de Matemรกticas Facultad de Ciencias Universidad de La Serena Cisternas 1200, La Serena, Chile http://www.dmatuls.cl edicionesdmatuls@userena.cl