Non-Uniform Drag Force on the Fermi Accelerator Model

Transcripción

Non-Uniform Drag Force on the Fermi Accelerator Model
Danila F. Tavares∗ and R. N. Costa Filho†
Departamento de Fı́sica, Universidade Federal do Ceará,
Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil
Edson D. Leonel‡
Departamento de Estatı́stica, Matemática Aplicada e Computação – UNESP–Universidade Estadual Paulista
Av. 24A, 1515 – CEP: 13506-900 – Rio Claro – SP – Brazil
(Dated: January 25, 2012)
Some dynamical properties for a particle suffering the action of a generic drag force are obtained
for a dissipative Fermi Acceleration model. The dissipation is introduced via a viscous drag force,
like a gas and is assumed to be proportional to any power of the velocity F ∝ −ηV γ . The dynamics
is described by a two dimensional nonlinear area-contracting mapping obtained via the solution of
the second Newton’s law of motion. We prove analytically that the decay of high energy is given by
a continued fraction which recovers the following expressions: (i) linear for γ = 1; (ii) exponential
for γ = 2 and (iii) second degree polynomial type for γ = 1.5. For any value of γ 6= 1, and γ 6= 2
the numerical results shows a polynomial behavior for the velocity decay. Our results are discussed
for both the complete and simplified versions. The procedure used in the present paper can be
extended to many different kinds of systems including a class of billiards problems.
∗ Electronic
address: [email protected]
address: [email protected]
‡ Electronic address: [email protected]
† Electronic
Dinámica del Cáncer y Teorı́a de Juegos: un
modelo extendido espacialmente
Rodrigo Toledo Hernández1 y Pablo Padilla Longoria2
1)Centro de Ciencias de la Complejidad (C3), CONACyT-UNAM,
Torre de Ingenierı́a 6◦ piso, CU; México D.F.
2)Instituto de Investigaciones en Matemáticas Aplicadas y en
Sistemas (IIMAS), UNAM, CU; México, D.F.
Abstract
Las aplicaciones de la teorı́a de juegos evolutiva (TJE) permean todas
las áreas de la biologı́a. Algunas ideas de mecanismos evolutivos pueden
ser formuladas en términos de ecuaciones matemáticas. La adecuación
en la Teorı́a Darwiniana puede depender de la abundancia relativa de
fenotipos dentro de las poblaciones (selección dependiente de la frecuencia). Este tipo de selección natural es descrito por la ecuación replicadora
de la TJE entre n diferentes fenotipos en una población infinitamente
grande, bien mezclada y sin mutaciones, por lo tanto es una herramienta
matemática apropiada para investigar estos sucesos.
dxi
= xi [f (~
x) − φ(~
x)]
dt
(1)
PN
para i = 1, 2, . . . , N donde xi = frecuencia del tipo i, f (x~i ) = 1 aij (x~j )
PN
es la función de adecuación y φ(x~i ) = 1 f (x~i )xi la adecuación promedio.
Por otro lado se puede argumentar en su contra que los efectos estocásticos
y espaciales son ignorados siendo que éstos tienen un papel importante en
los sistemas biológicos. “Bien mezclados” significa que la estructura espacial de la población es ignorada y que todos los individuos tienen la misma
probabilidad de interactuar con cualquier otro. En el presente trabajo se
incorporan explı́citamente aspectos de la estructura espacial del fenómeno
mediante su vecindad inmediata, ası́ como su naturaleza dinámica. Se usa
la ecuación replicadora para modelar la invasión de células de Mieloma
Múltiple al hueso, utilizando un juego de tres jugadores. Por lo que la
ecuación se modifica de la forma:
dxij
ij
i
= xij
x) − Φij (~
x)]
i [f (~
dt
(2)
P5
donde Φij = 15 1 fvi es el paramétro que acopla espacialmente a las
poblaciones. Se presenta el modelo y resultados tanto analı́ticos como
numéricos.
Conceptos clave: Teorı́a de Juegos Evolutiva, Ecuación Replicadora,
cáncer, Mieloma Múltiple.
Referencias:1) Nowak M, et al.(2004); Evolutionary Dynamics of Biological Games, Science 303:793. 2) Nowak M.; Ohtsuki H.(2006); The
replicator equation on graphs, J Theor Biol. 243(1): 86–97.
1
Anderson Localization in Ideal Metamaterials
with Random Refraction Index
E. J. Torres-Herrera, F. M. Izrailev, and N. M. Makarov
Benemérita Universidad Autónoma de Puebla,
Puebla, Pue., 72050, México
E-mail [email protected]
We consider one-dimensional periodic-on-average bi-layered structures with weak
randomness in refraction index (compositional disorder) of basic, a and b , slabs composing
the unit-cell. Main attention is paid to the comparison of two systems: the homogeneous stack
when both slabs are made from right-handed (RH) optic materials, and mixed stack when a
slabs contain RH-material while b layers are of left-handed (LH) material. The localization
length L loc , which is the key parameter of Anderson localization, is derived with the
Hamiltonian map method.
We show that when the thicknesses of basic layers are different, d a≠d b , the
localization length is described by an universal analytical expression that includes all possible
correlations within a and b disorders, as well as, between them. For both homogeneous RHRH and mixed RH-LH stack-structures, the inverse localization length obeys the conventional
frequency dependence
2
ω 0 .
L−1
for
(1)
loc∝ω
This expression remains valid also for the homogeneous RH-RH stack with d a =d b . In
addition, L loc vanishes at the bottom and top of spectral Bloch bands. However, specific
correlations in disorders may result in anomalous ω -dependence. In particular, the localization
length exhibits the Fabry-Perot resonances, resulting in its resonance increase, and
consequently, in the suppression of localization. If basic slabs are matched (have equal
impedances), the localization length diverges and the perfect transparency emerges, while a
disorder itself persists. This conclusion is general, and does not depend on the strength of
disorder.
The mixed RH-LH stack exhibits highly non-trivial properties originated from the
nonuniform distribution of wave phase, even in the case of white-noise disorder. We
analytically obtained that in this case the localization length diverges in the standard secondorder perturbation theory. Therefore, recently numerically revealed anomalies in L loc [3] are
due to the fourth order in disorder. Our further study confirmed this expectation and showed
that the correct limit has the following form
−1
8
ω 0 ,
L loc∝ ω
for
(2)
−1
6
and not L loc∝ ω as is claimed in Ref. [3]. Our numerical data manifest excellent agreement
with the analytical results.
[1] F.M. Izrailev, N.M. Makarov, E.J. Torres-Herrera, Physica B 405, 3022 (2010).
[2] E.J. Torres-Herrera, F.M. Izrailev, N.M. Makarov, Low Temp. Phys. 37, 1201 (2011).
[3] A.A. Asatryan et al., Phys. Rev. Lett. 99, 193902 (2007); Phys. Rev. B 81, 075124
(2010).
[4] E.J. Torres-Herrera, F.M. Izrailev, N.M. Makarov, (2012), to appear.
AMPLITUDE DEATH IN GLOBALLY COUPLED CHAOTIC SYSTEMS WITH
DELAY.
V. M. Rodríguez and M. G. Cosenza
Centro de Física Fundamental
Universidad de Los Andes
Mérida, Venezuela
Email: [email protected]
The phenomenon of amplitude death refers to a situation where individual oscillators
cease oscillating and become synchronized in a fixed point state when they are
coupled. In this work we investigate the emergence of amplitude death in systems of
chaotic oscillators, such as Rössler and Lorentz, subject to a global interaction with a
time delay. The appearance of synchronization in general, as well as the phenomenon
of amplitude death are studied on the space of parameters of the system, given by the
strength of the coupling and the amount of time delay. The regions of parameters
where amplitude death emerges on this space exhibit a complex structure, such as
islands of synchronization and islands of death. We have uncovered two distinct
scenarios for the occurrence of amplitude death: (i) synchronization and amplitude
death occur simultaneously, and (ii) synchronization precedes amplitude death. We
identify the conditions for the occurrence of both scenarios and propose a simple
geometric interpretation for them.
CRITICAL EXPONENTS VIA LOCALISATION OF INVARIANT SPANNING
CURVE FOR A FAMILY OF HAMILTONIAN SYSTEMS
Juliano A. de Oliveira and Edson D. Leonel
DEMAC and IGCE
Universidade Estadual Paulista - Rio Claro- SP - Brazil
Critical exponents that describe a transition from integrability to non-integrability in a
two-dimensional, nonlinear and area preserving map are obtained via localization of
the first invariant spanning curve in the phase space. In a general class of systems the
position of the first invariant tori is estimated by reducing the mapping of the system
to the standard mapping where a transition takes place from local to global chaos. The
phase space of the mapping shows a large chaotic sea surrounding periodic islands
and limited by a set of invariant tori whose position of the first of them depends on
the control parameters. The formalism leads us to obtain analytically critical
exponents that describe the behavior of the average variable (action) along the
chaotic sea. The result is compared to several models in the literature confirming the
approach is of large interest. The formalism used is general and the procedure can be
extended to many other different systems.
THE CONSEQUENCES OF WEAK DISSIPATION IN A FAMILY OF TWODIMENSIONAL MAPPING
Juliano A. de Oliveira and Edson D. Leonel
DEMAC and IGCE
The influence of weak dissipation and its consequences in a two-dimensional mapping
are studied. The mapping is parametrised by an exponent $\gamma$ in one of the
dynamical variables and by a parameter $\delta$ which denotes the amount of the
dissipation. It is shown that for different values of $\gamma$ the structure of the
phase space of the non-dissipative model is replaced by a large number of attractors.
The approach to the attracting fixed point is characterized both analytically and
numerically. The attracting fixed point exhibit a very complicate basin of attraction.
ANDERSON LOCALIZATION IN BI-LAYER DISORDERED ARRAYS WITH
ABSORPTION AND GAIN
O. Vazquez-Candanedo and F. M. Izrailev
Instituto de Física, Universidad Autónoma de Puebla,
Apt. Postal J-48, Puebla, Pue., 72570, México
We study Anderson localization in a bi-layer array composed by left and right-handed
materials with absorption/gain in basic slabs, and with a disorder incorporated in the
structure via random dielectric constants. Our main interest is in the transport
properties of such arrays as a function of frequency, and how these differ from cases
without absorption/gain Ref.[1]. We also analyze whether the prediction that the
inverse localization length can be expressed as a sum of two terms that separately
depend on the degree of gain/absorption and disorder obtained in Ref.[2], for models
with normal materials, are valid in the case of arrays with metamaterials. In the
numerical approach we use standard transfer matrix methods.
References:
[1]E.J. Torres-Herrera, F.M. Izrailev and N.M. Makarov, Non-conventional Anderson
Localization in Bilayered Structures with Metamaterials, (2011).
[2]Z.Q. Zhang, Phys. Rev. B, 52, 7960 (1995).
STEM CELL DYNAMICS OF THE ROOT MERISTEM NICHE
IN ARABIDOPSIS THALIANA.
María del Carmen Pérez Zarate, UACM, México
Alejandro López Malvez, ESCOM, México
Pablo Padilla Longoria, IIMAS, UNAM, D.F. México
Different studies in Arabidopsis thaliana have shown that meristems control the development
of organs in plants maintaining a balance between cell division and differentiation in the stem
cell niche.
In order to try to understand the dynamics of the root meristems, we propose a mathematical
model incorporating explicit space dependence that takes into account:


The pattern that is formed during early embryonic stages, in which the position of
meristematic cells plays a fundamental role in the determination of the corresponding
cell fate, due to both short and long range signals. The former come from neighboring
cells, while the latter are sent by mature cells from the rest of the plant and contribute
to the cell fate determination.
The quiescent center is considered as a reservoir of cells that replaces damaged cells
in the plant as well as a global organizing center of the cellular pattern.
With this model based on cellular automata and ordinary differential equations we intend to
show that the geometry and architecture of the root are coupled with the cell communication
processes mentioned above and that gives rise to a robust system.
STABILITY OF INJECTION BRINE MIXTURES
L. I. Ledesma Fosados (1), V. González Dávila (2),
J. A. Betancourt Mar (1) , E. J. Suárez Domínguez (1)
(1) Mexican Institute of Complex Systems. Tlaxcala 111 Col. Unidad Nacional, C.P.
89410, Cd. Madero, Tamps. México.
(2) Geo-Estratos S.A. de C.V, Calle 7 No 205-1, Col. Jardín 20 de Noviembre, C.P.
89440, Cd. Madero, Tamps. México .
Email:[email protected]
During the operation of an oil well, water is pumped to the reservoir and in this
way can maintain or even increase this pressure as a whole and thereby increase
the production of crude by physical displacement of the petroleum. However, the
most common problem with this method is the accumulation of sediments and the
formation of incrustations in the oil ducts due to the combination of ions of
connate water and the water introduced, affecting the production ducts and
injectors as well as the surface equipment. This represents increases in production
costs, for which we have sought to predict the amount of scale in pipes. These
predictions are quite limited by the number of variables that affect this system
such as temperature and ion concentration as well as heterogeneity of ions
present. In this work we determined the theoretical amount of precipitates from
experimental data on the concentration of ions in the mixture of connate water
and sea water in different percentages and determined on a gravimetric method.
These results were compared with hypothetical calculations of the concentration
of maximum possible precipitates and probable maximum precipitates expressed
in mg / L. It was found that the experimental results at low ion concentration
show a linear trend similar to the concentrations calculated at the same mixing
ratio. However, by increasing the concentration of ions, the experimentally
determined concentration loses its linearity.
ADAPTATIVE CONTROL OF ROBOT MANIPULATORS
Abraham Villatoro Tello, Dr. Fernando Reyes Cortés
Facultad de Ciencias De La Electrónica, BUAP,
Puebla, México.
Email: [email protected], [email protected]
In recent years, robotics has grown significantly due to advances in science such as
electronics, computers and communications. At the same time has grown the use
of control systems, which are increasingly sophisticated, as the tasks have become
more complex. This is why there is need for greater reliability and security of such
systems.
In response to these needs many proposals have emerged in the development of
controllers. Among them are the adaptive-type controllers, whose main feature is
the ease of handling the uncertainty created by changes in the system. That is,
the adaptive control is characterized by adapting to unpredictable environmental
changes, whether external or internal.
The position control of robot manipulators is a very important problem in the
industry, it can be interpreted as taking the robot from an initial position to a desired
position with the smallest error in the shortest time possible.
In this work we developed a new control structure, which was implemented in the last
link of a 3 degrees of freedom robot, which corresponds to the dynamic model of a
pendulum. We look for the tuning of the proportional gain to be automatic. This
means that the gain adjusts its values as the robot reaches the desired position.
This function was simulated for two types of controllers, and the results were very
favorable, which lead us to the next step, that is the experimental evaluation and
demonstrate Lyapunov stability.
ROBOTIC’S EXPERIMENTAL PLATAFORMA
L. Vargas Martínez, Dr. F. Reyes.
Grupo de robótica, FCE-BUAP, Apdo. Postal 62-570,
Av. San Claudio y 14 Sur, 72570, Puebla, Puebla, México.
[email protected], [email protected]
The robotic is a science is supported from various disciplines as: physic, mathematic,
electronic, control theory, the computational sciences and other. In the course of
time there’s been great progress as new control theories are proposed and as
technological advances allow, therefore, it is said that robotics is a new field of
modern technology.
This work addresses the implementation of a robot manipulator of 3 degrees of
freedom (d.o.f) as an experimental platform, this means, the mechanical part is linked
with the necessary electronics to enable the operation of the robot and software
development that allows position control under a proposed control law.
To know the physical parameters of the robot we use the method of parametric
identification based on the least squares algorithm and to validate the data was
performed a simulation of the dynamics of the robot based on the measured data
directly from the robot.
With the experimental platform we evaluate the proposed control algorithm in which
was showed a good performance that allow us it o say we achieve the goal of position
control.
That’s the importance of this type of platforms that are used for scientific research in
robotics.
DYNAMICAL PROPERTIES OF A FERMI-ULAM MODEL WITH EXTERNAL
PERTURBATION OF VAN DER POL TYPE
Tiago Botari
Departamento de Física, Universidade Estadual Paulista,UNESP,
Rio Claro - São Paulo, Brasil
Edson Denis Leonel
DEMAC and IGCE
The dynamics of a classical particle of mass m confined to bounce within two rigid
walls is considered. One of them is assumed to be fixed while the other one is time
dependent and its position is given by a Van der Pol oscillator. Two dynamical
regimes are considered: (i) the case where the moving wall is infinitely heavy and (ii)
the situation where the moving wall has a finite mass. For case (i), the statistical
properties of the average velocity of the particle reveals a scale invariance with
respect to the control parameters. For case (ii), a scenario of dissipation is observed
including attracting fixed points with a complicate basin
boundary geometry.
EVALUATION OF FORCE CONTROL STRATEGIES FOR ROBOT
MANIPULATORS
M.A. Limón-Díaz and F. Reyes-Cortés
Facultad de Ciencias de la Electrónica, BUAP, Puebla,
Puebla, México
Email: [email protected], [email protected]
The position and trajectory control have solved many problems in the industry, but in
several industrial applications such as polishing, surface cleaning, assembly, among
others, is not enough, because the robot manipulator enter in contact with the
environment, generating interaction forces between the robot manipulator and the
environment.
The force control is a technology that has been developed to all a gap in manufacture
automated processes such as machining, casting, forging, etc. This type of control
allows these industrial processes to obtain the desired result.
The force control is to ensure that the last link of the robot manipulator to apply a
desired force level during the task of interacting with the environment. An important
point to note is that a force control strategy only makes sense in those workspace
directions along which there are forces of interaction between the robot manipulator
and the environment.
Because of this, development new structures of force control is an important
contribution to the emerging needs of industry.
SYNCHRONIZATION ATTACK OF CHAOTIC COMMUNICATION
SYSTEMS
R. Sevilla-Escoboza (1), Massimiliano Zanin (3) , R. Jaimes-Reategui (1),
G. Huerta-Cuellar (1), A. N. Pisarchik (2), J. H. García-López (1)
(1) Centro Universitario de Los Lagos, Universidad de Guadalajara, Enrique Díaz de León 1144,
Paseo de la Montaña, Las de Moreno, Jalisco, México
(2)Centro de Investigaciones en Óptica A.C., Loma del Bosque 115, León, México.
(3) Universidad Autónoma de Madrid, 28049 Madrid, España
Synchronization of chaotic oscillators has an important application in cryptography.
When two identical oscillators are coupled, they can b e completely synchronized and
the chaotic output of the transmitter oscillator can b e used to mask a message.
Although the oscillator parameters are usually used as secret keys, the sensitivity of
such a cryptosystem to a parameter change has never been systematically analyzed.
To cryptanalize a communication system based on synchronization of chaotic
oscillators, we intro duce a synchronization attack which implies the recovering of
unknown parameters by searching a minimum synchronization error. Using the
synchronization attack we cryptanalize communication systems based on the Rossler
and Chua chaotic oscillators. We suggest to include this attack as a standard security
check for future works.
Keywords: Rossler oscillator, Chua oscillator, cryptanalysis, secure communication,
synchronization, chaos
NONLINEAR OPTICAL SYNAPSE
G. Huerta-Cuellar (1), R. Jaimes-Reategui (1), R. Sevilla-Escoboza (1),
A. N. Pisarchik (2), J. H. García-López (1), C. E. Castañeda-Hernandez (1),
and D. Lopez Mancilla (1)
(2) Centro de Investigaciones en Óptica A.C., Loma del Bosque 115, León, México
In this work we propose an optical synaptic sensor based on a fiber laser driven by a
neural FinzHung-Nagumo electronic circuit, to connect with another neuron. The
architecture of possible optical synaptic connections is described introducing
different kinds of opto-electronic coupling between neurons: laser cavity loss
modulation and pump laser modulation. The control parameters of the proposed
optical synapse provide additional degrees of flexibility to the neuron connection
traditionally controlled only by coupling strengths in artificial networks.
ROGUE WAVES IN A MULTISTABLE FIBER LASER
R. Jaimes-Reategui (1), R. Sevilla-Escoboza (1), G. Huerta-Cuellar (1)1,
A. N. Pisarchik (2), J. H. García-López (1), D. Lopez Mancilla (1),
and C. E. Castañeda-Hernandez (1)
(2) Centro de Investigaciones en Óptica A.C., Loma del Bosque 115, León, México
Clear evidence of rogue waves in a multistable system is revealed with an erbium-doped fiber laser
driven by harmonic pump modulation [1]. We demonstrate numerically and experimentally that a
low-pass noise filtering can control the probability for the appearance of a particular state. The
results of numerical simulations with the use of a three-level laser model display good agreement
with experimental results. The mechanism for the rogue wave formation lies in the interplay of
stochastic processes with multistable deterministic dynamics. Low-frequency noise applied to a
diode pump current induces rare jumps to coexisting subharmonic states with high-amplitude
pulses perceived as rogue waves. The probability of these events depends on the noise filtered
frequency and grows up when the noise amplitude increases. The probability distribution of spike
amplitudes confirms the rogue wave character of the observed phenomenon. We find the existence
of a particular noise amplitude for which a particular periodic orbit appears more frequently than
for other amplitudes.
1. A.N. Pisarchik, A.V. Kir’yanov, Yu.O. Barmenkov, R. Jaimes-Reátegui, Dynamics of an
erbium-doped fiber laser with pump modulation: theory and experiment, J. Opt. Soc. Am. B
22, 2107-2114 (2005).
ON DYNAMICS AND STATISTICS OF A CONSERVATIVE BOUNCER
MODEL: ACCELERATOR AND DECELERATOR MODES
André L. P. Livorati¹, Carl P. Dettmann², and Edson. D. Leonel³
¹ IFUSP - Univ. de São Paulo - USP - SP - Brazil
² Univ. of Bristol - England
³ DEMAC - Univ. Estadual Paulista - UNESP - SP - Brazil
The complete dynamics of a conservative Bouncer model is studied. The time
evolution of the system is given by a two dimensional nonlinear mapping in the
variables velocity of the particle and phase of the moving boundary. Fixed points were
found and characterized. A transition in the fixed points happens when the parameter
that controls the ratio of accelerations between the particle and the moving bounary,
called $\epsilon$ is bigger than the unity. In this transition, we were able to found
accelerator modes which leads the particle to experience the phenomenum of Fermi
acceleration with a time proportional, instead of $\sqrt(t)$. As well, we also found
decelerator modes in the anti-symmetric position of the accelerator ones, but they
shown to be very unstable. A whole numerical and statistical analyis were made in
order to identify, localize and understand the influence of these two modes in the
system dynamics concerning Fermi acceleration phenomenun.
EVOLUTION OF RNA VIRUSES WITH FREQUENCY-DEPENDENT
SELECTION.
Roberto Álvarez-Martínez
Centro de Ciencias de la Complejidad (C3),
Universidad Nacional Autónoma de México (UNAM),
D.F. México
RNA viruses show rapid adaptation both in laboratory and natural environments. The
enormous diversity generated by the high error rates is generally believed to be at the
root of such adaptive potential. Clearly, high mutation rates provide increased
variation for responding to strong selective pressures, hence accounting for the rapid
emergence of drug-resistant viruses, for instance HIV, hepatitis C, polio and inuenza.
During 30 years, quasi-species theory has become a standard conceptual frame to
study the RNA viruses evolution. However, one of the key problem with this theory is
its lack of frequency-dependent selection.
In other words, the fitness of a particular phenotype is set to a constant value
independently of the other competing 'players'. This can be solved by using the
replicator-mutator equations, which represents a general formulation for the
evolutionary dynamics. This work proposes, and solves computationally, a
modi_edquasi-species model in order to take this condition into account. This
modication allow us to study the Shannon entropy, the existence of an errorcatastrophe (a second-order phase transition), and the emergence of cooperation, we
interpret the results in terms of this more biologically realistic scheme.
"Diffusive properties of Lorentz gases via persistent
random walks"
Dr. David P. Sanders
Departamento de Física, Facultad de Ciencias, Universidad Nacional
Autónoma de México
Thomas Gilbert (1), Huu Chuong Nguyen (1) & *David P. Sanders* (2)
(1) Center for Nonlinear Phenomena and Complex Systems, Université
Libre de Bruxelles, C. P.
231, Campus Plaine, B-1050 Brussels, Belgium
(2) Departamento de Física, Facultad de Ciencias, Universidad Nacional
Autónoma de México,
04510 México DF, Mexico
We calculate [1] exact expressions for diffusion coefficients of persistent random walks on
cubic lattices. In these types of random walks, the direction of a walker at a given step
depends on the memory of its direction of motion in one or two previous steps. These
results are then applied to study billiard models, namely two- and three-dimensional
periodic Lorentz gases, with geometry chosen to exhibit normal diffusion. We calculate
numerically the transition probabilities between cells to compare the persistent randomwalk approximation with simulation results for the diffusion coefficient of the Lorentz
gases.
[1] Diffusive properties of persistent walks on cubic lattices with application to periodic
Lorentz gases. Thomas Gilbert, Huu Chuong Nguyen & David P. Sanders.
Normal modes of 1-D elastic networks
Francisco J. Martínez
IIMAS, UNAM, D.F. México
[email protected]
For a string of arbitrary length, where the interaction between particles
occurs within a fixed radius, the analysis is the relationship between mass
distribution and modes of vibration of this system, aiming at finding
localized vibrational modes in the regions higher bulk density. Using the
theory of normal functions and compared the results with those obtained
numerically.

Non-Uniform Drag Force on the Fermi Accelerator Model

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