STATISTICAL PHYSICS AND BEYOND: 2nd Mexican Meeting on Mathematical and Experimental Physics

Random and Deterministic Walks on Lattices
View Description Hide DescriptionRandom walks of particles on a lattice are a classical paradigm for the microscopic mechanism underlying diffusive processes. In deterministic walks, the role of space and time can be reversed, and the microscopic dynamics can produce quite different types of behavior such as directed propagation and organization, which appears to be generic behaviors encountered in an important class of systems. The various aspects of classical and not so classical walks on lattices are reviewed with emphasis on the physical phenomena that can be treated through a lattice dynamics approach.

Nonequilibrium Molecular Dynamics: Reversible Irreversibility from Symmetry Breaking, Thermostats, Entropy Production, and Fractals
View Description Hide DescriptionNonequilibrium Molecular Dynamics requires an extension of Newtonian and Hamiltonian mechanics. This new extended mechanics includes Gauss’ and Nosé’s thermostatted equations of motion. Here I review the past 20 years’ history of the various formulations, solutions, interpretations, and further extensions of these “new” motion equations. I emphasize the fractal nature of the resulting phase‐space distributions. I describe the connections of these fractal distributions to irreversibility, to time‐symmetry breaking (from reversible motion equations), and to entropy production and the Second Law, far from equilibrium.

Statistical Wave Scattering in Chaotic and Disordered Systems: Random Matrices and Maximum Entropy
View Description Hide DescriptionWe present a statistical theory of complex wave‐interference phenomena, applicable to systems where the complexity in wave scattering may derive from the chaotic nature of the underlying classical dynamics, as in microwave cavities and quantum dots, or from the quenched randomness of scattering potentials, as in disordered conductors. The resulting interference pattern is so complex that only a statistical treatment is meaningful. We follow a maximum‐entropy approach, in which Shannon’s information entropy is maximized, subject to the symmetries and constraints that are physically relevant. This is done in the framework of the powerful, non‐perturbative, approach known as random‐matrix theory.

Pure Fluids Confined by Chemically Nanostructured Solid Surfaces. Mean‐field Theory versus Monte Carlo Simulations
View Description Hide DescriptionWe focus on the phase behavior of fluids confined between planar, smooth solid surfaces that are endowed with a chemical structure (slit‐pore). The surfaces are separated by a distance comparable with typical ranges of fluid‐fluid and fluid‐substrate interactions which we model according to the square‐well potential. We employ a lattice model where fluid molecules are located at the nodes of a simple‐cubic lattice. The lattice constant is equal to the width of the attractive well so that fluid molecules interact only with their nearest neighbors. Based upon Bogoliubov’s variational approach combined with a mean‐field treatment of the system hamiltonian we arrive at a closed analytic expression for the grand potential Ω which we minimize numerically to compute the phase diagram of the confined fluid. We compare the mean‐field results with first‐principles grand canonical ensemble Monte Carlo (GCEMC) simulations where Ω can be computed via thermodynamic integration. Semi‐quantitative agreement between the two methods reveals that mean‐field theory is a powerful approach which is important because it is computationally far less demanding than GCEMC simulations.

Transient Currents in Nanoscopic Circuits
View Description Hide DescriptionQuantum transient currents are obtained after the sudden release of a particle initially confined in a nanoscopic circuit. We extend the 1D theory of diffraction in time to particles released from nanoscopic circuits with cylindrical cross‐sections. For a circular cross‐section, the particle’s free time evolution is described by cylindrical waves with their amplitudes showing diffraction in time. For large observation distances, the time‐dependent probability density current looks similar to the Fraunhofer diffraction pattern by a circular aperture. A temporal quantum Airy disk can be defined.

Non‐Local Effects in the Casimir Force
View Description Hide DescriptionAlthough the Casimir force, i.e., the force between the walls of a cavity due to the zero point and the thermal fluctuations of its electromagnetic field, was predicted half a century ago, it has only been measured with precision in the last decade. The possibility of comparing theory to experiment and the importance that Casimir forces might have on micro and nano machines has stimulated a renewed interest in their precise calculation for real materials. We show that the character of the cavity field is completely determined by the optical reflection amplitudes of the wall materials. Thus, we obtained an expression for the Casimir force which requires no assumption and no particular model for its walls; our results constitute a generalization of Lifshitz formula, applicable to a wide class of materials, which could be semi‐infinite or finite, local or spatially dispersive, homogeneous or layered, dissipative or dissipationless, isotropic or anisotropic, etc. As an application, we evaluate the force between two metallic slabs accounting for the spatial dispersion of the dynamical response of their conduction electrons. A self‐consistent jellium theory predicts a force that is significantly larger than that of a local theory at nanometric distances due to the fact that most of the screening charge at a metallic surface lies outside the nominal surface of the conductor and within vacuum.

Classical‐Quantum Competition Between two Capacitively Coupled Josephson Arrays with Micron‐Size Junctions
View Description Hide DescriptionWe have studied the helicity modulus and the inverse dielectric constant of two capacitively coupled Josephson junction arrays with charging energy, E_{c} , and Josephson coupling energy, E_{J} . The parameter that quantifies the quantum fluctuations in the i‐th array is defined by α_{ i } ≡ E_{ci }/E_{Ji } . Depending on the value of α_{ i }, each independent array may be in the semiclassical or in the quantum regime. Vortices are the dominant topological excitations in the semiclassical limit, while charge excitations are important in the quantum regime. We have extensively studied the interplay between vortex and charge dominated individual array phases. For each array we studied the behavior of their helicity modulus, Υ(α), and the inverse dielectric constant, ε^{−1}(α) as a function of temperature and inter‐layer capacitance. The two arrays are coupled via the interlayer capacitance C _{inter} at each site of the lattices. The results of extensive path integral Monte Carlo calculations indicate the existence of a reentrant transition in Υ(T, α), at low temperatures, when one of the arrays is in the semiclassical limit (i.e. α_{1} = 0.5) and the quantum array has 2.0 ⩽ α_{2} ⩽ 2.5, when 0.26087 ⩽ C _{inter} ⩽ 1.30435. Similar results were obtained for α_{2} = 4.0 with C _{inter} = 1.04348 and 1.30435. Nonetheless, for smaller values of C _{inter} the superconducting‐normal transition was not present. When array 2 is in the full quantum regime, 3.0 ⩽ α_{2} < 4.0, and for all values of C _{inter}, considered here, a novel reentrant phase transition occurs in the charge degrees of freedom, at low temperatures.

Formal Relationships Between the Bound States of Spatially Confined and Unconfined Quantum Systems
View Description Hide DescriptionFormal results about the properties of confined and unconfined quantum systems are discused. The results provide a rigorous theoretical foundation to many numerical results (that consider from one‐dimensional problems to atoms and molecules) which show that the properties of a system confined into a box Ω with impenetrable walls, converge to those of the unconfined system as Ω increases, independently of the shape of Ω. The analysis starts from the Schrödinger equation so that its extrapolation to density functional theory is immediate. The similarity beween the properties of confined and unconfined systems provides a powerful scheme to compute the bound states of the latter. This approach solves in a natural way several problems posed by the standard numerical methods to compute the eigenstates of a Schrödinger operator.

Effective Theory of Superconductivity in Carbon Nanotube Ropes
View Description Hide DescriptionWe summarize the main features of the effective low‐energy theory proposed to describe intrinsic superconductivity in Carbon nanotube ropes. A rope is modelled as an array of ballistic metallic nanotubes, with phonon‐mediated plus Coulomb interactions, and Josephson coupling between adjacent tubes. The Ginzburg‐Landau action including quantum fluctuations is derived and analyzed. Quantum phase slips are shown to cause a depression of the critical temperature T_{c} below the mean‐field value, and a temperature‐dependent resistance below T_{c} .

Applications of Statistical Physics in Cell Biology
View Description Hide DescriptionThe use of statistical physics and thermodynamics in cell biology is illustrated with examples relating to 1) membrane‐embedded, switchable ion transport channels and 2) clathrin coats, which play a central role in receptor‐mediated endocytosis and other cellular transport processes.

Ring Currents in Carbon Nanostructures: Magnetic Field Effects
View Description Hide DescriptionC_{60} polymerized carbon nanotori and haeckelites nanostructures are investigated when they are exposed to an external magnetic field (B). A π‐electron tight binding model in conjunction with the London approximation has been implemented in order to calculate the ring currents produced by the magnetic field B and thus, we determined the diamagnetic or paramagnetic response as a function of the carbon atomic arrangement. We have found that magnetic response depends strongly of the morphology of the carbon nanostructures. The results obtained for corrugated rings formed by C_{60} joined by 2‐fold, 3‐fold, and 5‐fold symmetry axis. In this case, we found that when C_{60} is joined, by the 5‐fold axis (by means of pentagonal rings), nanotori structures exhibit strong current fluxes that favor the generation of large magnetic moments perpendicular to the ring plane. When the C_{60} is joined trough the 2‐fold or 3‐fold axis of symmetry, the corrugated rings exhibit interesting ring patterns which depend of the number of C_{60} molecules. In addition, electronic and magnetic properties of haeckelite type tori are discussed.

Magnetism and Magneto Optics in Nanostructure Arrays
View Description Hide DescriptionRecent developments at the authors’ lab in the study of magnetic and magneto optical (MO) properties of nanostructure arrays are reviewed. The study focus on the effect on the magnetic and MO properties that patterning has in single layer ferromagnetic thin films, and thin films made of ferromagnetic insulator heterostructures. It is demonstrated the outmost importance of the magnetostatic coupling to describe the magnetization processes of the arrays, where fields and magnetization lay in plane. In single layer patterned arrays the magnetostatic coupling is responsible of the array behaving magnetically in a collective fashion below a certain threshold interelement separation distance. As expected, shining light on a periodic structure produces reflected and diffracted spots, and the significance of MO measurements in the diffracted beams is briefly addressed. For metal/insulator/metal array structures it is again the magnetostatic coupling, and the freedom of the magnetic flux to close now in the film growth direction, the determining factor in the measured magnetization processes. This produces and antiparallel alignment of the magnetizations of top and bottom electrode at zero field when the insulating barrier is thick enough to exchange decouple the electrodes. When the tunnel junction array is fabricated in such a way that instead of having physically separated junctions the junctions share the bottom electrode, the magnetostatic coupling is still the determining factor. This happens independently of the degree of overetching of the bottom electrode below the insulating tunnel barrier, and has allowed the fabrication of 1D and 2D periodic domain patterns in continuous flat ferromagnetic films. Fabricating this kind of structures (an array on top of a continuous film) on transparent substrates allows the MO measurements at either the patterned or the flat sides of the structures. At magnetic saturation the continuous film behaves as a mirror as expected, while at selected field values, where the periodic domain structure appears, it diffracts. This is a confirmation of a pure magneto optic diffraction due to a periodic domain structure.

Luminescent Photonics with Porous Silicon Nanostructures
View Description Hide DescriptionThe discovery of, fourteen years ago, the visible luminescence of porous silicon produced many expectations on the possibility of building optoelectronic devices. However, the difficulty in producing stable luminescence forced to look for other applications of this material. The possibility of constructing porous silicon layers with different refractive indices opened its use in the field of photonics. For this purpose, theoretical modeling of the refractive index is required. Many efforts were done in the past in order to combine the luminescence and the photonics in a single device. In this paper we discuss the origin of the luminescence, some theoretical drawbacks in calculating the dielectric function of this complex material, and we present some difficulties and solutions to fabricate photonic luminescent nanoestructure porous silicon multilayers.

Pressure Stimulated Electronic Transitions in Mats of Single‐Walled Carbon Nanotubes
View Description Hide DescriptionWe report high pressure studies in mats of single‐walled carbon nanotubes (SWNTs). Hydrostatic or quasi‐hydrostatic pressure can probe many electronic features. Resistance — Temperature measurements in mats of SWNTs with different chemical cleaning treatment and under quasi‐hydrostatic pressure, reveal some differences in the electronic characteristics: semiconducting‐like behavior, Kondo‐like features, due to magnetic impurities used to catalyse the nanotube formation, a metallic state and at high pressure a possible superconducting transition.

The Unfolding and Refolding Reactions of Triosephosphate Isomerase from Trypanosoma Cruzi Follow Similar Pathways. Guanidinium Hydrochloride Studies
View Description Hide DescriptionThe unfolding and refolding reactions of Trypanosoma cruzi triosephosphate isomerase (TcTIM) was studied under equilibrium conditions at increasing guanidinium hydrochloride concentrations. The changes in activity intrinsic fluorescence and far‐ultraviolet circular dichroism as a function of denaturant were used as a quaternary, tertiary and secondary structural probes respectively. The change in extrinsic ANS fluorescence intensity was also investigated. The results show that the transition between the homodimeric native enzyme to the unfolded monomers (unfolding), and its inverse reaction (refolding) are described by similar pathways and two equilibrium intermediates were detected in both reactions. The mild denaturant concentrations intermediate is active and contains significant amount of secondary and tertiary structures. The medium denaturant concentrations intermediate is inactive and able to bind the fluorescent dye. This intermediates are maybe related with those observed in the denaturation pattern of TIMs from other species; the results are discussed in this context.

Diffusion’s Study of Free Ligands Between Vesicle and Tubules Within the Endosome
View Description Hide DescriptionIn the mid 80’s, Linderman and Lauffenburger suggested that the sorting process of receptors and ligands in the endosome could be explained in terms of pure diffusion. With a reasonable choice of parameters in their model, they were led to the prediction that by the sorting time most of the receptors are in the tubule while the ligands apparently equilibrate throughout the vesicle and tubule volumes. The fraction of receptor predicted by their model is in excellent agreement with experimental observations. To calculate the mean capture time for the ligands they studied the case when the entrance of the tubule is an absorbing boundary. They solved a Poisson’s equation inside a sphere with mixed boundary conditions. This neglects the ligand return probability to the vesicle from the tubules, and vice versa. Under this assumptions they predict that by the sorting time around 70–60% of the ligands remain in the vesicle. In contrast with their prediction, the experimental observation is that most of the ligand molecules remain in the vesicle by the sorting time, typically degraded and routed to the lysosome. In this work we focus in ligand’s diffusion within the vesicle. We extend Linderman and Lauffenburger’s model including the effect of the presence of the tubules in the system, allowing the ligands to diffuse between the vesicle and the tubules. The principal biological implication of this extension, is that by the sorting time around 91% of the ligands remain in the vesicle. It means, that they can go many times from one chamber to the other before it is removed from the system. The validity of approximations was checked by simulations that indicated excellent agreement between analytical and numerical results.

Coherent Neuron Response in Ordered Exponential Networks
View Description Hide DescriptionWe construct networks of identical neurons using a special Hodgkin‐Huxley model. It is seen that a coherent spike response is generated robustly in some of the networks, depending on their connectivity. When the network has very short diameter and contains identical nodes, as in fully interconnected networks, or in ordered exponential networks, the signal is synchronous and it is sustained. For networks lacking these properties, the signal dies out very quickly. Signals like these have been found experimentally in real brain circuits. The present model seems a good one to simulate this collective behaviour.

Transient Situations in Traffic Flow: Modelling the Mexico City Cuernavaca Highway
View Description Hide DescriptionIn this paper a recent variable anticipation cellular automata model for single‐lane traffic flow is extended to analyze the situation of free and congested flow in the Highway from Mexico City to Cuernavaca. This highway presents free flow in standard days; but in the returning day of long weekends or holidays it exhibits congested flow and in rush hours jamming appears. We illustrate how our CA model for traffic flow can deal appropriately with transient situations and can be used to search new alternatives that allow to improve the traffic flow in Mexican highways.

The Informational Entropy in Traffic Flow
View Description Hide DescriptionThe description of the traffic flow problem has been done through different points of view, macrocopic, kinetic and microscopic approaches have been developed in the literature. In this work the informational entropy is introduced for a single lane traffic flow in which some macroscopic variables such as the concentration, velocity and variance are known. The maximization of the informational entropy allows the construction of a balance equation for it. Besides we introduce the Fisher information integral to obtain the distribution function for the homogeneous and steady flow. In this case a minimization principle provides us with multiple states determined by the restrictions imposed by the knowledge of experimental data. Its relationship with the informational entropy is given.