Towards a comprehensive characterization of spatio-temporal dependence of light-induced electromagnetic forces in dielectric liquids

The interaction of localized light with matter generates optical electrostriction within dielectric fluids, leading to a discernible change in the refractive index of the medium according to the excitation's light profile. This optical force holds critical significance in optical manipulation and plays a fundamental role in numerous photonic applications. In this study, we demonstrate the applicability of the pump-probe, photo-induced lensing (PIL) method to investigate optical electrostriction in various dielectric liquids. Notably, the thermal and nonlinear effects are observed to be temporally decoupled from the electrostriction effects, facilitating isolated observation of the latter. Our findings provide a comprehensive explanation of optical forces in the context of the recently introduced microscopic Ampère electromagnetic formalism, which is grounded in the dipolar approximation of electromagnetic sources within matter and characterizes electrostriction as an electromagnetic-induced stress within the medium. Here, the optical force density is re-obtained through a new Lagrangian approach.

Optical detection of the ultrasound-induced pulsed thermal lens close to the ice-water phase transition

Piezo-optic and thermo-optic coefficients are important material properties that play a critical role in the design and optimization of many optical devices. The ability to accurately measure and control these coefficients is essential for achieving high performance and reliability in a wide range of applications. In this article, we use the optical detection of the ultrasound-induced thermal lens effect to investigate these properties for water at low temperatures. The results show that the anomalous behavior of water around 4°C is easily observed. The thermal lens method is used to determine the temperature dependence of the piezo-optic and thermo-optic coefficients.

Unveiling bulk and surface radiation forces in a dielectric liquid

Precise control over light-matter interactions is critical for many optical manipulation and material characterization methodologies, further playing a paramount role in a host of nanotechnology applications. Nonetheless, the fundamental aspects of interactions between electromagnetic fields and matter have yet to be established unequivocally in terms of an electromagnetic momentum density. Here, we use tightly focused pulsed laser beams to detect bulk and boundary optical forces in a dielectric fluid. From the optical convoluted signal, we decouple thermal and nonlinear optical effects from the radiation forces using a theoretical interpretation based on the Microscopic Ampère force density. It is shown, for the first time, that the time-dependent pressure distribution within the fluid chiefly originates from the electrostriction effects. Our results shed light on the contribution of optical forces to the surface displacements observed at the dielectric air-water interfaces, thus shedding light on the long-standing controversy surrounding the basic definition of electromagnetic momentum density in matter.

Photoactivation of Erythrosine in simulated body fluids

Photodynamic Therapy (PDT) is a powerful technique for the treatment of cancer and non-cancerous diseases. The precise PDT treatment protocol definition must consider the performance difference between in vitroand in vivoapplications. This also occurs in other biological studies, and to partially overcome this difficulty, the simulated body fluids are generally applied as a prior understanding of the particularities of the different systems. However, in PDT these studies are scarce. In this work, we investigated the photoactivation of Erythrosine, a photosensitizer widely used in PDT, in different simulated body fluids. Differences in the photodegradation kinetics, triplet lifetime, and singlet oxygen generation were observed. The results can help to explain and to define PDT application protocols.

Induction and detection of pressure waves by pulsed thermal lens technique in water-ethanol mixtures

The mode-mismatched dual-beam thermal lens technique is widely applied in the characterization of optical and thermo-physical properties of solids and liquids. The technique has also been used to investigate transient acoustic waves induced by pulsed laser excitation at the nanosecond time scale. In this paper, we developed a semi-analytical model to describe the transient acoustic wave that allows a fitting procedure to get the physical properties of fluid samples. The method was used to investigate samples with different mixtures of ethanol and water, and quantitative information of piezo-optic coefficient and sound speed are evaluated for the fluid mixtures.

Studies of the early stages of the dynamic setting process of chemically activated restorative glass-ionomer cements

Objective

To evaluate the early stages of the setting process of chemically activated restorative glass-ionomer cements (GICs).

Material and methods

Five GICs were evaluated (n = 5): Equia Forte (GC), Equia Forte HT (GC), Ketac Universal (3M ESPE), Maxxion R (FGM) and Riva Self Cure (SDI) by Thermography, Fourier Transform Infrared Attenuated Total Reflectance Spectroscopy (FTIR-ATR) and Gillmore needle indentation mechanical testing. The FTIR-ATR spectra showed the formation of metal carboxylates within the cements and enabled the stabilization time (ST) to be determined and the thermographic camera measured the temperature field images in the sample. Data were statistically analyzed by ANOVA and Tukey–Kramer (α = 5%).

Results

The Gillmore needle test showed that the order of hardening was opposite to the order of ST values determined by FTIR. The results with the thermographic camera showed two stages of temperature variation, which coincided with the evolution of specific infrared bands. The exception was Maxxion R, which showed only a single step change in temperature.

Conclusion

The early stages of the GIC setting reaction show temperature changes, both endothermic and exothermic, at specific times, confirming the occurrence of individual chemical reactions. The early setting involves reactions other than carboxylate formation.

Significance: This study gives further detail of the early stages of the setting of GICs, and past research regarding the setting reaction of GIC.

The correlation of physicochemical properties of edible vegetable oils by chemometric analysis of spectroscopic data

This work aimed to investigate and compare the composition and the physicochemical properties of 18 different sources of edible vegetable oils. A systematic study on the correlation between composition and physical properties was performed using Fourier Transform Infrared (FTIR) Spectroscopy and fatty acid chromatographic analysis. Principal component analysis of FTIR spectra is performed to classify edible oils concerning their physical properties. The results demonstrate the potentiality of the method associated with multivariate statistics analysis as powerful, fast, and non-destructive tools for characterization and quality control of edible vegetable oils.

An Experimental Investigation of Sample-Fluid Heat Coupling Effect in Thermal Lens Technique

Laser-induced wavefront distortion is detectable by several techniques based on the photothermal effect. The effect is probed by monitoring the phase shift caused by the bulging of the heated area, the photoelastic effects, and the spatial distribution of the refractive index within the sample and in the fluid surrounding it. A simple analytical solution for the wavefront distortion was only possible for low absorbing materials, with the assumption that the stresses obey either the thin-disk or the long-rod type distributions. Recently, a unified theoretical description for the laser-induced optical path change was proposed to overcome part of this limitation for weakly absorbing materials, regardless of its thickness. In this work, we perform an experimental investigation taking the sample-fluid heat coupling effect into account using the thermal lens technique. The experimental investigation presented here validates the unified model. In addition, we show that the heat-coupling model provides an alternative method to obtain physical properties of non-absorbing fluid by using a reference solid sample.

Nanosecond pressure transient detection of laser-induced thermal lens

We use the thermal lens technique in the nanosecond time scale to describe the acoustic wave effect in liquids and the corresponding correlation with the speed of sound in the fluid, volumetric thermal expansion, and piezo-optic coefficient. These physical properties are found to be directly correlated to the anomalous effects observed in the transients at the nanosecond time scale, where acoustic waves dominate the thermal lens signal inducing an oscillating transient. Our results suggest the application of the thermal lens to study the generation and the detection of thermo-acoustic waves in liquids, which makes this method interesting for all-optoacoustic ultrasound detection and imaging.

Laser induced thermoelastic surface displacement in solids detected simultaneously by photothermal mirror and interferometry

We propose a combined pump-probe optical method to investigate heat diffusion properties of solids. We demonstrate single-shot simultaneous laser-induced thermoelastic surface displacement of metals detected by concurrent measurements using photothermal mirror and interferometry. Both methods probe the surface displacement by analyzing the wavefront distortions of the probe beams reflected from the surface of the sample. Thermoelastic properties are retrieved by transient analysis in combination with numerical description of the thermoelastic displacement and temperature rise in the sample and in the surrounding air. This technique presents a capability for material characterization that can be extended to experiments for quantitative surface mapping.

From Protohypericin to Hypericin: Photoconversion Analysis Using a Time-Resolved Thermal Lens Technique

Hypericin (Hyp) is a natural compound with interesting photophysical and pharmacological properties, which has been used in photodynamic therapy and photodynamic inactivation of microorganisms. Its synthesis is based on a series of chemical processes that ends with a light-drug interaction by the photoconversion of protohypericin (pHyp) to Hyp. Although this photosensitizer is used in a variety of medical applications, the photophysical and photochemical mechanisms involved in the final step related to the photo production of Hyp are not completely understood at the molecular level. Protohypericin concentration, solvents, light irradiation under different wavelengths, and a sort of variables could play an important role in predicting the yielding of this photoconversion process. Here, we used the high-sensitive and remote measurement characteristics of the time-resolved thermal lens technique to investigate the relation between the light-induced photoconversion rate of pHyp to Hyp and the initial concentration pHyp. The results show a linear dependence of the photoreaction rate with the concentration of pHyp, indicating that the overall reaction process includes steps comprising the formation of distinct intermediate species. We demonstrate the applicability of the thermal lens technique for the photochemical characterization of photosensitive drugs at low concentration levels.

Characterization of Heat Diffusion Properties of Rubberized Two-Layer Systems Using Open Photoacoustic Cell Spectroscopy

We applied the open photoacoustic cell method operating at high frequency as an efficient and highly precise tool for the measurement of thermal properties of rubberized two-layer systems. The heat-coupling between the two layers is treated using the analogy between thermal and electrical resistances widely used in heat transfer problems. The thermal resistance between the two layers is considered effective and the problem is decoupled for each layer. Measurements are performed in two-layer samples of aluminum foil coated with layers of rubberized paint with different thicknesses. Thermal diffusivity and thermal conductivity are determined for the paint. The results are retrieved from the thermal resistance model assuming the effective thermal diffusivity of the composite material.

Analysis of the Thermo-Reflectivity Coefficient Influence Using Photothermal Pump-Probe Techniques

Recent improvements in the modeling of photo-induced thermo-optical-mechanical effects have broadened the application of photothermal techniques to a large class of solids and fluids. During laser excitation, changes in optical reflectivity due to temperature variation may affect the photothermal signal. In this study, the influence of the reflectivity change due to heating is analyzed for two pump-probe photothermal techniques, thermal lens and thermal mirror. A linear equation for the temperature dependence of the reflectivity is derived, and the solution is tested using optical properties of semi-transparent and opaque materials. For semi-transparent materials, the influence of the reflectivity change in photothermal signals is less than 0.01%, while for opaque materials it is lower than 3%.

Investigation into photostability of soybean oils by thermal lens spectroscopy

Assessment of photochemical stability is essential for evaluating quality and the shelf life of vegetable oils, which are very important aspects of marketing and human health. Most of conventional methods used to investigate oxidative stability requires long time experimental procedures with high consumption of chemical inputs for the preparation or extraction of sample compounds. In this work we propose a time-resolved thermal lens method to analyze photostability of edible oils by quantitative measurement of photoreaction cross-section. An all-numerical routine is employed to solve a complex theoretical problem involving photochemical reaction, thermal lens effect, and mass diffusion during local laser excitation. The photostability of pure oil and oils with natural and synthetic antioxidants is investigated. The thermal lens results are compared with those obtained by conventional methods, and a complete set of physical properties of the samples is presented.

Modeling population and thermal lenses in the presence of Auger Upconversion for Nd(3+) doped materials

We present a theoretical model to thermal (TL) and population (PL) lenses effects in the presence of Auger upconversion (AU) for analysis of Nd(3+) doped materials. The model distinguishes and quantifies the contributions from TL and PL. From the experimental and theoretical results, the AU cannot be neglected because it plays an important role on the excited state population and therefore on the temperature and polarizability difference between excited and ground states. Considering the extensive use of these techniques, the model presented here could be useful for the investigation of materials and also to avoid misleading analysis of lenses transients.

Pulsed photothermal mirror technique: characterization of opaque materials

The time-resolved thermal mirror technique is developed under pulsed laser excitation for quantitative measurement of thermal and mechanical properties of opaque materials. Heat diffusion and thermoelastic equations are solved analytically for pulsed excitation assuming surface absorption and an instantaneous pulse. Analytical results for the temperature change and surface displacement in the sample are compared to all-numerical solutions using finite element method analysis accounting for the laser pulse width and sample geometry. Experiments are performed that validate the theoretical model and regression fitting is performed to obtain the thermal diffusivity and the linear thermal expansion coefficient of the samples. The values obtained for these properties are in agreement with literature data. The technique is shown to be useful for quantitative determinations of the physics properties of metals with high thermal diffusivity.

Role of photophysics processes in thermal lens spectroscopy of fluids: a theoretical study

Photophysics processes are ubiquitous in nature and difficult to be quantitatively characterized by conventional spectroscopy. Alternatively, pump-probe methods have been widely applied to study these complex processes. In this context, the thermal lens technique is a precise spectroscopic tool for material characterization and presents a wide range of applications in chemical analysis. Here, we present an all numerical approach to analyze the dynamics of photophysics processes and to identify the role of individual contributions of photoreaction and mass diffusion in the thermal lens experiments. The results are essential for a proper understanding of the dominant physical mechanisms in laser-induced photodegradation, which allow precise data analysis of the effects in photosensitive fluids.

Unravelling the effects of radiation forces in water

The effect of radiation forces at the interface between dielectric materials has been a long-standing debate for over a century. Yet there has been so far only limited experimental verification in complete accordance with the theory. Here we measure the surface deformation at the air-water interface induced by continuous and pulsed laser excitation and match this to rigorous theory of radiation forces. We demonstrate that the experimental results are quantitatively described by the numerical calculations of radiation forces. The Helmholtz force is used for the surface radiation pressure. The resulting surface pressure obtained is consistent with the momentum conservation using the Minkowski momentum density expression assuming that the averaged momentum per photon is given by the Minkowski momentum. Considering the total momentum as a sum of that propagating with the electromagnetic wave and that deposited locally in the material, the Abraham momentum interpretation also appears to be appropriate.

Discriminating the role of sample length in thermal lensing of solids

Thermal lens (TL) is a key effect in laser engineering and photothermal spectroscopy. The amplitude of the TL signal or its dioptric power is proportional to the optical path difference (OPD) between the center and border of the beam, which is proportional to the heat power (Ph). Due to thermally induced mechanical stress and bulging of end faces of the sample, OPD depends critically on the geometry of the sample. In this investigation, TL measurements were performed as a function of the sample length keeping the same Ph. It is experimentally demonstrated that for materials with positive ∂n/∂T OPD increases typically 30 to 50% with the decrease of sample length (from long rod to thin-disk geometry). For materials with negative ∂n/∂T, this variation is much larger due to the cancelation of the different contributions to OPD with opposite signs. Furthermore, the experimental investigation presented here validates a recently proposed unified theoretical description of the TL effect.

Combined photothermal lens and photothermal mirror characterization of polymers

We propose a combined thermal lens and thermal mirror method as concurrent photothermal techniques for the physical characterization of polymers. This combined method is used to investigate polymers as a function of temperature from room temperature up to 170 °C. The method permits a direct determination of thermal diffusivity and thermal conductivity. Additional measurements of specific heat, linear thermal expansion, and temperature-dependent optical path change are also performed. A complete set of thermal, optical, and mechanical properties of polycarbonate and poly (methyl methacrylate) samples are obtained. Methods presented here can be useful for in situ characterization of semitransparent materials, where fast and non-contacting measurements are required.

Resonant excited state absorption and relaxation mechanisms in Tb3+-doped calcium aluminosilicate glasses: an investigation by thermal mirror spectroscopy

Resonant excited state absorption (ESA) and relaxation processes in Tb(3+)-doped aluminosilicate glasses are quantitatively evaluated. A model describing the excitation steps and upconversion emission is developed and applied to interpret the results from laser-induced surface deformation using thermal mirror spectroscopy. The fluorescence quantum efficiency of level (5)D(4) was found to be close to unity and concentration independent while, for the level (5)D(3), it decreases with Tb(3+) concentration. Emission spectroscopy measurements supported these results. ESA cross sections are found to be more than three orders of magnitude higher than the ground state absorption cross section.

Pulsed-laser time-resolved thermal mirror technique in low-absorbance homogeneous linear elastic materials

A theoretical model for a time-resolved photothermal mirror technique using pulsed-laser excitation was developed for low absorption samples. Analytical solutions to the temperature and thermoelastic deformation equations are found for three characteristic pulse profiles and are compared to finite element analysis methods results for finite samples. An analytical expression for the intensity of the center of a continuous probe laser at the detector plane is derived using the Fresnel diffraction theory, which allows modeling of experimental results. Experiments are performed in optical glasses, and the models are fitted to the data. The parameters of the fit are in good agreement with previous literature data for absorption, thermal diffusion, and thermal expansion of the materials tested. The combined modeling and experimental techniques are shown to be useful for quantitative determination of the physical properties of low absorption homogeneous linear elastic material samples.

Modeling the population lens effect in thermal lens spectrometry

We report a theoretical model and experimental results for laser-induced lensing in solids. The model distinguishes and quantifies the contributions from population and thermal effects. Laser-induced lensing in ytterbium-doped fluorozirconate glass ZBLAN:Yb(3+) is measured, and the thermal and optical properties obtained from analyzing the data with the proposed model agree well with published values. Photothermal techniques are used extensively for the investigation of laser and laser-cooling materials, and the model developed here enables the interpretation of convoluted laser-induced lensing signals that have contributions from different sources.

A theoretical and experimental study of time-resolved thermal mirror with non-absorbing heat-coupling fluids

A theoretical and experimental study taking sample-fluid heat coupling into account in time-resolved photothermal mirror experiments is presented. Thermoelastic equations were solved to obtain a semi-analytical solution to the phase shift induced by the sample and the surrounding fluid. The solution was used to model the thermal mirror effects and found to be in excellent agreement with the finite element method analysis and experiment. Heat transferred to the air-coupling fluid did not introduce important effects in the phase shift when compared with the solution obtained, without considering heat flux. However, when using water as the fluid, heat coupling led to a significant effect in fluid phase shift. Experimental results using stainless steel in air and water were used to demonstrate the potentiality of the thermal mirror technique to determine the thermal properties of both the sample and the fluid.

Scale-invariant structure of size fluctuations in plants

A wide range of physical and biological systems exhibit complex behaviours characterised by a scale-invariant structure of the fluctuations in their output signals. In the context of plant populations, scaling relationships are typically allometric. In this study, we analysed spatial variation in the size of maize plants (Zea Mays L.) grown in agricultural plots at constant densities and found evidence of scaling in the size fluctuations of plants. The findings indicate that the scaling of the probability distribution of spatial size fluctuation exhibits non-Gaussian behaviour compatible with a Lévy stable process. The scaling relationships were observed for spatial scales spanning three orders of magnitude. These findings should provide additional information for the selection and development of empirically accurate models of pattern formation in plant populations.

Soret effect and photochemical reaction in liquids with laser-induced local heating

We report a theoretical model and experimental results for laser-induced local heating in liquids, and propose a method to detect and quantify the contributions of photochemical and Soret effects in several different situations. The time-dependent thermal and mass diffusion equations in the presence and absence of laser excitation are solved. The two effects can produce similar transients for the laser-on refractive index gradient, but very different laser-off behavior. The Soret effect, also called thermal diffusion, and photochemical reaction contributions in photochemically reacting aqueous Cr(VI)-diphenylcarbazide, Eosin Y, and Eosin Y-doped micellar solutions, are decoupled in this work. The extensive use of lasers in various optical techniques suggests that the results may have significance extending from physical-chemical to biological applications.

Time-resolved thermal lens and thermal mirror spectroscopy with sample-fluid heat coupling: a complete model for material characterization

This work presents a theoretical study of a heat transfer effect, taking into account the heat transfer within the heated sample and out to the surrounding medium. The analytical solution is used to model the thermal lens and thermal mirror effects and the results are compared with the finite element analysis (FEA) software solution. The FEA modeling results were found to be in excellent agreement with the analytical solutions. Our results also show that the heat transfer between the sample surface and the air coupling fluid does not introduce an important effect over the induced phase shift in the sample when compared to the solution obtained without considering axial heat flux. On the other hand, the thermal lens created in the air coupling fluid has a significant effect on the predicted time-dependent photothermal signals. When water is used as fluid, the heat coupling leads to a more significant effect in both sample and fluid phase shift. Our results could be used to obtain physical properties of low optical absorption fluids by using a reference solid sample in both thermal lens and thermal mirror experiments.

Thermal-lens study of photochemical reaction kinetics

We consider the time dependence of the absorption coefficient due to the photoinduced chemical reaction (PCR) and species diffusion to calculate the temperature rise in the thermal-lens (TL) effect. The TL signal at the detector plane is also calculated. This theoretical approach removes the restriction that the PCR time constant is much greater than the characteristic TL time constant, which was assumed in a previously published model. Hydrocarbon fuel and aqueous Cr(VI) samples are investigated, and quantitative experimental results for the thermal, optical, and PCR properties are obtained. While similar results were obtained for the Cr(VI) solution using the previous and present models, the relative difference between the PCR time constants extracted from the same experimental data for a hydrocarbon fuel sample is found to be more than 220%. This demonstrates the significant difference of the two models.

Time-resolved thermal mirror technique with top-hat cw laser excitation

A theoretical model was developed for time-resolved thermal mirror spectroscopy under top-hat cw laser excitation that induced a nanoscale surface displacement of a low absorption sample. An additional phase shift to the electrical field of a TEM(00) probe beam reflected from the surface displacement was derived, and Fresnel diffraction theory was used to calculate the propagation of the probe beam. With the theory, optical and thermal properties of three glasses were measured, and found to be consistent with literature values. With a top-hat excitation, an experimental apparatus was developed for either a single thermal mirror or a single thermal lens measurement. Furthermore, the apparatus was used for concurrent measurements of thermal mirror and thermal lens. More physical properties could be measured using the concurrent measurements.

Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials

Thermal lens spectroscopy is a highly sensitive and versatile photothermal technique for material analysis, providing optical and thermal properties. To use less expensive multimode non-Gaussian lasers for quantitative analysis of low-absorption materials, this Letter presents a theoretical model for time-resolved mode-mismatched thermal lens spectroscopy induced by a cw laser with a top-hat profile. The temperature profile in a sample was calculated, and the intensity of the probe beam center at the detector plane was also derived using the Fresnel diffraction theory. Experimental validation was performed with glass samples, and the results were found well consistent with literature values of the thermo-optical properties of the samples.

Nonlocal model for nematic liquid-crystal elastomers

We have developed a fully nonlocal model to describe the dynamic behavior of nematic liquid-crystal elastomers. The free energy, incorporating both elastic and nematic contributions, is a function of the material displacement vector and the orientational order parameter tensor. The free energy cost of spatial variations of these order parameters is taken into account through nonlocal interactions rather than through the use of gradient expansions. We also give an expression for the Rayleigh dissipation function. The equations of motion for displacement and orientational order are obtained from the free energy and the dissipation function by the use of a Lagrangian approach. We examine the free energy and the equations of motion in the limit of long-wavelength and small-amplitude variations of the displacement and the orientational order parameter. We compare our results with those in the literature. If the scalar order parameter is held fixed, we recover the usual viscoelastic theory for nematic liquid crystals.

General solution of the diffusion equation with a nonlocal diffusive term and a linear force term

We obtain a formal solution for a large class of diffusion equations with a spatial kernel dependence in the diffusive term. The presence of this kernel represents a nonlocal dependence of the diffusive process and, by a suitable choice, it has the spatial fractional diffusion equations as a particular case. We also consider the presence of a linear external force and source terms. In addition, we show that a rich class of anomalous diffusion, e.g., the Lévy superdiffusion, can be obtained by an appropriated choice of kernel.

A non-Gaussian model in polymeric network

We investigate a finite chain approximation, the non-Gaussian Tsallis distribution, to the polymeric network, which gives an improvement to the Gaussian model. This distribution presents some necessary characteristics, like a cutoff to the maximum chain length and a continuous limit to the Gaussian one for a large number of monomers. It also presents a simple quadratic structure that allows to generalize the Gaussian properties such as exact-moments calculation and Wick theorem. We obtain the free-energy density in its full tensorial structure.

Logarithmic diffusion and porous media equations: a unified description

In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a Lorentzian form, consequently this equation characterizes a superdiffusion like a Lévy kind. In addition an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension, is obtained. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.

N-dimensional nonlinear Fokker-Planck equation with time-dependent coefficients

An N-dimensional nonlinear Fokker-Planck equation is investigated here by considering the time dependence of the coefficients, where drift-controlled and source terms are present. We exhibit the exact solution based on the generalized Gaussian function related to the Tsallis statistics. Furthermore, we show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained by changing the time dependence of the coefficients.

Nonlinear anomalous diffusion equation and fractal dimension: exact generalized Gaussian solution

In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the N-dimensional nonlinear diffusion equation partial differential rho/ partial differential t=nabla.(Knablarho(nu))-nabla.(muFrho)-alpharho, where K=Dr(-theta), nu, theta, mu, and D are real parameters, F is the external force, and alpha is a time-dependent source. This equation unifies the O'Shaughnessy-Procaccia anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.

Average entropy of a subsystem from its average Tsallis entropy

In the non-extensive Tsallis scenario, Page's conjecture for the average entropy of a subsystem [Phys. Rev. Lett. 71, 1291 (1993)] as well as its demonstration are generalized, i.e., when a pure quantum system, whose Hilbert space dimension is mn, is considered, the average Tsallis entropy of an m-dimensional subsystem is obtained. This demonstration is expected to be useful to study systems where the usual entropy does not give satisfactory results.

q-exponential distribution in urban agglomeration

Usually, the studies of distributions of city populations have been reduced to power laws. In such analyses, a common practice is to consider cities with more than one hundred thousand inhabitants. Here, we argue that the distribution of cities for all ranges of populations can be well described by using a q-exponential distribution. This function, which reproduces the Zipf-Mandelbrot law, is related to the generalized nonextensive statistical mechanics and satisfies an anomalous decay equation.

Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solution

The nonlinear diffusion equation partial delta rho/delta t=D Delta rho(nu) is analyzed here, where Delta[triple bond](1/r(d-1))(delta/delta r)r(d-1-theta) delta/delta r, and d, theta, and nu are real parameters. This equation unifies the anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact point-source solution is obtained, enabling us to describe a large class of subdiffusion [ theta>(1-nu)d], "normal" diffusion [theta=(1-nu)d] and superdiffusion [theta<(1-nu)d]. Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.