We consider a colloidal particle immersed in an active bath and derive a Smoluchowski equation that governs the dynamics of the colloidal particle. We address this as active Smoluchowski equation. ...Our analysis based on this active Smoluchowski equation shows a short time superdiffusive behavior that strongly depends on the activity. Our model also predicts a non-monotonic dependence of the mean energy dissipation against time, a signature of activity-induced dynamics. By introducing a frequency-dependent effective temperature, we show that the mean rate of entropy production is time-dependent, unlike in a thermal system. The main reason for these anomalies is the absence of any fluctuation–dissipation theorem for the active noise. We also comment on how microscopic details of activity can reverse the trends for the mean energy dissipation and mean rate of entropy production.
•A Smoluchowski equation that governs the dynamics of a colloidal particle in an active bath is derived.•Frequency-dependent effective temperature has been used to quantify entropy production.•Quantification of energy dissipation has been made using the Harada-Sasa equality and force-position correlation.
In this work we consider the hydrodynamic behavior of a coupled electron–phonon fluid, focusing on electronic transport under the conditions of strong phonon drag. This regime occurs when the rate of ...phonon equilibration due to e.g. umklapp scattering is much slower than the rate of normal electron–phonon collisions. Then phonons and electrons form a coupled out-of-equilibrium state where the total quasi-momentum of the electron–phonon fluid is conserved. A joint flow-velocity emerges as a collective hydrodynamic variable. We derive the equation of motion for this fluid from the underlying microscopic kinetic theory and elucidate its effective viscosity and thermal conductivity. In particular, we derive decay times of arbitrary harmonics of the distribution function and reveal its corresponding super-diffusive relaxation on the Fermi surface. We further consider several applications of this theory to magneto-transport properties in the Hall-bar and Corbino-disk geometries, relevant to experiments. In our analysis we allow for general boundary conditions that cover the crossover from no-slip to no-stress flows. Our approach also covers a crossover from the Stokes to the Ohmic regime under the conditions of the Gurzhi effect. In addition, we consider the frequency dependence of the surface impedance and non-equilibrium noise. For the latter, we notice that in the diffusive regime, a Fokker–Planck approximation, applied to the electron–phonon collision integral in the Eliashberg form, reduces it to a differential operator with Burgers type nonlinearity. As a result, the non-equilibrium distribution function has a shock-wave structure in the energy domain. The consequence of this behavior for the Fano factor of the noise is investigated. In conclusion we discuss connections and limitations of our results in the context of recent electron–phonon drag measurements in Dirac and Weyl semimetals, and layout directions for further extensions and developments.
•Electron–phonon drag is described by distinct viscosity and thermal conductivity.•Phonon-mediated relaxation is super-diffusive on a Fermi surface.•Hydrodynamic magnetoresistance is inversely proportional to fluid viscosity.•Viscous skin effect has unconventional frequency dependence.•Burgers shock-wave distribution of nonequilibrium hot electrons determines noise.
•Fractional dual-phase-lag heat diffusion was considered from kinetic relaxation time.•Temperature distribution and the PA signal were obtained for periodic excitation.•Changes in fractional order ...derivative change the temperature profile.•Anomalous DPL heat diffusion shifts and attenuates the peak of the hyperbolic result.
We present the temperature distribution predictions for photothermal systems by considering an extension of dual-phase lag. It is an extension of the GCE-II and GCE-III models with a fractional dual-phase lag from kinetic relaxation time. Solving the one-dimensional problem considering a planar and periodic excitation, we obtained the temperature distribution and the Photoacoustic (PA) signal for the transmission setup. We also analyze the effects of fractional order derivatives and kinetic relaxation time. It is shown that the derived models have promising results that could be used to explain the experimentally observed behavior of PA signals measured on thin films with an inhomogeneous internal structure.
In this work, the solution of Riesz space fractional partial differential equations of parabolic type is considered. Since fractional-in-space operators have been applied to model anomalous diffusion ...or dispersion problems in the area of mathematical physics with success, we are motivated in this paper to model the standard Brownian motion with the fractional order operator in the sense of the Riesz derivative. We formulate two viable, efficient and reliable high-order approximation schemes for the Riesz derivative which incorporated both the left- and right-hand sides of the Riemann-Liouville derivatives. The proposed methods are analyzed for both stability and convergence. Finally, the methods are used to explore the dynamic richness of pattern formation in two important fractional reaction-diffusion equations that are still of recurring interest. Experimental results for different values of the fractional parameters are reported.
The infiltration of fluids into porous media frequently presents anomalous features, in which the fluid front displacement varies in time with an exponent ν different from the expected Fickean value ...of 1/2. A variety of transport models in fractal media predict the anomalies in this process and in related diffusion problems, but the associated exponents are non universal, i.e they depend not only on the fractal dimensions Df but also on other geometric properties. Here we study the horizontal infiltration in layered porous media where the matrices of higher conductivity have fractal distributions (Df<1) of lower conductivity inclusions. When the conductivity contrast is high, this process exhibits universal superdiffusive infiltration with ν=1/1+Df. This result is first demonstrated for inclusion patterns modeled by Cantor sets, but we argue that it extends to any fractal distributions of the inclusions under the condition of no spatial anisotropy in the rescaling of the observation size. Randomized versions of the Cantor sets are presented as possible realizations of disordered fractal patterns and confirm the universal relation. By considering properties of typical granular media and various soils, numerical calculations indicate that this universal superdiffusive infiltration could be observed in physically realizable laboratory and field settings.
•A model of pressure driven infiltration into horizontal layered media is developed.•Distributions of low conductivity layers (inclusions) in the media are fractal.•Infiltration is superdiffusive with exponent depending only on the fractal dimension.•The same results are applicable to regular or random fractal organizations.•The superdiffusive behavior is realizable in field and laboratory settings.
Metals in one spatial dimension are described at the lowest energy scales by the Luttinger liquid theory. It is well understood that this free theory, and even interacting integrable models, can ...support ballistic transport of conserved quantities including energy. In contrast, realistic one-dimensional metals, even without disorder, contain integrability-breaking interactions that are expected to lead to thermalization and conventional diffusive linear response. We argue that the expansion of energy when such a nonintegrable Luttinger liquid is locally heated above its ground state shows superdiffusive behavior (i.e., spreading of energy that is intermediate between diffusion and ballistic propagation), by combining an analytical anomalous diffusion model with numerical matrix-product–state calculations on a specific perturbed spinless fermion chain. Different metals will have different scaling exponents and shapes in their energy spreading, but the superdiffusive behavior is stable and should be visible in time-resolved experiments.
In this work, the superdiffusion equation with a Caputo derivative of order α∈(1,2) is considered. Some priori bounds on certain derivatives of the solution show that the solution exhibits a weak ...singularity at the initial time t=0. To resolve this initial singularity, we rewrite the superdiffusion equation as a coupled system by introducing a intermediate variable p:=Dtα/2(u−tu1), and adopt the L1 scheme and Alikhanov scheme on graded meshes in temporal direction. In spatial direction, the conforming finite element method is used. Furthermore, we derive the H1-norm stability result. It is worth noting that some priori bounds on certain derivatives of p are obtained, on basis of which, we derive an α-robust prior error estimate with optimal H1-norm convergence order. Finally, we provide the numerical experiment to further verify our theoretical analysis.
Generic scaling laws, such as Kolmogorov’s 5/3 law, are milestone achievements of turbulence research in classical fluids. For quantum fluids such as atomic Bose–Einstein condensates, superfluid ...helium, and superfluid neutron stars, turbulence can also exist in the presence of a chaotic tangle of evolving quantized vortex lines. However, due to the lack of suitable experimental tools to directly probe the vortex-tangle motion, so far little is known about possible scaling laws that characterize the velocity correlations and trajectory statistics of the vortices in quantum-fluid turbulence, i.e., quantum turbulence (QT). Acquiring such knowledge could greatly benefit the development of advanced statistical models of QT. Here we report an experiment where a tangle of vortices in superfluid 4He are decorated with solidified deuterium tracer particles. Under experimental conditions where these tracers follow the motion of the vortices, we observed an apparent superdiffusion of the vortices. Our analysis shows that this superdiffusion is not due to Lévy flights, i.e., long-distance hops that are known to be responsible for superdiffusion of random walkers. Instead, a previously unknown power-law scaling of the vortex–velocity temporal correlation is uncovered as the cause. This finding may motivate future research on hidden scaling laws in QT.
This paper analyzes the influence of the anomalous diffusive effects caused by micro-scale heterogeneity and kinetic and inertial thermal relaxations on the optically induced thermoelastic bending ...component of the photoacoustic response. We calculated the temperature distribution for a one-dimensional heat transfer problem with planar and periodic excitation, neglecting the influence of thermoelastic strains on the temperature profile. Thermoelastic bending was evaluated using a theoretical approximation of a thin plate, while pressure fluctuations in the photoacoustic cell were obtained by assuming adiabatic changes in the closed air. The model analysis shows that the relaxation processes could significantly affect the mechanical piston component of the photoacoustic response at frequencies higher than the minima of the inverse of two thermal relaxation times, while the influence of micro-scale heterogeneity is observable in the whole frequency range.
•The hyperbolic DPL model enhances accuracy of thermoelastic effect at higher frequencies.•Sensitivity to fractional order of DPL model emphasizes precision in TE control.•Anomalous DPL heat diffusion shifts and attenuates the peak of the hyperbolic result.•Maximum damping occurs to identical kinetic and thermal relaxation times.
Chimera states are a truly remarkable dynamical phenomenon that occur in systems of coupled oscillators. In this regime, regions of synchronized and unsynchronized elements are formed in the system. ...For many applied problems, especially in neuroscience, these states offer a rich potential for research. However, the plethora of models and the lack of a ”single simple principle” that leads to the development of chimeras makes it very difficult to understand their nature. In this work, we propose a three-component reaction-superdiffusion system based on a unified mechanism founded on the properties of the fractional Laplace operator and the nonlinear Hindmarsh-Rose model functions. In the proposed system, the non-local type of interaction forming the coupling between the elements depends significantly on the fractional Laplace operator exponents of the corresponding components. It is shown that in the framework of the superdiffusion type of interaction, chimera states are realized in the system. At the same time, many qualitative (shape, visual degree of inhomogeneity and area size) and quantitative characteristics of chimeras (synchronization factor, strength of incoherence, local order parameter, number of elements with a potential value exceeding a given one) depend significantly on the exponents of the fractional Laplace operator. In addition to classical chimeras and target-waves chimeras, the results of numerical simulations show the presence of mutually sustaining reaction processes of different scales in the system.
•Chimeras are found in the system of superdiffusion equations.•The origin and dynamics of chimeras depend on the fractional Laplacian exponent.•There are wave dynamics of activation in the space of fractional Laplacian exponents.•The number of activated elements during chimera development obeys a universal law.
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