Ultraviolet Catastrophe Research Articles

The Puzzling of Stefan–Boltzmann Law: Classical or Quantum Physics

Stefan–Boltzmann law, stating the fourth power temperature dependence of the radiation emission by a black-body, was empirically formulated by Stefan in 1874 by fitting existing experiments and theoretically validated by Boltzmann in 1884 on the basis of a classical physical model involving thermodynamics principles and the radiation pressure predicted by Maxwell equations. At first sight the electromagnetic (EM) gas assumed by Boltzmann and following Rayleigh (1900) identifiable as an ensemble of N classical normal-modes, looks like an extension of the classical model of the massive ideal-gas. Accordingly, for this EM gas the internal total energy, U, was assumed to be function of volume V and temperature T as [Formula: see text], and the equation of state was given by [Formula: see text], with P the radiation pressure. In addition, Boltzmann implicitly assumed that, for given values of V and T, U and the number of modes N would take finite values. However, from one hand these assumptions are not justified by Maxwell equations and classical statistics since, in vacuum (i.e., far from the EM sources), the values of N and U diverge, the so-called ultraviolet catastrophe introduced by Ehrenfest in 1911. From another hand, Boltzmann derivation of Stefan law is found to be macroscopically compatible with its derivation from quantum statistics announced by Planck in 1901. In this paper, we present a justification of this puzzling classical/quantum compatibility by noticing that the implicit assumptions made by Boltzmann is fully justified by Planck quantum statistics. Furthermore, we shed new light on the interpretation of recent classical simulations of a black body carried out by Wang, Casati, and Benenti in 2022 who found an analogous puzzling consistency between Stefan–Boltzmann law and their simulations to induce speculations on classical physics and black body radiation that are claimed to require a critical reconsideration of the role of classical physics for the understanding of quantum mechanics.

An integrative review of secondary school quantum physics curricula in Malaysia

<span lang="EN-US">As a relatively new subject in the physics curriculum of Malaysian secondary schools, quantum physics (QP) raises questions about its relevance to students and the best approach for teaching it. This paper aims to analyze the content of the QP curriculum to provide students with a meaningful learning experience and expose them to the nature of science (NOS). To accomplish this, the Malaysian standard curriculum document known as </span><em><span lang="EN-US">Dokumen Standard Kurikulum dan Pentaksiran</span></em><span lang="EN-US"> (DSKP) and the textbook were analyzed through integrative review. Frameworks and perspectives identified by Stadermann and co-workers on the common trust of the QP curriculum in 15 different countries were used as a benchmark for this analysis. It is found that the QP curriculum in Malaysia focuses on the fundamental principle of understanding the quantum energy of light and its interaction with matter. However, it is also found that there are specific NOS aspects that can be highlighted to help students develop their scientific literacy. These might include emphasizing the philosophy of complementarity in explaining the wave-particle duality principle, the ultraviolet catastrophe, and the contradiction of classical physics interpretation with QP. It is suggested that the QP curriculum be improved by including these and other relevant examples to be on par with other countries. Overall, this analysis provides insight into the relevance and content of QP in Malaysian secondary schools. The proposed changes may help improve students’ learning experience and provide a more comprehensive understanding of the NOS.</span>

The Metastable State of Fermi–Pasta–Ulam–Tsingou Models

Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan-Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a "metastable" state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi-Pasta-Ulam-Tsingou (FPUT) models. We treat both the α-FPUT and β-FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure-the spectral entropy (η)-and show that this measure has the power to quantify the distance from equipartition. For the α-FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state tm in the α-FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P1-Q1 plane. Applying this procedure gives us a power-law scaling for tm, with the important result that the power laws for different system sizes collapse down to the same exponent as Eα2→0. We examine the energy spectrum E(k) over time in the α-FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the "wave turbulence" theory. We next apply a similar approach to the β-FPUT model. Here, we explore in particular the different behavior for the two different signs of β. Finally, we describe a procedure for calculating tm in the β-FPUT model, a very different task than for the α-FPUT model, because the β-FPUT model is not a truncation of an integrable nonlinear model.

Revisiting the Boltzmann Derivation of the Stefan Law

The Stefan–Boltzmann (SB) law relates the radiant emittance of an ideal black-body cavity at thermal equilibrium to the fourth power of the absolute temperature [Formula: see text] as [Formula: see text], with [Formula: see text][Formula: see text]W[Formula: see text]m[Formula: see text][Formula: see text]K[Formula: see text] the SB constant, first estimated by Stefan to within [Formula: see text] of the present theoretical value. The law is an important achievement of modern physics since, following Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563], its microscopic derivation implies the quantization of the energy related to the electromagnetic field spectrum. Somewhat astonishing, Boltzmann presented his derivation in 1878 making use only of electrodynamic and thermodynamic classical concepts, apparently without introducing any quantum hypothesis (here called first Boltzmann paradox). By contrast, the Boltzmann derivation implies two assumptions not justified within a classical approach, namely: (i) the zero value of the chemical potential and (ii) the internal energy of the black body with a finite value and dependent from both temperature and volume. By using Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563] quantization of the radiation field in terms of a gas of photons, the SB law received a microscopic interpretation free from the above assumptions that also provides the value of the SB constant on the basis of a set of universal constants including the quantum action constant [Formula: see text]. However, the successive consideration by Planck [Uber die Begründung des Gesetzes der schwarzen Strahlung [On the grounds of the law of black body radiation], Ann. Phys. 6 (1912) 642–656] concerning the zero-point energy contribution was found to be responsible of another divergence of the internal energy for the single photon mode at high frequencies. This divergence is of pure quantum origin and is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum, given by the Rayleigh–Jeans law in 1900. As a consequence, from a rigorous quantum-mechanical derivation we would expect the divergence of the SB law (here called second Boltzmann paradox). Here, both the Boltzmann paradoxes are revised by accounting for both the quantum-relativistic photon gas properties, and the Casimir force.

Manipulating thermal light via displaced-photon subtraction

Thermal radiation played a pivotal role in the preliminary development of quantum physics where it helped resolve the apparent incongruity of the ultraviolet catastrophe. In contemporary physics, thermal state generation and manipulation finds new application in fields such as quantum imaging and quantum illumination and as a practical realization of Maxwell's demon. These applications often go hand in hand with photon subtraction operations which probabilistically amplify the mean photon number (MPN) of thermal light as a result of its super-Poissonian photon statistics. In this article, we introduce an operation for thermal states of light based on a generalized photon subtraction scheme. Displaced-photon subtraction (DPS) makes use of coherent state displacement followed by a subsequent anti-displacement in combination with single-photon detection to probe the MPN of a thermal state. We find regimes in which the output of a successful DPS is amplified, unchanged, or attenuated relative to the unconditioned output state. The regime of operation is controlled via the magnitude of the coherent displacement. A theoretical description of generalized photon subtraction of a displaced thermal state is derived via a two-mode moment-generating function (MGF) and used to describe generalized DPS. We perform a proof of principle experimental implementation of DPS for the case of a balanced beam splitter for which results demonstrate good agreement with the model.

Why even source-free gravity must be quantized

I note, on the most elementary grounds: its classical ultraviolet catastrophe, that even source-free General Relativity must be quantized, despite some eminent opinions to the contrary and with a little fine print.

On the Origins of the Oceanic Ultraviolet Catastrophe

Abstract We provide a first-principles analysis of the energy fluxes in the oceanic internal wave field. The resulting formula is remarkably similar to the renowned phenomenological formula for the turbulent dissipation rate in the ocean, which is known as the finescale parameterization. The prediction is based on the wave turbulence theory of internal gravity waves and on a new methodology devised for the computation of the associated energy fluxes. In the standard spectral representation of the wave energy density, in the two-dimensional vertical wavenumber–frequency (m–ω) domain, the energy fluxes associated with the steady state are found to be directed downscale in both coordinates, closely matching the finescale parameterization formula in functional form and in magnitude. These energy transfers are composed of a “local” and a “scale-separated” contributions; while the former is quantified numerically, the latter is dominated by the induced diffusion process and is amenable to analytical treatment. Contrary to previous results indicating an inverse energy cascade from high frequency to low, at odds with observations, our analysis of all nonzero coefficients of the diffusion tensor predicts a direct energy cascade. Moreover, by the same analysis fundamental spectra that had been deemed “no-flux” solutions are reinstated to the status of “constant-downscale-flux” solutions. This is consequential for an understanding of energy fluxes, sources, and sinks that fits in the observational paradigm of the finescale parameterization, solving at once two long-standing paradoxes that had earned the name of “oceanic ultraviolet catastrophe.” Significance Statement The global circulation models cannot resolve the scales of the oceanic internal waves. The finescale parameterization of turbulent dissipation, a formula grounded in observations, is the standard tool by which the energy transfers due to internal waves are incorporated in the global models. Here, we provide an interpretation of this parameterization formula building on the first-principles statistical theory describing energy transfers between waves at different scales. Our result is in agreement with the finescale parameterization and points out a large contribution to the energy fluxes due to a type of wave interactions (local) usually disregarded. Moreover, the theory on which the traditional understanding of the parameterization is mainly built, a “diffusion approximation,” is known to be partly in contradiction with observations. We put forward a solution to this problem, visualized by means of “streamlines” that improve the intuition of the direction of the energy cascade.

Heuristic determination of the cosmological constant

The vacuum catastrophe results from the disagreement between the theoretical value of the energy density of the vacuum in quantum field theory and the estimated one observed in cosmology. In a similar attempt in which the ultraviolet catastrophe was solved, we search for the value of the cosmological constant by brute-force through computation. We explore combinations of the fundamental constants in physics performing a dimensional analysis, in search of an equation resulting in the measured energy density of the vacuum or cosmological constant that is assumed to cause the accelerated expansion of the universe.

Nonlinear saturation of thermal instabilities

Recent low-frequency simulations of a one-layer model with lateral buoyancy inhomogeneity have revealed circulatory motions resembling quite closely submesoscale features on the ocean surface often visible in satellite observations. This model is known to lack a high-wavenumber instability cutoff and, thus, to possibly undergo ultraviolet catastrophe. However, the numerically observed instabilities, referred to as “thermal” due the ability of the above inhomogeneous-layer model to incorporate thermodynamic processes, are not seen to grow indefinitely. In this note, I show that the presence of a convex pseudo-energy–momentum integral of motion for the inviscid, unforced dynamics can arrest their nonlinear grow in the zonally symmetric case. Our result is an application of Arnold and Shepherd's methods.

A Breakdown in Potential Vorticity Estimation Delineates the Submesoscale‐to‐Turbulence Boundary in Large Eddy Simulations

AbstractMost submesoscale motions—fronts, eddies, filaments, and even large internal waves—are sufficiently rapidly rotating and stratified as to be strongly influenced by potential vorticity dynamics. Below the Ozmidov scale where turbulence overturns and isotropizes, potential vorticity is not commonly considered. Here, it is shown that in Large Eddy Simulations, the velocity gradients, buoyancy gradients, and potential vorticity are strongly influenced by grid‐scale processes. Grid‐scale processes in Large Eddy Simulations, as opposed to those in Direct Numerical Simulations, imply that spuriously noisy potential vorticity variance will become increasingly dominant as resolution increases—analogous to ultraviolet catastrophe. A solution, the prefiltered potential vorticity, is shown to be effective in linking the potential vorticity dynamics of the submesoscale to the nearly‐isotropic turbulent fluxes beyond the Ozmidov scale, and a derivation is provided for a set of closed conservation equations for use in interpreting potential vorticity dynamics in Large Eddy Simulations. This diagnostic approach is exceptional in that Large Eddy Simulation analysis and hydrostatic ocean modeling with parameterized turbulence analysis are harmonized.

Ultraviolet catastrophe of a fluctuating curved dislocation line

The authors demonstrate that the line tension expression for dislocation core energy and the regularization of elastic fields near the core fundamentally emerge from the periodicity of the discrete atomic lattice. A continuum model for the dislocation core is presented which addresses the problem of short wavelength instability of dislocation lines inherent to linear elasticity theory, and predicts configurational energies.

Beyond the Formulations of the Fluctuation Dissipation Theorem Given by Callen and Welton (1951) and Expanded by Kubo (1966)

The quantum formula of the fluctuation dissipation theorem (FDT) was given by Callen and Welton in 1951 for the case of conductors, and then expanded by Kubo in 1966. The drawback of these quantum relations concerns with the appearance of a zero-point contribution, ${\hbar \omega}/{2}$ with $\hbar$ the reduced Planck constant and $\omega$ the angular frequency of the considered photon, which implies a divergence of the fluctuation spectrum at increasing frequencies. This divergence is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum. As a consequence, the quantum formulation of the FDT as given by Callen-Welton and Kubo introduces a Field Grand Challenge associated with the existence or less of a vacuum-fluctuations catastrophe for the energy-density spectrum. Here we propose a solution to this challenge by taking into account of the Casimir energy that, in turns, is found to be responsible for a quantum correction of the Stefan-Boltzmann law.

Thermohaline layering on the microscale

A theoretical model is developed which illustrates the dynamics of layering instability, frequently realized in ocean regions with active fingering convection. Thermohaline layering is driven by the interplay between large-scale stratification and primary double-diffusive instabilities operating at the microscale – temporal and spatial scales set by molecular dissipation. This interaction is described by a combination of direct numerical simulations and an asymptotic multiscale model. The multiscale theory is used to formulate explicit and dynamically consistent flux laws, which can be readily implemented in large-scale analytical and numerical models. Most previous theoretical investigations of thermohaline layering were based on the flux-gradient model, which assumes that the vertical transport of density components is uniquely determined by their local background gradients. The key deficiency of this approach is that layering instabilities predicted by the flux-gradient model have unbounded growth rates at high wavenumbers. The resulting ultraviolet catastrophe precludes the analysis of such basic properties of layering instability as its preferred wavelength or the maximal growth rate. The multiscale model, on the other hand, incorporates hyperdiffusion terms that stabilize short layering modes. Overall, the presented theory carries the triple advantage of (i) offering an explicit description of the interaction between microstructure and layering modes, (ii) taking into account the influence of non-uniform stratification on microstructure-driven mixing, and (iii) avoiding unphysical behaviour of the flux-gradient laws at small scales. While the multiscale approach to the parametrization of time-dependent small-scale processes is illustrated here on the example of fingering convection, we expect the proposed technique to be readily adaptable to a wide range of applications.

The nature of the biological material and the irreproducibility problem in biomedical research.

Biomedical research has a reproducibility problem since various crucial landmark papers could not be independently reproduced. While there are many causes related to statistical analysis, methodology or insufficient reporting of experimental details, this commentary argues that the complexity of biological material itself is, until now, a largely ignored source of irreproducibility. By discussing examples from evolutionary biology, intrinsically disordered proteins and current biomedical research, it contends that some results are irreproducible because we do not have the knowledge, the tools or the analytical ability to understand biological complexity and how it can give rise to different results. Instead of casting irreproducible research out as bad or sloppy science, they should serve as an inspiration for pioneering research not just to develop such tools but also to attempt to explore what lies beneath our current inability to deal with complexity.

Numerical simulation of the geometrical-optics reduction of CE2 and comparisons to quasilinear dynamics

Zonal flows have been observed to appear spontaneously from turbulence in a number of physical settings. A complete theory for their behavior is still lacking. Recently, a number of studies have investigated the dynamics of zonal flows using quasilinear (QL) theories and the statistical framework of a second-order cumulant expansion (CE2). A geometrical-optics (GO) reduction of CE2, derived under an assumption of separation of scales between the fluctuations and the zonal flow, is studied here numerically. The reduced model, CE2-GO, has a similar phase-space mathematical structure to the traditional wave-kinetic equation, but that wave-kinetic equation has been shown to fail to preserve enstrophy conservation and to exhibit an ultraviolet catastrophe. CE2-GO, in contrast, preserves nonlinear conservation of both energy and enstrophy. We show here how to retain these conservation properties in a pseudospectral simulation of CE2-GO. We then present nonlinear simulations of CE2-GO and compare with direct simulations of quasilinear (QL) dynamics. We find that CE2-GO retains some similarities to QL. The partitioning of energy that resides in the zonal flow is in good quantitative agreement between CE2-GO and QL. On the other hand, the length scale of the zonal flow does not follow the same qualitative trend in the two models. Overall, these simulations indicate that CE2-GO provides a simpler and more tractable statistical paradigm than CE2, but CE2-GO is missing important physics.

An Oceanic Ultra-Violet Catastrophe, Wave-Particle Duality and a Strongly Nonlinear Concept for Geophysical Turbulence

There is no theoretical underpinning that successfully explains how turbulent mixing is fed by wave breaking associated with nonlinear wave-wave interactions in the background oceanic internal wavefield. We address this conundrum using one-dimensional ray tracing simulations to investigate interactions between high frequency internal waves and inertial oscillations in the extreme scale separated limit known as “Induced Diffusion”. Here, estimates of phase locking are used to define a resonant process (a resonant well) and a non-resonant process that results in stochastic jumps. The small amplitude limit consists of jumps that are small compared to the scale of the resonant well. The ray tracing simulations are used to estimate the first and second moments of a wave packet’s vertical wavenumber as it evolves from an initial condition. These moments are compared with predictions obtained from the diffusive approximation to a self-consistent kinetic equation derived in the ‘Direct Interaction Approximation’. Results indicate that the first and second moments of the two systems evolve in a nearly identical manner when the inertial field has amplitudes an order of magnitude smaller than oceanic values. At realistic (oceanic) amplitudes, though, the second moment estimated from the ray tracing simulations is inhibited. The transition is explained by the stochastic jumps obtaining the characteristic size of the resonant well. We interpret this transition as an adiabatic ‘saturation’ process which changes the nominal background wavefield from supporting no mixing to the point where that background wavefield defines the normalization for oceanic mixing models.

Dynamical thermalization of Bose-Einstein condensate in Bunimovich stadium

We study numerically the wave function evolution of a Bose-Einstein condensate in a Bunimovich stadium billiard being governed by the Gross-Pitaevskii equation. We show that for a moderate nonlinearity, above a certain threshold, there is emergence of dynamical thermalization which leads to the Bose-Einstein probability distribution over the linear eigenmodes of the stadium. This distribution is drastically different from the energy equipartition over oscillator degrees of freedom which would lead to the ultra-violet catastrophe. We argue that this interesting phenomenon can be studied in cold-atom experiments.

Turbulent diapycnal mixing in stratified shear flows: the influence of Prandtl number on mixing efficiency and transition at high Reynolds number

Motivated by the importance of small-scale turbulent diapycnal mixing to the closure of the large-scale meridional overturning circulation (MOC) of the oceans, we focus on a model problem which allows us to address the fundamental fluid mechanics that is expected to be characteristic of the oceanographic regime. Our model problem is one in which the initial conditions consist of a stably stratified parallel shear flow which evolves into the turbulent regime through the growth of a Kelvin–Helmholtz wave to finite amplitude followed by transition to turbulence. Through both linear stability analysis and direct numerical simulations (DNS), we investigate the secondary instabilities and the turbulent mixing at a fixed high Reynolds number and for a range of Prandtl numbers. We demonstrate that the oceanographically expected high value of the Prandtl number has a profound influence on the nature of the secondary instabilities that govern the transition process. Specifically through non-separable linear stability analysis, we discover new characteristics for the shear-aligned convective instability such that it is modified into a mixed mode that is driven both by static instability and by shear. The growth rate and ultimate strength of this mode are both strongly enhanced at higher $\mathit{Pr}$ while the growth rate and ultimate strength of the stagnation point instability (SPI), which may compete for control of the transition process, are simultaneously impeded. Of equal importance is the fact that, for higher $\mathit{Pr}$, the characteristic length scales associated with the dominant mixed mode of instability decrease and therefore there ceases to be a strong scale selectivity. In the limit of much higher $\mathit{Pr}$, we conjecture that a wide range of spatial scales become equally unstable so as to support an ‘ultraviolet catastrophe’, in which a direct injection of energy occurs into a broad range of scales simultaneously. We further establish the validity of these analytical results through a series of computationally challenging DNS analyses, and provide a detailed analysis of the efficiency of the turbulent mixing of the density field that occurs subsequent to transition and of the entrainment of fluid into the mixing layer from the high-speed flanks of the shear flow. We show that the mixing efficiency decreases monotonically with increase of the molecular value of the Prandtl number and the expansion of the shear layer is reduced as such entrainment diminishes.

Applicability and failure of the flux-gradient laws in double-diffusive convection

AbstractDouble-diffusive flux-gradient laws are commonly used to describe the development of large-scale structures driven by salt fingers – thermohaline staircases, collective instability waves and intrusions. The flux-gradient model assumes that the vertical transport is uniquely determined by the local background temperature and salinity gradients. While flux-gradient laws adequately capture mixing characteristics on scales that greatly exceed those of primary double-diffusive instabilities, their accuracy rapidly deteriorates when the scale separation between primary and secondary instabilities is reduced. This study examines conditions for the breakdown of the flux-gradient laws using a combination of analytical arguments and direct numerical simulations. The applicability (failure) of the flux-gradient laws at large (small) scales is illustrated through the example of layering instability, which results in the spontaneous formation of thermohaline staircases from uniform temperature and salinity gradients. Our inquiry is focused on the properties of the ‘point-of-failure’ scale ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H_{pof}$) at which the vertical transport becomes significantly affected by the non-uniformity of the background stratification. It is hypothesized that$H_{pof} $can control some key characteristics of secondary double-diffusive phenomena, such as the thickness of high-gradient interfaces in thermohaline staircases. A more general parametrization of the vertical transport – the flux-gradient-aberrancy law – is proposed, which includes the selective damping of relatively short wavelengths that are inadequately represented by the flux-gradient models. The new formulation is free from the unphysical behaviour of the flux-gradient laws at small scales (e.g. the ultraviolet catastrophe) and can be readily implemented in theoretical and large-scale numerical models of double-diffusive convection.

A Model for Staircase Formation in Fingering Convection

Fingering convection is a convective instability that occurs in fluids where two buoyancy-changing scalars with different diffusivities have a competing effect on density. The peculiarity of this form of convection is that, although the transport of each individual scalar occurs down-gradient, the net density transport is up-gradient. In a suitable range of non-dimensional parameters, solutions characterized by constant vertical gradients of the horizontally averaged fields may undergo a further instability, which results in the alternation of layers where density is roughly homogeneous with layers where there are steep vertical density gradients, a pattern known as "doubly-diffusive staircases". This instability has been interpreted in terms of an effective negative diffusivity, but simplistic parameterizations based on this idea, obviously, lead to ill-posed equations. Here we propose a mathematical model that describes the dynamics of the horizontally-averaged scalar fields and the staircase-forming instability. The model allows for unstable constant-gradient solutions, but it is free from the ultraviolet catastrophe that characterizes diffusive processes with a negative diffusivity.