Starting from a recent proposal of a nonlinear Maxwell-Cattaneo equation for the heat transport with relaxational effects at nanoscale, in a special case of thermal-wave propagation we derive a nonlinear Schrodinger equation for the amplitudes of the heatflux perturbation. The complete integrability of the obtained equation is investigated in order to prove the existence of infinite conservation laws, as well as the existence of infinite exact solutions. In this regards, we have considered the simplest nontrivial solutions, namely, the bright and dark (thermal) solitons, which may be interesting for energy transport and for information transmission in phononic circuits. (c) 2022 Elsevier B.V. All rights reserved.

We obtain some exact solutions in the context of solitons, for heat conduction with inertia along a cylinder whose heat exchange with the environment is a non-linear function of the difference of temperatures of the cylinder and the environment, due to a flux-limiter behavior of the exchange. We study the consequences of heat transfer and information transfer along the wire, and we compare the situation with analogous solitons found in nonlinear lateral radiative exchange studied in some previous papers. We also find further exact solutions in terms of Weierstrass elliptic functions for the sake of completeness.

The aim of this paper is to consider soliton propagation of heat signals along a cylinder whose heat exchange with the environment is a non-linear function of the difference of temperatures of the cylinder and the environment and whose heat transfer along the system is described by the Maxwell-Cattaneo equation. To find the soliton solutions we use the auxiliary equation method. Our motivation is to obtain and compare the speed of propagation, the maximum rate of information transfer, and the energy necessary for the transfer of one bit of information for different solitons, by assuming that a localized soliton may carry a bit of information. It is shown that a given total power (energy/time) may be used either to send a few bits in a fast way, or many bits in a slower way. This may be controlled by choosing the initial condition imposed at one end of the wire. (C) 2020 Elsevier Ltd. All rights reserved.

Turbulence is almost the rule in the flow of classical fluids. It is a complex nonlinear phenomenon for which the development of a satisfactory theoretical framework is still incomplete. Turbulence is often found in the flow of quantum fluids, especially superfluid Helium 4, known as liquid helium II (Donnelly, 1991), (Nemirovskii & Fiszdon, 1995), (Barenghi et al., 2001), (Vinen & Niemela, 2002). In recent years there has been growing interest in superfluid turbulence, because of its unique quantum peculiarities and of its similarity with classical turbulence to which it provides a wide range of new experimental possibilities at very high Reynolds numbers (Vinen, 2000), (Barenghi, 1999), and because of their influence in some practical applications, as in refrigeration by means of superfluid helium. We will consider here the turbulence in superfluid 4He, for which many detailed experimental techniques have been developed. The behavior of liquid helium, below the lambda point (Tc ≃ 2.17 K), is very different from that of ordinary fluids. One example of non-classical behavior is the possibility to propagate the second sound, a wave motion in which temperature and entropy oscillate. A second example of non-classical behavior is heat transfer in counterflow experiments. Using an ordinary fluid (such as helium I), a temperature gradient can bemeasured along the channel, which indicates the existence of a finite thermal conductivity. If helium II is used, and the heat flux inside the channel is not too high, the temperature gradient is so small that it cannot be measured, so indicating that the liquid has an extremely high thermal conductivity (three million times larger than that of helium I). This is confirmed by the fact that helium II is unable to boil. This effect explains the remarkable ability of helium II to remove heat and makes it important in engineering applications. The most known phenomenological model, accounting for many of the properties of He II, given by Tisza and Landau (Tisza, 1938), (Landau, 1941) is called the two-fluid model. The basic assumption is that the liquid behaves as a mixture of two fluids: the normal component with density ρn and velocity vn, and the superfluid component with density ρs and velocity vs, with total mass density ρ and barycentric velocity v defined by ρ = ρs + ρn and ρv = ρsvs + ρnvn. The second component is related to the quantum coherent ground state and it is an ideal fluid, which does not experience dissipation neither carries entropy. The superfluid component, which is absent above the lambda transition temperature, was originally considered to be composed by particles in the Bose-Einstein state and is an ideal 12

We generalize the K − (cid:15) model of classical turbulence to superfluid helium. In a classical viscous fluid the phenomenological eddy viscosity characterizing the effects of turbulence depends on the turbulent kinetic energy K and the dissipation function (cid:15) , which are mainly related to the fluctuations of the velocity field and of its gradient. In superfluid helium, instead, we consider the necessary coefficients for describing the effects of classical and quantum turbulence, involving fluctuations of the velocity, the heat flux, and the vortex line density of the quantized vortex lines. By splitting the several fields into a time-average part and a fluctuating part, some expressions involving the second moments of the turbulent fluctuations appear in the evolution equations for the average quantities. As in the K − (cid:15) model, a practical way of closing such equations is to tentatively express such fluctuating terms as a function of the average quantities. In this context we propose how the turbulent coefficients so introduced could depend on the second moments of the fluctuations of v , q and L (respectively denoted as K , K q and K L ), and on their respective dissipation functions (related to the second moments of their gradients) (cid:15) , (cid:15) q and (cid:15) L .

In the framework of weakly nonlocal thermodynamic theory, in this paper we derive a nonlocal and nonlinear heat-transport equation beyond the Fourier law by means of thermodynamic considerations in agreement with the second law. The obtained equation describes the transitions among different heat-transport regimes. The stability of the solution of that equation is also analyzed in a special case. (C) 2019 Elsevier Ltd. All rights reserved.

Motivated by the challenge of computer refrigeration, we study the limits set by the transition to quantum turbulence on the cooling of an array of heat-producing cylindrical nanosystems by means of superfluid-helium counterflow. The effective thermal conductivity in laminar counterflow superfluid helium is obtained in channels with rectangular cross section, through arrays of mutually parallel cylinders and in the combined situation of arrays of orthogonal cylinders inside the rectangular channel. The maximum cooling capacity is analyzed on the condition that turbulence is avoided and that the highest temperature does not exceed the lambda temperature. (C) 2016 Elsevier Ltd. All rights reserved.

Motivated by the experimental development of quasi-homogeneous Bose-Einstein condensates confined in box-like traps, we study numerically the dynamics of dark solitons in such traps at zero temperature. We consider the cases where the side walls of the box potential rise either as a power-law or a Gaussian. While the soliton propagates through the homogeneous interior of the box without dissipation, it typically dissipates energy during a reflection from a wall through the emission of sound waves, causing a slight increase in the soliton's speed. We characterise this energy loss as a function of the wall parameters. Moreover, over multiple oscillations and reflections in the box-like trap, the energy loss and speed increase of the soliton can be significant, although the decay eventually becomes stabilized when the soliton equilibrates with the ambient sound field.

Starting from a generalized black hole entropy with logarithmic area corrections, in this paper we obtain (for positive value of the coefficient of the correction term) a generalized equation of state for black holes with two dual branches. In one of them (the usual one for macro black holes) T ≃ 1/M, with T temperature and M mass. In the other one, for micro black holes, instead, T ≃ M. We compare the equilibrium and stability between macro black holes and electromagnetic radiation in a finite box with reflecting walls, with the dual situation corresponding to micro black holes and cosmic string loops, also in a finite box. In this model, the dual phenomenon of evaporation of unstable macro black holes into a stable gas of electromagnetic radiation would be the condensation of unstable gas of cosmic string loops into stable micro black holes.

The size-dependent and flux-dependent effective thermal conductivity of narrow channels filled with He II is analyzed. The classical Landau evaluation of the effective thermal conductivity of quiescent He II is extended to describe the transition to fully turbulent regime, where the heat flux is proportional to the cubic root of the temperature gradient (Gorter–Mellink regime). To do so, we use an expression for the quantum vortex line density L in terms of the heat flux considering the influence of the walls. From it, and taking into account the friction force of normal component against the vortices, we compute the effective thermal conductivity as a function of the heat flux, and we discuss in detail the corresponding size dependence.

We discuss the consequences of a duality-invariant Einstein-Planck relation on the equation of state of micro black holes. The results are analogous to those obtained from the "world crystal" model, but with some significative differences, as for instance a limiting vanishing value for temperature for very small black holes. The model leads to a total evaporation of micro black holes but with the final stage being very slow.

Physical objects with energy u(w) (l) similar to l(-3w) 3w, with l a characteristic length and w a dimensionless constant, lead to an equation of state p= w rho, with p the pressure and rho the energy density. Special entities with this property are, for instance, photons (u= hc/l, with l the wavelength) with w = 1/3, and some models of cosmic string loops (u =(c(4)/aG) l, with l the length of the loop and a a numerical constant), with w = -1 3. Here, we discuss some features of the spectral energy distribution of these systems and the corresponding generalization of Wien ' s law, which in terms of l has the form Tl-mp(3w) = constant, being l(mp) the most probable size of the mentioned entities.

We have developed a two-dimensional model of quantised vortices in helium II moving under the influence of applied normal fluid and superfluid in a counterflow channel. We predict superfluid and vortex-line density profiles which could be experimentally tested using recently developed visualization techniques.

We include the effects of anisotropy and polarization in the hydrodynamics of inhomogeneous vortex tangles, thus generalizing the well known Hall–Vinen–Bekarevich–Khalatnikov equations, which do not take them in consideration. These effects contribute to the mutual friction force Fns between normal and superfluid components and to the vortex tension force ρsT. These equations are complemented by an evolution equation for the vortex line density L, which takes into account these contributions. These equations are expected to be more suitable than the usual ones for rotating counterflows, or turbulence behind a cylinder, or turbulence produced by a grid of parallel thin cylinders towed across a superfluid, because in these situations polarization is expected to play a relevant role.

We discuss the thermodynamic aspects of a simple model of cosmic string loops, whose energy is nonlinearly related to their lengths. We obtain in a direct way an equation of state having the form p = -(1 + alpha)rho/3, with rho the energy density and 1 + alpha the exponent which relates the energy u(l) of a loop with its length l as u(l) similar to l(1+alpha). In the linear situation (alpha = 0) one has p = -rho/3, in the quadratic one (alpha = 1) p = -2 rho/3, and in the cubic case (alpha = 2) p = -rho. For all values of alpha the entropy goes as S similar to (2 - alpha)L-3/2 (L being the string length density). The expression of S is useful to explore the behavior of such string loops under adiabatic expansion of the Universe. Thermodynamic stability suggests that the gas of string loops must coexist with several long strings, longer than the horizon radius.

We discuss thermodynamics of electromagnetic radiation, with p = (1/3)rho and S proportional to (TV)-V-3, and of cosmic string loops, with p = -(1/3)rho and S proportional to T-3V, where p stands for pressure, T temperature, rho energy density, S entropy, and V volume. We write the thermodynamic formalisms under a common framework that illustrates their formal relationship and allows us to go from one to the other through a smooth transformation. From a microscopic perspective, these relations arise from the energy relations u(lambda) = hc/lambda for the photons of electromagnetic radiation, and u(l) = (c(4)/a(2)G)l for cosmic string loops, a being a numerical (dimensionless) constant and lambda and l the radiation wavelength and the length of a loop; G, c, and h are the gravitational constant, the speed of light in vacuo, and the Planck constant, respectively. The corresponding thermodynamic behaviors are seen to be connected through a related thermal duality corresponding to the change of T by T* = T-c(2)/T, with T-c a reference temperature related to h, c, and G.

By assuming a self-similar structure for Kelvin waves along vortex loops with successive smaller scale features, we model the fractal dimension of a superfluid vortex tangle in the zero temperature limit. Our model assumes that at each step the total energy of the vortices is conserved, but the total length can change. We obtain a relation between the fractal dimension and the exponent describing how the vortex energy per unit length changes with the length scale. This relation does not depend on the specific model, and shows that if smaller length scales make a decreasing relative contribution to the energy per unit length of vortex lines, the fractal dimension will be higher than unity. Finally, for the sake of more concrete illustration, we relate the fractal dimension of the tangle to the scaling exponents of amplitude and wavelength of a cascade of Kelvin waves.

In this paper we present some phenomenological ideas about the thermodynamics of quantized vortex loops arising in superfluid turbulence. The system of vortex loops may be seen as a dissipative structure, not existing on its own but only under the influence of an external heat flux. Starting from a simple definition of the temperature of the vortex tangle and from the relation between energy and vortex length, we obtain the entropy of the system, as well as the caloric and thermal equations of state, relating internal energy and pressure to temperature and volume. We discuss the connection between our macroscopic results and microscopic results on vortex length distribution function having a potential form.

The paper is devoted to propose a simple and a direct method to get solutions of a nonlinear evolution equation. In particular, we are interested to get solutions of the two-dimensional Boussinesq equation which has applications in dynamical fluid. The non integrability of this equation is not a limit for our method.