Modified Hamilton-Jacobi Equation Research Articles

Modified Hawking temperatures of black holes in Lorentz violation theory

In this paper, the tunneling of scalar particles near the event horizons of Riemann space time, BTZ black hole and Schwarzschild–de Sitter black hole are investigated by applying Hamilton–Jacobi equation with Lorentz violation theory in curved space time. The modified Hamilton–Jacobi equation is derived from Klein–Gordon equation of scalar particles induced by Lorentz violation theory in curved space time. The Hawking temperatures of Riemann space time and the BTZ black hole are modified due to the effect of Lorentz violation theory. Moreover, the Bekenstein–Hawking entropy near the event horizon of Schwarzschild–de Sitter black hole is also modified due to Lorentz violation theory. It is observed that the modified values of Hawking temperatures and change in Bekenstein–Hawking entropy depend upon the ether-like vectors [Formula: see text].

Diffusion Effect in Quantum Hydrodynamics

In this paper, we introduce (at least formally) a diffusion effect that is based on an axiom postulated by Werner Heisenberg in the early days of quantum mechanics. His statement was that—in quantum mechanics—kinematical quantities such as velocity must be treated as complex matrices. In the hydrodynamic formulation of quantum mechanics according to Madelung, the complex Schrödinger equation is rewritten in terms of two real equations—a continuity equation and a modified Hamilton–Jacobi equation. Considering seriously Heisenberg’s axiom, the velocity occurring in the continuity equation should be replaced by a complex one, thus introducing a diffusion term with an imaginary diffusion coefficient. Therefore, in quantum mechanics, there should be a diffusion effect showing up in the dynamics. This is the case in the time evolution of the free-motion wave packet under time reversal. The maximum returns to the initial position; however, the width of the wave packet does not shrink to its initial width. This effect is obvious but—as far as we know—it is not mentioned in any textbook. In our paper, we discuss this effect in detail and show the connection with a complex version of quantum hydrodynamics.

Modified Hawking temperature of Kerr–Newman black hole in Lorentz symmetry violation theory

In this paper, we study the quantum tunneling of scalar particles near the event horizon of Kerr–Newman black hole under Lorentz violation theory. First, the modified Hamilton–Jacobi equation under Lorentz violation theory in the curved spacetime is reviewed. The modified Hawking temperatures, heat capacities and entropies of Kerr–Newman black hole in frame dragging coordinates and Eddington coordinate are derived. The modified Hawking temperature, heat capacity and change in Bekenstein–Hawking entropy of Kerr–Newman black hole are found to be modified due to Lorentz violation theory and they depend not only on Lorentz violation parameter [Formula: see text] but also on ether-like vector [Formula: see text].

Conformal topology optimization of multi-material ferromagnetic soft active structures using an extended level set method

Ferromagnetic soft active structures using embedded ferromagnetic particles in the soft polymer matrix can generate flexible locomotion and change configurations remotely, rapidly, and biologically friendly with an applied magnetic field. To achieve the desired motion, these soft active structures can be designed by tailoring the layouts of the ferromagnetic soft polymer. Although many magnetic soft active structures have been designed and fabricated, they are limited by the developer’s intuition and experience. Structural topology optimization has become a promising method to achieve innovative structures by optimizing the material layout, opening a new path for architecting ferromagnetic-driven active structures. Given the widespread adoption of thin-shell structures for soft robots, the extended level set method (X-LSM) and conformal geometry theory are employed to perform topology optimization of the ferromagnetic soft active structures on manifolds. The boundary evolution on a free-form 3D surface can be transferred into a 2D rectangular plane by solving a modified Hamilton–Jacobi equation weighted by conformal factors. The reconciled level set (RLS) method is firstly implemented within the X-LSM framework in this paper to enable the design of multi-material ferromagnetic soft active structures on free-from surfaces. The design objective consists of a subobjective function for kinematic requirement and a subobjective function for minimum compliance. The shape sensitivity was analyzed using the material time derivative and the adjoint variable approach. The proposed method was applied to design several single and multi-material ferromagnetic soft active structures. Two topologically optimized designs have been printed using functional 3D printing technology, or the so-called 4D printing, to physically realize soft active structures with built-in functionalities. The results of the numerical verification and experimental validation demonstrate the effectiveness of the proposed design and fabrication framework.

The solution of a modified Hamilton–Jacobi equation with Lorentz-violating scalar field

A modified Hamilton–Jacobi equation with Lorentz-violating scalar field in the curved spacetime is used to research tunneling radiation from the horizons of Reissner–Nordstr $$\ddot{\mathrm{o}}$$ m de Sitter black hole. We investigate the influence of different etheric-like vector $$u^\alpha $$ on the solution of the modified Hamilton–Jacobi equation by using both the semi-classical approximation and beyond the semi-classical approximation. We find that the solution of the modified Hamilton–Jacobi equation strongly depends on the choices of $$u^\alpha $$ . As $$u^\alpha =\delta ^\alpha _t u^t$$ , the Lorentz-violating scalar field can enhances black holes’ Hawking temperature but reduces its entropy. However, the case will reverses as $$u^\alpha =\delta ^\alpha _r u^r$$ . Our conclusions are applicable to other spherically symmetric charged black holes.

Hawking radiation of Reissner–Nordström–de Sitter black hole with a global monopole derived from modified Hamilton–Jacobi equation

The tunneling characteristics at the cosmological and event horizons of Reissner–Nordström–de Sitter black hole with a global monopole are studied by using the modified Lorentz violation scalar field equation in curved space–time. Firstly, we get the modified Hamilton–Jacobi equation using the semi-classical approximation, then the Hawking radiation and thermodynamical properties of Reissner–Nordström–de Sitter black hole with a global monopole are computed based on the modified Hamilton–Jacobi equation. Our results show that Lorentz violation can lead to lower Hawking temperature and higher entropy of the black hole at the same time. This work improves the understanding of the physical nature of Lorentz violation in curved space–time.

Study on the Influence of Lorentz Dispersion Relation on Fermions Tunneling Radiation of Kerr-TAUB-NUT Black Hole

In this paper, by applying the correction of Lorentz dispersion relation, the fermion tunneling radiation of the stationary axisymmetric Kerr-TAUB-NUT black hole in curved space-time is studied. Based on the modified Lorentz dispersion relation, the revised Rarita-Schwinger equation has been obtained. Then, we get the dynamic equation of fermions via semi-classical theory, which is the modified Hamilton-Jacobi equation. So at the event horizon the tunneling radiation in the space-time of stationary axisymmetric Kerr-TAUB-NUT black hole has been effectively investigated. At last, the modified tunneling radiation and Hawking temperature have been obtained in the same space-time.

Some nonlinear extensions for the schrödinger equation

We investigate extensions for the Schrodinger equation, by considering different non-linear dependencies on the kinetic term, which admit the standard probabilistic interpretation for the wave-function. For one of the cases, we show that the Heisenberg equation is invariant in form and a modified Hamilton-Jacobi equation emerges from the semi-classical approximation for the wave-function. Furthermore, we find solutions assuming time scaling functions different from the standard case and observe nonstandard relaxation processes for the wave-function.

Modified Fermions Tunneling Radiation from Nonstationary, Axially Symmetric Kerr Black Hole

In this paper, by applying the deformed dispersion relation in quantum gravity theory, we study the correction of fermions’ tunneling radiation from nonstationary symmetric black holes. Firstly, the motion equation of fermions is modified in the gravitational space-time. Based on the motion equation, the modified Hamilton-Jacobi equation has been obtained by a semiclassical approximation method. Then, the tunneling behavior of fermions at the event horizon of nonstationary symmetric Kerr black hole is investigated. Finally, the results show that, in the nonstationary symmetric background, the correction of Hawking temperature and the tunneling rate are closely related to the angular parameters of the horizon of the black hole background.

Lorentz-violating theory and tunneling radiation characteristics of Dirac particles in curved spacetime of Vaidya black hole

In this paper, the modified Hawking radiation for Dirac particles via tunneling from the apparent horizon of Vaidya black hole is studied by using the Lorentz-violating Dirac field theory. We first extend the gamma matric from flat spacetime to the curved spacetime in the Lorentz-violating Dirac field theory, and generalize the general derivative to the covariant derivative. Then, by considering the commutative relation of the gamma matric, the Dirac equation in the Lorentz-violating Dirac field theory is obtained, which contains three correction terms related to the Lorentz-symmetry violation. In the semiclassical approximation, the modified Hamilton-Jacobi equation is obtained by using the commutative relation of gamma matric and treating the aether-like vector in the Lorentz-violating theory as a constant. We find that the modified Hamilton-Jacobi equation contains only two correction terms based on the Lorentz-symmetry violation, i.e. the corrected term containing the parameter <i>a</i> affects the mass term of the Dirac field, and the aether-like term containing the parameter <i>c</i> modifies the coefficient term of the action <i>S</i> of the separating variable. According to the modified Hamilton-Jacobi equation, we study the effect of Lorentz-symmetry violation on the characteristics of Hawking radiation for Dirac particles via tunneling from the apparent horizon <i>r</i><sub>a</sub> = 2<i>M</i>(<i>v</i>) of Vaidya black hole (the apparent horizon of Vaidya black hole coincides with the timelike limit surface, so the apparent horizon can be regarded as the boundary of Vaidya black hole). Since the Hawking tunneling radiation of black holes is the radial property at the horizon of black holes, we finally find that only the aether-like term containing the parameter <i>c</i> can modify the characteristics of Dirac particles’ tunneling radiation from the black hole. In addition, the corrected Hawking temperature of the black hole caused by considering the effect on the Lorentz-violating Dirac field theory has a small correction related to the aether-like term, which is consistent with the results obtained by studying the characteristics of Hawking tunneling radiation for scalar particles in the Lorentz-violating scalar field theory. The results suggest that the Lorentz-symmetry violation theory may provide a new method to further study the information loss paradox of black holes.

Reaction force surface for the hydrogen transfer reaction in malonaldehyde: A classical wavefront-based formulation

This paper investigates the simulation of a modified Hamilton–Jacobi equation in order to provide a classical wavefront-based formulation of reaction dynamics. Here, the wavefront is interpreted as minimizing a particular action functional. The method is rooted in the geometric optics and an eikonal equation is typically used for describing the light wave propagation. We have introduced a general formulation for reaction force surface (RFS) and a host of several force-based properties, which offer a reinvigorated understanding of the occurrence of a reaction event at molecular level. The calculation is performed on a model potential energy surface representing the hydrogen transfer reaction in malonaldehyde.

Quantum corrections to the thermodynamics of Schwarzschild–Tangherlini black hole and the generalized uncertainty principle

We investigate the thermodynamics of Schwarzschild-Tangherlini black hole in the context of the generalized uncertainty principle. The corrections to the Hawking temperature, entropy and the heat capacity are obtained via the modified Hamilton-Jacobi equation. These modifications show that the GUP changes the evolution of Schwarzschild-Tangherlini black hole. Specially, the GUP effect becomes susceptible when the radius or mass of black hole approach to the order of Planck scale, it stops radiating and leads to black hole remnant. Meanwhile, the Planck scale remnant can be confirmed through the analysis of the heat capacity. Those phenomenons imply that the GUP may give a way to solve the information paradox. Besides, we also investigate the possibilities to observe the black hole at LHC, the results demonstrate that the black hole can not be produced in the recent LHC.

Canonical quantisation via conditional symmetries of the closed FLRW model coupled to a scalar field

We study the classical, quantum and semiclassical solutions of a Robertson-Walker spacetime coupled to a massless scalar field. The Lagrangian of these minisuperspace models is singular and the application of the theory of Noether symmetries is modified to include the conditional symmetries of the corresponding (weakly vanishing) Hamiltonian. These are found to be the simultaneous symmetries of the supermetric and the superpotential. The quantisation is performed adopting the Dirac proposal for constrained systems. The innovation in the approach we use is that the integrals of motion related to the conditional symmetries are promoted to operators together with the Hamiltonian and momentum constraints. These additional conditions imposed on the wave function render the system integrable and it is possible to obtain solutions of the Wheeler-DeWitt equation. Finally, we use the wave function to perform a semiclassical analysis following Bohm and make contact with the classical solution. The analysis starts with a modified Hamilton-Jacobi equation from which the semiclassical momenta are defined. The solutions of the semiclassical equations are then studied and compared to the classical ones in order to understand the nature and behaviour of the classical singularities.

Symmetry Transformation in Extended Phase Space: the Harmonic Oscillator in the Husimi Representation

In a previous work the concept of quantum potential is generalized into extended phase space (EPS) for a particle in linear and harmonic potentials. It was shown there that in contrast to the Schrodinger quantum mechanics by an appropriate extended canonical transformation one can obtain the Wigner representation of phase space quantum mecha- nics in which the quantum potential is removed from dynamical equation. In other words, one still has the form invariance of the ordinary Hamilton-Jacobi equation in this repre- sentation. The situation, mathematically, is similar to the disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates. Here we show that the Husimi representation is another possible representation where the quantum potential for the harmonic potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form. This happens when the parameter in the Husimi transformation assumes a specific value corresponding to Q-function.

Quantum Potential and Symmetries in Extended Phase Space

The behavior of the quantum potential is studied for a particle in a linear and a harmonic potential by means of an extended phase space technique. This is done by ob- taining an expression for the quantum potential in momentum space representation followed by the generalization of this concept to extended phase space. It is shown that there exists an extended canonical transformation that removes the expression for the quantum poten- tial in the dynamical equation. The situation, mathematically, is similar to disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates that changes the physical potential to an effective one. The representation where the quantum potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form, is one in which the dynamical equation turns out to be the Wigner equation.

Incompleteness of trajectory-based interpretations of quantum mechanics

Trajectory-based approaches to quantum mechanics include the de Broglie-Bohm interpretation and Nelson's stochastic interpretation. It is shown that the usual route to establishing the validity of such interpretations, via a decomposition of the Schroedinger equation into a continuity equation and a modified Hamilton-Jacobi equation, fails for some quantum states. A very simple example is provided by a quantum particle in a box, described by a wavefunction initially uniform over the interior of the box. For this example there is no corresponding continuity or modified Hamilton-Jacobi equation, and the spacetime dependence of the wavefunction has a known fractal structure. Examples with finite average energies are also constructed.

The diffusion equation for a particle in a zero-point field: A free particle in a finite volume

The stochastic motion in one dimension of a free particle in a finite volume, subjected to a zero-point-field of random forces, is considered. The velocity of the particle is split, as usual, into a stochastic plus a stationary component, depending upon the coordinate only. The modified Hamilton-Jacobi equation for the deterministic component is then solved, and the resulting zeroth-order trajectories are computed. The diffusion equation can be written down in terms of the stationary drift component of the velocity, which turns out to be dependent upon a parameter γ, which characterizes the uncertainty in the velocity polarization, and the resulting flux through the walls. The zeroth-order trajectories can hence be classified following this parameter, which indicates a transition between two different regimes for some value lying between 0 and 1. For each type a diffusion equation in configuration space is obtained, together with an expression for the particle number flux in steady state. It is then briefly examined the effects of introducing a cut-off frequency into the zero-point spectrum, and a comparison with previous treatments of the same problem is made. It is found that the diffusion equations for non-zero values of γ are of a new type which had not been considered before.

Equation of motion for a charged bound particle in a zero-point field

A method previously proposed by the same author, of solving the equations of motion in the presence of friction and an external stochastic force, is applied to the nonrelativistic Dirac equation for a charged bound pointlike particle in a black-body radiation field. It separates the particle velocity field, under the assumtion of stationarity, into a position-dependent component and a randomly fluctuating component depending mainly upon the time. This description of the possible stationary states of the bound particle in the random force field is taken as a starting point for establishing a diffusion equation in configuration space. Since we identify the first component with the drift velocity, we show that it must be the solution to a modified Hamilton–Jacobi equation by using a term that is the counterpart of the quantum modification to the same equation, which has as a first approximation exactly the same form. This term gives results proportional to the diffusion coefficient and, thereby, to the spectral density of the external random force. The diffusion equation that is obtained has complex coefficients and therefore it defines a probability density for the complex values of the variable coordinate. We propose an interpretation of this probability density, based on a specific example.

A microscopic theory of vortices in superfluid helium

Abstract It is shown that the introduction of the depletion velocity field, the vortical part of the fluid velocity, is rooted on the nature of the field operators. This result is obtained by formulating the theory of vortices in terms of the vortex coordinates. The modified Hamilton-Jacobi equation leads to a generalization of the Schrodinger equation for a particle in the presence of a vortex line. It is shown that our theory is intimately related to the microscopic theory of Bose systems developed by Nishiyama. Mention is made of the close relation between the Higgs mechanism and a gauge-wheel in superfluid helium.

Hamilton's principal function for the Brownian motion of a particle and the Schrödinger–Langevin equation

The Brownian motion of a particle coupled to a heat bath can be derived from an exactly solvable many-body system. This motion is governed by the Langevin equation in Newtonian mechanics, and by an operator equation formally identical to the Langevin equation when the system composed of the particle plus the heat bath is quantized. To get this formal similarity between the classical and the quantal equations of motion of the particle, the classical dissipative system is formulated in terms of a modified Hamilton–Jacobi equation and is then quantized using the Schrödinger method. The result is identical with the Schrödinger–Langevin equation that has been obtained by quantizing the entire system and then isolating the motion of the particle. The non-linear wave equation describing the motion of a particle subject to conservative and time-dependent forces as well as frictional forces has been applied to the problems of motion of a wave-packet, and of the scattering and trapping of heavy-ions.