Madelung Equations Research Articles

Toward local Madelung mechanics in spacetime

It has recently been shown that relativistic quantum theory leads to a local interpretation of quantum mechanics wherein the universal wavefunction in configuration space is entirely replaced with an ensemble of local fluid equations in spacetime. For want of a fully relativistic quantum fluid treatment, we develop a model using the nonrelativistic Madelung equations, and obtain conditions for them to be local in spacetime. Every particle in the Madelung fluid is equally real, and has a definite position, momentum, kinetic energy, and potential energy. These are obtained by defining quantum momentum and kinetic energy densities for the fluid and separating the momentum into average and symmetric parts, and kinetic energy into classical kinetic and quantum potential parts. The two types of momentum naturally give rise to a single classical kinetic energy density, which contains the expected kinetic energy, even for stationary states, and we define the reduced quantum potential as the remaining part of the quantum kinetic energy density. We treat the quantum potential as a novel mode of internal energy storage within the fluid particles, which explains most of the nonclassical behavior of the Madelung fluid. For example, we show that in tunneling phenomena, the quantum potential negates the barrier so that nothing prevents the fluid from flowing through. We show how energy flows and transforms in this model, and that enabling local conservation of energy requires defining a quantum potential energy current that flows through the fluid rather than only flowing with it. The nonrelativistic treatment generally contains singularities in the velocity field, which undermines the goal of local dynamics, but we expect a proper relativistic treatment will bound the fluid particle velocities at c.

Instability and quantization in quantum hydrodynamics

We show how the quantum hydrodynamical formulation of quantum mechanics converts the nonlocality in the standard wave-like description of quantum systems by an instability of the quantum system, which opens the door to a new way for studying quantum systems based on known methodologies for studying the stability of fluids. As a second result, we show how the Madelung equations describe quantized energies without any external quantization conditions.

A Covariant Non-Local Model of Bohm's Quantum Potential.

Assuming that the energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm's quantum potential and the Madelung equation are obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics do admit a classical interpretation based on non-locality. Here, we generalize this approach imposing a finite speed of propagation of any perturbation, thus determining a covariant formulation of the Madelung equation.

The symplectic geometrical formulation of quantum hydrodynamics

Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.

A fluid description based on the Bernoulli equation of the one-body stationary states of quantum mechanics with real valued wavefunctions

A formalism is developed, and applied, that describes a class of one-body quantum mechanical systems as fluids where each stationary state is a steady flow state. The time-independent Schrödinger equation for one-body stationary states with real-valued wavefunctions is shown to be equivalent to a compressible-flow generalization of the Bernoulli equation of fluid dynamics. The mass density, velocity and pressure are taken as functions that are determined by the probability density. The generalized Bernoulli equation describes compressible, irrotational, steady flow with variable mass and a constant specific total energy, i.e, a constant energy per mass for each fluid element. The generalized Bernoulli equation and a generalized continuity equation provide a fluid dynamic interpretation of a class of quantum mechanical stationary states that is an alternative to the unrealistic, static-fluid interpretation provided by the Madelung equations and quantum hydrodynamics. The total kinetic energy from the Bernoulli equation is shown to be equal to the expectation value of the kinetic energy, and the integrand of the expectation value of the kinetic energy is given an interpretation. It is also demonstrated that variable mass is necessary for a satisfactory fluid model of stationary states. However, over all space, the flows conserve mass, because the rate of mass creation from the sources are equal to the rate of mass annihilation from the sinks. The following flows are examined: the ground and first excited-states of a particle in a one-dimensional box, the harmonic oscillator, and the hydrogen s states.

Towards a Theory of Quantum Gravity from Neural Networks.

Neural network is a dynamical system described by two different types of degrees of freedom: fast-changing non-trainable variables (e.g., state of neurons) and slow-changing trainable variables (e.g., weights and biases). We show that the non-equilibrium dynamics of trainable variables can be described by the Madelung equations, if the number of neurons is fixed, and by the Schrodinger equation, if the learning system is capable of adjusting its own parameters such as the number of neurons, step size and mini-batch size. We argue that the Lorentz symmetries and curved space-time can emerge from the interplay between stochastic entropy production and entropy destruction due to learning. We show that the non-equilibrium dynamics of non-trainable variables can be described by the geodesic equation (in the emergent space-time) for localized states of neurons, and by the Einstein equations (with cosmological constant) for the entire network. We conclude that the quantum description of trainable variables and the gravitational description of non-trainable variables are dual in the sense that they provide alternative macroscopic descriptions of the same learning system, defined microscopically as a neural network.

Emergent Quantumness in Neural Networks

It was recently shown that the Madelung equations, that is, a hydrodynamic form of the Schr\"odinger equation, can be derived from a canonical ensemble of neural networks where the quantum phase was identified with the free energy of hidden variables. We consider instead a grand canonical ensemble of neural networks, by allowing an exchange of neurons with an auxiliary subsystem, to show that the free energy must also be multivalued. By imposing the multivaluedness condition on the free energy we derive the Schr\"odinger equation with "Planck's constant" determined by the chemical potential of hidden variables. This shows that quantum mechanics provides a correct statistical description of the dynamics of the grand canonical ensemble of neural networks at the learning equilibrium. We also discuss implications of the results for machine learning, fundamental physics and, in a more speculative way, evolutionary biology.

A Non-local Phase Field Model of Bohm\u2019s Quantum Potential

Assuming that the free energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm’s quantum potential and the Madelung equation are identically obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics admit a classical interpretation based on non-locality.

Correlation of Bond Energy and Optical Band Energy of Annealed TiO2 Thin Films

We report on the correlation between bond energy and optical band energy of TiO2 thin films prepared through sol gel doctor-blade technique. The TiO2 films were deposited on doped fluorine tin IV oxide (SnO2:F) layer on glass substrates. UV-Vis-NIR spectroscopy was carried out on as-deposited and subsequent annealed films at different rates from room temperature up to 450°C. The average optical transmittance within the visible region was 73.5%, 73.4%, 70.5% and 69.9% for the as-deposited, 1-step annealed, 2°C/min, and 1°C/min films, respectively. FTIR spectroscopy confirmed presence of functional elements of Ti = O with a peak at 587.7 cm-1. Bond energy for the films was calculated using Madelung equation with inclusion of the second coordination sphere for crystalline state. The values of bond energies were found to be 3.99 eV, 4.02 eV, 4.12 eV and 4.16 eV for the as-deposited, 1-step annealed, 2°C/min, and 1°C/min films respectively. The analysis of the band gap was done using Tauc’s relation. Band gap energy ranged from 4.02 eV to 5.04 eV. A statistical correlation between bond energy and optical band energy was established. Films annealed at 1°C/min recorded the highest bond energy and lowest band gap energy. This is attributed to the process of nucleation and crystal growth which are governed by thermodynamic properties. Prolonged exposure to higher temperatures through low annealing rates led to formation of films with high bond energy and low band gap.

Quantum kinetic theory for unmagnetized and magnetized plasmas

Three different approaches to the treatment of quantum effects in plasmas are reviewed: quantum fluid theory (QFT), phase-space kinetic theory (PKT) and quantum plasmadynamics (QPD). The simplest form of QFT is analogous to a nonrelativistic fluid model for an unmagnetized plasma with a potential electric field, $$\phi $$ . The wave nature of the electron is included through the so-called Bohm term, as in the Madelung equations. The simplest form of PKT is based on the Wigner function and the kinetic equation that it satisfies in a quasi-classical $$\mathbf{p}$$ – $$\mathbf{x}$$ phase space. Further development of PKT involves including additional effects piecemeal. The electron spin is included through a classical model for a spin vector, $$\mathbf{s}$$ , with the phase space extended to include $$\mathbf{s}$$ . The inclusion of degeneracy is straightforward. Electromagnetic effects are described by including the vector potential, $$\mathbf{A}$$ , in Schrodinger’s equation. It is argued that further extensions, to a magnetized plasma and to include relativistic effects, raise conceptual difficulties concerning the phase-space approach. In a magnetic field, the electron (Landau) states are discrete, whereas the kinetic equation in PKT involves a derivative with respect to $$\mathbf{p}$$ . The relativistic case is based on Dirac’s equation, and to derive a Vlasov-like equation for electrons requires excluding the positron and virtual-pair contributions to the Dirac wavefunction, which cannot be achieved exactly. Moreover, specific spin states require a specific choice of spin operator in Dirac’s theory, whereas the Pauli matrices define the only (vectorial) spin operator in PKT. The Dirac wavefunctions for two spin operators are derived and shown to approximate the eigenstates of the Pauli theory only in the nonrelativistic limit. QPD is an exact theory, based on quantum electrodynamics, in which kinetic processes are described using Feynman diagrams. The presence of the plasma is taken into account through a statistical average of the electron propagator, analogous to the use of thermal Green functions. QPD is introduced for the unmagnetized case, and the generalization to include a background magnetic field is presented. It is shown how QPD is used to derive the linear response tensor for an electron gas in the relativistic quantum case for both unmagnetized and magnetized plasmas. A magnetized vacuum is shown to have response tensors analogous to a plasma, allowing processes (such as one-photon pair creation and photon splitting) that are forbidden in an unmagnetized vacuum. The various quantum effects that may be relevant to a plasma are summarized, and their possible application to laboratory and astrophysical plasmas are discussed briefly.

A model of emerging intelligence in Universe

AbstractThe paper proposes a scenario of origin and emerging of intelligent life in Universe based upon the mathematical discovery of a new class of dynamical systems described by ordinary differential equation (ODE) coupled with their Liouville equation. These systems called self-controlled since the role of actuators is played by the probability produced by the Liouville equation. Following the Madelung equation that belongs to this class, non-Newtonian and quantum-like properties such as randomness, entanglement and probability interference typical for quantum systems have been described. At the same time, these systems expose properties of livings: decomposition into motor and mental dynamics, the capability of self-identification and self-awareness, as well as self-supervision. But the most surprising discovery is the existence of a special sub-class, in which the dynamical systems can violate the second law of thermodynamics, and that makes them different from both Newtonian and quantum physics. This sub-class should be associated with intelligent livings due to capability to move from disorder to order without external help. Based upon the mathematical discovery described above, one can assume that there are good chances that similar dynamical systems representing intelligent livings exist in real physical world. This provides a reason for a ‘rehabilitation’ of the Maxwell demon and put it into physics of intelligent systems. Indeed, the Maxwell demon is implemented by the feedback from the Liouville equation to the original ODE, while this feedback is capable to rearrange the probability distribution against the second law of thermodynamics. In addition to that, the same feedback removes the entropy paradox by explaining high order in our surrounding by ‘intelligent life support’. Two-steps transition: from the Newtonian physics to the linear model of life, and from the latter to the model of intelligent life are analysed. The first transition is triggered by the Hadamard instability of the Newtonian physics with respect to small random disturbances in linear terms of the Liouville feedback. The second transition is triggered by instability of linear model of life with respect to small random disturbances of non-linear terms of Liouville feedback. This transition could be implemented by such physical phenomena as shock waves or negative diffusion in probability space. Both transitions can be associated with catastrophe theory, in which sudden shifts in behaviour arises from small changes in parameters of the model. In view of the proposed model, possible competition between artificial and human intelligence are discussed.

Modelling ‘Life’ against ‘heat death’

AbstractThis work is inspired by the discovery of a new class of dynamical system described by ordinary differential equations coupled with their Liouville equation. These systems called self-controlled since the role of actuators is played by the probability produced by the Liouville equation. Following the Madelung equation that belongs to this class, non-Newtonian properties such as randomness, entanglement and probability interference typical for quantum systems have been described. Special attention was paid to the capability to violate the second law of thermodynamics, which makes these systems neither Newtonian, nor quantum. It has been shown that self-controlled dynamical systems can be linked to mathematical models of living systems. The discovery of isolated dynamical systems that can decrease entropy in violation of the second law of thermodynamics, and resemblances of these systems to livings suggests that ‘Life’ can slow down the ‘heat death’ of the Universe and that can be associated with the Purpose of Life.

Analytic Solutions of the Madelung Equation

We present analytic self-similar solutions for the one, two and three dimensional Madelung hydrodynamical equation for a free particle. There is a direct connection between the zeros of the Madelung fluid density and the magnitude of the quantum potential.

On Entropy Production in the Madelung Fluid and the Role of Bohm’s Potential in Classical Diffusion

The Madelung equations map the non-relativistic time-dependent Schrodinger equation into hydrodynamic equations of a virtual fluid. Here we show that an increase of the Boltzmann entropy of this Madelung fluid is proportional to the expectation value of its velocity divergence. Hence, entropy growth is accompanied by expansion resulting from the ability of the Madelung fluid to be compressible. The compressibility itself reflects superposition of solutions of the Schrodinger equation. Thus, in unitary processes where the Madelung fluid expands and then shrinks, the Boltzmann entropy may, correspondingly, grow and then decrease. The notion of entropy growth due to expansion is common in diffusive processes, however in the latter the process is irreversible. Much unlike the Boltzmann entropy, the von Neumann entropy, does not vary with time. To elucidate the physical underpinning of the Boltzmann entropy, we examine several specific examples. We demonstrate that, for classical diffusive processes, the "force" accelerating diffusion has the form of the positive gradient of the quantum Bohm potential. In the Madelung fluid, the advective and the diffusive velocities correspond respectively to the the real and imaginary parts of the complex momentum. We find that the diffusion coefficient provides a lower bound of Heisenberg uncertainty type product between the gas mean free path and the Brownian momentum.

Games of entangled agents

AbstractThe paper extends the mathematical formalism of quantum physics to include games of intelligent agents that communicate due to entanglement. The novelty of the approach is based upon a human factor-based behavioral model of an intelligent agent. The model is quantum inspired: it is represented by a modified Madelung equation in which the gradient of quantum potential is replaced by a specially chosen information force. It consists of motor dynamics simulating actual behavior of the agent, and mental dynamics representing evolution of the corresponding knowledge base and incorporating this knowledge in the form of information flows into the motor dynamics. Due to feedback from mental dynamics, the motor dynamics attains quantum-like properties: its trajectory splits into a family of different trajectories, and each of those trajectories can be chosen with the probability prescribed by the mental dynamics; each agent is entangled (in a quantum way) to other agents and makes calculated predictions fo...

Interference of probabilities in dynamics

A new class of dynamical systems with a preset type of interference of probabilities is introduced. It is obtained from the extension of the Madelung equation by replacing the quantum potential with a specially selected feedback from the Liouville equation. It has been proved that these systems are different from both Newtonian and quantum systems, but they can be useful for modeling spontaneous collective novelty phenomena when emerging outputs are qualitatively different from the weighted sum of individual inputs. Formation of language and fast decision-making process as potential applications of the probability interference is discussed.

Quantum-inspired resonance for associative memory

A new kind of dynamics for simulations based upon quantum-classical hybrid is discussed. The model is represented by a modified Madelung equation in which the quantum potential is replaced by different, specially chosen potentials. As a result, the dynamics attains both quantum and classical properties: it preserves superposition and entanglement of random solutions, while allowing one to measure its state variables using classical methods. Such an optimal combination of characteristics is a perfect match for quantum-inspired information processing. In this paper, the retrieval of stored items from an exponentially large unsorted database is performed by quantum-inspired resonance using polynomial resources due to quantum-like superposition effect.

Quantum-inspired maximizer

The objective of this paper is to create a new kind of dynamical systems—a quantum-classical hybrid—that would preserve superposition and entanglement of random solutions while allowing one to measure their state variables by using classical methods. Such an optimal combination of characteristics is a perfect match for quantum-inspired computing. The model is represented by a modified Madelung equation in which the quantum potential is replaced by a different, specially chosen “computational” potential. As a result, the dynamics attains both quantum and classical properties. Similarities and differences of the proposed model with quantum systems are outlined. As an application, an algorithm for the global maximum of an arbitrary integrable function is proposed. The idea of the proposed algorithms is very simple: based on the quantum-inspired maximizer, introduce a positive function to be maximized as the probability density to which the solution is attracted. Then, the larger value of this function will have the higher probability to appear. Special attention is paid to the simulation of integer programming, NP-complete problems and information retrieval.

Isothermal Quantum Hydrodynamics: Derivation, Asymptotic Analysis, and Simulation

This article is devoted to the reformulation of an isothermal version of the quantum hydrodynamic model derived by Degond and Ringhofer in [J. Statist. Phys., 112 (2003), pp. 587–628] (which will be referred to as the quantum Euler system). We write the model under a simpler (differential) form. The derivation is based on an appropriate use of commutators. Starting from the quantum Liouville equation, the system of moments is closed by a density operator which minimizes the quantum free energy. Some properties of the model are then exhibited, and most of them rely on a gauge invariance property of the system. Several simplifications of the model are also written for the special case of irrotational flows. The second part of the paper is devoted to a formal analysis of the asymptotic behavior of the quantum Euler system in three situations: at the semiclassical limit, at the zero‐temperature limit, and at a diffusive limit. The remarkable fact is that in each case we recover a known model: respectively, the isothermal Euler system, the Madelung equations, and the entropic quantum drift‐diffusion model. Finally, we give in the third part some preliminary numerical simulations.

From reversible thermodynamics to life

Abstract The First Principle for dynamics of living systems has been proposed and discussed. It is based upon introduction of information potential that couples the Newtonian dynamics and the corresponding Liouville equation in the same way in which the quantum potential couples the Madelung equations in quantum mechanics. It has been demonstrated that the information potential can generate a negative diffusion that reverses the evolution towards decrease of entropy. Mathematical, physical and biological interpretations of the information potential are discussed. An algorithm for life-nonlife discrimination is proposed. An associated Boolean approach is introduced to simulate the evolution of intelligence.