arXiv subjects
Publications and source records attributed to Roumen Tsekov.
Entropy is critically examined as a fundamental concept in contemporary science and informatics. Although the typical Shannon entropy provides a proper framework for describing the canonical ensemble, it fails to represent adequately the microcanonical ensemble. This discrepancy manifests additionally in its inability to support a theoretical derivation of the Second Law of thermodynamics.
Advanced thermostats for molecular dynamics are proposed on the base of the rigorous Langevin dynamics. Because the latter accounts for the subsystem-bath interactions in details, the bath anisotropy and nonuniformity are described via the relevant friction tensor. The developed model reflects properly the relativistic dynamics of the subsystem evolution as well as the nonlinear friction, which can occur for fast particles with large momenta at elevated temperature.
Trailing the modern ideas of sociophysics, a minimalistic thermodynamic model of society is proposed, which consists of three social ingredients: people, economy, and entropy. Employing the universal van der Waals equation of state, many important relationships are discovered, including laws of econophysics. A paramount finding is that the Second Law of thermodynamics governs irreversible social evolution, revealing the power of social liberty and economic freedom.
The Schrodinger equation is solved for many free particles and their quantum entanglement is studied via correlation analysis. Converting the Schrodinger equation in the Madelung hydrodynamic-like form, the quantum mechanics is extended to open quantum systems by adding Ohmic friction forces. The dissipative evolution confirms the correlation decay over time, but a new integral of motion is discovered, being appropriate for storing everlasting quantum information.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
A nonlinear master equation is derived, reflecting properly the entropy of open quantum systems. In contrast to linear alternatives, its equilibrium solution is exactly the canonical Gibbs density matrix. The corresponding nonlinear equation for the Wigner function accounts rigorously for the thermo-quantum entropy. It reduces at large friction to the Smoluchowski-Bohm equation in the coordinate subspace, which reflects the stochastic Langevin-Bohm dynamics. The previously derived Maxwell-Heisenberg relation for the nonequilibrium momentum dispersion of quantum Brownian particles is confirmed as well as the related quantum generalization of the classical Einstein law of Brownian motion.
It is demonstrated how quantum mechanics is generated by stochastic momentum kicks from the force carriers, transmitting the fundamental interactions between the point particles. The picture is consistent with quantum field theory and points out that the force carriers are the only quantum particles. Since the latter are waves in the coordinate space, they are responsible for the wavy character of quantum mechanics.
A theoretical parallel between the classical Brownian motion and quantum mechanics is explored. It is shown that, in contrast to the classical Langevin force, quantum mechanics is driven by turbulent velocity fluctuations with diffusive behavior. In the case of simultaneous action of the two stochastic sources, the quantum Brownian motion takes place, which is theoretically described as well.
The quantum Schrodinger-Newton equation is solved for a self-gravitating Bose gas at zero temperature. It is derived that the density is non-uniform and a central hollow cavity exists. The radial distribution of the particle momentum is uniform. It is shown that a quantum black hole can be formed only above a certain critical mass. The temperature effect is accounted for via the Schrodinger-Poisson-Boltzmann equation, where low and high temperature solutions are obtained. The theoretical analysis is extended to a strong interacting gas via the Schrodinger-Yukawa equation, showing that the atomic nuclei are also hollow. Hollow self-gravitating Fermi gases are described by the Thomas-Fermi equation.
A stochastic Euler equation is proposed, describing the motion of a particle density, forced by the random action of virtual photons in vacuum. After time averaging, the Euler equation is reduced to the Reynolds equation, well studied in turbulent hydrodynamics. It is shown that the average pressure is nonlocal and the magnitude of the turbulent flow obeys the Fick law. Using the Madelung transformation, the Schrodinger equation is derived without any other assumptions.
We regard the non-relativistic Schrodinger equation as an ensemble mean representation of the stochastic motion of a single particle in a vacuum, subject to an undefined stochastic quantum force. The local mean of the quantum force is found to be proportional to the third spatial derivative of the probability density function, while its associated pressure is proportional to the second spatial derivative. The latter arises from the single particle diluted gas pressure, and this observation allows to interpret the quantum Bohm potential as the energy required to put a particle in a bath of fluctuating vacuum at constant entropy and volume. The stochastic force expectation value is zero and is uncorrelated with the particle location, thus does not perform work on average. Nonetheless it is anti-correlated with volume and this anti-correlation leads to an uncertainty relation. We analyze the dynamic Gaussian solution to the Schrodinger equation as a simple example for exploring the mean properties of this quantum force. We conclude with a few possible interpretations as to the origins of quantum stochasticity.
We consider for clarity the simple case of the one dimensional non-relativistic Schrödinger equation and regard it as an ensemble mean representation of the stochastic motion of a single particle in a vacuum, subject to an undefined stochastic quantum force. By analyzing the Bohm potential it is found that the imaginary part of the quantum momentum is the root mean square fluctuation of the particle around its mean velocity, where the latter is the real part of the quantum momentum. The local mean of the quantum force is found to be proportional to the third spatial derivative of the probability density function, and its associated pressure to the second spatial derivative. The latter is decomposed from the single particle diluted gas pressure, and this pressure partition allows the interpretation of the quantum Bohm potential as the energy required to put a particle in a bath of fluctuating vacuum at constant entropy and volume. The stochastic force expectation value is zero and is uncorrelated with the particle location, thus does not perform work on average. Nonetheless it is anti-correlated with volume and this anti-correlation leads to a new type of an Heisenberg like relation. We imply the dynamic Gaussian solution to the Schrödinger equation as a simple example for exploring the mean properties of this quantum force. Still an interesting remained open task is the identification of the stochastic law that leads to these obtained mean properties.
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.
Three existing interpretations of quantum mechanics, given by Heisenberg, Bohm and Madelung, are examined to describe dissipative quantum systems as well. It is found that the Madelung quantum hydrodynamics is the only correct approach. A new stochastic reinterpretation of the quantum mechanics is proposed, which represents the microscopic face of the Madelung hydrodynamics. The main idea is that the vacuum fluctuates permanently, which explains the probabilistic character of the quantum mechanics. Thus, it is an objective theory independent of the human beings and their measurements. The effect of the thermal fluctuations in the surrounding is also accounted for via a heuristic Langevin equation with two random forces. Some statistical characteristics of these quantum and thermal noises are determined by reproducing known results for the system phase-space dynamics.
A stochastic Langevin equation is derived, describing the thermal motion of a molecule immersed in a rested fluid of identical molecules. The fluctuation-dissipation theorem is proved and a number of correlation characteristics of the molecular Brownian motion are obtained.
Memory effect of Brownian motion in an incompressible fluid is studied. The reasoning is based on the Mori-Zwanzig formalism and a new formulation of the Langevin force as a result of collisions between an effective and the Brownian particles. Thus, the stochastic force autocorrelation function with finite dispersion and the corresponding Brownian particle velocity autocorrelation function are obtained.
The effect of nonlinear friction forces in quantum mechanics is studied via dissipative Madelung hydrodynamics. A new thermo-quantum diffusion equation is derived, which is solved for the particular case of quantum Brownian motion with a cubic friction. It is extended also by a chemical reaction term to describe quantum reaction-diffusion systems with nonlinear friction as well.
A Wigner-Klein-Kramers equation is proposed, which merges relativistic, quantum and thermo dynamics. The relativistic effect on quantum Brownian motion is studied via the Breit-Fermi Hamiltonian applied into a dissipative Madelung hydrodynamics. A new thermo-quantum Smoluchowski equation is derived, which accounts for the relativistic correction of the Bohm quantum potential.