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.
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.
where m is the particle mass at rest, c is the speed of light, p is the particle momentum and U is a potential. The root in Eq. (1) complicates substantially the way of quantizing the relativistic motion. In the literature there are two important solutions of this problem: the Klein-Gordon and Dirac equations. Their complex mathematical structures and some problems with definition of the probability density make it difficult to extend these equations for description of relativistic quantum Brownian motion. Moreover, it is not clear if the non-relativistic energy ˆ t E i and momentum p i operators, used in quantizing of Eq. (1), are valid in the relativistic quantum mechanics as well. The present paper proposes an alternative approach via a relativistic Wigner-Klein-Kramers equation. The quantum Brownian motion is described by the Madelung quantum hydrodynamics [4] coupled to the Breit-Fermi Hamiltonian. According to the relativistic dynamics the stochastic Langevin equation for a classical relativistic particle moving in a classical non-relativistic environment acquires the form
Einstein's Brownian motion of a quantum particle in a classical environment is studied via virial and equipartition theorems. The effect of continuous measurement in a strongly dissipative environment is accounted for and a quantum generalization of the classical Einstein law of Brownian motion is obtained. A thermo-quantum Smoluchowski diffusion equation is derived via a generalization of the Madelung quantum hydrodynamics. The latter is applied for description of the quantum tunneling at equilibrium and stationary states as well as of the motion of an electron in metals, i.e. the Smoluchowski-Poisson problem.
Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.
The quantum Brownian motion of a particle in an external potential V(x) is treated using the master equation for the Wigner distribution function W(x, p, t) in phase space (x, p). A heuristic method of determination of diffusion coefficients in the master equation is proposed. The time evolution equation so obtained contains explicit quantum correction terms up to o(4) and in the classical limit, → 0, reduces to the Klein–Kramers equation. For a quantum oscillator, the method yields an evolution equation for W(x, p, t) coinciding with that of Agarwal (1971 Phys. Rev. A 4 739). In the non-inertial regime, by applying the Brinkman expansion of the momentum distribution in Weber functions (Brinkman 1956 Physica 22 29), the corresponding semiclassical Smoluchowski equation is derived.
We generalize the classical theory of Brownian motion so as to reckon with non-Markovian effects on both Klein-Kramers and Smoluchowski equations. For a free particle and a harmonic oscillator, it is shown that such non-Markovian effects account for the differentiability of the Brownian trajectories as well as the breakdown of the energy equipartition of statistical mechanics at short times in some physical situations. This non-Markovian approach is also extended to look at anomalous diffusion. Next, we bring in the dynamical-quantization method for investigating open quantum systems, which does consist in quantizing the classical Brownian motion starting directly from our non-Markovian Klein-Kramers and Smoluchowski equations, without alluding to any model Hamiltonian. Accordingly, quantizing our non-Markovian Klein-Kramers in phase space gives rise to a non-Markovian quantum master equation in configuration space, whereas quantizing our non-Markovian Smoluchowski equation in configuration space leads to a non-Markovian quantum Smoluchowski equation in phase space. In addition, it is worth noticing that non-Markovian quantum Brownian motion takes place in presence of a generic environment (e.g. a non-thermal quantum fluid). As far as the special case of a heat bath comprising of quantum harmonic oscillators is concerned, a non-Markovian Caldeira-Leggett master equation and a thermal quantum Smoluchowski equation are derived and extended to bosonic and fermionic heat baths valid for all temperatures.
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.