We propose new equations of motion under the theory of the Brownian motion to connect the states of quantum, diffusion, soliton, and periodic localization. The new equations are nothing but the classical equations of motion with two additional terms and the one of them can be regarded as the the quantum potential. By choosing a parameter space, various important states are obtained. Further, the equations contain other interesting phenomena such as general dynamics of diffusion process, collapse of the soliton, the nonlinear extension of the Schrdinger equation, and the dynamics of phase transition.
We propose new equations of motion under the theory of the Brownian motion to connect the states of quantum, diffusion, soliton, and periodic localization. The new equations are nothing but the classical equations of motion with two additional terms and the one of them can be regarded as the the quantum potential. By choosing a parameter space, various important states are obtained. Further, the equations contain other interesting phenomena such as general dynamics of diffusion process, collapse of the soliton, the nonlinear extension of the Schrdinger equation, and the dynamics of phase transition.
We characterize the pointer states generated by the master equation of quantum Brownian motion and derive stochastic equations for the dynamics of their trajectories in phase space. Our method is based on a Poissonian unraveling of the master equation whose deterministic part exhibits soliton-like solutions that can be identified with the pointer states. In the semiclassical limit, their equations of motion in phase space turn into those of classical diffusion, yielding a clear picture of the quantum-classical transition induced by environmental decoherence.
We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of additional terms, for various polynomials forms of the coupling. We discuss the cases of linear and quadratic coupling in great detail and we derive, using Wigner function techniques, the stationary solutions of the master equation for a Brownian particle in a harmonic trapping potential. We predict quite generally Gaussian stationary states, and we compute the aspect ratio and the spread of the distributions. In particular, we find that these solutions may be squeezed (superlocalized) with respect to the position of the Brownian particle. We analyze various restrictions to the validity of our theory posed by non-Markovian effects and by the Heisenberg principle. We further study the dynamical stability of the system, by applying a Gaussian approximation to the time-dependent Wigner function, and we compute the decoherence rates of coherent quantum superpositions in position space. Finally, we propose a possible experimental realization of the physics discussed here, by considering an impurity particle embedded in a degenerate quantum gas.
Quantum Brownian motion in a periodic cosine potential is studied and a simple estimate of the tunneling effect is obtained in the frames of a semiclassical quasi-equilibrium approach. It is shown that the latter is applicable for heavy particles but electrons cannot be described properly since the quantum effects dominate over the thermal ones. The electron purely quantum diffusion is investigated at zero temperature and demonstrates that electrons do not obey the classical Einstein law of Brownian motion in the fi eld of periodic potentials, since the dispersion of the Gaussian wave packet increases logarithmically in time. Diffusion in the fi eld of periodic potentials is important for particle transport via ordered structures, typical in condensed matter. The underlying mechanism is Brownian motion. If the diffusing particles are quantum ones, e.g. electrons, protons, light atoms, etc., their migration is described via the theory of quantum Brownian motion [1, 2], which is signifi cantly affected by the tunneling effect [3]. Hereafter an alternative approach to the problem of quantum Brownian motion in periodic potentials is developed based on the Bohm quantum potential [4]. In contrast to the WKB theory valid in vacuum the present paper describes tunneling in a dissipative environment. The starting point is a nonlinear quantum Smoluchowski-like equation [5, 6] 0 [ ( ) ] / t x x b x V Q d b β ∂ ρ = ∂ ρ∂ + β + ∂ ρ β ∫ (1)
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 study of Schrödinger equation based on a hypothesis that every particle must move randomly in a quantum-sized volume has been done. In addition to random motion, every particle can do relative motion through the movement of its quantum-sized volume. On the other way these motions can coincide. In this proposed model, the random motion is one kind of intrinsic properties of the particle. The every change of both speed of randomly intrinsic motion and or the velocity of translational motion of a quantum-sized volume will represent a transition between two states, and the change of speed of randomly intrinsic motion will generate diffusion process or Brownian motion perspectives. Diffusion process can take place in backward and forward processes and will represent a dissipative system. To derive Schrödinger equation from our hypothesis we use time operator introduced by Nelson. From a fundamental analysis, we find out that, naturally, we should view the means of Newton’s Law F⃗ = ma⃗ as no an external force, but it is just to describe both the presence of intrinsic random motion and the change of the particle energy.
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.