Classical and Quantum 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.

We study the diffusion of a quantum Brownian particle in a one-dimensional periodic potential with substitutional disorder. The particle is coupled to a dissipative environment, which induces a frictional force proportional to the velocity. The dynamics for arbitrary temperature is studied by using Feynman's influence-functional theory. We calculate the mobility to lowest order in the disorder and strength of the periodic potential. It is shown that for weak dissipation the linear mobility, which vanishes atT=0 due to localization effects, may exhibit a maximum and a subsequent minimum with increasing temperature. The relation to the diffusion of heavy particles in metals or doped semiconductors is briefly discussed.

Quantum Finance Models

The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein‐Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology.

In this article, using conditional power, we extended the widely used classical Brownian motion (Bm) technique for monitoring clinical trial data to a larger class of stochastic processes fractional Brownian motion (fBm) with drift, and compare these results. We applied a Bayesian method to estimate parameters. The simulation study is presented as an example to illustrate scenarios under classical and fractional Brownian motion. We demonstrated the drift fractional Brownian motion in sequential monitoring of clinical trials and provided a formula for calculating conditional power under an alternative hypothesis for interim analysis. We investigated cases commonly used in clinical trials, with the Hurst parameter H = 0.5 and H > 0.5, respectively. Our simulation study suggests that the conditional power under the fBm assumption is generally higher than that under the Bm assumption when H > 0.5 and also matches better with our empirical results.

The theory of Brownian motion as described by nonlinear Langevin equations and the corresponding Fokker–Planck equations is discussed. The general problems of the theory of nonlinear Brownian motion considered are: Brownian motion in a medium with nonlinear friction; the critical analysis of three forms of the relevant Langevin and Fokker–Planck equations (Ito's form, Stratonovich's form, and the kinetic form); the Smoluchowski equations and master equations for different cases; two methods of transition from master equation to Fokker–Planck equation; master equations for one-step processes; traditional and nontraditional definition of transition probabilities; evolution of free energy and entropy in Brownian motion; Lyapunov functionals. The following particular examples are considered: Brownian motion in self-oscillatory systems; H-theorem for the van der Pol oscillator; S-theorem; oscillator with inertial nonlinearity; bifurcation of energy of the limiting cycle; oscillator with multistable stationary states; oscillators in discrete time; bifurcations of energy of the limiting cycle and the period of oscillations; criterion of instability upon transition to discrete time, based on the H-theorem; Brownian motion of quantum atoms oscillators in the equilibrium electromagnetic field; Brownian motion in chemically reacting systems; partially ionised plasmas; the Malthus–Verhulst process.

In this chapter we study some properties of systems of coupled quadratic quantum Hamiltonians. The well known example is coupled harmonic oscillators appearing in the Quantum Brownian Motion. An open system is a physical (S) interacting with the environment (E) such that the total system \((S\cup E)\) is isolated from the rest of the universe. So we have two independent Hamiltonians \(\hat{H}_S\), \(\hat{H}_E\) and an interaction \(\hat{V}\) between (S) and (E). The main problem discussed for open systems is to understand the dynamical action of the interaction \(\hat{V}\) with the environment on the time dependent evolution of (S) which is now irreversible because a part of its energy escapes to the environment. An important physical effect of the interaction of the environment on a quantum system is decoherence which give an explanation for the mechanism driving the quantum-to-classical transition. We restrict ourself to few mathematical problems related with this very active field of research in physics [155, 200, 231, 270]. In a first part we study the purity of a state of (S) (or the linear entropy) in a short time approximation. In a second part we discuss the separability of bipartite systems. In a third part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for coupled quadratic Hamiltonians by a bilinear inter-action. We give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time t is written as a convolution integral for the reduced density at initial time \(t_0=0\), with a Gaussian kernel, for \(0 \le t t_c\) if \(t_c 0\) small enough, are proved on our simple model which is similar to a representation of the Schrodinger cat.

Brownian motion theory is always challenging how to describe diffusion phenomena with the main issue is how to extend the classical theory of Brownian motion to the special relativity framework. In this study, we formulated dynamics and distribution of a Brownian particle in relativistic framework by using Langevin and Fokker-Planck equation. By representing Brownian particle dynamics by Langevin equation, the velocity curves were dependent on the presence of viscous friction coefficient (heat bath), and were used generalized in special relativity theory, A relativistic Langevin equation reduces to the classical theory at low velocities. Likewise, the distribution of Brownian particles is represented as a stationary solution of the relativistic Fokker-Planck equation. From numerical results, we found that the probability density in the relativistic Fokker-Planck equation for was reduced to the standard Fokker-Planck equation in Netownian classical theory. For the calculation result showed that the Hanggi-Klimontovich approach has a consistent result to the relativistic Maxwell distribution. This work could open a promising interpretation to formulate the diffusion phenomena into general relativity theory.

The dynamics of a Brownian particle in a constant magnetic field and time‐dependent electric field is studied in the limit of white noise, using a Langevin approach for the classical problem and the path‐integral Feynman‐Vernon and Caldeira‐Leggett framework for the quantum problem. A first goal of this study is to use the two‐dimensional problem of a Brownian particle in a magnetic field to show that a proper re‐formulation of the oscillator model of quantum Brownian motion allows one to recover the classical limit of the dynamics correctly. Furthermore, the probability distribution in configuration space of an initial pure state represented by an asymmetrical Gaussian wave function is worked out and its general time evolution is decomposed in the superposition of basic processes: (a) the classical motion of the center of mass, (b) a rotation around the mean position, and (c) spreading processes along the principal axes.

On the basis of the Wigner unitary representations of the covering group ISL(2,C) of the Poincaré group, we obtain spin-tensor wave functions of free massive particles with arbitrary spin. The wave functions automatically satisfy the Dirac–Pauli–Fierz equations. In the framework of the two-spinor formalism we construct spin-vectors of polarizations and obtain conditions that fix the corresponding relativistic spin projection operators (Behrends–Fronsdal projection operators). With the help of these conditions we find explicit expressions for relativistic spin projection operators for integer spins (Behrends–Fronsdal projection operators) and then find relativistic spin projection operators for half integer spins. These projection operators determine the numerators in the propagators of fields of relativistic particles. We deduce generalizations of the Behrends–Fronsdal projection operators for arbitrary space–time dimensions D>2.

Nonlinear friction forces are a problem in classical Brownian motion for a long time [1]. They are described either by Langevin or by Fokker-Planck equations [2-4]. However, the rigorous generalized Langevin equation is linear, which points out that nonlinear friction forces possess a macroscopic hydrodynamic origin [1]. For this reason, they are not present in the modern quantum theory of open systems [5]. The scope of the present paper is to investigate the nonlinear friction effect on quantum mechanics. The analysis is based on a dissipative Madelung quantum hydrodynamics. Thus, the nonlinear friction is a result of nonlinear non-equilibrium thermodynamics in contrast to the traditional mechanical nonlinear friction forces. A quantum particle in vacuum is described by the Schrodinger equation

Non-Markovian expansion in quantum Brownian motion

The quantum Brownian motion of a charged particle in the electromagnetic vacuum fluctuations is investigated near a perfectly reflecting flat boundary, taking into account the smooth switching process in the measurement. Constructing a smooth switching function by gluing together a plateau and the Lorentzian switching tails, it is shown that the switching tails have a great influence on the measurement of the Brownian motion in the quantum vacuum. Indeed, it turns out that the result with a smooth switching function and the one with a sudden switching function are qualitatively quite different. It is also shown that anticorrelations between the switching tails and the main measuring part plays an essential role in this switching effect. The switching function can also be interpreted as a prototype of a nonequilibrium process in a realistic measurement, so that the switching effect found here is expected to be significant in actual applications in vacuum physics.

We consider the Kyle-Back model for insider trading, with the difference that the classical Brownian motion noise of the noise traders is replaced by the noise of a fractional Brownian motion B H with Hurst parameter \({H>\frac{1}{2}}\) (when \({H=\frac{1}{2}, B^H}\) coincides with the classical Brownian motion). Heuristically, for \({H>\frac{1}{2}}\) this means that the noise traders has some “memory”, in the sense that any increment from time t on has a positive correlation with its value at t. (In other words, the noise trading is a persistent stochastic process). It also means that the paths of the noise trading process are more egular than in the classical Brownian motion case. We obtain an equation for the optimal (relative) trading intensity for the insider in this setting, and we show that when \({H\rightarrow\frac{1}{2}}\) the solution converges to the solution in the classical case. Finally, we discuss how the size of the Hurst coefficient H influences the optimal performance and portfolio of the insider.

We consider a point particle in one dimension initially confined to a finite spatial region whose state is frequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, the state never escapes from the region—this is the Zeno effect. In the corresponding classical problem, by contrast, the state diffuses out of the region with the frequent monitoring simply removing probability. The aim of this paper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We give a general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanism which suppresses interference between histories. We show how this works explicitly in two examples involving projections onto a one-dimensional subspace and identify the key time scales for the process. We extend this understanding to our main problem of interest, the case of a particle in a spatial region, by coupling it to a decohering environment. Smoothed projectors are required to give the problem proper definition and this implies the existence of a momentum cutoff and minimum length scale. We show that the escape rate from the region approaches the classically expected result, and hence the Zeno effect is suppressed, as long as the environmentally induced fluctuations in momentum are sufficiently large. We establish the time scale on which an arbitrary initial state develops sufficiently large fluctuations to satisfy this condition. We link our results to earlier work on the → 0 limit of the Zeno effect. We illustrate our results by plotting the probability flux lines for the density matrix (which are equivalent to Bohm trajectories in the pure-state case). These illustrate both the Zeno and anti-Zeno effects very clearly, and their suppression. Our results are closely related to our earlier paper [ Phys.Rev.A 88 , 022128 ( 2013 )], demonstrating the suppression of quantum-mechanical reflection by decoherence.