A universal master equation for the gravitational violation of quantum mechanics

In this paper we propose a master equation which contains a damping term universally violating the quantum mechanics of massive systems. From this equation follows that the quantum mechanical superposition principle breaks down if the states have radically different mass distributions. The damping of the coherence appears on a reasonable scale of masses and distances.     

Quantum statistical mechanics on stochastic phase space

Within the context of the theory of stochastic phase spaces, introduced in some earlier papers, a systematic mathematical procedure is developed for expressing quantum mechanical observables as generalized functions on a stochastic phase space. The states in such a theory are normalized, positive semidefinite, continuous functions of the phase space variables, satisfying marginality conditions appropriate to the stochastic nature of the underlying phase space. The action of a general quantum mechanical observable on the state space is then shown to lead in general to formal differential operators of finite or infinite order. Explicit computations of some typical operators are made to illustrate the theory. As a useful practical application, the theory is employed to derive a Bloch equation from which the Husimi transform of the canonical equilibrium state is then computed, after expressing it as an infinite series in powers of .Supported in part by a research grant from the National Research Council of Canada.     

Tomographic representation of quantum mechanics and statistical physics

The tomographic representation for operators dependent of continuous variables, as an example of star-product quantization, and the relationship between the tomographic representations of quantum mechanics and classical statistical mechanics are discussed.     

A causal multifractal stochastic equation and its statistical properties

Multiplicative cascades have been introduced in turbulence to generate random or deterministic fields having intermittent values and long-range power-law correlations. Generally this is done using discrete construction rules leading to discrete cascades. Here a causal log-normal stochastic process is introduced; its multifractal properties are demonstrated together with other properties such as the composition rule for scale dependence and stochastic differential equations for time and scale evolutions. This multifractal stochastic process is continuous in scale ratio and in time. It has a simple generating equation and can be used to generate sequentially time series of any length.Received: 15 April 2003, Published online: 23 July 2003PACS: 02.50.Ey Stochastic processes - 05.45.Df Fractals - 47.27.Eq Turbulence simulation and modeling     

A generalized stochastic liouville equation. Non-Markovian versus memoryless master equations

The interrelation between the well-known non-Markovian master equation and the new memoryless one used in the previous paper is clarified on the basis of damping theory. The latter equation is generalized to include cases in which the Hamiltonian or the Liouvillian is a random function of time, and is written in a form feasible for perturbational analysis. Thus, the existing stochastic theory in which those cases mentioned above are discussed is equipped with a more tractable basic equation. Two problems discussed in the previous paper, i.e., the random frequency modulation of a quantal oscillator and the Brownian motion of a spin, are treated from the viewpoint of the stochastic theory without such explicit consideration of external reservoirs as was taken in the previous paper.     

A stochastic derivation of the multivariate master equation describing reaction diffusion systems

We derive the multivariate master equation describing reaction diffusion systems from a discrete form master equation in phase space, assuming that the elastic collisions of the chemically active substances with the inert carrier gas have relaxed. In this state of collisional equilibrium the stochastic operator modelling the displacement of the particles between spatial cells reduces to the random wall operator and the reactive collision term yields the usual birth and death operator. Correlation functions are derived and their validity is discussed.     

Stochastic phase spaces and master Liouville spaces in statistical mechanics

The concept of probability space is generalized to that of stochastic probability space. This enables the introduction of representations of quantum mechanics on stochastic phase spaces. The resulting formulation of quantum statistical mechanics in terms of -distribution functions bears a remarkable resemblance to its classical counterpart. Furthermore, both classical and quantum statistical mechanics can be formulated in one and the same master Liouville space overL ²(). A joint derivation of a classical and quantum Boltzman equation provides an illustration of the practical uses of these formalisms.Supported in part by an NRC grant.     

Tomographic representation of kinetic equations in classical statistical mechanics

A tomographic representation of kinetic equations is constructed using the Radon transform. Liouville’s equation is considered for one and many particles. The reduced Liouville’s equation is obtained in the tomographic representation and the Bogolyubov chain is investigated in this representation. An example of the relativistic kinetic equation in the tomographic representation is considered.     

小系统的非平衡统计力学与随机热力学

热力学是一个古老的课题,古典热力学以宏观的具有大粒子数的系统为研究对象,自17世纪以来,科学家们构建了热力学的完备公理化体系。将热力学推广至小系统是近三十年来的研究前沿。文章介绍小系统的非平衡统计力学以及小系统的随机热力学。作为研究案例,利用时间依赖的谐振子势场控制单个粒子来构造随机热机的类卡诺循环,并发现该热机最大功率对应的效率等于1-,其中T_c和T_h分别对应于低温热库和高温热库的温度。     

Functional representation of a non-Markovian probability distribution in statistical mechanics

A path-integral representation of the statistical probability density function for a generalized non-linear and non-markovian Langevin equation is obtained.     

Poisson measure representation and cluster expansion in classical statistical mechanics

A new representation for distribution functions of the grand canonical ensemble by the Poisson measure functional integral is obtained. Due to the ultralocal nature of the measure, the construction of the cluster expansion is very simple. For the convergence of the cluster expansion, the requirement of exponential decay of the interaction potential is not necessary.Research partially supported by Ukrainian State Committee of Science and Technology     

An equation for the quantum dissipative system in statistical mechanics

A formalism for describing quantum dissipative systems in statistical mechanics is developed. A new equation of the Lindblad type with a quadratic superoperator consisting of Hermitian dissipative operators is derived from the Bloch equation for temperature density matrix using the Feynman integral over the trajectories with a modified Menskii weight functional. By way of example, this equation is solved for a one-dimensional quantum harmonic oscillator with linear dissipation. Applying the projection operator technique, an integral-differential equation for a reduced temperature statistical operator is obtained, which is analogous to the Zwanzig equation in statistical mechanics, and its formal solution is found as a convergent series. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 30–34, December, 2006.     

Generalized boltzmann equation in a manifestly covariant relativistic statistical mechanics

We consider the relativistic statistical mechanics of an ensemble of N events with motion in space-time parametrized by an invariant historical time . We generalize the approach of Yang and Yao, based on the Wigner distribution functions and the Bogoliubov hypotheses to find approximate dynamical equations for the kinetic state of any nonequilibrium system, to the relativistic case, and obtain a manifestly covariant Boltzmann- type equation which is a relativistic generalization of the Boltzmann-Uehling-Uhlenbeck (BUU) equation for indistinguishable particles. This equation is then used to prove the H-theorem for evolution in . In the equilibrium limit, the covariant forms of the standard statistical mechanical distributions are obtained. We introduce two-body interactions by means of the direct action potential V(q), where q is an invariant distance in the Minkowski space-time. The two- body correlations are taken to have the support in a relative O(2, 1)-invariant subregion of the full spacelike region. The expressions for the energy density and pressure are obtained and shown to have the same forms (in terms of an invariant distance parameter) as those of the nonrelativistic theory and to provide the correct nonrelativistic limit.     

A convenient expansion of the master equation

We present a simple and very convenient expansion of the master equation.     

A master equation description of local fluctuations

A theory of fluctuations of macrovariables in nonequilibrium systems based on a nonlinear master equation is outlined. This equation takes into account, via a “mean field” type of approximation, the effect of the spatial extension of fluctuations. A comparison with the birth and death formalism reveals several unsatisfactory features of the latter.     

Measurement master equation

We derive a master equation describing the evolution of a quantum system subjected to a sequence of observations. These measurements occur randomly at a given rate and can be of a very general form. As an example, we analyse the effects of these measurements on the evolution of a two-level atom driven by an electromagnetic field. For the associated quantum trajectories we find Rabi oscillations, Zeno-effect type behaviour and random telegraph evolution spawned by mini quantum jumps as we change the rates and strengths of measurement.     

Solving nonlinear master equation describing quantum damping by virtue of the entangled state representation

This paper solves the newly constructed nonlinear master equation dρ/dt=κ[2f (N) aρ (1/f (N-1))a+-a+aρ-ρa+a],where f(N) is an operator-valued function of N=a+a,for describing amplitude damping channel,and derives the infinite operator sum representation of quasi-Kraus operators for the density operator.It also shows that in this nonlinear process the initial pure number state density operator will evolve into the binomial field (a mixed state) when f (N)=1/(N+1)～(1/2).