A stochastic approach to generalized quantum mechanics with many-body long-range forces

A. Lavagno

Ying Oon Tan

Previously [2004, AIP Conf. Proc., 750, 361], we considered a stochastic mechanics in the form of a stochastic differential equation mdxdt = μ(t) where μ(t) is a stochastic process defined by the set of Bohmian momentum time histories from an ensemble of particles. In this paper, we consider another stochastic differential equation d2xdt2 = 1m[−dV(x)dx + ζ(t)] where the stochastic force ζ(t) is a stochastic process defined by the set of quantum force time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process characterized by an n‐th order joint probability density in the form of products of delta functions, then the stochastic mechanics with the stochastic force ζ(t) is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non‐purely random process, we show that the stochastic mechanics with the stochastic force ζ(t) is not equivalent to quantum mechanics. Therefore, this stochastic mechanics is also generally not equivalent to quantum mechanics.

Walter Lehr

S. Oryu

On the three-body kinematics, we investigate the threshold behavior which appears not only at the three-body break-up threshold (3BT), but also at the quasi two-body threshold (Q2T) for the reactions: $$A+(BC)\rightarrow A+B+C$$ , and $$(ABC)\rightarrow A+(BC)$$ , respectively. Recently, the author proposed a general particle $$ {transfer}$$ (GPT) potential which appears, not only at the 3BT, but also at the Q2T between A and (BC). The new potential indicates a Yukawa-type potential for short range, but a $$1/r^n$$ -type potential for long range. The long range part of the GPT potential for $$n=1$$ indicates an attractive Coulomb-like or a gravitation-like potential. While, $$n=2$$ indicates the Efimov-like potential between A and (BC). The three-body binding energy: $$E_n=\epsilon +\zeta _n$$ with the two-body binding energy $$\epsilon $$ , and the separation energy $$\zeta _n$$ for $$(ABC)\rightarrow A+(BC)$$ satisfies $$E_n/E_{n+1}=\zeta _n/\zeta _{n+1}$$ =const for $$\epsilon =0$$ or the two-body scattering length: $$a\rightarrow \infty $$ (i.e. the two-body unitary limit). At the Q2T, the condensation of the three-body binding energy is given by the GPT-potential in the form of $$E_n/E_{n+1}=(\zeta _n+\epsilon )/(\zeta _{n+1}+\epsilon )\rightarrow 1$$ (const) for $$n\rightarrow \infty $$ (with $$\zeta _n\rightarrow 0$$ ) which implies the existence of Efimov-like states at the Q2T in the hadron systems, thereby the possibility of “ultra low energy nuclear transformation”, where the three-body binding energies degenerate at zero energy. Finally, the origin of such a long range potential will be clarified.

Zi Cai, Claudius Hubig, Ulrich Schollwöck

Understanding the collective dynamics in many-particle systems is of important significance in nonequilibrium physics. In this paper, we systematically study the long-time behavior in stochastically driven interacting quantum systems. We find that even though the stochastic forces will inevitably drive a system into a featureless steady state, the asymptotic behavior to approach the steady state can be highly nontrivial and exhibits rich universal dynamics due to the interplay between the stochastic driving and quantum many-body effects. Special attention is devoted to the effect of the mode-coupling perturbations, which may have important consequence for the long-time dynamics of the stochastically driven quantum many-body systems.

Mikio Namiki, Satoshi Tanaka