Quantum stochastic trajectories: the Smoluchowski–Bohm equation

Abstract

Molecular systems are quantum systems, but the complete characterization of molecular motions within a fully quantum framework might appear to be an unfeasible task because it would require that the actual nuclear positions are established at any time. One would like to use a quantum molecular trajectory that defines the instantaneous nuclear positions and satisfies the predictions of quantum mechanics in terms of its statistical properties. Even though it can be proven that the single Bohm trajectory provides a representation of the quantum molecular trajectory, this solves the issue only on a theoretical ground: exact solutions of the Schrödinger-Bohm dynamical system are extremely computationally demanding. Therefore, we derive a stochastic equation of Smoluchowski type from the Schrödinger-Bohm dynamics, through projection operator techniques, in order to characterize the molecular motions of open quantum systems. The main quantum features of the motions emerge from the equilibrium distribution, i.e., the wave function's squared modulus integrated on the environment degrees of freedom. Furthermore, we verify the accuracy of the stochastic equation by comparing its predictions with those of the deterministic dynamics for a model system of six interacting harmonic oscillators. The indisputable advantage of this full quantum mechanical approach is that of representing the molecular dynamics, which controls important phenomena like vibrational relaxation, conformational transitions and activated processes, in a self consistent way and at the low computational cost of solving simple stochastic equations.

Quantum Stochastic Trajectories: The Fokker–Planck–Bohm Equation Driven by the Reduced Density Matrix

“…For the former case we have obtained the correlation time τ C = 1.87 2π/ω. On the other hand, the monoexponential decay of the Ornstein− Uhlenbeck process 18 with τ C =σ q S 2 /D is derived for the same autocorrelation function from the Smoluchowski−Bohm equation 38 if the normal (Gaussian) distribution of eq. 32 is used for the equilibrium probability density ρ eq S (q S ).…”

“…The objective of our theoretical analysis is the stochastic description of the Bohm coordinates for the open quantum system interacting with the environment. To this aim, the formal tools already introduced in ref and in ref are employed in order to derive the Fokker–Planck equation for the coordinates of the open quantum system. In the following three subsections, these tools are recalled before to examine the final Fokker–Planck description.…”

“…The same integration can be applied to the Liouville equilibrium distribution ρ eq Ω P ( x ) to obtain the reduced equilibrium distribution ρ eq R ( x R ). Furthermore, projection operator techniques allows the formal derivation of an independent equation for the evolution of the reduced distribution ρ R ( x R , t ) which is physically justified in the presence of a suitable time scale separation leading to a Markovian dynamics for the relevant variables. − In ref , we have described the projection procedure applied to the Liouville equation for the Schrödinger–Bohm dynamics and here we recall only the final results.…”

“…of the same equation. The former is driven by the reduced velocity field Λ R ( x R ) which is simply the set of the component of the deterministic velocity field Λ P ( x ) for the relevant variable x R after the average on the irrelevant ones: broadly speaking, Λ R ( x R ) represents the average drift for the relevant variables x R . The later term takes into account the fluctuating contributions to the dynamics of the relevant variable and, consequently, it describes the relaxation processes, such as the decay of a generic distribution ρ R ( x R , t ) toward the equilibrium distribution ρ eq R ( x R ).…”

“…By employing projection operator techniques, like Mori–Zwanzig procedures in classical mechanics, − we derive the stochastic equations for the set of relevant degrees of freedom from the Liouville representation of the Schrödinger–Bohm dynamics . By considering the open system coordinates as the only relevant variables, in ref , we have derived a Smoluchowski type equation, the so-called Smoluchowski–Bohm equation, for the distribution of the Bohm coordinates of the open system. In this framework the dynamics of the wave function does not appear explicitly besides its role as a source of the noise.…”