A branching random walk algorithm for many-body Wigner equations and its numerical applications for quantum dynamics in phase space are proposed and analyzed in this paper. Using an auxiliary function, the truncated Wigner equation and its adjoint form are cast into integral formulations, which can be then reformulated into renewal-type equations with probabilistic interpretations. We prove that the first moment of a branching random walk is the solution for the adjoint equation. With the help of the additional degree of freedom offered by the auxiliary function, we are able to produce a weighted-particle implementation of the branching random walk. In contrast to existing signed-particle implementations, this weighted-particle one shows a key capacity of variance reduction by increasing the constant auxiliary function and has no time discretization errors. Several canonical numerical experiments on the 2D Gaussian barrier scattering and a 4D Helium-like system validate our theoretical findings, and demonstrate the accuracy, the efficiency, and thus the computability of the proposed weighted-particle Wigner branching random walk algorithm.
A branching random walk algorithm for many-body Wigner equations and its numerical applications for quantum dynamics in phase space are proposed and analyzed in this paper. Using an auxiliary function, the truncated Wigner equation and its adjoint form are cast into integral formulations, which can be then reformulated into renewal-type equations with probabilistic interpretations. We prove that the first moment of a branching random walk is the solution for the adjoint equation. With the help of the additional degree of freedom offered by the auxiliary function, we are able to produce a weighted-particle implementation of the branching random walk. In contrast to existing signed-particle implementations, this weighted-particle one shows a key capacity of variance reduction by increasing the constant auxiliary function and has no time discretization errors. Several canonical numerical experiments on the 2D Gaussian barrier scattering and a 4D Helium-like system validate our theoretical findings, and demonstrate the accuracy, the efficiency, and thus the computability of the proposed weighted-particle Wigner branching random walk algorithm.
A branching process treatment for the nonlocal Wigner pseudo-differential operator and its numerical applications in quantum dynamics is proposed and analyzed. We start from the discussion on two typical truncations of the nonlocal term, i.e., the k-truncated and y-truncated models. After introducing an auxiliary function γ(x), the (truncated) Wigner equation is reformulated into the integral formulation as well as its adjoint correspondence, both of which can be regarded as the renewal-type equations and have transparent stochastic interpretation. We prove that the moment of a branching process happens to be the solution for the adjoint equation, which connects rigorously the Wigner quantum dynamics to the stochastic branching process, and thus a sound mathematical framework for the Wigner Monte Carlo methods is established. Within the framework, the branching process for the y-truncated model recovers the popular signed particle Monte Carlo method which needs a discretization of the momentum space beforehand, whereas the k-truncated branching method allows a continuous momentum sampling. More interestingly, with the help of the rigorous connection, we find that a constant auxiliary function γ(x) ≡ γ0 performs better from the point of view of both theoretical and numerical aspects. In theory, we obtain a simple but analytical calculation formula for the growth rate of particle number, i.e., e2γ0t with t being the time. In numerics, the constant auxiliary function may facilitate the random sampling from the exponential distribution and improve the accuracy. Typical numerical experiments on the Gaussian barrier scattering validate our theoretical findings, as well as demonstrate the accuracy, the efficiency and thus the computability of the Wigner branching process.
The Signed Particle Formulation provides a particle-based interpretation of quantum mechanics in phase space, where quantum dynamics are represented through the creation and evolution of signed particles. A central computational challenge in this framework is the evaluation of the Wigner kernel, which generally involves highly oscillatory integrals and can become computationally demanding in time-dependent simulations. This paper proposes an analytical approximation of the Wigner kernel for one-dimensional single body quantum systems by exploiting a series-based representation of the potential function. The resulting expression provides an efficient way to approximate the Wigner kernel and the associated Gamma function, which governs the particle-generation process in the Signed Particle Formulation framework. The proposed approximation is evaluated for several Gaussian-based potential profiles, including single, double, triple, and quadruple Gaussian potentials. Numerical comparisons between the approximated and directly computed Wigner kernels and Gamma functions show that the proposed method captures the main behavior of the exact quantities while significantly reducing the computational cost. These results indicate that the proposed approximation can serve as an efficient computational component for scalable Signed Particle Formulation based quantum simulations.
Abstract We present a simple method of discretely modeling solutions of the classical wave and Klein-Gordon equations using variations of random walks on a graph. Consider a collection of particles executing random walks on an undirected bipartite graph embedded in RD R D at discrete times Z Z , and assume those walks are “heat-like”, in the sense that a (conserved) density of particles obeys the heat equation in the continuum limit. If the particles possess a binary degree of freedom (so that they may be said to be either particles or “antiparticles”, i.e., “positive” or “negative), then there exist closely related branching random walks on the same graph that are “wave-like”, in the sense that their (also conserved) net density obeys the D-dimensional classical wave equation. Such wave-like paths can be generated even on random graphs. The transformation by which the heat-like random walks become wave-like branching random walks is as follows: at every time step, any “incoming” particle arriving at any node X of the graph along some edge eX creates a particle-antiparticle pair at that node, with the stipulation that the newly created particle with the opposite sign of the incoming particle must initially step (in “Huygens” back-propagation fashion) along eX in the reverse direction, while the other two take a step (in “Brownian” fashion) along any edge of X (including possibly eX) with equal likelihood, and chosen independently. An additional degree of freedom (resulting in “bra” and “ket” particles) leads to quasi-probability densities proportional to the square of the wave functions.
Abstract We present a simple method of discretely modeling solutions of the classical wave and Klein-Gordon equations using variations of random walks on a graph. Consider a collection of particles executing random walks on an undirected bipartite graph embedded in RD at discrete times Z, and assume those walks are “heat-like”, in the sense that a (conserved) density of particles obeys the heat equation in the continuum limit. If the particles possess a binary degree of freedom (so that they may be said to be either particles or “antiparticles”, i.e., “positive” or “negative), then there exist closely related branching random walks on the same graph that are “wave-like”, in the sense that their (also conserved) net density obeys the D-dimensional classical wave equation. Such wave-like paths can be generated even on random graphs. The transformation by which the heat-like random walks become wave-like branching random walks is as follows: at every time step, any “incoming” particle arriving at any node X of the graph along some edge eX creates a particle-antiparticle pair at that node, with the stipulation that the newly created particle with the opposite sign of the incoming particle must initially step (in “Huygens” back-propagation fashion) along eX in the reverse direction, while the other two take a step (in “Brownian” fashion) along any edge of X (including possibly eX) with equal likelihood, and chosen independently. An additional degree of freedom (resulting in “bra” and “ket” particles) leads to quasi-probability densities proportional to the square of the wave functions.
Recent advances in dynamical low-rank approximation (DLRA) have demonstrated its effectiveness in high-dimensional simulations. However, existing DLRA algorithms still face significant challenges when handling systems that involve complex collision terms, including the pseudo-differential operator (Ψ) in the Wigner equation, a representative operator characterized by nonlocality. It is deserving to carry out a series of works to develop the DLRA algorithms for solving the Wigner equation. As the first step in this series of works, we propose an efficient DLRA algorithm for the Wigner equation, using a separable decomposition of the difference potential. We combine this separable assumption with two often-used truncations of Ψ, namely K-truncation and Y-truncation, to obtain a kind of separated representation of Ψ. Complexity analysis and several challenging experiments, including harmonic oscillators, Gaussian barrier scattering, electron-electron scattering, and a Helium-like system, all of which satisfy the separable assumption, confirm that the proposed DLRA algorithm has significant advantages, achieving a reduction in computational effort by one to two orders of magnitude in both runtime and memory requirements compared to the full-grid approach. It is worth noting that, even in the absence of a predetermined low-rank structure for the solution, DLRA can still serve as a numerical scheme that balances efficiency and accuracy.
In the standard formulation of quantum mechanics states of a system are described by wave functions or density operators . However quantum description of the system can be given in many other ways , for example, in Wigner-Moyal or Feynman formulations . All these representation are equivalent in the the sense of physical results, but the form of presentation of quantum mechanics is different as wave functions, Wigner-Liouville functions or other representations of density operators differ in essential way from each other. In this work we are dealing with Wigner and probability representation of quantum mechanics. The probability representations or tomography representation was recently proposed in terms of marginal distribution functions (MDF). MDF describing the quantum states are positive distribution functions connected with wave functions or density matrices by known integral transformation. Sign conservation of the of the MDF can be valuable in computer simulations to overcome the "sign problem". Important advantage of the probability representation is that the quantum transitions between quantum states are descried by non-negative probabilities. This work is devoted to the development of a new stochastic approaches to numerical solution of the evolution equations for Wigner-Liouviile (WL) and marginal distribution functions. To solve the WL equation we combine both molecular dynamics and Monte Carlo methods and compute traces of the dynamical operators. The results obtained for a system of electrons and random scatterers clearly demonstrate that the many-particle interaction between the electrons leads to an enhancement of the conductivity compared to the case of noninteracting electrons. To obtain evolution of MDF we developed new stochastic approach to solution of the generalized Langevin equation (GLE). GLE is derived from Kolmogorov equations for Green function of evolution equation for MDF. We discuss the basic relations and main ideas of this approach, compare obtained numerical results with results of independent finite-difference calculations for quantum oscillator and quantum particles crossing the finite well and tunneling through the Gaussian barrier.