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.
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.
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.
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.
Abstract We build probabilistic process models of motions on a finite graph. For particle statistics, at first we select a particle. Then we build a path of the particle along consecutive edges. Particles are distinguished. Single particles perform Markov chains. For Bose statistics, at first we select an edge as request for a path of a particle along the edge. Particles are not distinguished. The edges are randomly selected by probability weights, or they are triggered by circuits of courses in small steps on circles. We present examples for these process models. We obtain some results, which follow from the balance equation, a refinement of the corresponding balance equation for Markov chains. Quantum mechanics inhibits these process models, because there the annihilation of the zero state is zero.
A different approach to wave mechanics is presented in accordance with de Broglie’s original hypothesis applying the case of photons to all material particles. It derives propagating matter waves conceptually from a theory of evolution, and not by the formal setting of eigenvalue equations. The quality of explanation is at issue, and not the mere description of phenomena. A monoenergetic particle transport along two directions is described by a partial differential equation and by a random walk model. The dual description is applied to the particle picture and the wave picture; it leads with identical initial values to identical causal and timelike solutions in either picture. The partial differential equations are a Telegrapher equation in 1‐d space and its analytic continuation, a modified Klein Gordon equation; the solutions represent a density distribution and an amplitude profile, respectively. However, only the corresponding random walk models — equivalent to Feynman’s integral over all paths — derive the solutions as path‐end distributions with the additional information about the flight direction. This allows following up momentum dissipation along two directions by means of two particle beams and their profiles. Thereby the total number of particles is divided up onto the two beams according to linear or squared fractions depending on the beam configuration putting up a 1‐d or a 2‐d space. This aspect escapes conventional descriptions but serves to describe the transport in the particle or in the wave picture in dependence on the knowledge/ignorance of the particle’s flight direction.
The theory of Schr¨ odinger bridges for diffusion processes is extended to discrete-time Markov chains, and to some problems for quantum discrete-time processes. Taking into account the past-future lack of symmetry of the discrete-time setting, results bear a striking resemblance to the classical ones. In particular, the solution of the path space maximum entropy problems is always obtained from the model by means of a suitable multiplicative functional transformation. Index Terms—Markov chain, maximum entropy problem, Schr¨ odinger bridge, time reversal evolution, space-time har- monic function, quantum operation. odinger considered, and basically solved, the following abstract probabilistic problem: Suppose a large number N of independent Brownian particles have been observed to have density ρ0(x) at time t0 and density ρ1(x) at some later time t1. Suppose the latter density considerably differs from what is predicted by the law of large numbers. It is apparent that the particles have been transported in an unlikely way. But of the many unlikely ways in which this could have happened, which one is the most likely? In modern terminology, this is a problem of large deviations of the empirical distribution. Using a coarse graining approach, Schrcomputed the most likely endpoint distribution under the prior tran- sition density of the Brownian motion p(s,x,t,y) and with the prescribed marginals. It turned out that the solution, namely the bridge from ρ0 to ρ1 over Brownian motion, has at each time a density q that factors as q(x,t )= ϕ(x,t)ˆ ϕ(x,t), where ϕ and ˆ ϕ are, in the language of Doob, a p-harmonic and a p-coharmonic functions, respectively. The existence and uniqueness of such a pair (ϕ, ˆ ϕ) satisfying the factorization above and the boundary conditions was guessed by Schron the basis of his intuition. He was later shown to be quite right in various degrees of generality by Fortet (9), Beurling (4), Jamison (11), F¨
We study the one-dimensional time-independent discrete Dirac equation on graphs with discrete edges. Both continuous and discrete formulations of the graphs are considered, with particular focus on star graphs with three edges as a simple example. Exact solutions for eigenstates and spectra are derived and compared with the continuum case, showing good agreement in tabulated results. This approach offers a systematic framework for analyzing relativistic quantum transport on branched structures relevant to nanostructures and quantum device design.