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arXiv:2401.01239

Phase space maximal entropy random walk: Langevin-like ensembles of physical trajectories

Jarek Duda

Published Jan 2, 2024• 0 citations
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Abstract

As written by statistician George Box "All models are wrong, but some are useful", standard diffusion derivation or Feynman path ensembles use nonphysical infinite velocity/kinetic energy nowhere differentiable trajectories - what seems wrong, might be only our approximation to simplify mathematics. This article introduces some basic tools to investigate this issue. To consider ensembles of more physical finite velocity trajectories, we can work in $(x,v)$ phase space like in Langevin equation with velocity controlling spatial steps, here also controlled with spatial potential $V(x)$. There are derived and compared 4 approaches to predict stationary probability distributions: using Boltzmann ensemble of steps/points in space (GRW - generic random walk) or in phase space (psGRW), and analogously Boltzmann ensemble of paths in space (MERW - maximal entropy random walk) and in phase space (psMERW), also generalized to L{é}vy flights. Path ensembles generally have much stronger Anderson-like localization, MERW has stationary distribution exactly as quantum ground state. Proposed novel MERW in phase space has some slight differences, which might be distinguished experimentally. For example for 1D infinite potential well: $ρ=1$ stationary distribution for step ensemble, $ρ\sim \sin^2$ for path ensemble (as in QM), and $ρ\sim \sin$ for proposed smooth path ensembles - more frequently approaching the barriers due to randomly gained velocity.

DOI
10.48550/arxiv.2401.01239

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As written by statistician George Box "All models are wrong, but some are useful", standard diffusion derivation or Feynman path ensembles use nonphysical infinite velocity/kinetic energy nowhere differentiable trajectories - what seems wrong, might be only our approximation to simplify mathematics. This article introduces some basic tools to investigate this issue. To consider ensembles of more physical finite velocity trajectories, we can work in (x,v)(x,v)(x,v) phase space like in Langevin equation with velocity controlling spatial steps, here also controlled with spatial potential V(x)V(x)V(x). There are derived and compared 4 approaches to predict stationary probability distributions: using Boltzmann ensemble of steps/points in space (GRW - generic random walk) or in phase space (psGRW), and analogously Boltzmann ensemble of paths in space (MERW - maximal entropy random walk) and in phase space (psMERW), also generalized to L{é}vy flights. Path ensembles generally have much stronger Anderson-like localization, MERW has stationary distribution exactly as quantum ground state. Proposed novel MERW in phase space has some slight differences, which might be distinguished experimentally. For example for 1D infinite potential well: ρ=1ρ=1ρ=1 stationary distribution for step ensemble, ρ∼sin⁡2ρ\sim \sin^2ρ∼sin2 for path ensemble (as in QM), and ρ∼sin⁡ρ\sim \sinρ∼sin for proposed smooth path ensembles - more frequently approaching the barriers due to randomly gained velocity.
We describe particles in a potential by a special diffusion process, the maximal entropy random walk (MERW) on a lattice. Since MERW originates in a variational problem, it shares the linear algebra of Hilbert spaces with quantum mechanics. The Born rule appears from measurements between equilibrium states in the past and the same equilibrium states in the future. Introducing potentials by the observation that time, in a gravitational field running in different heights with a different speed, MERW respects the rule that all trajectories of the same duration are counted with equal probability. In this way, MERW allows us to derive the Schrödinger equation for a particle in a potential and the Darwin term of the nonrelativistic expansion of the Dirac equation. Finally, we discuss why quantum mechanics cannot be simply a result of MERW, but, due to the many analogies, MERW may pave the way for further understanding.
There are mainly used two basic approaches for probabilistic modeling of motion: stochastic in which the object literally makes succeeding random decisions using arbi-trarily chosen by us probabilities or ergodic in which we usually assume some chaotic classical evolution and probabilities appear while averaging over infinite trajectories. Both approaches assume we know the exact way the system evolves. In contrast, in this paper we will focus on thermodynamical motion models: assuming maximal uncertainty. Specifically, in the space of possible choices of transition probabil-ities, we take the optimizing entropy or free energy one. Equivalent condition appears to be calculating transition probabilities as proportions between single steps in canonical ensemble of trajectories going through a given point. It makes that these probabilities depend on the whole space- the walker cannot directly use them. This model is ther-modynamical: only we use it to predict the most probable behavior. Standard diffusion models like Brownian motion can be seen as obtained by locally maximizing uncertainty. For regular space it agrees with fully maximizing entropy choice of transition probabili-
Surprisingly, the natural looking random walk leading to Brownian motion occurs to be often biased in a very subtle way: emphasizing some possibilities by only approximating maximal uncertainty principle. A new philosophy of stochastic modelling has been recently introduced, in which we use the only maximizing entropy choice of transition probabilities instead. Local behaviour of both approaches is similar, but they usually lead to completely different global situations. In contrast to Brownian motion leading to nearly uniform stationary density, this recent approach turns out in agreement with having strong localization properties, thermodynamical predictions of quantum mechanics, like thermalizing to dynamical equilibrium state of probability density as the quantum ground state: squares of coordinates of the lowest energy eigenvector of the Bose-Hubbard Hamiltonian for single particle in discrete case, or of the standard Schrödinger operator while including potential and making infinitesimal limit.
Same quantum density, different hidden electron paths - diffusion-like (Nelson/MERW), smooth with velocity memory, Bohm, psMERW. Diffusion-like paths give imaginary-time correlations and kinetic energy from increments, no smooth path can (TRK slope) - but smooth paths are valid samplers of response via Dalgarno-Lewis. Larmor clock on tunnelling Rb atoms: residence of stochastic paths agrees with weak value and measured decrease at lower energy if velocity memory << traversal, longer memory goes continuously to Bohm, 12-64% too long. psMERW splittings -0.956 ω/Δ from E_n-(D/2Δ)<V''>, at Compton gap it is minus Darwin term - excluded by Lamb shift. Code and data included.
In recent years, statistical physics tools have been used to study the dynamics of macroscopic quantum and classical integrable models. These studies have shown that, in the thermodynamic limit, long term time-averages of local observables equal statistical averages calculated with a Generalized Gibbs Ensemble (GGE). In the latter, the weight in phase space involves all constants of motion multiplied by Lagrange multipliers which are fixed by their initial values. We argue that the GGE can be interpreted as a proper Boltzmann measure of a different (non-standard) mechanical model. This measure can therefore be sampled with stochastic methods implemented, for example, as a Langevin process. We tailor the deterministic force in the Langevin equations so that at long times the stochastic trajectories sample the desired measure. We use the integrable Neumann Model -- the model of a classical particle constrained to move on a sphere under anisotropic harmonic potentials -- as a benchmark. We test the method in a setup where the Lagrange multipliers are fixed by the GGE governing the asymptotic dynamics of the isolated Neumann model following a quench. This problem has different phases depending on the initial conditions and quench protocol and, in all cases, with the stochastic evolution of an adequately tuned Langevin process we reproduce the asymptotic values of relevant observables.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.