Expansion Of Phase Space Research Articles

Numerical simulation of large-scale nonlinear open quantum mechanics

We introduce a method to solve nonlinear open quantum dynamics of a particle in situations where its state undergoes significant expansion in phase space while generating small quantum features at the phase-space Planck scale. Our approach involves simulating two steps. First, we transform the Wigner function into a time-dependent frame that leverages information from the classical trajectory to efficiently represent the quantum state in phase space. Next, we simulate the dynamics in this frame using a numerical method that implements this time-dependent nonlinear change of variables. To demonstrate the capabilities of our method, we examine the open quantum dynamics of a particle evolving in a one-dimensional weak quartic potential after initially being ground-state cooled in a tight harmonic potential. This approach is particularly relevant to ongoing efforts to design, optimize, and understand experiments targeting the preparation of macroscopic quantum superposition states of massive particles through nonlinear quantum dynamics. Published by the American Physical Society 2024

Contrasting the Implicit Method in Incoherent Lagrangian and the Correction Map Method in Hamiltonian

The equations of motion for a Lagrangian mainly refer to the acceleration equations, which can be obtained by the Euler–Lagrange equations. In the post-Newtonian Lagrangian form of general relativity, the Lagrangian systems can only maintain a certain post-Newtonian order and are incoherent Lagrangians since the higher-order terms are omitted. This truncation can cause some changes in the constant of motion. However, in celestial mechanics, Hamiltonians are more commonly used than Lagrangians. The conversion from Lagrangianto Hamiltonian can be achieved through the Legendre transformation. The coordinate momentum separable Hamiltonian can be computed by the symplectic algorithm, whereas the inseparable Hamiltonian can be used to compute the evolution of motion by the phase-space expansion method. Our recent work involves the design of a multi-factor correction map for the phase-space expansion method, known as the correction map method. In this paper, we compare the performance of the implicit algorithm in post-Newtonian Lagrangians and the correction map method in post-Newtonian Hamiltonians. Specifically, we investigate the extent to which both methods can uphold invariance of the motion’s constants, such as energy conservation and angular momentum preservation. Ultimately, the results of numerical simulations demonstrate the superior performance of the correction map method, particularly with respect to angular momentum conservation.

Convolutional Neural Network for Individual Identification Using Phase Space Reconstruction of Electrocardiogram

Electrocardiogram (ECG) biometric provides an authentication to identify an individual on the basis of specific cardiac potential measured from a living body. Convolutional neural networks (CNN) outperform traditional ECG biometrics because convolutions can produce discernible features from ECG through machine learning. Phase space reconstruction (PSR), using a time delay technique, is one of the transformations from ECG to a feature map, without the need of exact R-peak alignment. However, the effects of time delay and grid partition on identification performance have not been investigated. In this study, we developed a PSR-based CNN for ECG biometric authentication and examined the aforementioned effects. Based on a population of 115 subjects selected from the PTB Diagnostic ECG Database, a higher identification accuracy was achieved when the time delay was set from 20 to 28 ms, since it produced a well phase-space expansion of P, QRS, and T waves. A higher accuracy was also achieved when a high-density grid partition was used, since it produced a fine-detail phase-space trajectory. The use of a scaled-down network for PSR over a low-density grid with 32 × 32 partitions achieved a comparable accuracy with using a large-scale network for PSR over 256 × 256 partitions, but it had the benefit of reductions in network size and training time by 10 and 5 folds, respectively.

Quantum Kolmogorov-Sinai entropy and Pesin relation

We discuss a quantum Kolmogorov-Sinai entropy defined as the entropy production per unit time resulting from coupling the system to a weak, auxiliary bath. The expressions we obtain are fully quantum, but require that the system is such that there is a separation between the Ehrenfest and the correlation timescales. We show that they reduce to the classical definition in the semiclassical limit, one instance where this separation holds. We show a quantum (Pesin) relation between this entropy and the sum of positive eigenvalues of a matrix describing phase-space expansion. Generalizations to the case where entropy grows sublinearly with time are possible.

S-matrix formulation of thermodynamics with N-body scatterings

We apply a phase space expansion scheme to incorporate the N-body scattering processes in the S-matrix formulation of statistical mechanics. A generalized phase shift function suitable for studying the thermal contribution of N rightarrow N processes is motivated and examined in various models. Using the expansion scheme, we revisit how the hadron resonance gas model emerges from the S-matrix framework, and consider an example of structureless scattering in which the phase shift function can be exactly worked out. Finally we analyze the influence of dynamics on the phase shift function in a simple example of 3- and 4-body scattering.

Effective increase in beam emittance by phase-space expansion using asymmetric Bragg diffraction.

We propose an innovative method to extend the utilization of the phase space downstream of a synchrotron light source for X-ray transmission microscopy. Based on the dynamical theory of X-ray diffraction, asymmetrically cut perfect crystals are applied to reshape the position-angle-wavelength space of the light source, by which the usable phase space of the source can be magnified by over one hundred times, thereby "phase-space-matching" the source with the objective lens of the microscope. The method's validity is confirmed using SHADOW code simulations, and aberration through an optical lens such as a Fresnel zone plate is examined via matrix optics for nano-resolution X-ray images.

Entropy Evolution and Uncertainty Estimation with Dynamical Systems

This paper presents a comprehensive introduction and systematic derivation of the evolutionary equations for absolute entropy H and relative entropy D, some of which exist sporadically in the literature in different forms under different subjects, within the framework of dynamical systems. In general, both H and D are dissipated, and the dissipation bears a form reminiscent of the Fisher information; in the absence of stochasticity, dH/dt is connected to the rate of phase space expansion, and D stays invariant, i.e., the separation of two probability density functions is always conserved. These formulas are validated with linear systems, and put to application with the Lorenz system and a large-dimensional stochastic quasi-geostrophic flow problem. In the Lorenz case, H falls at a constant rate with time, implying that H will eventually become negative, a situation beyond the capability of the commonly used computational technique like coarse-graining and bin counting. For the stochastic flow problem, it is first reduced to a computationally tractable low-dimensional system, using a reduced model approach, and then handled through ensemble prediction. Both the Lorenz system and the stochastic flow system are examples of self-organization in the light of uncertainty reduction. The latter particularly shows that, sometimes stochasticity may actually enhance the self-organization process.

Quantum corrections to the semiclassical Hartree-Fock theory of a harmonically trapped Bose gas

Using the phase-space expansion of the thermodynamical distribution functions we provide a general and systematic method for including effects beyond the local-density approximation to the semiclassical Hartree-Fock theories. We illustrate the method by applying it to the case of a strictly two-dimensional, harmonically trapped Bose gas. Thereby, we address the ambiguous prediction of the Hartree-Fock approximation, namely, whether a fixed number of trapped atoms undergoes Bose-Einstein condensation or not. We also investigate the dependence of the critical temperature on the interaction strength.

From Kadanoff-Baym dynamics to off-shell parton transport

The description of strongly interacting quantum fields is based on two-particle irreducible (2PI) approaches that allow for a consistent treatment of quantum systems out-of-equilibrium as well as in thermal equilibrium. As a theoretical test case the quantum time evolution of Φ4-field theory in 2+1 space-time dimensions is investigated numerically for out-of-equilibrium initial conditions on the basis of the Kadanoff-Baym-equations including the tadpole and sunset self energies. In particular we address the dynamics of the spectral (‘off-shell’) distributions of the excited quantum modes and the different phases in the approach to equilibrium described by Kubo-Martin-Schwinger relations for thermal equilibrium states. A detailed comparison of the full quantum dynamics to approximate schemes like that of a standard kinetic (on-shell) Boltzmann equation is performed. We find that far off-shell 1↔3 processes are responsible for chemical equilibration, which is not included in the Boltzmann limit. Furthermore, we derive generalized transport equations for the same theory in a first order gradient expansion in phase space thus explicitly retaining the off-shell dynamics as inherent in the time-dependent spectral functions. The solutions of these equations compare very well with the exact solutions of the full Kadanoff-Baym equations with respect to the occupation numbers of the individual modes, the spectral evolution as well as the chemical equilibration process. Furthermore, the proper equilibrium off-shell distribution is reached for large times contrary to the quasiparticle Boltzmann limit. We additionally present a direct comparison of the solution of the generalized transport equations in the Kadanoff-Baym (KB) and Botermans-Malfliet (BM) form; both solutions are found to agree very well with each other. The off-shell transport equation in the BM scheme allows for an explicit solution within an extended dynamical quasiparticle Ansatz. This leads to generalized equations of motion for dynamical quasiparticles that exceed the classical Hamilton equations and allow for the description of dynamical spectral functions. The dynamical quasiparticle model (DQPM) is used to extract partonic spectral functions from lattice QCD in the temperature range 0.8≤T/Tc ≤10. By consideration of time-like and space-like sectors of ‘observables’ such as number densities, energy densities etc. mean-field potentials as well as effective interactions are extracted at different temperature T and quark chemical potentials μq. The latter determine the off-shell dynamics in the Parton-Hadron-String Dynamics (PHSD) transport approach. Illustrative examples for off-shell dynamics in the hadron and parton sector are presented for e+e- production from nucleus-nucleus collisions from SIS to RHIC energies. The generalized transport approach, furthermore, qualifies for the description of hadronization without leading to the problem of entropy reduction as in conventional coalescence models.

Dynamics of resonantly interacting equatorial waves

In this paper we explore some dynamical features on the non-linear interactions among equatorial waves. The shallowwater equation model with the equatorial β-plane approximation is used for this purpose. The Galerkin method is applied to the governing equations with the basis functions given by the eigensolutions of the linear problem. From the phase space expansion of two particular integrals of motion of the system, quadratic to lowest order, some constraints are obtained which the coupling coefficients must satisfy in order to ensure the invariance of such integrals. From the numerical evaluation of the coupling coefficients, these constraints are used to determine the possible resonant triads among equatorial waves. Numerical integrations of the resonant three-wave problem show that the energy of the waves in a resonant triad evolves periodically in time, with the period and amplitude of the energy oscillations dependent on the magnitude of the initial amplitudes of the waves and the way in which the initial energy is distributed among the triad components. The high-frequency modes are found to be energetically more active than the low-frequency modes. The latter tend to act as ‘catalytic’ components in a resonant triad. Integrations of the problem of two resonant triads coupled by a single mode point out the importance of gravity waves in the intertriad energy exchanges, suggesting the significance of these modes in the redistribution of energy throughout the atmospheric motion spectrum. The results also show that the intertriad energy exchanges provided by the highest frequency mode of two triads occur in a longer time-scale than the intratriad interactions. Therefore, these results also suggest the importance of the high-frequency modes in the generation of the low-frequency variability (intraseasonal and even longer term) of the atmospheric flow.

Eaters of the lotus: Landauer's principle and the return of Maxwell's demon

Landauer's principle is the loosely formulated notion that the erasure of n bits of information must always incur a cost of k ln n in thermodynamic entropy. It can be formulated as a precise result in statistical mechanics, but for a restricted class of erasure processes that use a thermodynamically irreversible phase space expansion, which is the real origin of the law's entropy cost and whose necessity has not been demonstrated. General arguments that purport to establish the unconditional validity of the law (erasure maps many physical states to one; erasure compresses the phase space) fail. They turn out to depend on the illicit formation of a canonical ensemble from memory devices holding random data. To exorcise Maxwell's demon one must show that all candidate devices—the ordinary and the extraordinary—must fail to reverse the second law of thermodynamics. The theorizing surrounding Landauer's principle is too fragile and too tied to a few specific examples to support such general exorcism. Charles Bennett's recent extension of Landauer's principle to the merging of computational paths fails for the same reasons as trouble the original principle.

The Nearest-Neighbor Self-Avoiding Walk with Complex Killing Rates

The Green’s function for the nearest-neighbor self-avoiding walk on a hypercubic lattice in d > 2 dimensions is constructed and shown to be analytic for values of the killing rate $ a \in \textbf{C} $ satisfying $ \vert a \vert > \epsilon, \vert \textrm{arg}\,\, a \vert $ \lt; $ 3\pi/4 - b $ with $ \epsilon > 0 $ and 0 \lt; b \lt; $ \pi/4 $. We restrict $ \vert a \vert > \epsilon > 0 $ in order to use the killing rate as an infrared cutoff, which allows us to construct Green’s function using a single scale cluster expansion. The presence of non-real killing introduces complications that we resolve through the use of an appropriate choice of decoupling scheme and a subsidiary expansion. Our methods can be used to control a single momemtum slice in a phase-space expansion.

Intramolecular Vibrational Relaxation Seen as Expansion in Phase Space. 4. Generic Relaxation Laws for a Spectroscopic Clump Profile

We examine the volume of phase space sampled by a nonstationary wave packet when the spectral function consists of a single clump or of a series of them. The relaxation laws are expressed in terms of reduced time variables τ, whose definition involves either the average density of states (for a single clump) or appropriately weighted average densities of states (when the spectrum consists of many clumps). Introducing reasonable approximations, very simple generic relaxation laws are derived for the ratio N(τ)/N ∞ , which measures the fraction of available phase space that has been sampled by time τ. Under certain assumptions, these laws are found to depend neither on the number nor on the individual features (shapes and widths) of the clumps. However, they strongly depend on the nature (regular or chaotic) of the underlying dynamics. When the dynamics is regular, the relaxation law is expressed in terms of τ - 1 , whereas the corresponding equation in the chaotic limit is slightly more complicated and involves terms in τ - 2 and τ - 2 In τ. Phase space is thus explored according to essentially different relaxation laws in the regular and chaotic limits, the difference being appreciable during the entire relaxation. These laws reflect in the time domain the difference in the distribution of nearest-neighbor level spacings observed in the energy domain (Poisson or Wigner statistics).

A semiclassical theory of quantum noise in open chaotic systems

We consider the quantum evolution of classically chaotic systems in contact with the surroundings. Based on ℏ-scaling of an equation for time evolution of the Wigner's quasi-probability distribution function in presence of dissipation and thermal diffusion we derive a semiclassical equation for quantum fluctuations. This identifies an early regime of evolution dominated by fluctuations in the curvature of the potential due to classical chaos and dissipation. A stochastic treatment of this classical fluctuations leads us to a Fokker-Planck equation which is reminiscent of Kramers' equation for thermally activated processes. This reveals an interplay of three aspects of evolution of quantum noise in weakly dissipative open systems; the reversible Liouville flow, the irreversible chaotic diffusin which is characteristic of the system itself, and irreversible dissipation induced by the external reservoir. It has been demonstrated that in the dissipation-free case a competition between Liouville flow in the contracting direction of phase space and chaotic diffusion sets a critical width in the Wigner function for quantum fluctuations. We also show how the initial quantum noise gets amplified by classical chaos and ultimately equilibrated under the influence of dissipation. We establish that there exists a critical limit to the expansion of phase space. The limit is determined by chaotic diffusion and dissipation. Making use of appropriate quantum-classical correspondence we verify the semiclassical analysis by the fully quantum simulation in a chaotic quartic oscillator.

Intramolecular vibrational relaxation seen as expansion in phase space. III. The long-time limit

Asymptotic formulas that describe the behavior of the function N(T) measuring the phase space volume sampled by a nonstationary wave packet during its time evolution are derived. It is shown that, in the long-time limit, N(T)∼T−1 when the dynamics is regular, whereas N(T)∼T−2 ln T for the chaotic case.

Intramolecular vibrational relaxation seen as expansion in phase space. II. Reference ergodic systems

The aim of the paper is to estimate the volume of phase space that is, in principle, available to a nonstationary wave packet during its intramolecular vibrational relaxation. For that purpose, use is made of the maximum entropy method, together with the concept of constrained ergodicity to construct two so-called reference ergodic systems. The first one concerns thermal excitation processes. In that case, the only two constraints that are imposed on the intramolecular dynamics arise from the normalization of the wave function and from the conservation of energy. These constraints affect the zeroth and first moments of the spectrum. The second reference system concerns a situation where, as an additional constraint, use is made of the information that the system has been prepared spectroscopically, i.e., by a specific excitation process, consisting in the coherent excitation of an initial pure state. Then, the second moment of the spectrum, denoted σ, is shown to provide the appropriate additional constraint. Translated into the time domain, the prior knowledge of the dynamics used as a constraint is limited to an infinitesimally brief period of time [0,dt] with the remaining evolution determined by the maximum entropy method. The spectroscopic reference system constructed in that way can be understood as the one that samples the maximal volume of phase space available to a wave packet having a specified average energy and being put in motion by a specified initial force. Closed-form expressions are obtained for the phase space volumes occupied by these two reference systems for various simple parametrizations of the function D(E) that expresses the density of states as a function of the internal energy (power laws or exponential increase). Thermal reference systems are found to sample a larger volume of phase space than their spectroscopic counterparts. The difference between these two cases depends critically on the value of σ, and also on the symmetry characteristics of the excitation process. In general, the volumes occupied by the reference systems, thermal as well as spectroscopic, can be expressed as ηEavD(Eav), where Eav is the (conserved) average energy of the wave packet and η is a correcting factor that depends on the functional form of D(E) and on the nature of the imposed constraints. In all cases studied, the value of η was found not to greatly differ from 1. The method has been applied to the analysis of three experimental photoelectron spectra presenting different spectral characteristics (X̃ 2A1 state of NH+3, X̃ 2B3 state of C2H+4, and the X̃ 2A″ state of C2H3F+). The fractional occupancy index F defined by Heller as the fraction of the available phase space eventually explored up to the break time TB could be determined. After a time of the order of 100 fs, F was found to be of the order of a few percent for thermal excitation. When the molecule presents some symmetry, the expansion of the wave packet is restricted to that part of phase space spanned by the totally symmetric wave functions. The use of this additional a priori knowledge increases the fractional index F.

Remarks on phase space formalism and beam transport

In this paper we discuss the phase space formalism within the framework of charged- and optical-beam transport. We explore the phase space expansion when the beam undergoes a multiple scattering and in the case of «phase space transfer» as it happens in synchrotron radiation problems.

Intramolecular vibrational energy relaxation seen as expansion in phase space. I. Some experimental results for H2O+(X̃ 2B1), C2H+4(X̃ 2B3), and HCN+(B̃ 2∑+)

It has been shown by Heller that a nonstationary wave packet resulting from a Franck–Condon transition evolves on the potential energy surface of the final electronic state and propagates through phase space at a rate which can be determined from the autocorrelation function ↓C(t)↓2=↓〈(0)‖(t)〉↓2. Since C(t) can be obtained by Fourier transformation of an optical spectrum S(E), i.e., from an observable quantity, it is possible to derive from an experimental measurement information concerning the density operator of a so-called dynamical statistical ensemble (DSE). This density operator, denoted ρav, represents a statistical mixture of the eigenstates of the system with weights determined by the dynamics of the system. It becomes diagonal after a so-called break time 𝒯B. Its measure, according to a definition due to Stechel, can be interpreted as an effective number of states (denoted 𝒩) that significantly contribute to the dynamics. The break time 𝒯B represents the finite period of time allowed to expand in the phase space and after which no further progress can be made. Therefore, the number 𝒩∞ of phase space cells which are accessed after a very long interval of time (or in practice after the break time) remains limited. Information on the validity of statistical theories of unimolecular reactions is contained in the fraction ℱ of the available phase space which is eventually explored. In order to assess the representativity of the sampling, it is necessary to account for the selection rule which requires all the states counted in 𝒩∞ to belong to the totally symmetric representation. It is also appropriate to estimate the role played by Fermi resonances and similar vibrational interactions which bring about energy flow into zero-order antisymmetric modes. A method to carry out the necessary partitionings is suggested. The functions 𝒩T and ℛT, and the quantities 𝒯B, 𝒩∞, 𝒩 *, and ℱ have been determined from experimental data in three cases. In each case, the rate ℛT=d𝒩T/dT starts from an initial value of zero, increases up to a maximum which is reached after a time of the order of 10−14 s, and then exhibits an overall decrease upon which oscillations are superimposed. For state X̃ 2B1 of H2O+, 𝒯B≂2.4×10−14 s and ℱ≂0.3. The wave packet never accesses that part of the phase space that corresponds to the excitation of antisymmetric vibrations. For state X̃ 2B3u of C2H+4, 𝒯B≂1.6×10−13 s and ℱ≂5×10−4. This fraction raises to 6×10−3 if measured with respect to the effectively available phase space. When the spectrum consists of a discrete part followed by a dissociation continuum, the method can be extended to study the behavior of the bound part of the wave packet only. This has been applied to state B̃ 2∑+ of HCN+ which is characterized by a very irregular spectrum. This case offers an example of complete occupation of phase space after a break time which is of the order of 2×10−13 s.

Construction and Borel summability of infrared ? 4 4 by a phase space expansion

We construct the thermodynamic limit of the critical (massless) φ4 model in 4 dimensions with an ultraviolet cutoff by means of a “partly renormalized” phase space expansion. This expansion requires in a natural way the introduction of effective or “running” constants, and the infrared asymptotic freedom of the model, i.e. the decay of the running coupling constant, plays a crucial role. We prove also that the correlation functions of the model are the Borel sums of their perturbation expansion.

Phase space analysis and particle structure in field theory

Progress in the last few years in constructive field theory has included (1) the construction of three-dimensional, non-Abelian gauge theories in a finite volume, (2) the construction of models that are asymptotically free or have a nontrivial fixed point, (3) a new presentation of perturbation theory yielding old and new large-order estimates, and (4) asymptotic completeness for renormalizable models in the two-particle region; the existence of multiparticle structure equations in the Euclidean region, provided that the coupling constant goes to zero as the energy increases; these equations yield formally the asymptotic completeness. In obtaining most of these results the phase space expansion pioneered by Glimm and Jaffe plays a crucial role. The first part of this review is a description of the phase space expansion in the “Ecole Polytechnique” version; the second part is devoted to the construction of models, the third part treats perturbation theory; and the last part deals with asymptotic completeness.