Phase Space Path Research Articles

Phase-space path integral approach to the kinetics of black hole phase transition in massive gravity

The dynamics of the state-switching process of black holes in dRGT massive gravity theory is presented using free energy landscape and stochastic Langevin equations. The free energy landscape is constructed using the Gibbons-Hawking path integral method. The black hole phases are characterized by taking its horizon radius as the order parameter. The free energy landscape provides three black hole phases: small, intermediate, and large. The small and large black holes are thermodynamically stable whereas the intermediate one is unstable. The Martin–Siggia–Rose–Janssen–de Dominicis (MSRJD) functional describes the stochastic dynamics of black hole phase transition. The Hamiltonian flow lines are obtained from the MSRJD functional and are used to analyze the stability and the phase transition properties. The dominant kinetic path between different phases is discussed for various configurations of the free energy landscape. We discuss the effect of black hole charge and the graviton mass on the critical behavior of black hole phase transition.

Phase space path integral approach to the kinetics of black hole phase transition

Phase space path integral approach to the kinetics of black hole phase transition

Spectral signatures of coexisting isolas and four-wave mixing under the effect of noise in photonic oscillators

The optical power spectrum is the prime observable to dissect, understand, and design the long- time behavior of small and large arrays of optically coupled semiconductor lasers. A long-standing issue has been identified within the literature of injection locking in photonic oscillators: first how the thickness of linewidth and the lineshape spectral envelope correlates with the deterministic evolution of the monochromatic injected laser oscillator and second how the presence of noise and the typically dense proximity in phase space of coexisting limit cycles of the coupled system are shaping and influencing the overall spectral behavior. In addition, we are critically interested in the regions where the basin of attraction has a fractal-like structure, still, the long-time orbits are P1 (period 1) and/or P3 (period 3) limit cycles. Numerically computed evidence shows that, when the coupled system lives in the regions of coexisting isolas and four-wave mixing (FWM) limit cycles, the overall optical power spectrum is deeply imprinted by a strong influence from the underlying noise sources. A particularly intriguing observation in this region of parameter space that we examine is that the isolas draw most of the trajectories on its phase space path.

Visualization of fast ion phase-space flow in plasmas well-below, near and well-above Alfvén eigenmode stability threshold in tokamak

Transport of fast ions along certain local phase space paths, referred to as fast ion phase-space flow, has been systematically measured by an imaging neutral particle analyzer (INPA) in three plasma regimes, which are well-below, near, and well-above the Alfvén eigenmode stability threshold (Du et al 2021 Phys. Rev. Lett. 127 235002). (1) In plasmas well-below the Alfvén eignenmodes (AE) stability threshold, fast ions are well-confined on passing particle orbits without noticeable transport over the phase space. The observed INPA images agree well with the synthetic INPA images, using the fast ion distribution predicted by neoclassical theory. (2) In plasmas near the AE stability threshold, INPA images in the presence of AE activity moderately deviate from those without AE activity. The image difference can be well interpreted by AE-driven, phase-space fast ion flow. Paths of this flow over the velocity space (or streamlines) are reconstructed by the intersection lines of curved Eʹ and µ surfaces, referred to as Eʹ and µ line (where ; E, P ζ and µ are the energy, canonical toroidal momentum and magnetic moment of ions; ω and n are the angular frequency and toroidal mode number of AEs, respectively). Resonant fast ions move radially inward by gaining energy and move radially outward by losing energy and the trajectory well aligns with the Eʹ and µ lines that pass through the mode resonances near the injection energy of neutral beams. (3) In plasmas well-above the AE stability threshold, fast ion phase-space dynamics shows additional features. Fast ions are transported out of the birth positions so promptly along the streamlines that the slowing-down process from the injection energy is not observable, exhibiting strong critical gradient behavior at local phase space. As a result, the increase of electron temperature is very small, in spite of an increase of beam power by . It should be emphasized that the directions of phase-space transport, induced by AEs with different frequencies, structures and mode numbers, do not largely differ.

Energy transitions driven by phase space reflection operators

Phase space reflection operators lie at the core of the Wigner–Weyl representation of density operators and observables. The role of the corresponding classical reflections is known in the construction of semiclassical approximations to Wigner functions of pure eigenstates and their coarsegrained microcanonical superpositions, which are not restricted to classically integrable systems. In their active role as unitary operators, they generate transitions between pairs of eigenstates specified by transition Wigner functions (or cross-Wigner functions): the square modulus of the transition Wigner function at each point in phase space is the transition probability for the reflection through that point.Coarsegraining the initial and final energies provides a transition probability density as a phase space path integral. It is here investigated in the simplest classical approximation involving microcanonical Wigner functions. A reflection operator generates a transition between a pair of energy shells with a probability density given by the integral of the inverse modulus of a Poisson bracket over the intersection of a shell with the reflection of its pair. The singularity of the pair of Wigner functions at their dominant caustics is nicely integrable over their intersection, except for a single degree of freedom. Even though this case is not directly relevant for future investigations of chaotic systems, it is shown here how the improved approximation of the spectral Wigner functions in terms of Airy functions resolves the singularity.

Phase space Feynman path integrals of parabolic type on the torus as analysis on path space

We provide general sets of functionals for which parabolic phase space Feynman path integrals on the torus \({\mathbb {T}}^{d} \equiv ({\mathbb {R}} / 2\pi {\mathbb {Z}})^{d}\) have a mathematically rigorous meaning. More exactly, for each functional belonging to each set, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of \({\mathbb {T}}^{d} \times {\mathbb {Z}}^{d}\) to some function of the ending point of position paths and the starting point of momentum paths. Each set of functionals is closed under addition, multiplication, translation, invertible integer linear transformation, and functional differentiation. As a result, we can create a large number of path integrable functionals. Though we must exercise caution when using phase space path integrals, several properties comparable to those of conventional integrals are applicable.

Path integrals as analysis on path space by time slicing approximation

We explain the Lagrangian (configuration space) path integrals in Section 1 and the Hamiltonian (phase space) path integrals in Section 2. The two sections are independent of each other. Readers can read from either section.

Inertial particle under active fluctuations: Diffusion and work distributions.

We study the underdamped motion of a passive particle in an active environment. Using the phase space path integral method we find the probability distribution function of position and velocity for a free and a harmonically bound particle. The environment is characterized by an active noise which is described as the Ornstein-Uhlenbeck process (OUP). Taking two similar, yet slightly different OUP models, it is shown how inertia along with other relevant parameters affect the dynamics of the particle. Further we investigate the work fluctuations of a harmonically trapped particle by considering the trap center being pulled at a constant speed. Finally, the fluctuation theorem of work is validated with an effective temperature in the steady-state limit.

Path integral formulation for position–time-dependent mass systems

The problem of position-time dependent mass with general time-dependent potential is treated in the framework of the phase space path integral approach. By means of an appropriate point canonical transformation, the problem is converted into that relative to a constant mass with stationary potential. Particular examples are also considered.

Phase space path integral for modified pöschl-teller potential

The phase space path integral for a particle inside the modified Pöschl-Teller potential hole (MPT) is exactly evaluated by using the delta functional technique. The bound-state energy-levels and the corresponding suitably normalized wave functions are derived.

Compression-induced solidification of shock-melted cerium

Compression-induced solidification has been observed in cerium on nanosecond timescales. A series of experiments was conducted in Sandia National Laboratories' Z Facility in which cerium was shock melted and subsequently shocklessly, or ramp, loaded across the melt line inducing solidification. The signature of solidification manifested in the recovery of material strength and the propagation of waves at the local elastic sound velocity. Density functional theory simulations of cerium along the experimental phase-space path exhibit spontaneous freezing to a tetragonal phase at the same pressure and closely predict the observed physical properties of solid and liquid cerium near melt.

Kinetic equation for particle transport in turbulent flows

With a new approach based on the ensemble average over particle state transition paths in phase space, a kinetic equation for particles transported in turbulent flows is derived. The probability density function (PDF) for particles is defined as an ensemble average of a special fine-grained PDF, referred to as the local path density operator. The kinetic equation is derived from a Taylor series expansion of the PDF in terms of the cumulants with respect to particle paths in phase space and leads to a closed expression for its diffusion terms. It shows that the random forcing of eddy fluctuations, non-stationarity of turbulence, and inertia of particles are explicitly presented in the diffusion coefficient, which could help us to understand how particles are diffused by these underlying mechanisms. The kinetic equation is applicable to non-Markovian, non-Gaussian, and non-stationary stochastic processes, while for Markovian processes, it recovers the classical Fokker–Planck equation. The macroscopic equations for particle phase are derived based on the kinetic equation and compared with the direct numerical simulation of particles transported in turbulent flows.

Phase space path integral on torus for the fundamental solution of higher-order parabolic equations

We give a rigorous formulation of phase space path integrals on the torus $${\mathbb {T}}^{d} = ({\mathbb {R}} / 2\pi {\mathbb {Z}})^{d}$$ for fundamental solutions of higher-order parabolic equations. Especially, by using the pseudo-differential operators on the torus, we prove that the time slicing approximation of the phase space path integral converges on compact subsets of $${\mathbb {T}}^{d} \times {\mathbb {R}}^{d}$$ .

Cessation and reversals of large-scale structures in square Rayleigh–Bénard cells

We consider direct numerical simulations of turbulent Rayleigh–Bénard convection inside two-dimensional square cells. For Rayleigh numbers $Ra=10^{6}$ to $Ra=5\times 10^{8}$ and Prandtl numbers $Pr=3$ and $Pr=4.3$, two types of flow regimes are observed intermittently: consecutive flow reversals (CR), and extended cessations (EC). For each regime, we combine proper orthogonal decomposition (POD) and statistical tools on long-term data to characterise the dynamics of large-scale structures. For the CR regime, centrosymmetric modes are dominant and display a coherent dynamics, while non-centrosymmetric modes fluctuate randomly. For the EC regime, all POD modes follow Poissonian statistics and a non-centrosymmetric mode is dominant. To explore further the differences between the CR and EC regimes, an analysis based on a cluster partition of the POD phase space is proposed. This data-driven approach confirms the successive mechanisms of the generic reversal cycle in CR as proposed in Castillo-Castellanos et al. (J. Fluid Mech., vol. 808, 2016, pp. 614–640). However, these mechanisms may take one of multiple paths in the POD phase space. Inside the EC regime, this approach reveals the presence of two types of coherent time sequences (weak reversals and actual cessations) and more rarely intense plume crossings. Finally, we analyse within a range of Rayleigh numbers up to turbulent flow, the relation between dynamical regimes and the POD energetic contents as well as the residence time in each cluster.

Diffusion caused by two noises—active and thermal

The diffusion of colloids inside an active system—e.g. within a living cell or the dynamics of active particles itself (e.g. self-propelled particles) can be modeled through overdamped Langevin equation which contains an additional noise term apart from the usual white Gaussian noise, originating from the thermal environment. The second noise is referred to as ‘active noise’ as it arises from activity such as chemical reactions. The probability distribution function (PDF or the propagator) in space-time along with moments provides essential information for understanding their dynamical behavior. Here we employ the phase-space path integral method to obtain the propagator, thereby moments and PDF for some possible models for such noise. At first, we discuss the diffusion of a free particle driven by active noise. We consider four different possible models for active noise, to capture the possible traits of such systems. We show that the PDF for systems driven by noises other than Gaussian noise largely deviates from normal distribution at short to intermediate time scales as a manifestation of out-of-equilibrium state, albeit converges to Gaussian distribution after a long time as a consequence of the central limit theorem. We extend our work to the case of a particle trapped in a harmonic potential and show that the system attains steady state at long time limit. Also, at short time scales, the nature of distribution is different for different noises, e.g. for particle driven by dichotomous noise, the probability is mostly concentrated near the boundaries whereas a long exponential tail is observed for a particle driven by Poissonian white noise.

Work fluctuation relations for a dragged Brownian particle in active bath

We study the work distribution of a Brownian particle diffusing in an environment of active particles and being trapped in a harmonic potential, the center of which is subjected to a time-dependent protocol. Employing phase space path integral technique we find an expression of work distribution for any generic model of active noise. Here we consider two active noise models — Gaussian correlated and Poisson white, each of which can represent some physical systems. For both the cases, it is found that transient fluctuation relation of work is not applicable though at steady state it holds by defining a renormalized temperature τr in place of bath temperature. Interestingly, τr is the same for both the models and can be expressed in terms of diffusivities of active and thermal noises. For correlated Gaussian bath, an alternative approach is presented. Analogous to the formalism given by Hatano and Sasa (2001), we obtain a work like quantity from nonequilibrium potential with the inclusion of a new stationary parameter Ω. With proper choice of Ω, a steady-state fluctuation relation, namely Jarzynski equality is satisfied.

Phase space Feynman path integrals of parabolic type with smooth functional derivatives

We give two general sets of functionals for which the phase space path integrals of m-th order parabolic type (m>0) have a mathematically rigorous meaning. More precisely, for any functional belonging to each set, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the final point of position paths and to the initial point of momentum paths. Each set of functionals is closed under addition, multiplication, translation, invertible linear transformation and functional differentiation. Therefore, we can produce many functionals which are phase space path integrable. Furthermore, though we need to pay attention for use, we ensure that the following operations are valid for the phase space path integrals: (1) interchange of the order with integrals with respect to time (2) interchange of the order with some limits (3) invariance under orthogonal transformations (4) invariance under translations with respect to momentum paths (5) integration by parts with respect to momentum paths.

Free Energy Calculation of Transmembrane Ion Permeation: Sample with a Single Reaction Coordinate and Analysis along Transition Path.

A single reaction coordinate is designed to perform umbrella sampling for the water-chain-assist transmembrane permeation processes of ions and highly polar groups. In these permeation processes, there are three types of configuration changes including deformation of both leaflets of the lipid bilayer, formation of a water chain across the bilayer, and translocation of the ion through the bilayer. Due to the complexity of the permeation process, multiple reaction coordinates and expensive high dimensional free energy calculations have been used in previous studies. Our single reaction coordinate can be used to obtain sample data of the entire permeation process, which significantly reduces the computational cost in free energy calculations. We show that one-dimensional umbrella sampling along the new reaction coordinate can be used to obtain continuous sample data roughly along the minimum free energy path (MFEP) on the phase plane of the two components of the new reaction coordinate. Along the MFEP, a one-dimensional bin-segmentation of the sample data is then used to calculate the potential of mean force accurately by the newly developed free energy analysis method: weighted least-square analysis method (Welsam). Our work not only proposes a useful reaction coordinate to obtain sample data for water-chain assist transmembrane permeation processes but also establishes a general methodology for one-dimensional free energy calculations with bin-segmentation along the transition path in multiple dimensional phase space.

Heat fluctuation of a harmonically trapped particle in an active bath.

We study the heat fluctuation of an overdamped Brownian particle trapped in a harmonic potential and driven by active noise. Employing the phase-space path integral method we derive a general formula for the probability distribution of heat exchange in a generic model of an active bath. The work has been extended by considering two particular models of active noise and computing an exact analytical expression for distribution in Gaussian colored noise and a semianalytical result in the Poissonian bath. We corroborate the fluctuation theorem with our analytical findings by introducing the familiar concept of effective temperature and as a corollary the total entropy production is calculated.

Relationship between the Track and Structural Evolution of Hurricane Sandy (2012) Using a Regional Ensemble

Abstract An ensemble of 72 Weather Research and Forecasting (WRF) Model simulations is evaluated to examine the relationship between the track of Hurricane Sandy (2012) and its structural evolution. Initial and boundary conditions are obtained from ECMWF and GEFS ensemble forecasts initialized at 0000 UTC 25 October. The 5-day WRF simulations are initialized at 0000 UTC 27 October, 48 h into the global model forecasts. Tracks and cyclone phase space (CPS) paths from the 72 simulations are partitioned into 6 clusters using regression mixture models; results from the 4 most populous track clusters are examined. The four analyzed clusters vary in mean landfall location from southern New Jersey to Maine. Extratropical transition timing is the clearest difference among clusters; more eastward clusters show later Sandy–midlatitude trough interaction, warm seclusion formation, and extratropical transition completion. However, the intercluster variability is much smaller when examined relative to the landfall time of each simulation. In each cluster, a short-lived warm seclusion forms and contracts through landfall while lower-tropospheric potential vorticity concentrates at small radii. Despite the large-scale similarity among the clusters, relevant intercluster differences in landfall-relative extratropical transition are observed. In the easternmost cluster the Sandy–trough interaction is least intense and the warm seclusion decays the most by landfall. In the second most eastward cluster Sandy retains the most intact warm seclusion at landfall because of a slightly later (relative to landfall) and weaker trough interaction compared to the two most westward clusters. Nevertheless, the remarkably similar large-scale evolution of Sandy among the four clusters indicates the high predictability of Sandy’s warm seclusion extratropical transition before landfall.