Markovian Dynamics Research Articles

Hydrodynamic limit for a 2D interlaced particle process

The Markov dynamics of interlaced particle arrays, introduced by A. Borodin and P. L. Ferrari in (Comm. Math. Phys. 325 (2014) 603–684), is a classical example of (2+1)-dimensional random growth model belonging to the so-called Anisotropic KPZ universality class. In (Comm. Pure Appl. Math. 72 (2018) 620–666), a hydrodynamic limit—the convergence of the height profile, after space/time rescaling, to the solution of a deterministic Hamilton–Jacobi PDE with nonconvex Hamiltonian—was proven when either the initial profile is convex, or for small times, before the solution develops shocks. In the present work, we give a simpler proof, that works for all times and for all initial profiles for which the limit equation makes sense. In particular, the convexity assumption is dropped. The main new idea is a new viewpoint about “finite speed of propagation” that allows to bypass the need of a priori control of the interface gradients, or equivalently of inter-particle distances.

Variation of quantum speed limit under Markovian and non-Markovian noisy environment

The time required for a given quantum system to evolve is known as the quantum speed limit (QSL), which can be used to characterize the speed of evolution of a quantum system. We study QSL time for Markovian and non-Markovian dynamics. Investigating the effect of QSL under non-unitary evolution is of fundamental interest. In this work, we derive QSL time for maximally entangled state and apply its results under various noise channels. We discuss measure of QSL, how correlations affect the rate of QSL. Further, we carried an investigation on connection between non-Markovianity and QSL. We show that the non Markovian evolution can speed up quantum evolution, therefore lead to smaller QSL time. We have established the connections between correlated channels, information loss and non-Markovianity. This theory may suggest some further connections to retrieve back information from surrounding and preserves decoherence.

Evolution of Gaussian Rényi-2 quantum correlations in a squeezed thermal environment

In this paper, we investigate, in the framework of the theory of open quantum systems, based on completely positive dynamical semigroups, the Markovian dynamics of Gaussian Rényi-2 correlations — quantum entanglement, quantum discord, mutual information and classical correlations in a system composed of two bosonic modes, interacting with a squeezed thermal bath. We show that the time evolution of the Rényi-2 correlations strongly depends on the parameters of the initial Gaussian squeezed thermal state of the considered system and on the parameters characterizing the squeezed thermal bath. It is shown that while Gaussian Rényi-2 entanglement is suppressed in a finite time, due to the interaction with the squeezed thermal bath, the correlations beyond entanglement — Gaussian Rényi-2 discord, classical correlations and mutual information undergo a freezing-like behavior, namely they decay only asymptotically, in the limit of large times. We also illustrate a fundamental hierarchy for bipartite Gaussian correlations.

Master equation for non-Markovian quantum Brownian motion: The emergence of lateral coherences

Understanding the behavior of a quantum system coupled to its environment is of fundamental interest in the general field of quantum technologies. It also has important repercussions on foundational problems in physics, such as the process of decoherence and the so-called quantum measurement problem. There have been many approaches to explore Markovian and non-Markovian dynamics within the framework of open quantum systems, but the richness of the ensuing dynamics is still not fully understood. In this paper we develop a non-Markovian extension of the standard Caldeira-Leggett model, based on expanding the dynamics of the reduced system at high temperature in inverse powers of the high-frequency cutoff of the Ohmic spectral density of the environment and derive a non-Markovian master equation for the reduced density matrix for the case of a general potential. We also obtain a fully analytical solution in the free particle case. While the short-time behavior of this solution does not diverge substantially from the Markovian behavior, at intermediate times we find a resurgence of coherence, which we name lateral coherence. We identify this with a corresponding transient negative entropy production rate, which is understood to be characteristic of non-Markovian dynamics. We also analyze the positivity of the reduced density matrix and derive the corresponding Fokker-Planck equation in the classical limit.

Stochastic embeddings of dynamical phenomena through variational autoencoders

System identification in scenarios where the observed number of variables is less than the degrees of freedom in the dynamics is an important challenge. In this work we tackle this problem by using a recognition network to increase the observed space dimensionality during the reconstruction of the phase space. The phase space is forced to have approximately Markovian dynamics described by a Stochastic Differential Equation (SDE), which is also to be discovered. To enable robust learning from stochastic data we use the Bayesian paradigm and place priors on the drift and diffusion terms. To handle the complexity of learning the posteriors, a set of mean field variational approximations to the true posteriors are introduced, enabling efficient statistical inference. Finally, a decoder network is used to obtain plausible reconstructions of the experimental data. The main advantage of this approach is that the resulting model is interpretable within the paradigm of statistical physics. Our validation shows that this approach not only recovers a state space that resembles the original one, but it is also able to synthesize new time series capturing the main properties of the experimental data.

Dynamics of Open Quantum Systems II, Markovian Approximation

A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperatureT>0. We show that for fixed, small values of the coupling constantλ, the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics isO(|λ|1/4)for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracyO(|λ|1/4)for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as1/t4or faster, which is a significant improvement relative to the previously required exponential decay.

Shaken dynamics on the 3d cubic lattice

On the space of ±1 spin configurations on the 3d square lattice, we consider the shaken dynamics, a parallel Markovian dynamics that can be interpreted in terms of Probabilistic Cellular Automata. The transition probabilities are defined in terms of pair ferromagnetic Ising-type Hamiltonians with nearest neighbor interaction J, depending on an additional parameter q, measuring the tendency of the system to remain locally in the same state. Odd times and even times have different transition probabilities. We compute the stationary measure of the shaken dynamics and we investigate its relation with the Gibbs measure for the 3d Ising model. It turns out that the two parameters J and q tune the geometry of the underlying lattice. We conjecture the existence of unique line of critical points in J−q plane. By a judicious use of perturbative methods we delimit the region where such curve must lie and we perform numerical simulation to determine it. Our method allows us to find in a unified way the critical values of J for Ising model with first neighbors interaction, defined on a whole class of lattices, intermediate between the two-dimensional hexagonal and the three-dimensional cubic one, such as, for example, the tetrahedral lattice. Finally we estimate the critical exponents of the magnetic susceptibility and show that our model captures a dimensional transition in the geometry of the system at q=0.

Modeling of biomolecular machines in non-equilibrium steady states.

Numerical computations have become a pillar of all modern quantitative sciences. Any computation involves modeling-even if often this step is not made explicit-and any model has to neglect details while still being physically accurate. Equilibrium statistical mechanics guides both the development of models and numerical methods for dynamics obeying detailed balance. For systems driven away from thermal equilibrium, such a universal theoretical framework is missing. For a restricted class of driven systems governed by Markov dynamics and local detailed balance, stochastic thermodynamics has evolved to fill this gap and to provide fundamental constraints and guiding principles. The next step is to advance stochastic thermodynamics from simple model systems to complex systems with tens of thousands or even millions of degrees of freedom. Biomolecules operating in the presence of chemical gradients and mechanical forces are a prime example for this challenge. In this Perspective, we give an introduction to isothermal stochastic thermodynamics geared toward the systematic multiscale modeling of the conformational dynamics of biomolecular and synthetic machines, and we outline some of the open challenges.

Topology by Dissipation: Majorana Bosons in Metastable Quadratic Markovian Dynamics.

Majorana bosons, that is, tight bosonic analogs of the Majorana fermionic quasiparticles of condensed-matter physics, are forbidden for gapped free bosonic matter within a standard Hamiltonian scenario. We show how the interplay between dynamical metastability and nontrivial bulk topology makes their emergence possible in noninteracting bosonic chains undergoing Markovian dissipation. This leads to a distinctive form of topological metastability, whereby a conserved Majorana boson localized on one edge is paired, in general, with a symmetry generator localized on the opposite edge. We argue that Majorana bosons are robust against disorder and identifiable by signatures in the zero-frequency steady-state power spectrum. Our results suggest that symmetry-protected topological phases for free bosons may arise in transient metastable regimes, which persist over practical timescales.

Upper bounds on the leakage of private data and an operational approach to Markovianity

We quantify the consequences of a private key leakage and private randomness generated during quantum key distribution. We provide simple lower bounds on the one-way distillable key after the leakage has been detected. We also show that the distributed private randomness does not drop by more than twice the number of qubits of the traced-out system. We further focus on irreducible private states, showing that their two-way distillable key is non-lockable. We then strengthen this result by referring to the idea of recovery maps. We further consider the action of special case of side-channels on some of the private states. Finally, we connect the topic of (non)markovian dynamics with that of hacking. In particular, we show that an invertible map is non-CP-divisible if and only if there exists a state whose the key witnessed by a particular privacy witness increases in time. This complements the recent result of J. Kolody\'nski et al. [Phys. Rev. A 101, 020303(R) (2020)] where the log-negativity was connected with the (non)markovianity of the dynamics.

Orbital diamagnetism of two-dimensional quantum systems in a dissipative environment: Non-Markovian effect and application to graphene.

The non-Markovian dynamics of a charged particle confined in the harmonic oscillator and linearly coupled to a neutral bosonic heat bath is investigated in the external uniform magnetic field. The analytical expressions are derived for the time-dependent and asymptotic orbital angular momenta. The transition from non-Markovian dynamics to Markovian dynamics and the transition from a confined charge particle to a free charge particle are considered. The orbital diamagnetism of graphene in a dissipative environment and an external uniform magnetic field is studied and compared with existing experimental data. The results are presented for the electric conductivity and resonance behavior of the mass magnetization in graphene.

Re-examination of risk-return dynamics in international equity markets and the role of policy uncertainty, geopolitical risk and VIX: Evidence using Markov-switching copulas

This study re-examines the empirical relationship between risk and return from 1994m12 to 2020m08 in fifteen international equity markets employing the novel time-varying Markov switching copula models. We provide first-time insightful evidence of time-varying Markov tail dependence structure and dynamics between risk and return in international equity markets. Results show that the dependence structure is positive for USA, UK, Germany, Italy, Brazil, Australia, Taiwan, Canada, Mexico, Japan, France and South Africa and negative for Singapore, India, Japan and China. Finally, we document the effects of policy uncertainty, geopolitical risk and VIX conditional on different markets states.

Quantum master equation approach to heat transport in dielectrics and semiconductors

We report on the derivation of the heat transport equation for nonmetals using a quantum Markovian master equation in Lindblad form. We first establish the equations of motion describing the time variation of the on-site energy of atoms in a one dimensional periodic chain that is coupled to a heat reservoir. In the continuum limit, the Fourier law of heat conduction naturally emerges, and the heat conductivity is explicitly obtained. It is found that the effect of the heat reservoir on the lattice is described by a heat source density that depends on the diffusion coefficients of the atoms. We show that the Markovian dynamics is equivalent to the long wavelength approximation for phonons, which is typical for the case of elastic solids. The high temperature limit is shown to reproduce the classical heat conduction equation.

Enhanced quantumness via non-Markovianity

In this paper we address the issue of the interplay between non-Markovianity and nonclassicality of a system. We consider the action of non-Markovian environments on a qubit in terms of two models, a classical noise model and a quantum microscopic collision model. For both models we use the violation of the Leggett-Garg inequality (LGI) as a criterion of quantumness, and use the conditional past-future (CPF) correlation to measure non-Markovianity. Besides for the collision model, we also use violation of the nondiscord generating and detecting (NDGD) dynamics to indicate the nonclassicality. And LGI, NDGD, and CPF are probed sequentially by three projective measurements of some observable. In this paper we do not use the assumption of the quantum regression theorem and especially for the collision model the measurement backaction on the environment is explicitly considered. We find that compared with the Markovian dynamics the LGI can be violated for a longer time interval, i.e., non-Markovianity can preserve the quantumness of the system. For the collision model we find that for a specific measurement process, although the non-Markovianity cannot be detected, the effect of non-Markovianity of the system dynamics on nonclassicality still exists, and for some specific measurement operator the violation of both LGI and NDGD will never occur for the Markovian dynamics while with the enhancement of non-Markovianity the quantumness can appear. We also find that the NDGD can be violated for a much wider parameter regime than the LGI.

Non-Markovian dynamics control of spin-1/2 system interacting with magnets

The unavoidable interaction of an open quantum system with its surrounding environment may follow non-Markovian dynamics behavior, which causes its memory effects to play a key role in many quantum technologies. In this work, we first consider the dynamics control of a spin-1/2 system that simultaneously interacts with two magnets. Subsequently, we study in detail the evolution of a classically driven spin-1/2 system that is coupled with a magnet. Two dynamical crossovers of the spin system, namely, from Markovian dynamics to non-Markovian dynamics and from no-speedup evolution to speedup evolution, can be controlled in these two models. Slightly different from the previous quantum speedup scenarios with controllable non-Markovianity, the stronger non-Markovianity does not necessarily lead to the quantum speedup dynamics process of the spin system in our spin-magnet schemes.

Continuous time reversal and equality in the thermodynamic uncertainty relation

We introduce a continuous time-reversal operation which connects the time-forward and time-reversed trajectories in the steady state of an irreversible Markovian dynamics via a continuous family of stochastic dynamics. This continuous time-reversal allows us to derive a tighter version of the thermodynamic uncertainty relation (TUR) involving observables evaluated relative to their local mean value. Moreover, the family of dynamics realizing the continuous time-reversal contains an equilibrium dynamics halfway between the time-forward and time-reversed dynamics. We show that this equilibrium dynamics, together with an appropriate choice of the observable, turns the inequality in the TUR into an equality. We demonstrate our findings for the example of a particle diffusing in a tilted periodic potential.

Stochastic dynamics, large deviation principle, and nonequilibrium thermodynamics.

By examining the deterministic limit of a general ε-dependent generator for Markovian dynamics, which includes the continuous Fokker-Planck equations and discrete chemical master equations as two special cases, the intrinsic connections among mesoscopic stochastic dynamics, deterministic ordinary differential equations or partial differential equations, large deviation rate functions, and macroscopic thermodynamic potentials are established. Our result not only solves the long-lasting question of the origin of the entropy function in classical irreversible thermodynamics, but also reveals an emergent feature that arises automatically during the deterministic limit, through its large deviation rate function, with both time-reversible dynamics equipped with a Hamiltonian function and time-irreversible dynamics equipped with an entropy function.

An Algorithm for Making Regime-Changing Markov Decisions

In industrial applications, the processes of optimal sequential decision making are naturally formulated and optimized within a standard setting of Markov decision theory. In practice, however, decisions must be made under incomplete and uncertain information about parameters and transition probabilities. This situation occurs when a system may suffer a regime switch changing not only the transition probabilities but also the control costs. After such an event, the effect of the actions may turn to the opposite, meaning that all strategies must be revised. Due to practical importance of this problem, a variety of methods has been suggested, ranging from incorporating regime switches into Markov dynamics to numerous concepts addressing model uncertainty. In this work, we suggest a pragmatic and practical approach using a natural re-formulation of this problem as a so-called convex switching system, we make efficient numerical algorithms applicable.

Many-server asymptotics for join-the-shortest-queue: Large deviations and rare events

The join-the-shortest-queue routing policy is studied in an asymptotic regime where the number of processors n scales with the arrival rate. A large deviation principle (LDP) for the occupancy process is established, as n→∞, in a suitable infinite-dimensional path space. Model features that present technical challenges include, Markovian dynamics with discontinuous statistics, a diminishing rate property of the transition probability rates, and an infinite-dimensional state space. The difficulty is in the proof of the Laplace lower bound which requires establishing the uniqueness of solutions of certain infinite-dimensional systems of controlled ordinary differential equations. The LDP gives information on the rate of decay of probabilities of various types of rare events associated with the system. We illustrate this by establishing explicit exponential decay rates for probabilities of long queues. In particular, denoting by Ejn(T) the event that there is at least one queue with j or more jobs at some time instant over [0,T], we show that, in the critical case, for large n and T, P(Ejn(T))≈exp[−n(j−2)24T].

A hidden Markov model reliably characterizes ketamine-induced spectral dynamics in macaque local field potentials and human electroencephalograms.

Ketamine is an NMDA receptor antagonist commonly used to maintain general anesthesia. At anesthetic doses, ketamine causes high power gamma (25-50 Hz) oscillations alternating with slow-delta (0.1-4 Hz) oscillations. These dynamics are readily observed in local field potentials (LFPs) of non-human primates (NHPs) and electroencephalogram (EEG) recordings from human subjects. However, a detailed statistical analysis of these dynamics has not been reported. We characterize ketamine’s neural dynamics using a hidden Markov model (HMM). The HMM observations are sequences of spectral power in seven canonical frequency bands between 0 to 50 Hz, where power is averaged within each band and scaled between 0 and 1. We model the observations as realizations of multivariate beta probability distributions that depend on a discrete-valued latent state process whose state transitions obey Markov dynamics. Using an expectation-maximization algorithm, we fit this beta-HMM to LFP recordings from 2 NHPs, and separately, to EEG recordings from 9 human subjects who received anesthetic doses of ketamine. Our beta-HMM framework provides a useful tool for experimental data analysis. Together, the estimated beta-HMM parameters and optimal state trajectory revealed an alternating pattern of states characterized primarily by gamma and slow-delta activities. The mean duration of the gamma activity was 2.2s([1.7,2.8]s) and 1.2s([0.9,1.5]s) for the two NHPs, and 2.5s([1.7,3.6]s) for the human subjects. The mean duration of the slow-delta activity was 1.6s([1.2,2.0]s) and 1.0s([0.8,1.2]s) for the two NHPs, and 1.8s([1.3,2.4]s) for the human subjects. Our characterizations of the alternating gamma slow-delta activities revealed five sub-states that show regular sequential transitions. These quantitative insights can inform the development of rhythm-generating neuronal circuit models that give mechanistic insights into this phenomenon and how ketamine produces altered states of arousal.