Coarse-grained Master Equation Research Articles

Non-additive large deviation function for the particle densities of driven systems in contact

We investigate the non-equilibrium large deviation function of the particle densities in two steady-state driven systems exchanging particles at a vanishing rate. We first derive through a systematic multi-scale analysis the coarse-grained master equation satisfied by the distribution of the numbers of particles in each system. Assuming that this distribution takes for large systems a large deviation form, we obtain the equation (similar to a Hamilton–Jacobi equation) satisfied by the large deviation function of the densities. Depending on the systems considered, this equation may satisfy or not the macroscopic detailed balance property, i.e., a time-reversibility property at large deviation level. In the absence of macroscopic detailed balance, the large deviation function can be determined as an expansion close to a solution satisfying macroscopic detailed balance. In this case, the large deviation function is generically non-additive, i.e., it cannot be split as two separate contributions from each system. In addition, the large deviation function can be interpreted as a non-equilibrium free energy, as it satisfies a generalization of the second law of thermodynamics, in the spirit of the Hatano–Sasa relation. Some of the results are illustrated on an exactly solvable driven lattice gas model.

Completely positive master equation for arbitrary driving and small level spacing

Markovian master equations are a ubiquitous tool in the study of open quantum systems, but deriving them from first principles involves a series of compromises. On the one hand, the Redfield equation is valid for fast environments (whose correlation function decays much faster than the system relaxation time) regardless of the relative strength of the coupling to the system Hamiltonian, but is notoriously non-completely-positive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with a finite level spacing, which makes it incompatible with arbitrarily fast time-dependent driving. Here we show that a recently derived Markovian coarse-grained master equation (CGME), already known to be completely positive, has a much expanded range of applicability compared to the Davies equation, and moreover, is locally generated and can be generalized to accommodate arbitrarily fast driving. This generalization, which we refer to as the time-dependent CGME, is thus suitable for the analysis of fast operations in gate-model quantum computing, such as quantum error correction and dynamical decoupling. Our derivation proceeds directly from the Redfield equation and allows us to place rigorous error bounds on all three equations: Redfield, Davies, and coarse-grained. Our main result is thus a completely positive Markovian master equation that is a controlled approximation to the true evolution for any time-dependence of the system Hamiltonian, and works for systems with arbitrarily small level spacing. We illustrate this with an analysis showing that dynamical decoupling can extend coherence times even in a strictly Markovian setting.

Accuracy assessment of perturbative master equations: Embracing nonpositivity

The reduced dynamics of an open quantum system obtained from an underlying microscopic Hamiltonian can in general only approximately be described by a time local master equation. The quality of that approximation depends primarily on the coupling strength and the structure of the environment. Various such master equations have been proposed with different aims. Choosing the most suitable one for a specific system is not straight forward. By focusing on the accuracy of the reduced dynamics we provide a thorough assessment for a selection of methods (Redfield Equation, Quantum Optical Master Equation, Coarse-Grained Master Equation, a related dynamical map approach and a partial-secular approximation). Whether or not an approach guarantees positivity we consider secondary, here. We use two qubits coupled to a Lorentzian environment in a spin-boson like fashion modeling a generic situation with various system and bath time scales. We see that, independent of the initial state, the simple Redfield Equation with time dependent coefficients is significantly more accurate than all other methods under consideration. We emphasize that positivity violation in the Redfield formalism becomes relevant only in a regime where any of the perturbative master equations considered here are rendered invalid anyway. This implies that the loss of positivity should in fact be welcomed as an important feature: it indicates the breakdown of the weak coupling assumption. In addition we present the various approaches in a self-contained way and use the behavior of their errors to provide further insight into the range of validity of each method.

Nonequilibrium dynamics with finite-time repeated interactions.

We study quantum dynamics in the framework of repeated interactions between a system and a stream of identical probes. We present a coarse-grained master equation that captures the system's dynamics in the natural regime where interactions with different probes do not overlap, but it is otherwise valid for arbitrary values of the interaction strength and mean interaction time. We then apply it to some specific examples. For probes prepared in Gibbs states, such channels have been used to describe thermalization: while this is the case for many choices of parameters, for others one finds out-of-equilibrium states including inverted Gibbs and maximally mixed states. Gapless probes can be interpreted as performing an indirect measurement, and we study the energy transfer associated with this measurement.

Effective formalism for open-quantum-system dynamics: Time-coarse-graining approach

We formulate an effective-description framework for the dynamics of open quantum systems by extending the time-coarse-graining formalism to open systems. Our coarse-graining procedure efficiently removes high-frequency processes which are responsible for coherences between lower- and upper-manifold states and are irrelevant when considering only low-energy dynamics. We investigate the regime of validity of the resulting coarse-grained master equation by applying it to multi-level atoms driven by far-detuned lasers. Except for such high-frequency coherences, we find good agreement between the exact and coarse-grained dynamics unless the driving lasers are too strong or the initial high-frequency coherences are sizable.

Nanopore electric snapshots of an RNA tertiary folding pathway

The chemical properties and biological mechanisms of RNAs are determined by their tertiary structures. Exploring the tertiary structure folding processes of RNA enables us to understand and control its biological functions. Here, we report a nanopore snapshot approach combined with coarse-grained molecular dynamics simulation and master equation analysis to elucidate the folding of an RNA pseudoknot structure. In this approach, single RNA molecules captured by the nanopore can freely fold from the unstructured state without constraint and can be programmed to terminate their folding process at different intermediates. By identifying the nanopore signatures and measuring their time-dependent populations, we can “visualize” a series of kinetically important intermediates, track the kinetics of their inter-conversions, and derive the RNA pseudoknot folding pathway. This approach can potentially be developed into a single-molecule toolbox to investigate the biophysical mechanisms of RNA folding and unfolding, its interactions with ligands, and its functions.

A photonic Carnot engine powered by a spin-star network

We propose a spin-star network, where a central spin-(1/2), acting as a quantum fuel, is coupled to N outer spin-(1/2) particles. If the network is in thermal equilibrium with a heat bath, the central spin can have an effective temperature, higher than that of the bath, scaling nonlinearly with N. Such temperature can be tuned with the anisotropy parameter of the coupling. Using a beam of such central spins to pump a micromaser cavity, we determine the dynamics of the cavity field using a coarse-grained master equation. We find that the central-spin beam effectively acts as a hot reservoir to the cavity field and brings it to a thermal steady state whose temperature benefits from the same nonlinear enhancement with N and results in a highly efficient photonic Carnot engine. The validity of our conclusions is tested against the presence of atomic and cavity damping using a microscopic master equation method for typical microwave cavity-QED parameters. The role played by quantum coherence and correlations on the scaling effect is pointed out. An alternative scheme where the spin-(1/2) is coupled to a macroscopic spin- particle is also discussed.

Coarse-grained kinetic Monte Carlo simulation of diffusion in alloys

We present a coarse-grained Monte Carlo method designed for the simulation of vacancy-mediated diffusion phenomena. A coarse-grained master equation is derived from the atomic scale master equation using an assumption of local equilibrium within each coarse-grained cell. Atomic kinetic Monte Carlo (AKMC) simulations are used to perform both the thermodynamic and the kinetic parametrization of the coarse-grained simulations. Quantitative reproduction of flux couplings is achieved in the coarse-grained simulation. The ability of the CKMC method to simulate kinetics controlled by diffusion such as a precipitate dissolution and the decay of sinusoidal modulations of the concentration field is illustrated on body-centered-cubic (bcc) model alloys with a clustering tendency. It is shown on the case of a model of the Fe-Cu alloy previously developed that the use of this method can reduce the computational cost of a kinetic simulation by several orders of magnitude.

Time-reversal symmetry relation for nonequilibrium flows ruled by the fluctuating Boltzmann equation

A time-reversal symmetry relation is established for out-of-equilibrium dilute or rarefied gases described by the fluctuating Boltzmann equation. The relation is obtained from the associated coarse-grained master equation ruling the random numbers of particles in cells of given position and velocity in the one-particle phase space. The symmetry relation concerns the fluctuating particle and energy currents of the gas flowing between reservoirs or thermalizing surfaces at given particle densities or temperatures.

Critical Dynamics of Dimers: Implications for the Glass Transition

The Adam-Gibbs view of the glass transition relates the relaxation time to the configurational entropy, which goes continuously to zero at the so-called Kauzmann temperature. We examine this scenario in the context of a dimer model with an entropy-vanishing phase transition and stochastic loop dynamics. We propose a coarse-grained master equation for the order parameter dynamics which is used to compute the time-dependent autocorrelation function and the associated relaxation time. Using a combination of exact results, scaling arguments, and numerical diagonalizations of the master equation, we find nonexponential relaxation and a Vogel-Fulcher divergence of the relaxation time in the vicinity of the phase transition. Since in the dimer model the entropy stays finite all the way to the phase transition point and then jumps discontinuously to zero, we demonstrate a clear departure from the Adam-Gibbs scenario. Dimer coverings are the "inherent structures" of the canonical frustrated system, the triangular Ising antiferromagnet. Therefore, our results provide a new scenario for the glass transition in supercooled liquids in terms of inherent structure dynamics.

A quasi-linear response theory of statistical heavy-ion collisions: (I). Basic formulation : Transport equations and roles of Green functions

We develop a time-dependent theory of heavy-ion collisions which consistently treats the relative and the intrinsic motions by coupled equations derived from the many-body von Neumann equation. The structure of the equations determining the mean trajectory and the fluctuations of the relative motion is the same as that of the corresponding equations in the known linear response theory. The present theory differs, however, from the linear response theory, in that it presumes neither weak coupling between the relative motion and the intrinsic excitations, nor the canonical distribution function for the density operator of the intrinsic motions. We apply the theory especially to deep inelastic collisions, where the relative motion couples to intrinsic excitations through a stochastic hamiltonian. Based on the stochastic assumption, we study the properties of the Green functions that take into account the higher order effects of the coupling hamiltonian. We then discuss, in particular, the effects of the Green functions on the time evolution of the intrinsic state, which is described in terms of a coarse-grained master equation, the friction tensor and fluctuation dissipation theorems.

Kinetics of clusters in a binary linear system

We consider the stochastically time-dependent behaviour of a binary linear chain of N units at temperature T and in an external field H. The kinetics is described in terms of clusters (sequences) of specified numbers of units in the same state. A coarse-grained master equation for the cluster densities on the chain is derived by an approximation due to Felderhof. We calculate the relaxation time of fluctuations, and the zero-field frequency-dependent susceptibility, of the fraction of units in a given state. Our results are compared to the phenomenological theory of irreversible thermodynamics.

Irreversible Statistical Mechanics of Polymer Chains. I. Fokker–Planck Diffusion Equation

A theory of the irreversible statistical mechanics of flexible polymer chains is developed on the basis of new ideas. The Brownian motion of polymer chains is assumed to be a Markoff random transition among their rotational isomeric states. The theory is described for ring polymer chains, for which the “normal coordinates” can be determined by the consideration of their symmetry alone. First, we derive the master equation which describes the discrete Brownian motion of a ring polymer chain. The master equation is averaged over all the configurations, fixed several normal coordinates to certain values. This averaging process is called “coarse graining.” By Taylor expansion of the coarse-grained master equation, we get a Fokker–Planck diffusion equation which is specified by two kinds of molecular constants, the diffusion constant Dα and the expansion parameter γα, both of which depend on the suffix α of the normal coordinates. For slow relaxation phenomena, the diffusion equation is reduced to that of the spring-bead model theory. The general behaviors of Dα and γα are discussed: The reduced diffusion constant D̃α(= Dα / D0) decreases rapidly as α increases, whereas γα2 / α2 remains roughly constant.