Master Equation For System Research Articles

Raman transitions between hyperfine clock states in a magnetic trap

We present our experimental investigation of an optical Raman transition between the magnetic clock states of $^{87}$Rb in an atom chip magnetic trap. The transfer of atomic population is induced by a pair of diode lasers which couple the two clock states off-resonantly to an intermediate state manifold. This transition is subject to destructive interference of two excitation paths, which leads to a reduction of the effective two-photon Rabi-frequency. Furthermore, we find that the transition frequency is highly sensitive to the intensity ratio of the diode lasers. Our results are well described in terms of light shifts in the multi-level structure of $^{87}$Rb. The differential light shifts vanish at an optimal intensity ratio, which we observe as a narrowing of the transition linewidth. We also observe the temporal dynamics of the population transfer and find good agreement with a model based on the system's master equation and a Gaussian laser beam profile. Finally, we identify several sources of decoherence in our system, and discuss possible improvements.

Rovibrational energy transfer and dissociation in O2-O collisions.

A set of state-specific transition rates for each rovibrational level is generated for the O2(X(3)Σ(g)(-))-O(3)P system using the quasi-classical trajectory method at temperatures observed in hypersonic flows. A system of master equations describes the relaxation of the rovibrational ensemble to thermal equilibrium under ideal heat bath conditions at a constant translational temperature. Vibrational and rotational relaxation times, obtained from the average internal energies, exhibit a pattern inherent in a chemically reactive collisional pair. An intrinsic feature of the O3 molecular system with a large attractive potential is a weak temperature dependence of the rovibrational transition rates. For this reason, the quasi-steady vibrational and rotational temperatures experience a maximum at increasing translational temperature. The energy rate coefficients, that characterize the average loss of internal energy due to dissociation, quickly diminish at high temperatures, compared to other molecular systems.

Master Equation Study of Vibrational and Rotational Relaxations of Oxygen

Transition rates in collisions for each vibrational and rovibrational state are generated by means of the quasi-classical trajectory method using potential energy surfaces of different fidelities. The first potential energy surface, obtained via the double many-body expansion method, is adopted to obtain vibrational transition rates assuming transrotational equilibrium. The trajectory simulation on a simpler potential surface, based on the two-body pairwise interaction, generates a complete set of rates for each internal state. Vibrational and rotational relaxations of oxygen in a heat bath of parent atoms are modeled by a system of master equations at translational temperatures between 1000 and 20,000 K. It is shown that the vibrational relaxation becomes less efficient at high temperatures, in contrast with the conventional equation for relaxation time proposed by Millikan and White (“Systematics of Vibrational Relaxation,” Journal of Chemical Physics, Vol. null, No. null, 1963, Paper 3209). Rotational and vibrational relaxation times are the same order of magnitude in the range of temperatures observed in hypersonic flows. The excitation of the vibrational mode in collisions under typical postshock conditions occurs earlier than in other molecular systems. The exchange channel plays an important role in the internal energy randomization.

Mean-field approximation of counting processes from a differential equation perspective

Deterministic limit of a class of continuous time Markov chains is considered based purely on differential equation techniques. Starting from the linear system of master equations, ordinary differential equations for the moments and a partial differential equation, called Fokker–Planck equation, for the distribution is derived. Introducing closures at the level of the second and third moments, mean-field approximations are introduced. The accuracy of the mean-field approximations and the Fokker–Planck equation is investigated by using two differential equation-based and an operator semigroup-based approach.

High fidelity modeling of thermal relaxation and dissociation of oxygen

A master equation study of vibrational relaxation and dissociation of oxygen is conducted using state-specific O2–O transition rates, generated by extensive trajectory simulations. Both O2–O and O2–O2 collisions are concurrently simulated in the evolving nonequilibrium gas system under constant heat bath conditions. The forced harmonic oscillator model is incorporated to simulate the state-to-state relaxation of oxygen in O2–O2 collisions. The system of master equations is solved to simulate heating and cooling flows. The present study demonstrates the importance of atom-diatom collisions due to the extremely efficient energy randomization in the intermediate O3 complex. It is shown that the presence of atomic oxygen has a significant impact on vibrational relaxation time at temperatures observed in hypersonic flow. The population of highly-excited O2 vibrational states is affected by the amount of atomic oxygen when modeling the relaxation under constant heat bath conditions. A model of coupled state-to-state vibrational relaxation and dissociation of oxygen is also discussed.

Master Equation Analysis of Post Normal Shock Waves of Nitrogen

One-dimensional post normal shock flow calculations are carried out using state-of-the-art thermochemical nonequilibrium models. Two-temperature, four-temperature, and electronic master equation coupling models are adopted in the present work. In the four-temperature model, the rotational nonequilibrium is described by Parker and modified Park models. In the electronic master equation coupling model, recently evaluated electron and heavy-particle impacts and radiative transition cross-sections are employed in constructing the system of electronic master equations. In analyzing the shock-tube experiments, the results calculated by the state-of-the-art thermochemical nonequilibrium models are compared with existing shock-tube experimental data. The four-temperature and electronic master equation coupling models with rotational nonequilibrium described by the modified Park model approximately reproduce the measured rotational, vibrational, and electronic temperatures.

Thermochemical nonequilibrium analysis of O2+Ar based on state-resolved kinetics

The thermochemical nonequilibrium of the three lowest lying electronic states of molecular oxygen, O2(X3Σg-,a1Δg,b1Σg+), through interactions with argon is studied in the present work. The multi-body potential energy surfaces of O2+Ar are evaluated from the semi-classical RKR potential of O2 in each electronic state. The rovibrational states and energies of each electronic state are calculated by the quantum mechanical method based on the present inter-nuclear potential of O2. Then, the complete sets of the rovibrational state-to-state transition rate coefficients of O2+Ar are calculated by the quasi-classical trajectory method including the quasi-bound states. The system of master equations constructed by the present state-to-state transition rate coefficients are solved to analyze the thermochemical nonequilibrium of O2+Ar in various heat bath conditions. From these studies, it is concluded that the vibrational relaxation and coupled chemical reactions of each electronic state needs to be treated as a separate nonequilibrium process, and rotational nonequilibrium needs to be considered at translational temperatures above 10,000K.

State-resolved master equation analysis of thermochemical nonequilibrium of nitrogen

Using a NASA database of state-to-state transition rates for N+N2, master equation studies are performed for various nonequilibrium heat bath conditions. In these master equation studies, relaxation of the rotational and vibrational modes, time variation of chemical composition, reaction rate coefficients, and average rotational and vibrational energy losses due to dissociation are each considered in strong and weak nonequilibrium conditions. A system of master equations is coupled with one-dimensional flow equations to analyze the relaxation of N2 in post-normal shock flows. From the results of master equations and the post-normal shock calculations, it is recommended that the rotational nonequilibrium of N2 should be treated as a nonequilibrium mode in hypersonic re-entry calculations.

State-resolved thermochemical nonequilibrium analysis of hydrogen mixture flows

The complete sets of state-to-state transition rate coefficients for both target and projectile molecules of hydrogen are derived from the predicted response surface designed by the ordinary Kriging model. A system of master equations is constructed for bound-bound and bound-free transitions with these designed transition rate coefficients, and the rovibrational number densities are numerically evaluated by implicitly integrating a system of master equations. In these master equation studies, relaxation of rotation and vibration modes, number density relaxation, reaction rate coefficients, and average rotational and vibrational energy losses due to dissociation are each considered in strong nonequilibrium conditions. A system of master equations is coupled with one-dimensional flow equations to analyze the relaxations of hydrogen in post-normal shock and nozzle expanding flows. In post-normal shock flows, at high temperature, the relaxation of the rotational mode is similar to the relaxation of the vibrational mode. In nozzle expanding flows, the relaxations of both rotational and vibrational modes appear to be frozen.

Splitting of the rate matrix as a definition of time reversal in master equation systems

Motivated by recent progress in nonequilibrium fluctuation relations, we present a generalized time reversal for stochastic master equation systems with discrete states, which is defined as a splitting of the rate matrix into irreversible and reversible parts. An immediate advantage of this definition is that a variety of fluctuation relations can be attributed to different matrix splittings. Additionally, we find that the accustomed total entropy production formula and conditions of the detailed balance must be modified appropriately to account for the reversible rate part, which was previously ignored.

Macroscopic constraints for the minimum entropy production principle

In an essential and quite general setup, based on networks, we identify Schnakenberg's observables as the constraints that prevent a system from relaxing to equilibrium, showing that, in the linear regime, steady states satisfy a minimum entropy production principle. The result is applied to master equation systems, opening a new path to a well-known version of the principle regarding invariant states. Moreover, with the aid of a simple example, the principle is shown to conform to Prigogine's original formulation. Finally, we discuss analogies and differences with a recently proposed maximum entropy production principle.

Where is the pseudoscalar glueball?

The pseudoscalar mesons π(1300), K(1460), η(1295), η(1405) and η(1475) are assumed to form the meson decuplet which includes the glueball as the basis state supplementing the standard SU(3)F nonet of light states (q = u, d, s). The decuplet is investigated by using the algebraic approach based on the hypothesis of vanishing exotic commutators of SU(3)F ‘charges’ and their time derivatives. This leads to a system of master equations determining: (a) octet contents of the physical isoscalar mesons, (b) the mass formula relating all masses of the decuplet and (c) the mass ordering rule. The states of the physical isoscalar mesons η(1295), η(1405), η(1475) are expressed as superpositions of the ‘ideal’ (N and S) states and the glueball G one. The ‘mixing matrix’ realizing transformation from the unphysical states to the physical ones follows from the octet contents and is totally expressed by the decuplet meson masses. Among four one-parameter families of the resulting mixing matrices (multitude of the solutions arising from bad quality of data on the π(1300) and K(1460) meson masses) there is a family attributing the glueball-dominated composition to the η(1405) meson. The pseudoscalar decuplet is similar in some respects to the scalar one: both are composed of the excited states and G; the mass ordering of their N, S, G—dominated isoscalars is the same. Contrary to the lattice QCD and other predictions, the mass of the pseudoscalar pure glueball state is smaller than the scalar one.

Thermal dissipation effects of the entangled photon generated by four-wave mixing process in a nonlinear ring resonator

We firstly analyze the thermal dissipation effects of the entangled photons generated by a nonlinear optical ring resonator. To obtain the corresponding equation of motion of the entangled photons generated by a four-wave mixing process within the system, we propose the Markov approximation to repel the reservoir operators. The system master equation in the interaction picture for both degenerate and non-degenerate cases is analyzed and obtained. The established system can be used to characterize the optimum entangled photons in some cases where the thermal dissipation effects may be introduced noise into the system. In this work, the entangled photons can be generated into two forms, firstly, the two entangled photon states is generated, the other, the four entangled photon states can be easily obtained. Results obtained have shown that the optimum entangled photon in term of entangled photon visibility can be compensated (i.e. unchanged) under thermal dissipation effects.

Master Equation Study and Nonequilibrium Chemical Reactions for Hydrogen Molecule

The state-to-state cross sections and rates for collisional transitions of rotational and vibrational states were first calculated by using quasi-classical trajectory calculations for H 2 + H 2 collisions. Accuracy of these calculations was verified by comparing them with the results of quantum mechanical and other quasi-classical trajectory calculations. A system of master equations was constructed for a total of 348 ro-vibrational states with these rate coefficients. Unlike in existing works, the internal states of the colliding particles were assumed to be distributed under a Boltzmann distribution specified by a nonequilibrium temperature. The nonequilibrium temperature was in turn determined from the ro-vibrational energy contents. From the results of the master equation calculation, the relaxation rate of vibrational and rotational modes, rate of relaxation of number densities, the average ro-vibrational energy transferred during dissociation, the quasi-steady state rate coefficients, and two-temperature rate coefficients were derived. The present results are different from those of Sharma (Rotational Relaxation of Molecular Hydrogen at Moderate Temperatures, Journal of Thermophysics and Heat Transfer, Vol. 8, No. 1,1994, pp. 35-39) and of Furudate et al. (Coupled Rotational-Vibrational Relaxation of Molecular Hydrogen at High Temperatures, Journal of Thermophysics and Heat Transfer, Vol. 20, No. 3, 2006, pp. 457-464), and agree closer with existing experimental data.

A Stochastic Approach to Coarsening of Cellular Networks

We prove existence for a system of master equations modeling grain growth proposed in [K. Barmak et al., Remarks on a multiscale approach to grain growth in polycrystals, in Variational Problems in Materials Science, Birkhäuser, Basel, 2006, pp. 1–11]. The method used to prove existence for these nonlinear, integro-PDEs employs tools from both functional analysis and probability. The stochastic interpretation for the evolution of the mesoscopic properties of this large scale metastable system we provide here is general enough to apply to other cellular networks. This approach leads to an internally consistent theory and opens a new discussion on asymptotics for the distribution of grains in a two-dimensional network, which is the subject of future work.

Random dynamics of gene transcription activation in single cells

The recent measurements of gene transcription activity at single cell resolution revealed that genes are often transcribed randomly and discontinuously. In order to elucidate how the environmental signals contribute to the stochasticity of gene transcription, a random transition model was recently proposed [M. Tang, The mean and noise of stochastic gene transcription, J. Theor. Biol. 253 (2008) 271–280; M. Tang, The mean frequency of transcriptional bursting and its variation in single cells, J. Math. Biol. (2009) doi:10.1007/s00285-009-0258-7, in press; published online: March 10, 2009]. In this model it is assumed that the transcription system transits randomly between three different functional states, quantifying the timing and strength of gene transcription by a sequence of probability functions P n x ( t ) , coupled in an infinite differential system of master equations. Here n ⩾ 1 are integers and x specifies each of the three functional states. In this work we further study this model aiming to understand the stochastic dynamics of gene transcription. When n ⩽ 3 , the exact form of P n x ( t ) is found analytically by solving the system of master equations. For larger n however, it is unfeasible to find P n x ( t ) explicitly, so we explore the properties of probability functions by analyzing the master operator L that transforms P ( n − 1 ) x ( t ) to P n x ( t ) . We prove that L “mollifies” the behavior of P ( n − 1 ) x ( t ) by increasing its order of differentiability and by flattening its growth globally. We also show that the n-th cycle of transcription activity condenses at distinct peak instants, with a decreasing peak strength with respect to n. The timings of these peak instants are estimated and several further open questions toward a general theory are discussed.

Kinetics of proton pumping in cytochrome c oxidase

We propose a simple model of cytochrome c oxidase, including four redox centers and four protonable sites, to study the time evolution of electrostatically coupled electron and proton transfers initiated by the injection of a single electron into the enzyme. We derive a system of master equations for electron and proton state probabilities and show that an efficient pumping of protons across the membrane can be obtained for a reasonable set of parameters. All four experimentally observed kinetic phases appear naturally from our model. We also calculate the dependence of the pumping efficiency on the transmembrane voltage at different temperatures and discuss a possible mechanism of the redox-driven proton translocation.

Exact reductions of Markovian dynamics for ion channels with a single permissive state

We show that the cooperative model for the kinetics of a tetrameric potassium ion channel derived in Nekouzadeh et al. (Biophys J 95(7):3510-3520, 2008) is an invariant manifold reduction of the full master equation for the channel kinetics. We further establish the validity of this reduction for ion channel models consisting of multiple independent subunits with cooperative transitions from a single permissive state to a conducting state. Finally, we conclude that solutions of the reduced model are globally asymptotically stable solutions of the full master equation system.

Listening to the noise: random fluctuations reveal gene network parameters

The cellular environment is abuzz with noise originating from the inherent random motion of reacting molecules in the living cell. In this noisy environment, clonal cell populations show cell-to-cell variability that can manifest significant phenotypic differences. Noise-induced stochastic fluctuations in cellular constituents can be measured and their statistics quantified. We show that these random fluctuations carry within them valuable information about the underlying genetic network. Far from being a nuisance, the ever-present cellular noise acts as a rich source of excitation that, when processed through a gene network, carries its distinctive fingerprint that encodes a wealth of information about that network. We show that in some cases the analysis of these random fluctuations enables the full identification of network parameters, including those that may otherwise be difficult to measure. This establishes a potentially powerful approach for the identification of gene networks and offers a new window into the workings of these networks.

Thermal dissipation effects of the entangled photon generated by a nonlinear ring resonator

We firstly analyze the thermal dissipation effects of the entangled photons generated by a nonlinear optical ring resonator. To obtain the corresponding equation of motion of the entangled photons generated by a four-wave mixing process within the system, we propose the Markov approximation to repel the reservoir operators. The system master equation in the interaction picture for both degenerate and non-degenerate cases is analyzed and obtained. The established system can be used to characterize the optimum entangled photons in some cases where the thermal dissipation effects may introduce noise into the system. In this work, the entangled photons can be generated into two forms, firstly, the two entangled photon states are generated and, the other, the four entangled photon states, can be easily obtained. Results obtained have shown that the optimum entangled photon in terms of entangled photon visibility can be compensated (i.e. unchanged) under thermal dissipation effects.