Detailed Balance Condition Research Articles

Spectral Convergence Bounds for Classical and Quantum Markov Processes

We introduce a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum Markov chains and they do not require additional assumptions like detailed balance, irreducibility or aperiodicity. We use the method in order to derive convergence bounds that improve significantly upon known spectral bounds. The core technical observation is that power-boundedness of transition maps of Markov chains enables a Wiener algebra functional calculus in order to upper bound any norm of any holomorphic function of the transition map. Finally, we discuss how general detailed balance conditions for quantum Markov processes lead to spectral convergence bounds.

Fast algorithm for relaxation processes in big-data systems.

Relaxation processes driven by a Laplacian matrix can be found in many real-world big-data systems, for example, in search engines on the World Wide Web and the dynamic load-balancing protocols in mesh networks. To numerically implement such processes, a fast-running algorithm for the calculation of the pseudoinverse of the Laplacian matrix is essential. Here we propose an algorithm which computes quickly and efficiently the pseudoinverse of Markov chain generator matrices satisfying the detailed-balance condition, a general class of matrices including the Laplacian. The algorithm utilizes the renormalization of the Gaussian integral. In addition to its applicability to a wide range of problems, the algorithm outperforms other algorithms in its ability to compute within a manageable computing time arbitrary elements of the pseudoinverse of a matrix of size millions by millions. Therefore our algorithm can be used very widely in analyzing the relaxation processes occurring on large-scale networked systems.

Nonequilibrium Statistical Mechanics for Adiabatic Piston Problem

We consider the dynamics of a freely movable wall of mass $M$ with one degree of freedom that separates a long tube into two regions, each of which is filled with rarefied gas particles of mass $m$. The gases are initially prepared at equal pressure but different temperatures, and we assume that the pressure and temperature of gas particles before colliding with the wall are kept constant over time in each region. We elucidate the energetics of the setup on the basis of the local detailed balance condition, and then derive the expression for the heat transferred from each gas to the wall. Furthermore, by using the condition, we obtain the linear response formula for the steady velocity of the wall and steady energy flux through the wall. Using perturbation expansion in a small parameter $\epsilon\equiv\sqrt{m/M}$, we calculate the steady velocity up to order $\epsilon$.

On Uniform Decay of the Entropy for Reaction–Diffusion Systems

This work provides entropy decay estimates for classes of nonlinear reaction–diffusion systems modeling reversible chemical reactions under the detailed-balance condition. We obtain explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the Log-Sobolev estimate and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: (i) vanishing diffusion constants in some chemical components and (ii) usage of different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.

Measuring the Irreversibility of Numerical Schemes for Reversible Stochastic Differential Equations

For a Markov process the detailed balance condition is equivalent to the time-reversibility of the process. For stochastic differential equations (SDE's) time discretization numerical schemes usually destroy the property of time-reversibility. Despite an extensive literature on the numerical analysis for SDE's, their stability properties, strong and/or weak error estimates, large deviations and infinite-time estimates, no quantitative results are known on the lack of reversibility of the discrete-time approximation process. In this paper we provide such quantitative estimates by using the concept of entropy production rate, inspired by ideas from non-equilibrium statistical mechanics. The entropy production rate for a stochastic process is defined as the relative entropy (per unit time) of the path measure of the process with respect to the path measure of the time-reversed process. By construction the entropy production rate is nonnegative and it vanishes if and only if the process is reversible. Crucially, from a numerical point of view, the entropy production rate is an {\em a posteriori} quantity, hence it can be computed in the course of a simulation as the ergodic average of a certain functional of the process (the so-called Gallavotti-Cohen (GC) action functional). We compute the entropy production for various numerical schemes such as explicit Euler-Maruyama and explicit Milstein's for reversible SDEs with additive or multiplicative noise. Additionally, we analyze the entropy production for the BBK integrator of the Langevin processes. We show that entropy production is an observable that distinguishes between different numerical schemes in terms of their discretization-induced irreversibility. Furthermore, our results show that the type of the noise critically affects the behavior of the entropy production rate.

The role of non-equilibrium fluxes in the relaxation processes of the linear chemical master equation.

We propose a non-equilibrium thermodynamical description in terms of the Chemical Master Equation (CME) to characterize the dynamics of a chemical cycle chain reaction among m different species. These systems can be closed or open for energy and molecules exchange with the environment, which determines how they relax to the stationary state. Closed systems reach an equilibrium state (characterized by the detailed balance condition (D.B.)), while open systems will reach a non-equilibrium steady state (NESS). The principal difference between D.B. and NESS is due to the presence of chemical fluxes. In the D.B. condition the fluxes are absent while for the NESS case, the chemical fluxes are necessary for the state maintaining. All the biological systems are characterized by their "far from equilibrium behavior," hence the NESS is a good candidate for a realistic description of the dynamical and thermodynamical properties of living organisms. In this work we consider a CME written in terms of a discrete Kolmogorov forward equation, which lead us to write explicitly the non-equilibrium chemical fluxes. For systems in NESS, we show that there is a non-conservative "external vector field" whose is linearly proportional to the chemical fluxes. We also demonstrate that the modulation of these external fields does not change their stationary distributions, which ensure us to study the same system and outline the differences in the system's behavior when it switches from the D.B. regime to NESS. We were interested to see how the non-equilibrium fluxes influence the relaxation process during the reaching of the stationary distribution. By performing analytical and numerical analysis, our central result is that the presence of the non-equilibrium chemical fluxes reduces the characteristic relaxation time with respect to the D.B. condition. Within a biochemical and biological perspective, this result can be related to the "plasticity property" of biological systems and to their capabilities to switch from one state to another as is observed during synaptic plasticity, cell fate determination, and differentiation.

On the flux-force partitioning and non-equilibrium thermodynamics near a steady state

The flux-force linear relationship is a basic building block in the development of irreversible thermodynamics near equilibrium. Here, we explore the fate of this relationship near a steady state in a finer detail by partitioning the fluxes and the forces into time-independent and time-dependent components. To this end, we use a master equation approach without assuming the condition of detailed balance. The connection of the flux-force components with various state functions and path functions provides a detailed picture of the variations of such quantities in terms of the deviation of probabilities from the steady state. Pilot calculations on an exactly-solvable case furnish insights into the energy-balance mechanism for non-equilibrium systems, revealing additionally how an out-of-equilibrium scenario can be favorable in realizing a minimized free energy state.

Population dynamics in presence of state dependent fluctuations

We discuss a model of a system of interacting populations for the case when: (i) the growth rates and the coefficients of interaction among the populations depend on the populations densities; and (ii) the environment influences the growth rates and this influence can be modelled by a Gaussian white noise. The system of model equations for this case is a system of stochastic differential equations with: (i) deterministic part in the form of polynomial nonlinearities; and (ii) state-dependent stochastic part in the form of multiplicative Gaussian white noise. We discuss both the cases when the formal integration of the stochastic differential equations leads: (i) to integrals of Itô kind; or (ii) to integrals of Stratonovich kind. The systems of stochastic differential equations are reduced to the corresponding Fokker–Planck equations. For the Itô case and for the case of 1 population analytic results are obtained for the stationary p.d.f. of the population density. For the case of more than one population and for both the Itô case and Stratonovich case the detailed balance conditions are not satisfied. As a result the exact analytic solutions of the corresponding Fokker–Planck equations for the stationary p.d.f.s for the population densities are not known. We obtain approximate solutions for this case by the method of adiabatic elimination.

Quantum fluctuations in modulated nonlinear oscillators

We use a modulated oscillator to study quantum fluctuations far from thermal equilibrium. A simple but important nonequilibrium effect that we discuss first is quantum heating, where quantum fluctuations lead to a finite-width distribution of a resonantly modulated oscillator over its quasienergy (Floquet) states. We also discuss the recent observation of quantum heating. We analyze large rare fluctuations responsible for the tail of the quasienergy distribution and switching between metastable states of forced vibrations. We find the most probable paths followed by the quasienergy in rare events, and in particular in switching. Along with the switching rates, such paths are observable characteristics of quantum fluctuations. As we show, they can change discontinuously once the detailed balance condition is broken. A different kind of quantum heating occurs where oscillators are modulated nonresonantly. Nonresonant modulation can also cause oscillator cooling. We discuss different microscopic mechanisms of these effects.

2P006 レプリカ置換分子動力学法の詳細釣り合い条件の有無に対する検証と生体分子への応用(01A. 蛋白質:構造,ポスター,第52回日本生物物理学会年会(2014年度))

2P006 レプリカ置換分子動力学法の詳細釣り合い条件の有無に対する検証と生体分子への応用(01A. 蛋白質:構造,ポスター,第52回日本生物物理学会年会(2014年度))

An irreversible Markov-chain Monte Carlo method with skew detailed balance conditions

An irreversible Markov-chain Monte Carlo (MCMC) method based on a skew detailed balance condition is discussed. Some recent theoretical works concerned with the irreversible MCMC method are reviewed and the irreversible Metropolis-Hastings algorithm for the method is described. We apply the method to ferromagnetic Ising models in two and three dimensions. Relaxation dynamics of the order parameter and the dynamical exponent are studied in comparison to those with the conventional reversible MCMC method with the detailed balance condition. We also examine how the efficiency of exchange Monte Carlo method is affected by the combined use of the irreversible MCMC method.

Energy consumption and entropy production in a stochastic formulation of BCM learning

Biochemical processes in living cells are open systems, therefore they exchange materials with their environment and they consume chemical energy. These processes are molecular-based and for that reason the role of fluctuations can not be ignored and the stochastic description is the most appropriate one. The chemical master equation describes in exact way the probabilistic dynamics of a given discrete set of states and helps us to understand and clarify the differences between closed and open systems. A closed system is related to a condition of detailed balance (DB), i.e. an equilibrium state. After a sufficiently long period, an open system will reach a non-equilibrium steady state (NESS) that is sustained by a flux of external energy. We demonstrate that two implementations of the BCM learning rule (BCM82) and (BCM92) are, respectively, always in DB, and never in DB. We define a one parameter parametrization of the BCM learning rule that interpolates between these two extremes. We compute thermodynamical quantities such as internal energy, free energy (both Helmholtz and Gibbs) and entropy. The entropy variation in the case of open systems (i.e. when DB does not hold) can be divided into internal entropy production and entropy exchanged with surroundings. We show how the entropy variation can be used to find the optimal value (corresponding to increased robustness and stability) for the parameter used in the BCM parametrization. Finally, we use the calculation of the work to drive the system from an initial state to the steady state as the parameter of the plasticity of the system.

A Parallel Tempering algorithm for probabilistic sampling and multimodal optimization

AbstractNon-linear inverse problems in the geosciences often involve probabilistic sampling of multimodal density functions or global optimization and sometimes both. Efficient algorithmic tools for carrying out sampling or optimization in challenging cases are of major interest. Here results are presented of some numerical experiments with a technique, known as Parallel Tempering, which originated in the field of computational statistics but is finding increasing numbers of applications in fields ranging from Chemical Physics to Astronomy. To date, experience in use of Parallel Tempering within earth sciences problems is very limited. In this paper, we describe Parallel Tempering and compare it to related methods of Simulated Annealing and Simulated Tempering for optimization and sampling, respectively. A key feature of Parallel Tempering is that it satisfies the detailed balance condition required for convergence of Markov chain Monte Carlo (McMC) algorithms while improving the efficiency of probabilistic sampling. Numerical results are presented on use of Parallel Tempering for trans-dimensional inversion of synthetic seismic receiver functions and also the simultaneous fitting of multiple receiver functions using global optimization. These suggest that its use can significantly accelerate sampling algorithms and improve exploration of parameter space in optimization. Parallel Tempering is a meta-algorithm which may be used together with many existing McMC sampling and direct search optimization techniques. It's generality and demonstrated performance suggests that there is significant potential for applications to both sampling and optimization problems in the geosciences.

Reformulation and Solution of the Master Equation for Multiple-Well Chemical Reactions

We consider an alternative formulation of the master equation for complex-forming chemical reactions with multiple wells and bimolecular products. Within this formulation the dynamical phase space consists of only the microscopic populations of the various isomers making up the reactive complex, while the bimolecular reactants and products are treated equally as sources and sinks. This reformulation yields compact expressions for the phenomenological rate coefficients describing all chemical processes, i.e., internal isomerization reactions, bimolecular-to-bimolecular reactions, isomer-to-bimolecular reactions, and bimolecular-to-isomer reactions. The applicability of the detailed balance condition is discussed and confirmed. We also consider the situation where some of the chemical eigenvalues approach the energy relaxation time scale and show how to modify the phenomenological rate coefficients so that they retain their validity.

Entropic Fluctuations of Quantum Dynamical Semigroups

We study a class of finite dimensional quantum dynamical semigroups exp(tL) whose generators L are sums of Lindbladians satisfying the detailed balance condition. Such semigroup arise in the weak coupling (van Hove) limit of Hamiltonian dynamical systems describing open quantum systems out of equilibrium. We prove a general entropic fluctuation theorem for this class of semigroups by relating the cumulant generating function of entropy transport to the spectrum of a family of deformations of the generator L. We show that, besides the celebrated Evans-Searles symmetry, this cumulant generating function also satisfies the translation symmetry recently discovered by Andrieux et al., and that in the linear regime near equilibrium these two symmetries yield Kubo's and Onsager's linear response relations.

Violation of detailed balance accelerates relaxation

Recent studies have experienced the acceleration of convergence in Markov chain Monte Carlo methods implemented by the systems without detailed balance condition (DBC). However, such advantage of the violation of DBC has not been confirmed in general. We investigate the effect of the absence of DBC on the convergence toward equilibrium. Surprisingly, it is shown that the DBC violation always makes the relaxation faster. Our result implies the existence of a kind of thermodynamic inequality that connects the nonequilibrium process relaxing toward steady state with the relaxation process which has the same probability distribution as its equilibrium state.

Guaranteeing total balance in Metropolis algorithm Monte Carlo simulations

The condition of detailed balance has long been used as a proxy for the more difficult-to-prove condition of total balance, which along with ergodicity is required to guarantee convergence of a Markov Chain Monte Carlo (MCMC) simulation to the correct probability distribution. However, some simple-to-program update schemes such as the sequential and checkerboard Metropolis algorithms are known not to satisfy detailed balance for such common systems as the Ising model.It has been an open question whether these update schemes satisfy the weaker condition of total balance. In this work, we show that under fairly broad conditions, a large class of update schemes for the Metropolis algorithm, including the sequential and checkerboard schemes, do indeed satisfy total balance for important distributions. We also show that detailed balance itself can be satisfied by straightforward modifications to these schemes.

Entropy production along nonequilibrium quantum jump trajectories

For classical nonequilibrium systems, the separation of the total entropy production into the adiabatic and nonadiabatic contributions is useful for understanding irreversibility in nonequilibrium thermodynamics. In this paper, we formulate quantum analogues for driven open quantum systems describable by quantum jump trajectories by applying a quantum stochastic thermodynamics. Our main conclusions are based on a quantum formulation of the local detailed balance condition.

On the flexibility of the design of multiple try Metropolis schemes

The Multiple Try Metropolis (MTM) method is a generalization of the classical Metropolis-Hastings algorithm in which the next state of the chain is chosen among a set of samples, according to normalized weights. In the literature, several extensions have been proposed. In this work, we show and remark upon the flexibility of the design of MTM-type methods, fulfilling the detailed balance condition. We discuss several possibilities and show different numerical results.

Enhanced exchange algorithm without detailed balance condition for replica exchange method

The replica exchange method (REM) is a powerful tool for the conformational sampling of biomolecules. In this study, we propose an enhanced exchange algorithm for REM not meeting the detailed balance condition (DBC), but satisfying the balance condition in all considered exchanges between two replicas. Breaking the DBC can minimize the rejection rate and make an exchange process rejection-free as the number of replicas increases. To enhance the efficiency of REM, all possible pairs--not only the nearest neighbor--were considered in the exchange process. The test simulations of the alanine dipeptide confirmed the correctness of our method. The average traveling distance of each replica in the temperature distribution was also increased in proportion to an increase in the exchange rate. Furthermore, we applied our algorithm to the conformational sampling of the 10-residue miniprotein, chignolin, with an implicit solvent model. The results showed a faster convergence in the calculation of its free energy landscape, compared to that achieved using the normal exchange method of adjacent pairs. This algorithm can also be applied to the conventional near neighbor method and is expected to reduce the required number of replicas.