Mathematical formalism for isothermal linear irreversibility

Qian, Hong

doi:10.1098/rspa.2001.0811

Cited by 43 publications

(46 citation statements)

References 25 publications

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“…This realization led to the introduction of detailed balance (DB) as a necessary condition for Markov kinetic models of closed system [8]. The celebrated fluctuation-dissipation relation [9] is in fact an explicit expression of detailed balance in the Langevin dynamics [10]. When a molecular system is in an open environment with sustained chemical input and dissipation, detailed balance breaks down [2, 11], which gives rise to free energy transduction, such as in c EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx…”

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confidence: 99%

Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

Qian

Wang

2006

Europhys. Lett.

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Abstract. -Continuous time random walk (CTRW) is studied with a new dynamic equation based on the age-structure of states. For a CTRW in a closed molecular system, two necessary conditions for microscopic reversibility are introduced: 1) independence of transition direction and waiting time for every state and 2) detailed balance among the transition probabilities. Together they are also sufficient condition. For a CTRW in an open system with explicit chemical energy input 1) still holds while 2) breaks down. Hence, CTRW models not satisfying 1) are either inconsistent with thermodynamics or cannot attain equilibrium due to hidden dissipation in non-Markovian states. Each CTRW defines a unique corresponding Markov process (cMP). The steady-state distribution of a CTRW equals that of the corresponding Markov process, and the two systems have the same steady-state flux, the same exit probabilities and the same mean trapping times. Mechanicity is discussed; a paradox observed by Kolomeisky and Fisher (J. Chem. Phys., 113 (2000) 10867) is resolved.Stochastic models are the theoretical basis for describing dynamics of biological, chemical and physical processes at the single-molecule level [1,2]. Recent studies in single-molecule spectroscopy and enzymology have invigorated a classic subject, the continuous time random walks (CTRW), which was developed and extensively studied many years ago by Montroll and coworkers [3,4]. See [5][6][7] for the recent work that motivated CTRW in connection to single motor proteins and single-enzyme kinetics.Applying stochastic models to molecular procsses requires serious considerations of the microscopic reversibility. When a closed molecular system with fluctuations reaches its stationary state, it is necessarily a chemical equilibrium with zero flux in any part of the system. This realization led to the introduction of detailed balance (DB) as a necessary condition for Markov kinetic models of closed system [8]. The celebrated fluctuation-dissipation relation [9] is in fact an explicit expression of detailed balance in the Langevin dynamics [10]. When a molecular system is in an open environment with sustained chemical input and dissipation, detailed balance breaks down [2, 11], which gives rise to free energy transduction, such as in c EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx

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confidence: 99%

Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

Qian

Wang

2006

Europhys. Lett.

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“…The process is reversible and thus e p (t) ≡ 0 for all t ≥ 0, if and only if θ = 0 [34]. From now on, we only consider the case…”

Section: The Stochastic Process In Magnetic Fields and Its Eprmentioning

confidence: 99%

The large deviation principle and steady-state fluctuation theorem for the entropy production rate of a stochastic process in magnetic fields

Chen

Xiong

et al. 2016

Journal of Mathematical Physics

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Fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades. Steadystate fluctuation theorem of sample entropy production rate in terms of large deviation principle for diffusion processes have not been rigorously proved yet due to technical difficulties. Here we give a proof for the steady-state fluctuation theorem of a diffusion process in magnetic fields, with explicit expressions of the free energy function and rate function.The proof is based on the Karhunen-Loéve expansion of complex-valued 1 LDP of entropy production rate in magnetic fields 2 Ornstein-Uhlenbeck process.

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“…Starting from the work of Onsager (9), there has been extensive literature on dynamical behavior near a stable fixed point (21)(22)(23). The new construction clearly works for this important situation and has indeed offered a new angle.…”

Section: Stating the Problemmentioning

confidence: 99%

Structure of stochastic dynamics near fixed points

Kwon

Thouless

2005

Proc. Natl. Acad. Sci. U.S.A.

156

222

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We analyze the structure of stochastic dynamics near either a stable or unstable fixed point, where the force can be approximated by linearization. We find that a cost function that determines a Boltzmann-like stationary distribution can always be defined near it. Such a stationary distribution does not need to satisfy the usual detailed balance condition but might have instead a divergence-free probability current. In the linear case, the force can be split into two parts, one of which gives detailed balance with the diffusive motion, whereas the other induces cyclic motion on surfaces of constant cost function. By using the Jordan transformation for the force matrix, we find an explicit construction of the cost function. We discuss singularities of the transformation and their consequences for the stationary distribution. This Boltzmann-like distribution may be not unique, and nonlinear effects and boundary conditions may change the distribution and induce additional currents even in the neighborhood of a fixed point.Boltzmann distribution ͉ cost function ͉ detailed balance ͉ cyclic motion I n equilibrium statistical mechanics, the principle of detailed balance and the related fluctuation-dissipation theorem play important roles. Einstein used the principle that the excess energy that is put into each mode of an equilibrium system in the course of thermal fluctuations is also removed from the same mode by dissipative forces. This principle is implicit in his work on Brownian movement (1), and explicit in later works on the photoelectric effect (2), and on the relation between spontaneous and induced emission of electromagnetic radiation (3). It was formulated as the principle of detailed balance by Bridgman (4) and used to explain Johnson noise in electrical circuits by Nyquist (5). It is related to the fact that the same processes that drive fluctuations in the neighborhood of a typical equilibrium configuration also drive the configuration back towards a typical equilibrium or steady-state configuration when it is displaced from equilibrium by an amount that is small, but large compared with the fluctuations in thermal equilibrium. In this situation, the equilibrium distribution in phase space is just the Boltzmann distribution, proportional to e Ϫ␤E , where ␤ is inversely proportional to temperature, and E is the energy of the point in phase space. Configurations that differ significantly from those that contribute to the minimum of the free energy are driven back to the neighborhood of this minimum by dissipative effects such as thermal or electrical conduction or viscosity, and the magnitude of these effects is related to the equilibrium fluctuations of related variables.In many situations, there is no thermodynamic equilibrium, but external steady and fluctuating forces drive the system into a steady or very slowly varying state for which the principle of detailed balance does not hold. A light bulb powered by an external battery or a chemical reaction in which the reactants are introduced at a steady ra...

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Mathematical formalism for isothermal linear irreversibility

Abstract: We prove the equivalence among symmetricity, time reversibility, and zero entropy production of the station-

Cited by 43 publications

References 25 publications

Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

The large deviation principle and steady-state fluctuation theorem for the entropy production rate of a stochastic process in magnetic fields

Structure of stochastic dynamics near fixed points

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