Fluctuation theorem for nonequilibrium reactions

A fluctuation theorem is derived for stochastic nonequilibrium reactions ruled by the chemical master equation. The theorem is expressed in terms of the generating and large-deviation functions characterizing the fluctuations of a quantity which measures the loss of detailed balance out of thermodynamic equilibrium. The relationship to entropy production is established and discussed. The fluctuation theorem is verified in the Schl?gl model of far-from-equilibrium bistability.     

Entropy production of a mechanically driven single oligomeric enzyme: a consequence of fluctuation theorem

In this work we have shown how an applied mechanical force affects an oligomeric enzyme kinetics in a chemiostatic condition where the statistical characteristics of random walk of the substrate molecules over a finite number of active sites of the enzyme plays important contributing factors in governing the overall rate and nonequilibrium thermodynamic properties. The analytical results are supported by the simulation of single trajectory based approach of entropy production using Gillespie’s stochastic algorithm. This microscopic numerical approach not only gives the macroscopic entropy production from the mean of the distribution of entropy production which depends on the force but also a broadening of the distribution by the applied mechanical force, a kind of power broadening. In the nonequilibrium steady state (NESS), both the mean and the variance of the distribution increases and then saturates with the rise in applied force corresponding to the situation when the net rate of product formation reaches a limiting value with an activationless transition. The effect of the system-size and force on the entropy production distribution is shown to be constrained by the detailed fluctuation theorem.     

Influence of the reaction mechanism on the time course of the entropy production during reversible polymerization

The paper shows the influence of the reaction mechanism on the time course of the entropy production sigma in a closed system during reversible polymerization. We consider two different reaction mechanisms with (a) being a polycondensation and (b) a chain growth mechanism. For both mechanisms explicit expressions for the entropy production sigma as a function of time t are derived. To demonstrate the application of these general expressions we consider two different polymerization experiments where the reaction starts (i) from monomer molecules polymerizing to a defined number average chain length x(n,eq) and (ii) from monodisperse polymer molecules reacting with each other under the constraint that x(n) is the same at the beginning and the end of the reaction. In both cases we treat the system to be ideal and describe the kinetics of the reversible polymerization reactions using two kinetic constants for the forward and backward reactions, respectively. Under these assumptions the difference in the curvature of the entropy production sigma between a polycondensation and a chain growth mechanism is only marginal if the reaction starts from monomer molecules polymerizing to a defined number average chain length x(n,eq).     

Potential and flux decomposition for dynamical systems and non-equilibrium thermodynamics: curvature, gauge field, and generalized fluctuation-dissipation theorem

The driving force of the dynamical system can be decomposed into the gradient of a potential landscape and curl flux (current). The fluctuation-dissipation theorem (FDT) is often applied to near equilibrium systems with detailed balance. The response due to a small perturbation can be expressed by a spontaneous fluctuation. For non-equilibrium systems, we derived a generalized FDT that the response function is composed of two parts: (1) a spontaneous correlation representing the relaxation which is present in the near equilibrium systems with detailed balance and (2) a correlation related to the persistence of the curl flux in steady state, which is also in part linked to a internal curvature of a gauge field. The generalized FDT is also related to the fluctuation theorem. In the equal time limit, the generalized FDT naturally leads to non-equilibrium thermodynamics where the entropy production rate can be decomposed into spontaneous relaxation driven by gradient force and house keeping contribution driven by the non-zero flux that sustains the non-equilibrium environment and breaks the detailed balance. On any particular path, the medium heat dissipation due to the non-zero curl flux is analogous to the Wilson lines of an Abelian gauge theory.     

Theory for non-equilibrium statistical mechanics

This paper reviews a new theory for non-equilibrium statistical mechanics. This gives the non-equilibrium analogue of the Boltzmann probability distribution, and the generalization of entropy to dynamic states. It is shown that this so-called second entropy is maximized in the steady state, in contrast to the rate of production of the conventional entropy, which is not an extremum. The relationships of the new theory to Onsager's regression hypothesis, Prigogine's minimal entropy production theorem, the Langevin equation, the formula of Green and Kubo, the Kawasaki distribution, and the non-equilibrium fluctuation and work theorems, are discussed. The theory is worked through in full detail for the case of steady heat flow down an imposed temperature gradient. A Monte Carlo algorithm based upon the steady state probability density is summarized, and results for the thermal conductivity of a Lennard-Jones fluid are shown to be in agreement with known values. Also discussed is the generalization to non-equilibrium mechanical work, and to non-equilibrium quantum statistical mechanics. As examples of the new theory two general applications are briefly explored: a non-equilibrium version of the second law of thermodynamics, and the origin and evolution of life.     

Fluctuation theorem and mesoscopic chemical clocks

The fluctuation theorems for dissipation and the currents are applied to the stochastic version of the reversible Brusselator model of nonequilibrium oscillating reactions. It is verified that the symmetry of these theorems holds far from equilibrium in the regimes of noisy oscillations. Moreover, the fluctuation theorem for the currents is also verified for a truncated Brusselator model.     

Fluctuation theorem and Onsager reciprocity relations

The Onsager and higher-order reciprocity relations are derived from a fluctuation theorem for nonequilibrium reactions ruled by the chemical master equation. The fluctuation theorem is obtained for the generating function of the macroscopic fluxes between chemiostats maintaining the system in a nonequilibrium steady state. The macroscopic affinities associated with the fluxes are identified by graph theory. The Yamamoto-Zwanzig formulas for the reaction constants are also derived from the fluctuation theorem.     

受高斯白噪声影响的有限化学反应体系的随机热力学——外噪声对化学反应体系影响的有效热力学量度

通过分析噪声对跃迁概率的不同影响,借助Novikov定理及MSR理论,建立了受多源白噪声影响的有限化学反应体系的有效主方程及有效熵平衡方程,导出非平衡定态时这类噪声体系熵产生的一般表达式,揭示涨落熵产生的统计内涵及噪声贡献,并针对非宏观量级的外噪声,借助扰动按分布参数分离法及有效主方程的Kramers-Moyal展开,进一步对简单加合性噪声建立了非平衡定态宏观稳定性判据的随机模拟,论证了噪声对化学反应体系定态稳定性的弱化作用.     

The rectified second law of thermodynamics

Equilibrium thermodynamics is combined with Jarzynski's irreversible work theorem to quantify the excess entropy produced by irreversible processes. The resulting rectified form of the second law parallels the first law, in the sense that it facilitates the experimental measurement of excess entropy changes resulting from irreversible work and heat exchanges, just as the first law quantifies energy changes produced by either reversible or irreversible work and heat exchanges. The general form of the rectified second law is further applied to a broad class of quasi-static irreverisble (QSI) processes, for which all of the thermodynamic functions of both the system and surroundings remain continuously well-defined, thus facilitating excess entropy measurements by integrating exact differential functions along QSI paths. The results are illustrated by calculating the mechanical and thermal excess entropy produced by the irreversible unfolding of an RNA molecule.     

Stochastic Thermodynamics of Mesoscopic Electrochemical Reactions

In this work, we discussed the stochastic thermodynamics of mesoscopic electron transfer reactions between ions and electrodes. With a relationship between the reaction rate constant and the electrode potential, we find that the heat dissipation βq equals to the dynamic irreversibility of the reaction system minus an internal entropy change term. The total entropy change Δs_t is defined as the summation of the system entropy change Δs and the heat dissipation βq such that Δs_t=Δs+βq. Even though the heat dissipation depends linearly on the electrode potential, the total entropy change is found to satisfy the fluctuation theorem < (e)^-Δs_t>=1, and hence a second law-like inequality reads <Δs_t>≥0. Our study provides a practical methodology for the stochastic thermodynamics of electrochemical reactions, which may find applications in biochemical and electrochemical reaction systems.     

Entropy production during reversible polymerization in nonideal systems

A general route is shown to calculate the entropy production sigma as function of time t in a closed system during reversible polymerization. We treat the polymer molecules to behave nonideal and apply exemplarily the classical Flory-Huggins theory to get explicit expressions for the activity coefficient. At the beginning of the polymerization the system is in a nonequilibrium state where chemical reactions take place that irreversibly drive the system towards equilibrium with sigma approaching zero in the limit t-->infinity. The time-dependent course of the entropy production is explicitly calculated for two cases where the reaction starts (i) from monomer molecules polymerizing to a defined number average chain length xn,eq and (ii) from monodisperse polymer molecules reacting with each other under the constrain that xn is the same at the beginning and the end of the reaction. In both cases we find that the nature of the activity coefficient has an important effect on the curvature of sigma which may considerably differ from that of an ideal behavior.     

Photosynthetic models with maximum entropy production in irreversible charge transfer steps

Steady-state bacterial photosynthesis is modelled as cyclic chemical reaction and is examined with respect to overall efficiency, power transfer efficiency, and entropy production. A nonlinear flux–force relationship is assumed. The simplest two-state kinetic model bears complete analogy with the performance of an ideal (zero ohmic resistance of the P–N junction) solar cell. In both cases power transfer to external load is much higher than the 50% allowed by the impedance matching theorem for the linear flux–force relationship. When maximum entropy production is required in the transition with a load, one obtains high optimal photochemical yield of 97% and power transfer efficiency of 91%. In more complex photosynthetic models, entropy production is maximized in all irreversible electron/proton (non-slip) transitions in an iterative procedure. The resulting steady-state is stable with respect to an extremely wide range of initial values for forward rate constants. Optimal proton current increases proportionally to light intensity and decreases with an increase in the proton-motive force (the backpressure effect). Optimal affinity transfer efficiency is very high and nearly perfectly constant for different light absorption rates and for different electrochemical proton gradients. Optimal overall efficiency (of solar into proton-motive power) ranges from 10% (bacteriorhodopsin) to 19% (chlorophyll-based bacterial photosynthesis). Optimal time constants in a photocycle span a wide range from nanoseconds to milliseconds, just as corresponding experimental constants do. We conclude that photosynthetic proton pumps operate close to the maximum entropy production mode, connecting biological to thermodynamic evolution in a coupled self-amplifying process.     

Entropy-energy decomposition from nonequilibrium work trajectories

We derive expressions for the equilibrium entropy and energy changes in the context of the Jarzynski equality relating nonequilibrium work to equilibrium free energy. The derivation is based on a stochastic path integral technique that reweights paths at different temperatures. Stochastic dynamics generated by either a Langevin equation or a Metropolis Monte Carlo scheme are treated. The approach enables the entropy-energy decomposition from trajectories evolving at a single-temperature and does not require simulations or measurements at two or more temperatures. Both finite difference and analytical formulae are derived. Testing is performed on a prototypical model system and the method is compared with existing thermodynamic integration and thermodynamic perturbation approaches for entropy-energy decomposition. The new formulae are also put in the context of more general, dynamics-independent expressions that derive from either a fluctuation theorem or the Feynman-Kac theorem.     

On the fluctuation theorem for the dissipation function and its connection with response theory

Recently, there has been considerable interest in the fluctuation theorem (FT). The Evans-Searles FT shows how time reversible microscopic dynamics leads to irreversible macroscopic behavior as the system size or observation time increases. We show that the argument of this FT, the dissipation function, plays a central role in nonlinear response theory and derive the dissipation theorem, giving exact relations for nonlinear response of classical N-body systems that are more widely applicable than previous expressions. These expressions should be verifiable experimentally. When linearized they reduce to the well-known Green-Kubo expressions for linear response.     

物理化学教学内容研究交流研讨会纪要

2015年4月16~17日,高等教育出版社在山东青岛组织召开了物理化学教学内容研究交流研讨会,就自发变化和可逆过程的定义、可逆与平衡、自发与不可逆的关系、环境熵和总熵、环境的热力学界定和总熵判据、反应的方向与限度等问题进行了研讨。本文是此次会议的纪要,希望对纠正物理化学教学中的一些偏颇甚至错误的认识有所裨益。     

Extending the fluctuation theorem to describe reaction coordinates

The fluctuation theorem describes the distribution of work done on small systems which have been pushed out of equilibrium in response to an external field. The theorem has recently been a subject of much interest for describing single-molecule experiments and simulations. In this communication, it is shown how the fluctuation theorem can be extended to describe fluctuations not only in the work done on a system, but also in a reaction coordinate. The extension explored in this work allows for a generalized derivation of Hummer and Szabo's expression (G. Hummer and A. Szabo, Proc. Natl. Acad. Sci. 98, 3658 (2001)) for reconstructing the potential of mean force from nonequilibrium trajectories. The derivation demonstrates how implementation of this expression can be more easily facilitated. Atomistic simulations of a biomolecular system are presented which support these results.     

Specific reversible melting of polymers

Temperature‐modulated differential scanning calorimetry can detect a certain amount of reversible latent heat in flexible macromolecules. In short, one can identify a reversible melting in such polymers earlier thought to exhibit only fully irreversible crystallization and melting. Details of the reversible melting of isotactic polypropylene and ethylene‐1‐octene copolymers of low and medium densities have newly been measured and linked to the crystallization, annealing, or melting temperature. It is possible to assign the experimental reversibility of melting to specific crystal fractions that ultimately melt irreversibly at higher temperatures; that is, it is suggested that reversible melting mainly occurs only between the temperatures of their formation and their zero‐entropy‐production melting temperature, at which they change to a melt of the same degree of metastability. This is supported by the almost complete absence of reversibility below the temperature of crystal formation and the observation of a distinct relationship between the amount of irreversibly by annealing reorganized material and reversibility in the case of isotactic polypropylene. A given crystal fraction, characterized by its formation temperature and zero‐entropy‐production melting temperature, has a specific reversibility of the melt‐to‐crystal transition, which is represented by the ratio of the reversible latent heat to the total enthalpy change when the crystal fraction of interest ultimately melts. This specific reversibility is, for ethylene‐1‐octene copolymers, at least 25% at temperatures in the primary crystallization range, and this indicates that the reversible contribution to the total of the melting processes is much larger than expected from simple calculations by the excess apparent reversible heat capacity being referred to the heat of fusion of the polymer, as is commonly done. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2039–2051, 2003     

Thermodynamic criteria for estimating the kinetic parameters of catalytic reactions

Kinetic parameters are estimated using two criteria in addition to the traditional criterion that considers the consistency between experimental and modeled conversion data: thermodynamic consistency and the consistency with entropy production (i.e., the absolute rate of the change in entropy due to exchange with the environment is consistent with the rate of entropy production in the steady state). A special procedure is developed and executed on a computer to achieve the thermodynamic consistency of a set of kinetic parameters with respect to both the standard entropy of a reaction and the standard enthalpy of a reaction. A problem of multi-criterion optimization, reduced to a single-criterion problem by summing weighted values of the three criteria listed above, is solved. Using the reaction of NO reduction with CO on a platinum catalyst as an example, it is shown that the set of parameters proposed by D.B. Mantri and P. Aghalayam gives much worse agreement with experimental values than the set obtained on the basis of three criteria: the sum of the squares of deviations for conversion, the thermodynamic consistency, and the consistency with entropy production.     

Exact solutions for the entropy production rate of several irreversible processes

We investigate thermal conduction described by Newton's law of cooling and by Fourier's transport equation and chemical reactions based on mass action kinetics where we detail a simple example of a reaction mechanism with one intermediate. In these cases we derive exact expressions for the entropy production rate and its differential. We show that at a stationary state the entropy production rate is an extremum if and only if the stationary state is a state of thermodynamic equilibrium. These results are exact and independent of any expansions of the entropy production rate. In the case of thermal conduction we compare our exact approach with the conventional approach based on the expansion of the entropy production rate near equilibrium. If we expand the entropy production rate in a series and keep terms up to the third order in the deviation variables and then differentiate, we find out that the entropy production rate is not an extremum at a nonequilibrium steady state. If there is a strict proportionality between fluxes and forces, then the entropy production rate is an extremum at the stationary state even if the stationary state is far away from equilibrium.