Nonequilibrium steady state systems

Thus, the maximum entropy on a limited spectrum of parameters (which are considered as pseudoconstant or variable with short gradients) is proved for nonequilibrium steady-state systems. 

Holian B L 1996 The character of the nonequilibrium steady state beautiful formalism meets ugly reality Monte Carlo and Molecular Dynamics of Condensed Matter Systems, vol 49, ed K Binder and G Ciccotti (Bologna Italian Physical Society) pp 791-822... 

One shortcoming of Schild analysis is an overemphasized use of the control dose-response curve (i.e., the accuracy of every DR value depends on the accuracy of the control EC o value). An alternative method utilizes nonlinear regression of the Gaddum equation (with visualization of the data with a Clark plot [10], named for A. J. Clark). This method, unlike Schild analysis, does not emphasize control pECS0, thereby giving a more balanced estimate of antagonist affinity. This method, first described by Lew and Angus [11], is robust and theoretically more sound than Schild analysis. On the other hand, it is not as visual. Schild analysis is rapid and intuitive, and can be used to detect nonequilibrium steady states in the system that can corrupt... 

In this section, we consider a system in a nonequilibrium steady state, such as a conductor between two particle reservoirs at different chemical potentials (see Fig. 14). The state co of the system at a given time can be represented by the numbers of particles in the different cells composing the... 

Beside the work performed on the system, interesting quantities are the currents crossing a system in a nonequilibrium steady state. Recently, we have... 

In nonequilibrium steady states, the mean currents crossing the system depend on the nonequilibrium constraints given by the affinities or thermodynamic forces which vanish at equihbrium. Accordingly, the mean currents can be expanded in powers of the affinities around the equilibrium state. Many nonequilibrium processes are in the linear regime studied since Onsager classical work [7]. However, chemical reactions are known to involve the nonlinear regime. This is also the case for nanosystems such as the molecular motors as recently shown [66]. In the nonlinear regime, the mean currents depend on powers of the affinities so that it is necessary to consider the full Taylor expansion of the currents on the affinities ... 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

Nonequilibrium Steady State (NESS). The system is driven by external forces (either time dependent or nonconservative) in a stationary nonequilibrium state, where its properties do not change with time. The steady state is an irreversible nonequilibrium process that cannot be described by the Boltzmann-Gibbs distribution, where the average heat that is dissipated by the system (equal to the entropy production of the bath) is positive. 

There are still other categories of NESS. For example, in nonequilibrium transient steady states the system starts in a nonequilibrium steady state but is driven out of that steady state by an external perturbation to finally settle in a new steady state. 

The nonequilibrium aging state (NBAS, see Section III.A) is a nonstationary state characterized by slow relaxation and a very low rate of energy dissipation to the surroundings. Aging systems fail to reach equilibrium unless one waits an exceedingly large amount of time. For this reason, the NEAS is very different from either the nonequilibrium transient state (NETS) or the nonequilibrium steady state (NESS). 

Under those conditions P behaves as a Lagrangian in mechanics. Furthermore, as P is a nonnegative function for any positive value of the concentrations X,, by a theorem due to Lyapounov, the asymptotic stability of nonequilibrium steady states is ensured (theorem of minimum entropy production.1-23 These steady states are thus characterized by a minimum level of the dissipation in the linear domain of nonequilibrium thermodynamics the systems tend to states approaching equilibrium as much as their constraints permit. Although entropy may be lower than at equilibrium, the equilibrium type of order still prevails. The steady states belong to what has been called the thermodynamic branch, as it contains the equilibrium state as a particular case. 

There are two more important advantages of these models. One is that it is possible under some conditions to carry out an exact stability analysis of the nonequilibrium steady-state solutions and to determine points of exchange of stability corresponding to secondary bifurcations on these branches. The other is that branches of solutions can be calculated that are not accessible by the usual approximate methods. We have already seen a case here in which the values of parameters correspond to domain 2. This also happens when the fixed boundary conditions imposed on the system are arbitrary and do not correspond to some homogeneous steady-state value of X and Y. In that case Fig. 20 may, for example,... 

Finally, we want to mention that Prigogine [1] has shown that for a coupled system of flows and associated driving forces, the rate of entropy generation is at a minimum when the system is in the nonequilibrium steady state all non-steady states are associated with higher rates of entropy generation. ... 

H. Qian. Open-system nonequilibrium steady state statistical thermodynamics, fluctuations and chemical oscillations. J. Phys. Chem. B., 110 15063-15074, 2006. 

The equality of this equation represents a system at equilibrium where JT = A = 0. The work done by the controlling system dissipates as heat. This is in line with the first law of thermodynamics. The inequality in Eq. (11.4) represents the second law of thermodynamics. The cyclic chemical reaction in nonequilibrium steady-state conditions balances the work and heat in compliance with the first law and at the same time transforms useful energy into entropy in the surroundings in compliance with the second law. The dissipated heat related to affinity A under these conditions is different from the enthalpy difference AH° = (d(Aii°/T)/d(l/Tj). The enthalpy difference can be positive if the reaction is exothermic or negative if the reaction is endothermic. On the other hand, the A contains the additional energy dissipation associated with removing a P molecule from a solution with concentration cP and adding an S molecule into a solution with concentration cs. 

This simple example may be generalized to biochemical reaction cycles in which there are a number of reactions and boundary flows that add and remove substrates. The controlled concentrations and boundary flows maintain the system in a nonequilibrium steady-state condition. 

By relating the fluctuations to the reaction coordinate v, de = SNh Eqs. (12.10) and (12.12) imply that if a system is stable to fluctuations in diffusion, it is also stable to fluctuations in chemical reactions, which is called the Duhem-Jougeut theorem (Kondepudi and Prigogine, 1999). However, a nonequilibrium steady state involving chemical reactions may be unstable even if the system is stable with respect to diffusion. 

However, nonequilibrium steady states may be unstable even if the system is stable with respect to diffusion. For a nonequilibrium state, the stability condition for a chemical reaction in terms of excess entropy production is... 

Fig. 6.1. Approach to steady states from different initial states. The nonequilibrium steady state that the system reaches depends on the system s initial state. Under the conditions listed below there are two stable steady states, at x = 0.08 and 3.70. There is an unstable state at 1.22. If the initial concentration of x is greater than 1.22, the system goes to the higher steady state. If the initial concentration is less than 1.22, the lower steady state is occupied...

The distribution of Eq. [137] is canonical in laboratory momentum and positions for a general strain rate tensor Vu this is the expected form for a system subject to an external field. Equation [137] is the first distribution function to be derived for SLLOD-type dynamics and has provided impetus for studies concerning the nature of the distribution function in the nonequilibrium steady state. 

To obtain the friction coefficient X as the transport coefficient in LRT, we would like to take a nonequilibrium steady state average of the frictional force. Thus we identify F with the phase variable B. The frictional force in our system is the force exerted by the surface on the fluid. At equilibrium, the average of this force will be zero, because there is equal likelihood of fluid particles flowing in any given direction. Thus, the first term in Eq. [210] will be zero. Under shear, the surface will exert on average a nonzero force on the fluid due to the directionality of the flow. The frictional force is given by... 

Another approach towards a thermodynamics of steady-state systems is presented by Santamaria-Holek et al.193 In this formulation a local thermodynamic equilibrium is assumed to exist. The probability density and associated conjugate chemical potential are interpreted as mesoscopic thermodynamic variables from which the Fokker-Planck equation is derived. Nonequilibrium equations of state are derived for a gas of shearing Brownian particles in both dilute and dense states. It is found that for low shear rates the first normal stress difference is quadratic in strain rate and the viscosity is given as a simple power law in the strain rate, in contrast to standard mode-coupling theory predictions (see Section 6.3). 

When a constant force is imposed on an open system, the system will eventually reach a nonequilibrium steady state where the response to the force appears as a time-independent flux. (A closed system in the same situation will reach a new equilibrium state, as discussed above.)... 

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