Current nonequilibrium

Chapter 14 - Superoperator many-body theory of molecular currents Nonequilibrium Green functions in real time. Pages 373-396, Upendra Harbola and Shaul Mukamel... 

The current density j is, according to linear nonequilibrium thermodynamics, proportional to the gradient of a chemical potential difference... 

An electrochemical reaction is said to be polarized or retarded when it is limited by various physical and chemical factors. In other words, the reduction in potential difference in volts due to net current flow between the two electrodes of the corrosion cell is termed polarization. Thus, the corrosion cell is in a state of nonequilibrium due to this polarization. Figure 4-415 is a schematic illustration of a Daniel cell. The potential difference (emf) between zinc and copper electrodes is about one volt. Upon allowing current to flow through the external resistance, the potential difference falls below one volt. As the current is increased, the voltage continues to drop and upon completely short circuiting (R = 0, therefore maximum flow of current) the potential difference falls toward about zero. This phenomenon can be plotted as a polarization diagram shown in Figure 4-416. 

Considering the similarity between Figs. 1 and 2, the electrode potential E and the anodic dissolution current J in Fig. 2 correspond to the control parameter ft and the physical variable x in Fig. 1, respectively. Then it can be said that the equilibrium solution of J changes the value from J - 0 to J > 0 at the critical pitting potential pit. Therefore the critical pitting potential corresponds to the bifurcation point. From these points of view, corrosion should be classified as one of the nonequilibrium and nonlinear phenomena in complex systems, similar to other phenomena such as chaos. 

Figure 38. Classification of nonequilibrium fluctuations. (Reprinted from M. Asanuma and R. Aogaki, Non-equilibrium fluctuation theory on pitting dissolution. I. Derivation of dissolution current equations." J. Chem. Phys. 106,9938,1997. Copyright 1997, American Institute of Physics.)...

Figure 40. Plot of the fluctuation-diffusion current J vs. iwr.91 id is the slope of the fluctuation-diffusion current given by Eq. (115). Solid and dotted lines correspond to the theoretical and experimental results, respectively. (NiCljJ = 0.1 mol nT3. [NsCl] = 7 mol m 3. V = 0.1 V, T= 300 K. (Reprinted from M. Asanuma and R. Aogaki, Nonequilibrium fluctuation theory on pitting dissolution, n. Determination of surface coverage of nickel passive film, J. Chem. Phys. 106, 9938, 1997, Fig. 8. Copyright 1997, American Institute of Physics.)...

Two cases where equihbrium is lacking must be distinguished that which is unrelated to current flow and can be observed even for a nonworking electrode, and that which is related exclusively to the passage of current through the electrode. The former is the case of nonequilibrium open-circuit potentials (nonequihbrium OCP), and the latter is that of electrode polarization. 

Equilibrium potentials can be calculated thermodynamically (for more details, see Chapter 3) when the corresponding electrode reaction is known precisely, even when they cannot be reached experimentally (i.e., when the electrode potential is nonequilibrium despite the fact that the current is practically zero). The open-circuit voltage of any galvanic cell where at least one of the two electrodes has an nonequilibrium open-circuit potential will also be nonequilibrium. Particularly in thermodynamic calculations, the term EMF is often used for measured or calculated equilibrium OCV values. 

In electrochemical systems, a steady state during current flow implies that a time-invariant distribution of the concentrations of ions and neutral species, of potential, and of other parameters is maintained in any section of the cell. The distribution may be nonequilibrium, and it may be a function of current, but at a given current it is time invariant. 

While quantum chemistry cannot be used to work with the Landauer/Buttiker/Imry formula as it stands, a very different approach based on nonequilibrium Green s functions yields a different formula (sometimes called the Caroli formula [68], or the NEGF formula in the Landauer/Imry/Buttiker limit). It is, for the current I ... 

A logical extension of the spectral equilibrium model for e would be to consider nonequilibrium spectral transport from large to small scales. Such models are an active area of current research (e.g., Schiestel 1987 and Hanjalic et al. 1997). Low-Reynolds-number models for e typically add new terms to correct for the viscous sub-layer near walls, and adjust the model coefficients to include a dependency on Re/,. These models are still ad hoc in the sense that there is little physical justification - instead models are validated and tuned for particular flows. 

The effect of thermal pion fluctuations on the specific heat and the neutrino emissivity of neutron stars was discussed in [27, 28] together with other in-medium effects, see also reviews [29, 30], Neutron pair breaking and formation (PBF) neutrino process on the neutral current was studied in [31, 32] for the hadron matter. Also ref. [32] added the proton PBF process in the hadron matter and correlation processes, and ref. [33] included quark PBF processes in quark matter. PBF processes were studied by two different methods with the help of Bogolubov transformation for the fermion wave function [31, 33] and within Schwinger-Kadanoff-Baym-Keldysh formalism for nonequilibrium normal and anomalous fermion Green functions [32, 28, 29],... 

For example, thermodynamic calculations will provide a value for the maximum voltage of a storage battery—that is, the voltage that is obtained when no current is drawn. When current is drawn, we can predict that the voltage will be less than the maximum value, but we cannot predict how much less. Similarly, we can calculate the maximum amount of heat that can be transferred from a cold environment into a building by the expenditure of a certain amount of work in a heat pump, but the actual performance will be less satisfactory. Given a nonequilibrium distribution of ions across a cell membrane, we can calculate the minimum work required to maintain such a distribution. However, the actual process that occurs in the cell requires much more work than the calculated value because the process is carried out irreversibly. 

These new methods of nonequihbrium statistical mechanics can be applied to understand the fluctuating properties of out-of-equilibrium nanosystems. Today, nanosystems are studied not only for their structure but also for their functional properties. These properties are concerned by the time evolution of the nanosystems and are studied in nonequilibrium statistical mechanics. These properties range from the electronic and mechanical properties of single molecules to the kinetics of molecular motors. Because of their small size, nanosystems and their properties such as the currents are affected by the fluctuations which can be described by the new methods. 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

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

In a nonequilibrium steady state, the entropy production is the sum of the affinities multiplied by the mean currents ... 

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 ... 

Figures 6.5 and 6.6 illustrate the RedOx and the metal/metal-ion electrodes in a nonequilibrium state, respectively. When current flows through an electrode, its potential A(/>(/) deviates from the equilibrium value A(/>eq by the amount ry, which was defined by Eq. (6.1a). Thus, the nonequilibrium potential difference A(p in Eqs. (6.33) and (6.34) can be substituted by the nonequilibrium value A(/)(/) ...

---