After establishing the local systems (see Figures 1 and 2), a model for the interactions between systems represented by the arrows in Figures 1 and 2 is needed to calculate the mass and heat fluxes as well as the reactions. These interactions result in entropy generation that is captured within the SEAQT framework by its equation of motion, which is used to model the kinetics and dynamics of the chemical and electrochemical reaction and diffusion pathways directly. This first-principles approach incorporates quantum as well as classical microscopic information directly into the model via the system energy eigenstructure, which plays a key role in the equation of motion.

The underlying theory of our approach is introduced here using an isolated, finite energy eigenlevel quantum system. A discussion of how the theory applies more broadly to other kinds of systems is included in [32, 33, 34]. A quantum system is well defined by a group of energy eigenstates and associated eigenvalues $\{{e}_{i},i=1,2,\cdots \}$. A given thermodynamic (nonequilibrium or equilibrium) state of the system is uniquely defined by a probability distribution among the energy eigenstates denoted by $\{{p}_{i},i=1,2,3,\cdots \}$, and all possible distributions form the state space of the system. In particular, the stable equilibrium state of an isolated system has a probability distribution described by the Maxwellian distribution. An evolution of state occurs when the system in a state of nonequilibrium evolves to one at stable equilibrium; and it is the given set of probability distributions predicted by the SEAQT equation of motion, which define the unique thermodynamic path that the system takes during the evolution.

For an isolated system, energy and mass conservations require $\sum {e}_{i}{p}_{i}=E$ and $\sum {p}_{i}=1$.

The SEAQT equation of motion is constructed so that it takes the direction in state space, which coincides with that of steepest entropy ascent constrained by energy and mass conservations. At this level of description, the entropy of the system takes the form $S=-{k}_{b}\sum {p}_{i}ln{p}_{i}$.

The equation of motion for an isolated system comprised of a single elementary constituent (i.e. a single particle, a single assembly of indistinguishable particles, or a single field) is then written for the case when the state operator commutes with the Hamiltonian operator of energy eigenstates as
$\frac{d{p}_{j}}{dt}=-\frac{1}{\tau}D=-\frac{1}{\tau}\left|\begin{array}{ccc}{p}_{j}ln{p}_{j}& \phantom{\rule{thickmathspace}{0ex}}{p}_{j}& \phantom{\rule{thickmathspace}{0ex}}{e}_{j}{p}_{j}\\ \sum {p}_{i}ln{p}_{i}& \phantom{\rule{thickmathspace}{0ex}}1& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}{p}_{i}\\ \sum {e}_{i}{p}_{i}ln{p}_{i}& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}{p}_{i}& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}^{2}{p}_{i}\end{array}\right|/\left|\begin{array}{cc}1& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}{p}_{i}\\ \sum {e}_{i}{p}_{i}& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}^{2}{p}_{i}\end{array}\right|$(5)

where the only generators of the motion are the identity and Hamiltonian operators. Unlike traditional equations of motion such as, for example, the classical [37] and quantum [13, 38] versions of the Boltzmann equation, eq. (5) requires no near-equilibrium assumption. It can be used in the nonlinear, far-from-equilibrium realm without any theoretical ambiguities or violations of the laws of thermodynamics or of quantum mechanics. The only parameter, which must be estimated, is the relaxation time $\tau $ associated with the dissipation term $D$ in eq. (5). The relaxation time can be a constant or a function of the local system eigenstructure and state [39, 40]. The dissipation term predicts the rate of entropy generation for a given system evolving in time. The relaxation time can be realistically estimated using experimental data, a more microscopic model (e.g., from kinetic theory) such as that for a reaction rate constant [30], diffusion coefficient, and heat transfer coefficient [33, 34], or even *ab initio* [40].

The solution of the SEAQT equation of motion exhibits a number of convenient features, which lead to a complete description of each nonequilibrium state and a fundamental as opposed to phenomenological definition of nonequilibrium temperature. More discussion is presented in reference [32], and an example is provided below. The energy eigenlevels of the system $\{{e}_{i},i=1,2,\cdots \}$ can be divided into to two sets $\{{e}_{iA},i=1,2,\cdots \}$ and $\{{e}_{iB},i=1,2,\cdots \}$ so that the system state space (Hilbert space) ${H}$ can be expressed by ${H}={{H}}_{A}\oplus {{H}}_{B}$, which is the sum of the two subspaces ${{H}}_{A}$ and ${{H}}_{B}$.

The system state can then be represented by the distributions for the two subspaces’ energy eigenlevels $\{{p}_{iA},i=1,2,3,\cdots \}$ and $\{{p}_{iB},i=1,2,\cdots \}$. If the distribution in subspace *A* yields to the canonical distribution of temperature ${T}_{A}$, the temperature of subspace $A$ is defined to be ${T}_{A}$. Given the division of the state space, if the system probability distributions in the two subspaces are both canonical distributions, the state of the system is called a second-order hypoequilibrium state [32], which can be described uniquely by the total probability in each subspace (${p}^{A}=\sum _{i}{p}_{iA}$ and ${p}^{B}=\sum _{i}{p}_{iB}$) and the temperature of each subspace ( ${T}_{A}$ and ${T}_{B}$). If the initial state of the system is a second-order hypoequilibrium state, then
$\begin{array}{rl}{p}_{iA(B)}^{initial}& =\frac{{p}^{A(B)}}{{Z}^{A(B)}({T}_{A(B)})}{e}^{\frac{-{e}_{iA(B)}}{{k}_{b}{T}_{A(B)}}},\phantom{\rule{1em}{0ex}}i=1,2,\cdots ,{n}_{A(B)}\end{array}$(6)

where ${k}_{b}$ is Boltzmann’s constant, ${n}_{A}$ and ${n}_{B}$ are the number of energy eigenlevels in each subspace, respectively, ${Z}^{A}({T}_{A})$ is the partition function of subspace $A$ at temperature ${T}_{A}$, and ${Z}^{B}({T}_{B})$ the partition function of subspace $B$ at temperature ${T}_{B}$. When constant relaxation times are used for local systems, Li and von Spakovsky [32] have proven using eq. (5) that the system remains in a second-order hypoequilibrium state throughout the nonequilibrium state evolution so that
$\begin{array}{rl}{p}_{iA(B)}(t)& =\frac{{p}^{A(B)}(t)}{{Z}^{A(B)}({T}_{A(B)}(t))}{e}^{\frac{-{e}_{iA(B)}}{{k}_{b}{T}_{A(B)}(t)}},\phantom{\rule{1em}{0ex}}i=1,2,\cdots ,{n}_{A(B)}\end{array}$(7)

Thus, each subspace has temperature fundamentally (i.e., canonically) defined throughout the entire nonequilibrium process of relaxation since the hypoequilibrium concept assumes that the system in a nonequilibrium state (one even far-from-equilibrium) is composed of more than one subsystem (or subspace), each with a different temperature. This temperature is a “nonequilibrium temperature” because the system need not be in stable equilibrium. Furthermore, when relaxation times are not constant, temperature can still be fundamentally defined using a generalized form of hypoequilibrium state [39]. One further point is that it would be of interest to investigate how this concept and these temperatures relate to the “constraint potentials” [41, 42] of the Rate-Controlled Constrained-Equilibrium (RCCE) model in which the RCCE constraints slowly vary and hence are rate controlling, representing the bottlenecks of the dynamics.

In calculating the interaction between local systems, a combined system is set up for each link (Figure 3a for a reaction and Figure 3b for diffusion). The energy eigenlevels of the combined system are a combination of the energy eigenlevels of the two linked local systems, and the state space is the sum of the Hilbert spaces of the two linked local systems. If the distribution in each linked local system is canonical, the nonequilibrium state of the combined system is in a second-order hypoequilibrium state even when far-from equilibrium. Using the equation of motion, eq. (5), for the combined system, the nonequilibrium evolution of the combined system can be determined so that the nonequilibrium interactions between the two linked local systems can be modeled.

Figure 3: Behavior of the mass flux of Figure 1 for (a: left) a chemical reaction, (b: middle) diffusion, and (c: right) more than one interaction.

The state space of the reactive system is spanned by the eigenstates of the reactants and those of the products. Intermediate states between reactants and products are represented by a mixture of reactant states and product states. This state space can be represented by the Hilbert space ${H}={{H}}_{1}\oplus {{H}}_{2}$, which is the sum of reactant and product subspaces.

Each subspace of the state is generated from single molecular state spaces by the cross product. For example, the reaction $A+BC\leftrightarrow AC+B$

is modeled so that the two subspaces are given by ${{H}}_{1}={{H}}_{BC}\otimes {{H}}_{A}$ and ${{H}}_{2}={{H}}_{AC}\otimes {{H}}_{B}$.

The eigenstate of each particle is generated from the one-particle eigenstates available to each molecule. For example, the state space of *BC* is expanded to include degrees of freedom for translation, rotation, and vibration such that
${{H}}_{BC}=\text{span}\{|{t}_{x}{t}_{y}{t}_{z}{\u3009}_{bc}\otimes |lm{\u3009}_{bc}\otimes |\nu {\u3009}_{bc}\}$(8)

where $|{t}_{x}{t}_{y}{t}_{z}\u3009$ represents the translational eigenstates, $|lm\u3009$ the rotational ones, and $|\nu \u3009$ the vibrational ones. Inside each ket ($|\cdots \u3009$) are the quantum numbers for the one-particle eigenstate. Starting from these one-particle eigenstates, one can then form the system-level eigenstate of the reactants and products, which spans the system state space. The thermodynamic state of the system at any given instant of time is then represented by a probability distribution among the system eigenstates. More detailed descriptions of this novel approach describing the chemical and electrochemical kinetics of reacting quantum systems can be found in [29, 30, 43]. Given the energy eigenlevels of the system and each nonequilibrium state represented by a probability distribution $\{{p}_{i},i=1,2,\cdots \}$, the SEAQT equation of motion is used to model the chemical and electrochemical reaction pathways. In the SEAQT SOFC model, the reactant subspace and the product subspace are formed by two local systems linked by a chemical reaction (see Figure 3b).

The state space of the diffusion system is spanned by the eigenstates of the particle at position $x$ and position $x+\mathrm{\Delta}x$. Strictly speaking, $\mathrm{\Delta}x$ needs to be much larger than the de Broglie wavelength of the particle, which is about ${10}^{-11}$ m for the problem studied here where $\mathrm{\Delta}x=1$ $\text{\hspace{0.17em}}\text{\mu}$m. This state space can be represented by the Hilbert space ${H}={{H}}_{1}\oplus {{H}}_{2}$, which is the sum of

the subspaces of the particle at positions $x$ (${{H}}_{1}$) and $x+\mathrm{\Delta}x$ (${{H}}_{2}$). The eigenstate of the particle at $x$ and $x+\mathrm{\Delta}x$ can be generated in the same way as is done for the reactants and products described in the previous section. The set of system eigenstates is composed of the eigenstates of the particle at $x$ and $x+\mathrm{\Delta}x$ and is represented by a probability distribution among the system eigenstates. Given the energy eigenstates of the system and each nonequilibrium state represented by a probability distribution $\{{p}_{i},i=1,2,\cdots \}$, the SEAQT equation of motion is used to model the interaction between two neighboring particle systems via the diffusion process. In the SEAQT SOFC model, subspace 1 and subspace 2 are formed by two local systems linked by diffusion (see Figure 3b).

The equation of motion eq. (5) uses probability distributions to represent system state, and this can be generalized to systems with more than one particle. Consider the interaction (either mass diffusion or reaction) between two linked local systems. Local system $1$ has ${m}_{1}$ particles and has the distribution among its energy eigenstates of $\{{p}_{i,1},i=1,2,\cdots ,{n}_{1}\}$ where ${n}_{1}$ stands for the number of eigenstates. Local system $2$ has ${m}_{2}$ particles and has the distribution among its energy eigenstates of $\{{p}_{i,2},i=1,2,\cdots ,{n}_{2}\}$ where ${n}_{2}$ stands for the number of eigenstates. The combined system has ${m}_{1}+{m}_{2}$ particles with a new distribution $\{{p}_{i,1}^{new},i=1,2,\cdots ,{n}_{1},{p}_{i,2}^{new},i=1,2,\cdots ,{n}_{2}\}$ where
$\begin{array}{rl}{p}_{i,1(2)}^{new}& ={m}_{1(2)}{p}_{i,1(2)}/({m}_{1}+{m}_{2}),\phantom{\rule{1em}{0ex}}i=1,2,\cdots ,{n}_{1(2)}\end{array}$(9)

Using these new distributions as input to the equation of motion, eq. (5) leads to a prediction of their time rate of change, i.e., $\{\frac{d{p}_{i,1}^{new}}{dt},\phantom{\rule{thinmathspace}{0ex}}i=1,2,\cdots ,{n}_{1}\}$ and $\{\frac{d{p}_{i,2}^{new}}{dt},\phantom{\rule{thinmathspace}{0ex}}i=1,2,\cdots ,{n}_{2}\}$.

The mass flow for one energy eigenlevel of the system then is
$\begin{array}{rl}\frac{d{m}_{i,1(2)}^{new}}{dt}& =({m}_{1}+{m}_{2})\frac{d{p}_{i,1(2)}^{new}}{dt},\phantom{\rule{1em}{0ex}}i=1,2,\cdots ,{n}_{1(2)}\end{array}$(10)

The mass flow rate for one local system from the interaction is then given by
$\begin{array}{rl}\frac{d{m}_{1(2)}}{dt}& =({m}_{1}+{m}_{2})\sum _{i}\frac{d{p}_{i,1(2)}^{new}}{dt}\end{array}$(11)

Thus, one can calculate the mass flow along a given link for every single eigenstate, as well as the total mass flow to a local system. If a local system has multiple links (e.g., as in Figure 3c), the total flow to one energy eigenlevel of the local system is the sum of the flows through all the links. Recall that each link maintains the local system’s distribution canonical. Thus, the total effect from multiple links is to as well keep the local system’s distribution canonical [44] so that the temperature is well defined for local systems even though it undergoes a nonequilibrium interaction.

A local system can also interact with a temperature reservoir with the result that only energy and no mass is transferred. The effect of the energy flow is a redistribution of particle number among the available energy eigenlevels. This heat diffusion between the reservoir and local system is modeled by forming a composite system of a given local system (local system $1$) and its duplicate (local system $2$) so that the systems have the same mass and energy eigenstructure but different temperatures. Unlike the case for mass diffusion and that for reaction, which yield to one probability conservation law and one energy conservation law, the equation of motion for the case of heat diffusion yields to two probability conservation laws and one energy conservation law. For more details, the reader is referred to [33, 44].

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