{'fraction_non_alphanumeric': 0.042094092677750265, 'fraction_numerical': 0.008754863813229572, 'mean_word_length': 4.675533249686324, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Relaxation of Chemical Reactions to Stationary States in the Chemical Affinities Space

Using the mass balance equations for chemical reactions, we show how the system relaxes towards a steady state in and out of the Onsager region. In the chemical affinities space, after fast transients, the relaxation process is a straight line when operating in the Onsager region, while out of this regime, the evolution of the system is such that the projections of the evolution equations for the forces and the shortest path on the flows coincide. For spatially-extended systems, similar results are valid for the evolution of the thermodynamic mode (i.e., the mode with wave-number k = 0). These results allow us to obtain the expression for the affine connection of the space covered by the thermodynamic forces, close to the steady states. Through the affine connection, the nonlinear closure equations are derived.


I. INTRODUCTION
It is known that the evolution equations for the moments of the distribution functions we are confronted with the familiar difficulty appearing so often in statistical physics: an infinity hierarchy of moment equations. When we truncate the hierarchy, the resulting set of dynamical moments equations are not closed. Thus, a theory for the closure equations, (or the transport theory) should be formulated and combined with the truncated moment equations. A thermodynamical field theory (TFT) has been established addressing the closure problem for thermodynamic systems out of Onsager's region [1]. This theory allows to determine the nonlinear corrections to the Onsager theory so that the thermodynamic theorems for systems far from equilibrium are respected. Clearly, transport theory without a knowledge of microscopic dynamical laws can not be developed. Transport theory is only but an aspect of non-equilibrium statistical mechanics, which provides the link between micro and macro-levels of descriptions. This link appears indirectly in the "unperturbed" Onsager coefficients used as an input in the equations. The Onsager coefficients, which depend on the material under consideration, have to be calculated in the usual way by kinetic theory.
The nonlinear contributions are derived starting from the De Donder-Prigogine principle [also refereed to as the Thermodynamic Covariant Principle (TCP)] stating that: "Thermodynamic systems are thermodynamically equivalent if, under transformation of fluxes and forces, the bilinear form of the entropy production remains unaltered". Flux-forces transformations leaving invariant the expression of the entropy production, are referred to as the Thermodynamic Coordinate Transformations (TCT). The TCP provides the possibility of defining an abstract space, the thermodynamical forces space, covered by the r independent thermodynamic forces X i , whose metric is identified with the symmetric part of the transport matrix. The law of evolution is not the dynamical law of particle motion, or the set i.e., of two-fluid macroscopic equations of plasma dynamics. It is rather the evolution of the thermodynamic forces of the system in the thermodynamical forces space.
The nonlinear transport equations have been successively derived from two assumptions: 1. The thermodynamic theorems valid when a generic thermodynamic system relaxes out of equilibrium are satisfied. In particular, the Glansdorff-Prigogine Universal Criterion of Evolution (UCE)] is fulfilled; 2. There exists a thermodynamic action, scalar under TCT, which is stationary for general variations in the transport coefficients and the affine connection of the thermodynamical forces space.
However, the determination of the thermodynamic action requires the knowledge of the affine connection. As we shall see, this expression can be deduced by analyzing several examples of relaxation. This theory has been successfully applied for studying transport processes in non equilibrium systems such as magnetically confined plasmas, materials submitted to temperature and electric potential gradients or chemical reactions. In particular, in tokamak-plasmas, the thermodynamic forces and the conjugate flows are the generalized frictions and the Hermitian moments, respectively. Hence, we can determine the particle fluxes (electrons and ions) and energy losses as well as the (nonlinear) particle distribution functions. This has been accomplished in Ref. [2] and in the paper reported in the footnote [15].
In this paper, starting from the mass balance equations applied to chemical reaction-diffusion systems, we establish the following analytical results: (i) After fast transients, in the Onsager region of the thermodynamical forces space, a homogeneous system relaxes towards a stable steady state along a straight line.
(ii) After fast transients, out of the Onsager region, a homogeneous system where r chemical reactions take place simultaneously, relaxes towards a stable steady state such where U i (X, ̺) and U i (X s.p. , ̺) are the evolution equation for the chemical affinities and the shortest path in the chemical affinities space, respectively. J i denotes the vector of the chemical reaction rates. The trajectory traced out by the system and the shortest past are parametrized by ̺. This parameter is defined in the Subsection III B [see Eq. (35)].
Similar theorems for the relaxation of the thermodynamic mode (i.e., the mode with wavenumber k = 0) to the steady-state, can also be derived for spatially-extended systems.
These theorems allow us to obtain the expression for the affine connection of the space covered by the thermodynamic forces, close to the steady-states. Through the affine connection, the nonlinear transport relations are derived.
The paper is organized as follows. In Section II, the simplest cases of isothermal and homogeneous chemical systems, relaxing in the Onsager region, are analyzed. A detailed study of chemical systems out of the linear region can be found in Section III. Since we are interested in the solution close to the stationary states, all expressions and matrices, entering in the calculations, are be developed in terms of a small parameter ǫ measuring the "distance" of the system from the steady state. We conclude in Section (IV).
Such examples enable us to determine the expression of the affine connection for the thermodynamical forces space, close the non-equilibrium steady-states. In Ref. [1], the nonlinear closure equations have been derived from assumption 2.

II. CHEMICAL REACTIONS IN THE ONSAGER REGIME
This section is devoted to the analysis of chemical systems relaxing towards a steadystate in Onsager's region of the thermodynamical forces space (i.e., the space of the chemical affinities) [3]. For the sake of simplicity, we start by considering the relaxation of isothermal and homogeneous chemical reactions. We begin by studying in detail the relaxation of a triangular chemical system, maintained out of equilibrium by keeping constant the concentrations of two (among the three) chemical species. Our analysis is successively generalized by considering systems where r independent chemical reactions take place simultaneously.
Spatially-extended thermodynamic systems will be analyzed in the next section.

A. Unimolecular Triangular Chemical Reactions
Let us consider an ideal homogeneous system where the chemical reactions involve three substances B, C and D obeying to the following chemical scheme [4] Indicating with A i , J i and C i the chemical affinities, the chemical reaction rate of reaction i, and the concentrations of species i, respectively, we have where k i are the direct (→) and the inverse (←) kinetic constants of the reactions and µ i the chemical potentials. Note that, from Eqs (1) we find A 1 + A 2 + A 3 = 0, so only two affinities, say A 1 and A 2 , are independent variables. At equilibrium, each chemical reaction rate vanishes, so the equilibrium concentrationsC B,C,D satisfy the conditions We maintain the system out of equilibrium by keeping constant the concentrations of species B and C: The dimensionless parameter δ measures the distance from equilibrium. This parameter is supposed to be small so that the linear analysis applies. Due to constraints Eqs (3), the concentration of species D varies. Its variation can be described by introducing a small dimensionless parameter ε The mass-balance equation for C D reads or, in terms of parameter ε dε dt This system admits the non equilibrium steady-state given by Close to equilibrium, we can write where R is the gas constant and T is temperature of the system. By performing the linear we obtain where σ indicates the entropy production. The expressions for the Onsager matrix L ij and the thermodynamic forces X i are Since the entropy production remains unaltered under transformations (9), according to the general formulation, the systems (A i , J i ) and (A ′ i , J ′ i ) are thermodynamically equivalent [1], [5], [6]. We can check that the thermodynamic forces satisfy the evolution equations The relaxation of the thermodynamic forces in terms of the arc-parameter ς is obtained.
From the definition of ς where ]. As seen from Eq. (14), in the Onsager region of the thermodynamic space, the system relaxes to the steady-state along a straight line.
Let us now quote the Glansdorff-Prigogine dissipative quantity P. Noting that after a little algebra, we have in agreement with the general results reported in Ref. [1]. It is worthwhile to mention that the compact formalism provided by the differential geometry allows to obtain results (17) directly, in only one line of calculation.

B. Several Independent Chemical Reactions
Let us now analyze the relaxation of r coupled chemical reactions taking place simultaneously in a uniform system. In the Onsager region, the balance equations for the n independent thermodynamic forces X α read [see Eqs (12)] where A α β is a matrix independent of the thermodynamic forces (but it may depend on time) and b α is a constant vector. It is useful to write the solutions of Eqs (18) as whereÃ α βst. is the inverse of matrix A α β evaluated when t → +∞ and the steady-states, X α st. , satisfy the condition where δ α β denoting the Kronecker delta. The expression of the arc-parameter reads where "dot" indicates the time derivative. The second principle of thermodynamics ensures that dς 2 ≥ 0. Substituting Eq. (21) into Eqs (19), close to the steady-state, ǫ = (l−ς)/l ≪ 1, we have Now, taking into account the symmetry we finally obtain We may then summarize our results as follows: In the Onsager region of the thermodynamical forces space, after fast transients, the thermodynamic system relaxes towards a stable steady state along a straight line.

III. RELAXATION IN THE NONLINEAR REGION
In the following sections we shall be concerned with the relaxation of chemical reactions towards a steady-state in the nonlinear region of the thermodynamical forces space i.e., out of Onsager's region. We first study the relaxation of isothermal and homogeneous chemical reactions and then we shall extend our analysis to spatially-distributed chemical systems. The study of these examples will enable us to establish the validity of the statements mentioned in the introduction.

A. Chemical Reactions in the Nonlinear Region
In the first part of this section, we shall prove that, close to the steady-state, after where ρ α is the density (mass per unit volume) of α, v α is the velocity of α, and ν i α J i the production of α per unit volume in the i th chemical reaction. The quantity ν i α divided by the molecular mass M α of component k is proportional to the stoichiometric coefficient with which α appears in the chemical reaction i. The coefficients ν i α are counted positive when components α appear in the second, negative when they appear in the first member of the reaction equations. Since the mass law is conserved in each separate chemical reaction we The closure equations for Eqs (25) read where we have introduced the chemical affinities A i of the reaction i (i = 1, 2, · · · , r) and the transport coefficients τ ij (which may depend on the chemical affinities). In general, τ ij is an asymmetric matrix. The chemical affinities are linked to the chemical potential of component α, denoted by µ k , through the De Donder relation [7] A where X i denote the thermodynamic force of the reaction i (X i ≡ A i /T ). We shall now analyze the relaxation towards stationary states. We shall firstly study the simpler case of isothermal and homogeneous chemical systems and, successively, we shall extend our treatment to space-dependent chemical systems.

B. Isothermal and Homogeneous Chemical Reactions
In this case the mass balance equations, Eqs (25), simplify tȯ From Eq. (27-29), we obtain the equation for the thermodynamic forceṡ By inverting matrix M ij (X) (see also the footnote [16]), the equations for the thermodynamic forces can be brought into the form τ ij (X)X j (i = 1, 2, · · · , r) (out of the steady state) (31) BeingM ij (X) a positive-definite matrix (for the demonstration see, for example, Ref. [8]), the dissipative quantity P is always negative throughout the evolution of the system. The inequality is only saturated at the steady-state. In literature, inequality Eq. (32) is referred to as the Universal Criterion of Evolution (UCE) [9].
In Ref. [1] it is demonstrated the validity of the following theorem: If a thermodynamic system relaxes towards a steady-state along the shortest path in the space of the thermodynamical forces then the UCE is automatically satisfied and where ς denotes the arc-parameter, defined as and ̺ is a convenient parameter allowing to cast the equation for the shortest path in the space of the thermodynamic forces into the form Parameter ̺ increases monotonically as the thermodynamic system evolves in time. It can also be chosen so that it vanishes when the thermodynamic system begins to evolve and it takes the (positive) value, say l, when the system reaches the steady-state [1]. Γ i jl denotes the affine connection of the chemical affinities space [1]. For easy reference, the expression of Γ i jl is reported in the Appendix of this paper. We want to check that, after fast transients, close to the steady state, Eq. (32) reduces indeed to the general expression, Eq. (33). To this purpose, we re-consider the forces equation, Eq. (30), the closure equation, Eq. (27), and constraint (26). Defining Y i ≡ X i − X i st. , close to the steady-state X st. , we may write Taking into account that We note that Eqs (38) are manifestly invariant under transformations Recalling the definition of matrix M ij [see Eq. (30)], we also find that, under Eqs (39), Hence, close to the steady-state [i.e., for ǫ = (̺ − l)/l ≪ 1], we may adopt the following expansions (see also the footnote [17]) By inserting expansions Eqs (41) into the mass balance equation, Eqs (38), at the first order we obtain̺ or, by inverting the matrix M ji Multiplying both sides of Eq. (43) by (̺ − l)X i (1) and summing over the index "i", we finḋ However, from expansions Eqs (41), we have Eq. (44) may then be rewritten aṡ where identity Eq. (34) has been taken into account. Close to the steady-state, we finally in agreement with Eq. (33) and Ref. [1].
We are now interested in the evolution of the system near the steady-state. To this end, we reconsider the equation for the thermodynamic forces, Eq. (30), rewritten in terms of We note that this balance equation transforms in a covariant way under the following thermodynamic coordinate transformations (TCT) [1] Indeed, by inserting transformations (49) into the Eq. (48) we finḋ where comma (, ) stands for partial differentiation with respect to the thermodynamic forces.
Eq. (51) may also be rewritten as By using expansions (41), close to the steady-state, we get (see Appendix) Near the steady-states, Eq. (52) satisfies the UCE. Indeed, multiplying Eq. (52) with the thermodynamic flows J i and summing, we obtain d̺ and Eq. (53) has been taken into account. In the Appendix, it is shown the validity of the following identity r i,j,l=1 From this equation and Eq. (34), we obtain the expression Integrating from the initial condition to the steady-state, we get where Eq. (31) has been taken into account. So, after fast transients, we finally obtain The inequality is only saturated at the steady-state. The main conclusion of our analysis is thus: After fast transients, in the chemical affinities space, the system relaxes towards the steady-state so that Moreover, the Glansdorff-Prigogine dissipative quantity P, differs from the one evaluated assuming that the system relaxes along the shortest path, by a factor of the order O(ǫ 3 ). We have also verified that, during the relaxation, the UCE is satisfied. Note that in Onsager's region, ̺ ∝ ς (see Ref. [1]) and dτ ij /dς = 0. Hence, Eq. (51) reduces to In agreement with Eq. (24).

C. Spatially-extended chemical systems
Previous results can easily be generalized for spacial chemical systems. In this section we shall show a general result valid for the relaxation of the thermodynamical mode (i.e., the mode with wave-number k = 0). For this we shall adopt the following strategy: firstly we develop the space-time dependent thermodynamic forces (X i ), flows (J i ) and transport coefficients in (spatial) Fourier's series. Then, we perform the same calculations as in the homogeneous case by taking into account the slaving principle [10] (see the footnote [18]).
Close to the steady-state, we have dv denotes a (spatial) volume element of the system, and the integration is over the entire space V occupied by the system in question. Suffix "(0)" denotes the thermodynamic mode.
The evolution equations for ρ (k)α readṡ where symbol " ⋆ " indicates the convolution andμ α ≡ µ α /T . At the thermodynamic mode we have˙ where O.T.(k, k ′ ) k =k ′ =0 stands for other contributions from different wave-numbers. Due to the slaving principle, near the steady-state, the evolution equations for ρ (0)α reduce tȯ g il,p + g ip,l dX l d̺ dX p d̺ = r j,l,p,q=1 From Eq. (A13) (see the Appendix) we have r j,l,p,q=1 So, close to the steady-state, we obtain At the lowest order, previous equation may be rewritten as where g ip denotes the inverse of the matrix g jp (i.e., r p=1 g ip g jp = δ i j ). It is not difficult to show that Eq. (70) satisfies the UCE but it is not covariant under the TCT. This because the sum of the terms 1 2 r l,p=1 g ij g jl,p + g jp,l , does not transform as an affine connection under the TCT. This condition is, however, satisfied by adding the Levi-Civita term −1/2 r j=1 g ij g lp,j . Now, if one wants the Universal Criterion of Evolution satisfied also when the system relaxes along a shortest path, without imposing a priori any restrictions on transport coefficients, an extra term to the Levi-Civita affine connection should be added. It can be checked that the most general expression for this extra term is 1/(2σ) r j=1 X i X j g lp,j [1]. Hence, the affine connection my be written as In the examples examined in this paper, the small parameter ǫ measures the distance of the system from the steady state. In the article cited in the footnote [19], it is analyzed the relaxation of magnetically confined plasmas in a toroidal geometry. The characteristic feature of these equations is the presence of a small parameter, the drift parameter, defined as the Larmor radius over a macroscopic length [11]. In this situation, the long-time behaviour of the solution, describing the evolution of the system near the steady state, may be obtained by using the multiple time-scale perturbation expansion (see, for example, the book cited in Ref. [12]). The results obtained are valid generally because the dynamics include all relevant moment equations and the parameter ǫ is not related to the distance of the system from the stationary states. It is possible to show that, even in this case, the results (i) and (ii) (see the Introduction) remain valid.
A non-Riemannian geometry may successively be constructed out of the components of the affine connections [1], [13].
The main conclusion of the present analysis is thus: Close to steady-states, the geometry of the thermodynamic space is non-Riemannian with affine connection given by Eq. (A1). The knowledge of the expression for the affine connection is a fundamental prerequisite for the construction of the (nonlinear) closure theory on transport processes. In Ref. [1], the curvature tensor and the nonlinear transport equations have successively been derived from Eq. (A1) and by introducing the following assumption: There exists a thermodynamic action, scalar under T CT , which is stationary with respect to arbitrary variations in the transport coefficients and the affine connection.
From this principle, a set of closure equations, constraints, and boundary conditions have been derived. These equations determine the nonlinear corrections to the linear ("Onsager") transport coefficients. The validity of the nonlinear transport equations has been largely tested by analyzing several symmetric transport processes, such as the thermoelectric effect and magnetically confined plasmas [14]. Transport processes in tokamak plasmas are of particular interest. In this case, even in absence of turbulence, the state of the plasma remains close to (but, it is not in) a state of local equilibrium. This prevents the transport relations from being linear. The nonlinear transport equations have been used for computing the particle and energy losses in tokamak-plasmas and for deriving the distribution density functions for the species α in several collisional regimes [2].
where identities (A5) have been taken into account.
Let us now prove that Taking into account definition (A3) and expansions (41), we have Thus, at the lowest order Multiplying Eq. (52) by J i and summing over the index "i", we obtain where Eq. (A6) has been taken into account. Eq. (A16) may also be rewritten as However, from Eq. (31) and Eq. (43), and taking into account that r i=1 J i (X st. )X i (1) = 0, we have Hence, at the lowest order, we finally get low-collisional regimes, submitted to publication in the review Physics of Plasmas (2010).
[18] The slaving principle establishes that in a relaxation process, contributions from different wave-numbers are negligible with respect to those with same wave-numbers.
[19] G. Sonnino, Relaxation of Magnetically Confined Tokamak-Plasmas to Mechanical Equilibria, to be submitted for publication in the review Physics of Plasmas (2010).