Entropy production and Onsager symmetry in neoclassical transport processes of toroidal plasmas

Entropy production and Onsager symmetry in neoclassical transport processes of magnetically confined plasmas are studied in detail for general toroidal systems, including nonaxisymmetric configurations. It is found that the flux surface average of the entropy production defined from the linearized collision operator and the gyroangle‐averaged distribution function coincides with the sum of the inner products of the thermodynamic forces and the conjugate fluxes consisting of the Pfirsch–Schluter, banana‐plateau, nonaxisymmetric parts of the neoclassical radial fluxes and the parallel current. It is proved from the self‐adjointness of the linearized collision operator that the Onsager symmetry is robustly valid for the neoclassical transport equations in the cases of general toroidal plasmas consisting of electrons and multi‐species ions with arbitrary collision frequencies. It is shown that the Onsager symmetry holds whether or not the ambipolarity condition is used to reduce the number of the conjugate pa...


I. INTRODUCTION
Transport processes caused by binary Coulomb collisions between charged particles in toroidal magnetic configurations are described by the neoclassical transport theory. [1][2][3] Particle and heat transport fluxes observed in most fusion devices exceed the predictions of the neoclassical transport theory, and thus are called anomalous transport. 4 The anomalous transport is considered to result from the turbulent fluctuations driven by various plasma instabilities, which are not taken into account by the neoclassical theory. However, the neoclassical theory is regarded as a standard model with a well-established framework, which gives lower limits of the transport fluxes in quiescent plasmas close to thermal equilibria, and it also has practical use for predicting the parallel transport such as a bootstrap current with more accuracy than the predictions for the radial transport. The characteristics of the neoclassical theory lie in the inclusion of the effects of the global magnetic field geometry due to the long mean free path of the hot particles. In less collisional regimes such as plateau and banana regimes, the processes in deriving the neoclassical transport equations, which relate the neoclassical transport fluxes to the thermodynamic forces, are more complicated than the derivation of the classical transport equations in more collisional regimes. Since the neoclassical transport coefficients contain the parameters relating to both the collisionality and the magnetic geometry, it is less trivial than in the case of the classical transport to show the validity of the Onsager symmetry 5 for the neoclassical transport. In the cases of axisymmetric systems, the analytical expressions of the full neoclassical transport coefficients for all collisionality regimes have been obtained, and it is well-known that the Onsager symmetry holds for the axisymmetric neoclassical transport matrix.
The neoclassical transport equations for nonaxisymmetric systems [6][7][8][9][10][11] are more complicated due to the geometrical complexities, and they involve the nonambipolar parts. The absence of symmetry may also cause the breakup of magnetic surfaces into islands and ergordic regions, 12 although such problems are beyond the scope of this work. Here we assume the existence of toroidal nested magnetic surfaces as in other previous works. [6][7][8][9][10][11] Balescu and Fantechi derive the full neoclassical transport coefficients for the nonaxisymmetric plasma in the plateau regime, and claim that the Onsager symmetry partly breaks down in that case. 11 Contrary to Ref. 11, we show in the present work that the Onsager symmetry is robustly valid for the neoclassical transport equations in the cases of general toroidal plasmas consisting of electrons and multi-species ions with arbitrary collision frequencies.
Concerning the Onsager relation, it is important to discuss the entropy production resulting from the transport processes since the Onsager relation, if it holds, is satisfied only by the transport matrix connecting the conjugate pairs of transport fluxes and thermodynamic forces, which should be specified through the entropy production. 5 According to the terminology in Ref. 3, we expect that the kinetic form of the entropy production defined from the collision operator should coincide with its thermodynamic form, in which the entropy production is expressed as the sum of the products of the conjugate pairs of the fluxes and forces. In Chap. 17 of Ref. 3, Balescu presented detailed analyses on the kinetic and thermodynamic forms of the entropy production in the classical and neoclassical transport processes for the axisymmetric case. Using the Hermitian moment representation, he confirmed that the kinetic form of the entropy production includes the thermodynamic form given by the products of the thermodynamic forces and their conjugate classical and neoclassical Pfirsch-Schlüter fluxes, although he did not identify the thermodynamic form corresponding to the neoclassical banana-plateau transport. In this work, before the proof of the Onsager symmetry, we show for general toroidal geometry the complete coincidence between the kinetic form of the entropy production and its full thermodynamic form, including all contributions from the classical and neoclassical Pfirsch-Schlüter, banana-plateau, nonaxisymmetric fluxes. The main difference between our treatment and that in Ref. 3 is that we use directly the distribution function and the drift kinetic equations instead of the Hermitian moment expansion.
The proof of the Onsager symmetry for the neoclassical transport equations in general toroidal configurations is given in the similar manner to that in the Appendix of Ref. 2. The proof uses the self-adjointness of the linearized collision operator and the formal solution of the linearized drift kinetic equation, although neither axisymmetry of the magnetic configuration nor any condition for collisionality is required. We also derive the full neoclassical transport coefficients in the nonaxisymmetric system for collision frequencies in the Pfirsch-Schlüter and plateau regimes, from which the Onsager symmetry of the full neoclassical transport matrix is directly confirmed.
In the axisymmetric configurations, the ambipolarity is automatically satisfied by the neoclassical transport and the radial electric field does not affect the transport fluxes. On the other hand, in the nonaxisymmetric configurations, the radial electric field is determined through the particle transport equations if the ambipolarity condition is imposed. In both cases with and without the ambipolarity condition, we give the neoclassical transport equations and check their Onsager symmetry.
This work is organized as follows. In Sec. II, the entropy production defined from the collision operator is divided into the two parts, which are derived from the gyroangleaveraged and gyroangle-dependent parts of the distribution function. The entropy production from the gyroangledependent distribution function is given by the sum of the inner products of the classical radial particle and heat fluxes and the radial gradient thermodynamic forces. The transport matrix relating these classical fluxes and forces is shown to satisfy the Onsager symmetry. We find that the entropy production from the gyroangle-averaged distribution function is written as the sum of the inner products of the thermodynamic forces and the corresponding conjugate fluxes which consist of the Pfirsch-Schlüter, banana-plateau, nonaxisymmetric parts of the neoclassical radial fluxes and the parallel current. In Sec. III, using the formal solution of the linearized drift kinetic equation and the self-adjointness of the linearized collision operator, we prove that the Onsager symmetry is satisfied by the neoclassical transport equations for arbitrary collision frequencies in general toroidal systems, including nonaxisymmetric cases. The effects of the ambipolarity on the neoclassical transport coefficients are examined for both axisymmetric and nonaxisymmetric cases to show the robust validity of the Onsager symmetry independent of the use of the ambipolarity condition. In Sec. IV, the full transport coefficients are derived for the banana-plateau and nonaxisymmetric parts, separately, and their symmetry properties are investigated. We derive the nonaxisymmetric transport coefficients for arbitrary collision frequencies in the Pfirsch-Schlüter and plateau regimes, and directly confirm that the total banana-plateau and nonaxisymmetric transport equations satisfy the Onsager symmetry. Finally, conclusions and discussions are given in Sec. V. There, we discuss the reason why our two main results, i.e., the complete correspondence between the kinetic and thermodynamic forms of the entropy production, and the Onsager symmetry in the nonaxisymmetric case, were not confirmed in Ref. 3

II. ENTROPY PRODUCTION IN CLASSICAL AND NEOCLASSICAL TRANSPORT PROCESSES
Here, we show that the thermodynamic form of the entropy production is equivalent to its kinetic form defined from the collision operator for the classical and neoclassical transport in the general case of magnetically confined plasma with arbitrary toroidal geometry. For that purpose, we first describe several properties of the collision operator which is denoted for species a by where f a ( f b ) is the distribution function of the species a (b) and C ab represents the contribution from the collision between the particles a and b. The collision operator conserves the particles' number, momentum, and kinetic energy, which is written as Furthermore, the collision operator is invariant under arbitrary translational and rotational transform of the velocity variable v of distribution functions, which is expressed by where T f and Rf denote functions f with arbitrary translational and rotational transform of the velocity variable v, respectively.
The entropy production for the species a is defined from the collision operator by The second law of the thermodynamics or the positive definiteness of the entropy production is given by where the total entropy production a Ṡ a vanishes if and only if the distribution functions for all species are the Maxwellian with the same temperature and the same mean velocity. However, even if two particle species with much different masses such as for electrons and ions have the Maxwellian distributions with different temperatures, the collisional heat exchange between the two species are negligibly slow and the entropy production due to their collisions is so small that we hereafter neglect it based on the small mass ratio ordering.
For magnetically confined plasmas, the lowest-order distribution function for the species a with respect to the drift ordering is given by the Maxwellian with no mean velocity as where v Ta 2 2T a /m a and x a v/v Ta are defined from the temperature T a . The distribution functions and the collision operator are perturbatively expanded in the drift ordering parameter /L (: the thermal gyroradius, L: the equilibrium scale length as where the linearized collision operator C ab L is defined by The linearized collision operator also has the conservation and symmetry properties, which are expressed by Eqs. 2 and 3 with C ab replaced by C ab L . The self-adjointness of the linearized collision operator 13 is written as which is exactly valid for T a T b and is approximately satisfied for T a T b when m a /m b 1 or m b /m a 1. For example, in the case of collisions between ions (ai) and electrons (be) with T i T e , C ie L contains a part which breaks the complete self-adjointness although it is neglected to the lowest order in (m e /m i ) 1/2 see Sec. IV of Ref. 13. Concerning the positive definiteness of the entropy production described in Eq. 5, we find the positive definiteness of the quadratic form associated with the linearized collision operator as which is valid for T a T b to the lowest order of the small mass ratio m a /m b 1 or m b /m a 1) as Eq. 9. Using in Eq. 4 ln f a ln f a0 f a1 f a0 the entropy production Ṡ a up to O ( 2 ) is given by Instead of Eq. 5, we have from Eq. 10 Now, let us divide the first-order distribution function f a1 into the gyroangle-averaged part f a1 and the gyroangledependent part f a1 as Due to the rotational symmetry of the collision operator in the velocity space shown in Eq. 3, the entropy production a separates into the corresponding parts as a a a , 14 First, we consider the entropy production a due to the gyroangle-dependent part of the distribution function. The gyroangle-dependent part f a1 is given from the lowest-order distribution function f a0 by where nB/B is the unit vector along the magnetic field B and a e a B/m a c is the gyrofrequency of the particle with the mass m a and the charge e a . In Eq. 17, f a0 is regarded as a function of (V,E,) (V: the volume inside the flux surface, E 1 2 m a v 2 e a : the particle's energy, and m a v 2 /2B: the magnetic moment, and we have used where the thermodynamic forces X a1 and X a2 are defined from the radial gradients of the pressure p a , the electrostatic potential , and the temperature T a as X a1 p a n a e a , X a2 T a S V D , 19 in terms of which the perpendicular components of the fluid velocity and the heat flow are Here the friction forces F a1 and F a2 are given by It should be noted that, from the rotational symmetry of the collision operator, that f a1 does not contribute to the perpendicular friction forces F a j while f a1 does not contribute to the parallel friction forces F i a j . Equation 21 shows that the entropy production a defined from the gyroangle-dependent part of the distribution function is caused by the classical particle and heat transport, and that the classical fluxes J a1 cl and J a2 cl are conjugate to the thermodynamic forces X a1 and X a2 , respectively. The momentum conservation due to the collision operator gives a F a1 0, 25 which in turn causes the classical particle fluxes to satisfy the ambipolarity as a e a a cl 0. 26 We have the relations between the perpendicular friction forces and flows from Eqs. 17 and 24 as where the coefficients l jk ab are the same ones defined in Ref. 2 and are given by Here, L 0 (3/2) (x 2 )1, L 1 (3/2) (x 2 ) 5 2 x 2 , •••, are the Laguerre polynomials of order 3 2 . In Eqs. 27 and 28, the rotational symmetry of the linearized collision operator is used. From the self-adjointness of the linearized collision operator shown in Eq. 9, the coefficients l jk ab have the following symmetry : The momentum conservation property described in Eq. 25 imposes another constraint on the coefficients l jk ab : Here we have chosen a certain particle species denoted by I. We hereafter regard I as the ion species with the smallest particle number density. If a plasma consists of electrons and a single ion species i, we take Ii.
Equation 31 shows that the number of the conjugated pairs of the classical fluxes and thermodynamic forces is reduced by employing the new pairs (J a1 ,X a1 * ) aI ,(J a2 ,X a2 ) instead of (J a1 ,X a1 ),(J a2 ,X a2 ). We also find that the radial electric field does not appear in the new set of the thermodynamic forces. The transport equations which relate the classical fluxes (J a1 ,J a2 ) to the thermodynamic forces (X a1 ,X a2 ) are obtained from Eqs. 20, 22, 23, and 27 as

33
where the classical transport coefficients (L cl ) jk ab are given by which shows that the transport coefficients (L cl ) jk ab are the same as in Eq. 33, except for the limitation (a, j),(b,k) (I,1), and that the Onsager symmetry is valid for both of the conjugate pairs. We should note that the ambipolarity condition 36 reduces the number of the thermodynamic forces required for determining the classical fluxes by one, and that the radial electric field does not enter the reduced set of the thermodynamic forces (X a1(aI) * ,X a2 ). Next, let us consider the entropy production a due to the first-order gyroangle-averaged distribution function f a1 , which satisfies the linearized drift kinetic equation: [1][2][3][7][8][9][10][11]14,15 v where v da is the sum of the EB, B and curvature drift velocities, and C a Here, it should be noted that the electric drift term v E -f a1 is sometimes retained in the linearized drift kinetic equation 16 which causes the nonlinear radial electric field dependence of the neoclassical transport coefficients and of the ambipolarity condition for a nonaxisymmetric system. There are two main reasons why the electric drift term However, if the radial electric field much larger than assumed by the drift ordering is generated by some techniques such as neutral beam injections, the electric drift term v E -f a1 should be retained in the drift kinetic equation as treated in Ref. 16. Another important reason is as follows.
The radial electric field is regarded as one of the thermodynamic forces as seen from Eq. 19. When the ambipolarity condition is imposed for nonaxisymmetric systems, the radial electric field is regarded as a function of the other thermodynamic forces as shown later. If the electric drift term is added in the left-hand side of the linearized drift kinetic equation, the radial electric field dependence appears in the neoclassical transport coefficients as mentioned before, and accordingly we obtain the nonlinear transport equations of the form JL(X)-X in which the transport fluxes J are nonlinear functions of the thermodynamic forces X. On the other hand, as shown in detail in Ref. 5, the Onsager symmetry is relevant to linear transport equations of the form JL-X with the transport matrix L independent of the thermodynamic forces X. Since, in this work, we are concerned with the Onsager symmetry for the transport matrix in the linear neoclassical transport equations, the electric drift term causing the nonlinear dependence on the thermodynamic forces should be neglected. We obtain from Eqs. 15 and 38, where the parallel flow velocity u i a is defined by Here, the flux surface average of the first term in the right-hand side vanishes. In taking the flux surface average of the second term, we use the following two equations for the radial particle and heat fluxes: where the Hamada coordinates (V,,) are employed to represent the poloidal and toroidal components of the magnetic fieldssee Appendix A and Eq. A2 is used. Here, we have used the definitions p a1 The neoclassical particle and heat fluxes are given by which consist of the Pfirsch-Schlüter (J a j PS ), the bananaplateau (J a j bp ), and nonaxisymmetric (J a j na ) parts defined by Here the Pfirsch-Schlüter fluxes are written as parts of the neoclassical fluxes as in Refs. 1-3 where the term ''neoclassical'' is used for the transport due to guiding center motions in a toroidal magnetic configuration affected by collisions, which is a contrast to the ''classical'' transport caused by particle gyro-motions with collisions. However, the term ''neoclassical'' is sometimes used in a narrower sense for referring to the transport fluxes due to particles with long mean free paths such as the banana-plateau fluxes, which exclude the Pfirsch-Schlüter fluxes. Then, we find that the flux surface average of the second term in the right-hand side of Eq. 39 is given by the products of the neoclassical radial fluxes and the thermodynamic forces as The flux surface average of the third term in the right-hand side of Eq. 39 is given by where Eq. A7 is used. Then, we finally obtain from Eqs. 39, 44, and 45 the thermodynamic form of the fluxsurface-averaged entropy production a as where the parallel flux J a3 and the parallel force X a3 are defined by J a3

47
Taking the species summation of Eq. 46, we have a T a a a J a1 where J E and X E are defined from the total parallel current J i and the parallel electric field E i as

49
Thus the flux surface average of the entropy production due to the gyroangle-averaged distribution functions is given in the thermodynamic form, in which the neoclassical radial fluxes J a1 ncl , J a2 ncl and the parallel current J E are conjugate to the radial gradient forces X a1 , X a2 and the parallel electric field X E , respectively. It should be noted that the neoclassical thermodynamic form of the entropy production can be obtained only through the magnetic surface average as in Eq. 48, which is a remarkable contrast to the classical thermodynamic form 21 defined locally in the configuration space. Now, let us consider the ambipolarity condition for the neoclassical particle fluxes. Using the momentum conservation 25 by collisions and the charge neutrality condition a n a e a 0, we obtain the flux surface average of the total parallel momentum balance equation as a B-a 0.
Then, we find from the definitions in Eq. 43 that the intrinsic ambipolarity holds for both of the Pfirsch-Schlüter and banana-plateau particle fluxes in the same way as for the classical particle fluxes, which implies that the ambipolar conditions, a e a J a1 PS a e a J a1 bp 0, 50 are valid for arbitrary values of the thermodynamic forces (X a1 ,X a2 ,X E ). On the other hand, the nonaxisymmetric particle fluxes J a1 na and accordingly the total neoclassical particle fluxes J a1 ncl are not ambipolar generally. If the ambipolarity condition for the total neoclassical particle fluxes, a e a J a1 ncl a e a J a1 na 0, 51 is imposed, we find in the similar way as in Eq. 31 that Eq. 48 is rewritten as a T a a aI J a1 In axisymmetric toroidal systems, the nonaxisymmetric fluxes J a j na ( j1,2) vanish and therefore the neoclassical particle fluxes J a1 ncl are intrinsically ambipolar. As discussed in the next section, in nonaxisymmetric systems, the ambipolarity condition 51 combined with the neoclassical transport equations gives a constraint on the thermodynamic forces (X a1 ,X a2 ,X E ) from which the radial electric field is expressed by a linear form in the pressure and temperature gradients and the parallel electric field. Then, independent thermodynamic forces for nonaxisymmetric systems are given not by the set (X a1 ,X a2 ,X E ) but by the reduced one (X a1(aI) * ,X a2 ,X E ). In the present work, we show the neoclassical transport equations for both cases with (X a1 ,X a2 ,X E ) and with (X a1(aI) * ,X a2 ,X E ) used as the forces, in order to elucidate the relation of the ambipolarity to the axisymmetry and to the Onsager symmetry.

III. ONSAGER SYMMETRY OF NEOCLASSICAL TRANSPORT EQUATIONS FOR GENERAL TOROIDAL SYSTEMS
In this section, it is proved that the Onsager symmetry is satisfied by the neoclassical transport equations for general toroidal systems, including nonaxisymmetric cases. For that, it is convenient to define the distribution function ḡ a by where l dl denotes the integral along the magnetic field line. Then, Eq. 38 is rewritten as where the functions S a j ( j1,2,3) are defined by Here, it is worthwhile making some remarks on the case where, just as in Ref. 16, the electric drift term Here, in v E -ḡ a , the spatial derivative is taken with (x,v,) as independent phase space variables defined in Ref. 16, and the electric drift velocity is given by v E cEB/B 2 to satisfy an important phase space conservative property see Ref. 16. In that case, nonlinear neoclassical transport equations are derived due to the nonlinear dependence on the radial electric field as mentioned after Eq. 38. Here, let us artificially regard the radial electric field E V /V added in the left-hand side of Eq. 54 as an independent parameter, although E V is already contained as a part of the thermodynamic forces in the right-hand side of Eq. 54. By doing this, the resulting transport equations are written in the apparently linear form JL(E V )-X with E V as a parameter in the transport matrix L. Then, it is shown that the proof of the Onsager symmetry for L(E V ) in this section is still valid even if A E ḡ a is retained. This follows from the fact that the operator A E has the same properties as v i nwhich are given by A E f a0 0 and d 3 vA E F0 for an arbitrary function F on the phase space. The neoclassical radial fluxes J a j ncl ( j1,2) and the parallel flux J a3 are expressed in terms of ḡ a and S a j ( j1,2,3) as where we have used the following formula for an arbitrary function F(x): 57 Noting in Eq. 54 that the left-hand side is linear with respect to ḡ a and that X a j ( j1,2,3) occur in the right-hand side as parameters, we find that the solution ḡ a of Eq. 54 is given by Then, using Eqs. 56 and 58, we obtain the transport equations relating J a j ncl ( j1,2) and J a3 to X a j ( j1,2,3) as where the transport coefficients L jk ab are given by

61
Equation 60 is rewritten as the transport equations relating J a j ncl ( j1,2) and J E to X a j ( j1,2) and X E : where the coefficients L jE a , L E j a ( j1,2) and L EE are given by In order to show the Onsager symmetry of the transport coefficients, it is useful to separate Eq. 59 into even and odd parts with respect to v i as where the superscripts and denote the even and odd parts, respectively. Noting that S a j are even for j1,2 and odd for j3, and using Eqs. 61 and 64, we obtain

66
We find from the self-adjointness of the linearized collision operator given by Eq. 9 that the right-hand side of Eq. 66 is invariant under the permutation of the subscripts (a, j)↔(b,k) and that Equations 67 and 68 show that the Onsager symmetry is satisfied by the transport matrix which combines the conjugate pairs of the fluxes (J a1 ncl ,J a2 ncl ,J E ) and the forces (X a1 ,X a3 ,X E ) for general toroidal systems.
Here, let us discuss the relation between the transport equations and the ambipolarity condition. In axisymmetric systems, intrinsic ambipolarity holds for the neoclassical particle fluxes and is expressed in terms of the relation between the transport coefficients as a e a L 1 j ab a e a L 1E a 0 j1,2,3. 69 Then, the number of the conjugate pairs of fluxes and forces is reduced by one as shown in Eq. 52 using X a1(aI) * instead of X a1 . We find from Eqs. 67, 68, and 69 that the transport equations relating the fluxes (J a1(aI) ncl ,J a2 ncl ,J E ) to (X a1(aI) * ,X a2 ,X E ) in the axisymmetric case are given by In the transport equations 70, the transport coefficients L jk ab are the same as in Eq. 62 except for the limitation (a, j),(b,k) (I,1), and the Onsager symmetry is still valid.
In nonaxisymmetric systems, the ambipolarity condition 51 gives a relation between the thermodynamic forces which is used to express one thermodynamic force X I in terms of the other thermodynamic forces (X a1(aI) * ,X a2 ,X E ) as where the transport coefficients L jk ab , L jE a , L E j a , and We see from Eqs. 67 and 68 that the Onsager symmetry still holds for the transport coefficients given by Eq. 73: Thus, we have established for nonaxisymmetric systems with no net radial current that the radial electric field is determined by the pressure, temperature gradients, and the parallel electric field, and that transport satisfies the Onsager symmetry.

IV. BANANA-PLATEAU AND NONAXISYMMETRIC TRANSPORT COEFFICIENTS
In the previous section, we have shown the Onsager symmetry for the transport coefficients relating the neoclassical radial fluxes and the parallel current (J a1 ncl ,J a2 ncl ,J E ) to the radial gradient forces and the parallel electric field (X a1 ,X a2 ,X E ) in general toroidal systems. The neoclassical radial particle and heat fluxes (J a1 ncl ,J a2 ncl ) consist of the Pfirsch-Schlüter, banana-plateau, and nonaxisymmetric parts while, as shown in Appendix B, it is well known that the transport equations for the Pfirsch-Schlüter fluxes (J a1 PS ,J a2 PS ) and the radial gradient forces (X a1 ,X a2 ) satisfy the Onsager symmetry. Thus, it is clear that the sum of the banana-plateau and nonaxisymmetric radial fluxes, and the parallel current (J a1 bp J a1 na ,J a2 bp J a2 na ,J E ) are related to the forces (X a1 ,X a2 ,X E ) by the transport coefficients with the Onsager symmetry. In this section, using the 13 moment 13M approximation, 3 we derive the transport equations for the banana-plateau fluxes and those for the nonaxisymmetric fluxes separately, in the case of general toroidal plasma consisting of electrons and one ion species. Then, the symmetry properties are investigated for each of the transport equations, and the Onsager symmetry for the total transport is directly confirmed.
The parallel momentum balance equations combined with the friction-flow relations are given in the 13M approximation by where the momentum conservation 25 in collisions and the charge neutrality condition a e a n a 0 are used.
Solving the linearized drift kinetic equation gives the equations for the parallel viscosities, which have the following form for all collision frequencies in the Pfirsch-Schlüter, plateau, and banana regimes: where c a and a j ( j1,2,3) are the dimensionless parameters for the viscosity coefficients, and G a represents the geometrical factor which measures the deviation from the axisymmetric configuration. These parameters c a , a j ( j1,2,3) and G a are given in Appendix C and all of them are generally dependent on the collision frequencies although, in the axisymmetric case, the geometrical factor is given by G a B /B for both species and for all the collision frequencies.
Using Eqs. 76-78, the ion parallel flows and viscosity are written as G .

82
We also find from Eq. 75 that the parallel current is divided into the classical part J E cl and the neoclassical part J E bp due to the electron parallel viscosities as Finally, we obtain from Eqs. 77 and 81-83 the banana-plateau transport equations which relate

84
where the banana-plateau transport coefficients are given by

97
Here, the force X e1 ** is defined by When G e G i , X e1 ** coincides with X e1 * , which is proportional to the total pressure gradient p( p e p i ), and the radial electric field never affects the bananaplateau transport. However, if the electrons and the ions belong to different collisionality regimes in the nonaxisymmetric case, the banana-plateau radial particle and heat fluxes, and the bootstrap current depend on the radial electric field through X e1 ** since G e G i .
Next, let us derive the nonaxisymmetric transport equations. For the derivation, it is essential to note that the toroidal viscosities are given in the following form for both of the Pfirsch-Schlüter and plateau regimes:

99
where c ta and G ta are given in Appendix C and a j   ( j1,2,3) are the same as in Eq. 78. Here and hereafter, we consider the toroidal viscosities and the nonaxisymmetric fluxes only for the Pfirsch-Schlüter and plateau regimes since the expressions similar to Eq. 99 have not been obtained yet for the banana regime.
Appendix C shows that the ratio between the toroidal and parallel viscosities c ta /c a is related to the geometrical factor G a in Eq. 78 by the following equation: , 100 which is an essential relation for showing the Onsager symmetry of the total neoclassical transport. Using Eqs. 78 and 99, the toroidal viscosities are written in terms of the parallel viscosities and the thermodynamic forces as Then, from Eqs. 43, 84, and 101, we obtain the nonaxisymmetric transport equations relating (J e1 where the nonaxisymmetric transport coefficients are given by

104
It is found that (L na ) jk aa (L na ) k j aa is always valid although (L na ) jk ei (L na ) k j ie is satisfied only when c te /c e c ti /c i . We see from Eq. 100 that the condition c te /c e c ti /c i is equivalent to G e G i , which holds if the electrons and the ions belong to the same collisionality regime.
where the total transport coefficients are given by Thus, from the above definitions and the symmetry properties given by Eq. 94, we can directly confirm that the total banana-plateau and nonaxisymmetric transport coefficients satisfy the following Onsager symmetry: L bn jk ab L bn k j ba , L bn E j a L bn jE a a,be,i; j,k1,2. 109 Here, we should note that these transport coefficients contain terms of different orders in (m e /m i ) 1/2 . As seen from Eqs.
85-93 and Eqs. 106-108, the ion-ion coefficients (L bn ) jk ii are O (m i /m e ) 1/2 larger than the other coefficients. When the ambipolarity condition is imposed in the nonaxisymmetric case, we obtain, in the similar way as in Eqs. 71-73, the radial electric field: where the transport coefficients are defined by The Onsager symmetry still holds for the above coefficients: L bn jk ab L bn k j ba , L bn jE a L bn E j a . 113 As mentioned earlier, terms of different orders in (m e /m i ) 1/2 are included in the transport coefficients (L bn ) jk ab . Therefore, the coefficients (L bn ) jk ab for the reduced transport equations also contain different order terms. To the lowest order of (m e /m i ) 1/2 , Eq. 110 is approximated by

V. CONCLUSIONS AND DISCUSSION
In this work, we have investigated the entropy production, the full transport equations, and their Onsager symmetry for the neoclassical transport processes in magnetically confined plasmas with general toroidal configurations. It was clearly shown that, for both the classical and neoclassical transport processes, the kinetic form of the entropy production defined from the linearized collision operator is equivalent to its thermodynamic form written as the inner product of the thermodynamic forces and their conjugate transport fluxes. The entropy production from the gyroangledependent distribution function corresponds to the sum of the products of the classical radial particle and heat fluxes and the radial gradient thermodynamic forces, while the magnetic surface average of the entropy production from the gyroangle-averaged distribution function is given by the sum of the products of the thermodynamic forces and their conjugate fluxes which consist of the Pfirsch-Schlüter, bananaplateau, nonaxisymmetric parts of the neoclassical radial fluxes and the parallel current. This equivalence between the kinetic and thermodynamic forms of the entropy production for the full neoclassical transport fluxes were not confirmed by Balescu in Chap. 17 of Ref. 3. The reason is now discussed.
In deriving the thermodynamic form of the entropy production, we used the linearized drift kinetic equation without employing the Hermitian moment expansion of the distribution function. Balescu expressed the kinetic form of the entropy production as the quadratic form of only the vector Hermitian moment part of the distribution function which corresponds to the l1 part of the Legendre polynomial expansion. However, since the neoclassical banana-plateau and nonaxisymmetric fluxes are caused by the parallel and toroidal viscosities, the tensor Hermitian moment part or the l2 part of the Legendre polynomial expansion needs to be included for deriving the neoclassical thermodynamic form of the entropy production. Furthermore, as shown in Appendix D, we find that it is necessary to include all the tensor moments including higher order parts with l3,4,5,..., in the kinetic form of the entropy production to derive the neoclassical thermodynamic form in the banana and plateau regimes. This is intuitively understandable by considering the resonant particles responsible for the neoclassical fluxes in the plateau regime. The resonant particle distribution is highly anisotropic in velocity space and is approximated by a delta function in pitch angle so that the all Hermitian moments or all lth-order Legendre polynomials are required. As shown in Appendix D, the operator v i nin the drift kinetic equation 38 introduces the anisotropic distribution in the velocity space and it connects the lth-order moment with (l1)th-order moments in contrast with the linearized collision operator which is isotropic in the velocity space and connects the lth-order moment with the same lth-order moment alone. In the Pfirsch-Schlüter regime, the collision operator dominates v i nand the distribution function has small contributions from higher-order moments representing the anisotropy. Then, in the Pfirsch-Schlüter regime, the l1 vector moment is enough to express the entropy production as in Ref. 3, while the negligibly small viscosity-induced neoclassical fluxes are included in the l2 tensor moment part. On the other hand, as the collision frequency decreases, the operator v i nis comparable to, and then dominates, the collision operator and all l-order moments are required in the kinetic entropy production functional to obtain its neoclassical thermodynamic form in the banana and plateau regimes.
We also proved from the formal solution of the linearized drift kinetic equation with the self-adjoint linearized collision operator that the Onsager symmetry is robustly valid for the neoclassical transport equations for general toroidal plasmas consisting of electrons and multi-species ions with arbitrary collision frequencies. Furthermore, we derived in Sec. IV, in the case of a single ion species, the full bananaplateau transport coefficients for all collisionality regimes and the full nonaxisymmetric transport coefficients for the Pfirsch-Schlüter and plateau regimes. The symmetry properties of these transport matrices were separately examined and the Onsager symmetry for their total transport equations was confirmed. We discussed the effects of the ambipolarity condition on the transport equations in detail for both axisymmetric and nonaxisymmetric configurations. The ambipolarity condition reduces by one the number of the conjugate pairs of the transport fluxes and the thermodynamic forces. In the axisymmetric case, the intrinsic ambipolarity holds and the radial electric field does not affect the transport. On the other hand, for broken toroidal symmetry, a radial current is a function of the thermodynamic forces (X a j ,X E ) in which the radial electric field is included. When the ambipolarity condition is imposed in the nonaxisymmetric case, the radial electric field is given by a linear combination of the other thermodynamic forces. We showed that the Onsager symmetry is satisfied whether the conjugate pairs of the fluxes and forces are reduced by the ambipolarity condition or not.
Balescu and Fantechi derived the full neoclassical transport coefficients for the plateau regime in the nonaxisymmetric configuration and claimed that the Onsager symmetry is slightly broken by the nonaxisymmetry. They showed the transport equations only for the reduced pairs of the fluxes and forces, in which the radial electric field is eliminated by the ambipolar condition. There, terms of O (m e /m i ) 1/2 were neglected in expressing the radial electric field in terms of the other forces as in Eq. 114. Then, the resultant transport coefficients in Ref. 11 deviate from those in Eq. 112 and the Onsager symmetry is broken in them since part of O (m e /m i ) 1/2 terms necessary for the symmetry are dropped.
The banana-plateau and nonaxisymmetric transport equations obtained here are valid whether electrons and ions belong to the same collisionality regime or not. When both species are in the same collisionality regime, we find that the geometrical factors for electrons and ions coincide with each other G e G i and that, as far as the radial fluxes and the radial forces are concerned, the Onsager symmetry is separately valid for the banana-plateau transport matrix (L bp ) jk ab and for the nonaxisymmetric transport matrix (L na ) jk ab . When electron and ion collisionality regimes are different, G e G i and the mixed electron-ion coefficients in each matrix are not symmetric, although the total matrix L bn are sym-metric. In the latter case, the radial electric field appears in the thermodynamic forces for the banana-plateau radial particle and heat fluxes and the bootstrap current in the nonaxisymmetric systems.
Since we proved the robust validity of the Onsager symmetry for the neoclassical transport equations, even in the nonaxisymmetric cases, this symmetry property can be utilized for the calculation of the nonaxisymmetric transport coefficients in the banana regime which were not given in Sec. IV. For example, from the banana-plateau transport coefficients (L bp ) jE a and (L bp ) E j a (ae,i; j1,2) for the banana regime given in Sec. IV, we can immediately obtain part of the nonaxisymmetric transport coefficients for the banana regime as (L na ) jE a (L bp ) jE a (L bp ) E j a (ae,i; j1,2). In the previous work, we investigated the neoclassical and anomalous transport in weakly turbulent plasma and described the entropy production and the Onsager symmetry in electrostatic turbulence. 17 We are also investigating a unified description of the transport equations, the entropy production, and the Onsager symmetry for neoclassical and turbulent processes with both electrostatic and electromagnetic fluctuations, which we will report on in a future work.

ACKNOWLEDGMENTS
The author HS thanks Professor M. Okamoto for his encouragement of this work, and Dr. N. Nakajima for useful discussions.
This work is supported in part by the Grant-in-Aid from the Japanese Ministry of Education, Science, and Culture, and in part by U.S. Department of Energy Grant No. DE-FG05-80ET-53088.

APPENDIX A: HAMADA COORDINATES AND INCOMPRESSIBLE FLOWS
In general toroidal configurations, the magnetic field is written in the contravariant form : where and are the poloidal and toroidal angle variables, respectively, and corresponding basis vectors for the contravariant representation are given by x/ and x/.
where B B-x/. When a solenoidal vector field U (-U0) is tangential to magnetic surfaces U-V0 and satisfies UBK(V) with some flux quantity K(V), it is written in the Hamada coordinates as where the both contravariant components U Uand U Uare flux surface quantities. Since both of the flow velocity u a and the heat flow q a are incompressible -u a -q a 0 see Eq. D12 and satisfies the above conditions see Eq. 20 to the lowest order in , all of their contravariant components u a u a -, u a u a -, q a q a -, q a q aare flux surface quantities. Also, u a and q a are separated into the parallel and perpendicular components as where the perpendicular components u a and q a are given by the thermodynamic forces X a1 and X a2 as in Eq. 20. Then, the contravariant poloidal and toroidal flow components are given by the linear combinations of the parallel flow components and the thermodynamic forces as

APPENDIX B: PFIRSCH-SCHLÜ TER TRANSPORT EQUATIONS
The relations between the parallel friction forces and the parallel flows are given by the 13M approximation in the same form as in Eq. 27, and are written as where the coefficients l jk ab are defined in Eq. 28. Then, the Pfirsch-Schlüter particle and heat fluxes (J a1 PS ,J a2 PS ) defined in Eq. 43 are rewritten as Substituting Eq. A7 into Eq. B2, we obtain the Pfirsch-Schlüter transport equations: where the Pfirsch-Schlüter transport coefficients (L PS ) jk ab are given by which shows that the transport coefficients (L PS ) jk ab are the same as in Eq. B3, except for the limitation (a, j),(b,k) (I,1) and that the Onsager symmetry is valid for both of the conjugate pairs.

APPENDIX C: PARALLEL AND TOROIDAL VISCOSITIES
It is shown from the solution of the linearized drift kinetic equation that the parallel viscosities in general toroidal configurations for all collision frequencies in the Pfirsch-Schlüter, plateau, and banana regimes are given in the following form : where c a and a j ( j1,2,3)   The geometrical factor G a is written as LG for the Pfirsch-Schlüter regime, F m,n ulnB mn u 2 umB nB u G 1 m,n ulnB mn u 2 mB nB mB nB umB nB u for the plateau regime.

C4
The geometrical factor G a for the banana regime is given in Ref. 7 or in Ref. 9 as G b . When we put (lnB) mn 0 for all n 0 in Eqs. C3 and C4, the expressions for axisymmetric systems are reproduced. In the axisymmetric case, G a is independent of the collision frequency and is given by G a B /B for all particle species.
It is shown that the toroidal viscosities are given in the following form for both of the Pfirsch-Schlüter and plateau regimes:

C7
The expressions similar to Eq. C5 have not been obtained yet for the banana regime. We find from Eqs. C3, C4, and C6 that the ratio between the toroidal and parallel viscosities c ta /c a is related to the geometrical factor G a by the following equation: which is an essential relation for showing the Onsager symmetry of the total neoclassical transport. We find from Eqs. C1 and C5 that the toroidal viscosities are written in terms of the parallel viscosities and the thermodynamic forces as c e a F a1 a2 a2 a3 GF X a1 X a2 G . C9

APPENDIX D: DRIFT KINETIC EQUATION AND LEGENDRE POLYNOMIAL EXPANSION
The linearized drift kinetic equation is written as where f a1 and f a0 are regarded as functions of the phase space variables (x,E,) (E 1 2 m a v 2 e a : the particle's energy; m a v 2 /2B: the magnetic moment). Using (x,v,) (v i /v) as the phase space variables instead of (x,E,), we consider the expansion by the Legendre polynomials P l () P 0 ()1,P 1 (), P 2 () 3 2 2 1 2 ,•••] for an arbitrary function F(x,v,) as We have the following formulas for the Legendre polynomials: lP l1 l1 P l1 , D4 1 2 dP l d ll1 2l1 P l1 P l1 .
We find from Eqs. D3 and D4 that the operator A v i nwith (x,E,) as the phase space variables in the drift kinetic equation D1 transforms the lth Legendre component to the linear combination of the (l1)th components: Contrastively, from the velocity space isotropy of the collision operator described in Eq. 3, the operator C a L transforms the lth component to only the lth one. The second and third terms in the right-hand side of Eq. D1 are rewritten from Eqs. 18 and 20 as F 2 3 1 3 and e a E i T a v f a0 , D7 respectively. The former is proportional to the divergence of the diamagnetic flows and contains only the zeroth and second Legendre polynomial components while the latter is proportional to the parallel electric field and contains only the first Legendre component. Now, let us write the drift kinetic equation D1 by each Legendre component, separately. The zeroth Legendre component or the scalar moment part of Eq. D1 is given by Here, the l1 Legendre component f a1 (l1) of the distribution function is expanded by the Laguerre polynomials

D9
where f a1 (l1,j2) denotes the sum of the jth Laguerre polynomial components with j2, which is neglected in the 13M approximation. The first term of Eq. D8 is rewritten as Af a1 l1 l0 2 3 x a 2 F -u i a n S x a 2 5 2 D 2 5 p a -q i a n G f a0 Af a1 l1,j2 l0 . D10 Then, Eq. D8 reduces to 2 3 x a 2 F -u a S x a 2 5 2 D 2 5p a -q aG f a0 Af a1 l1,j2 l0 C a L f a1 l0 . D11 Integrating Eq. D11 multiplied by 1 and x a 2 over the velocity space, we obtain the incompressibility of u a and q a : -u a -q a 0, D12 where we used the particle number and energy conservation by the collision operator described in Eq. 2. Exactly speaking, -q a 0 is valid to the lowest order of the small mass ratio as in Eq. 9. Thus, we have Af a1 l1,j2 l0 C a L f a1 l0 , D13 from which the contribution from the scalar moment part f a1 (l0) of the distribution function to the kinetic form of the entropy production is given by In the 13M approximation, a (l0) vanishes. Taking the first Legendre component or the vector moment part of Eq. D1, we have A f a1 l0 f a1 l2 l1 e a E i a T a v f a0 C a L f a1 l1 .

D15
We obtain the following parallel momentum balance equations from velocity integration of Eq. D15 multiplied by m a v and m a v(x a 2 5 2 ): n-p a1a n a e a E i F i a1 , D16 n-a1a F i a2 .