S.R. de Groot

Coefficients of viscosity for a fluid in a magnetic field or in a rotating system

Synopsis The linear equations between the elements of the viscous pressure tensor and the rates of deformation are investigated for the case of an isotropic fluid in an external magnetic field or for the equivalent case of a rotating fluid. Since these equations can be incorporated within the thermodynamics of irreversible processes, the Onsager reciprocity relations hold for the scheme of phenomenological coefficients. For the present case the viscous behaviour is seen to be described by 8 coefficients between which one Onsager relation exists. The remaining 7 independent coefficients can be combined in a linear way so as to yield 5 coefficients of ordinary viscosity, the other 2 coefficients then describing the volume viscosity and a cross-effect between the ordinary and the volume viscosity, respectively. For the special case of vanishing volume viscosity the equations are compared with those derived from kinetic theory by Chapman and Cowling for an ionized gas in a magnetic field.

On the theory of thermocouples and thermocells

Synopsis The thermodynamical theory of irreversible processes is applied to thermocouples and thermocells, and a general expression for the thermopotential is derived. In the derivations absolute material flows and a reduced heat flow have been used. The thermopotential consists of four parts, viz. two “heterogeneous” pontential differences (between the electrodes and the electrolyte) and two “homogeneous” potential differences (in the wire and in the electrolyte.) The general expression for the thermopotential can be split up into two parts, one containing essentially entropies of transfer and one containing known quantities (entropies and transference numbers). This latter part can be calculated and, from comparison with experimental results, conclusions as to the magnitude of the first part can be drawn.

On pressure and ponderomotive force in a dielectric Statistical mechanics of matter in an electromagnetic field II

Synopsis A statistical mechanical derivation of the expressions for both pressure and ponderomotive force in a dielectric is given in terms of averages over microscopic quantities. The existence of microscopic long-range interactions leads to the possibility of defining pressure and ponderomotive force in different ways. A natural way is indicated of dividing the average of the microscopic force into long and short range contributions, which yields the form of the ponderomotive force proposed by Kelvin. It is also shown that, in statistical equilibrium, one may obtain Helmholtzs expression for the ponderomotive force. In the two cases pressure must be defined differently. The relation between these pressures is a predictd by thermodynamic theory.

Phenomenological equations and onsager relations : the case of dependent fluxes or forces

Synopsis The influence of linear dependencies between the fluxes or between the forces occurring in the expression for the entropy production on the phenomenological coefficients and the Onsager reciprocal relations is investigated. If both sets of variables are dependent the phenomenological coefficients are not uniquely defined. It is shown that they always can be chosen such as to satisfy the Onsager relations.

Relativistic thermodynamics of irreversible processes. II: Heat conduction and diffusion; physical part

Synopsis The results of the relativistic thermodynamics of irreversible processes in an isotropic mixture of an arbitrary number of chemical components are reformulated, so far as heat conduction and diffusion are concerned, in three-dimensional tensor form with the help of quantities which are used in the non-relativistic theory. The transformation properties of the vectors J su(j) = ϱsu(j) ( v su(j) v ), where ϱsu(j) = density of rest mass of component j , v su(j) = velocity of component j , v = barycentric velocity, are examined. It is seen that J su(j) = J ′ su(j) + J ′ ∥ / su(j) (1 - v 2 / c 2 ) 1/2 , where J ′ su(j) is the vector J su(j) measured by an observer in the Lorentz frame where v = 0 and J ′ su(j) and J ′ ∥ / su(j) are the components of J ′ su(j) perpendicular and parallel to v respectively. As a consequence of the developed formalism it appears that the density of entropy, multiplied by a constant, is the fourth component of a four-vector. It is seen that in the absence of heat conduction and diffusion the entropy in a small element of volume is a Lorentz invariant quantity. Further, it appears that in all practical cases if there is heat conduction and diffusion the change of the entropy in an element of volume-is only very small under Lorentz transformations. The connection between different sets of heats of transfer occurring in the literature is derived. The heats of transfer are exactly or almost Lorentz invariant. Explicit formulae are given from which the difference between the results of the relativistic and the non-relativistic theory may easily be surveyed. The concentrations of the chemical components appear to be almost Lorentz invariant quantities.

Relativistic thermodynamics of irreversible processes. V: The energy-momentum tensor of the macroscopic electromagnetic field, the macroscopic forces acting on the matter and the first and second laws of thermodynamics

Synopsis A symmetric energy-momentum tensor for the macroscopic electromagnetic field in media which are isotropic as far as polarization and magnetization are concerned is derived from the general expressions, obtained in a previous paper, for the forces acting on tne matter. The non-diagonal elements of this tensor are the same as those of Abrahams tensor. Explicit expressions are given for the macroscopic forces acting on the matter and for the first and second laws of thermodynamics. It is shown that Abrahams tensor leads to an equivalent formalism and that from the point of view of the developed theory this tensor is preferable to Minkowskis tensor.

Thermodynamical theory of galvanomagnetic and thermomagnetic phenomena. I: Reciprocal relations in anisotropic metals

Synopsis The thermodynamic theory of galvanomagnetic and thermomagnetic phenomena in anisotropic metals is developed. With the method of De Groot and Mazur, which allows to treat vectorial phenomena, the following reciprocal relations are derived from microscopic reversibility L s s ( B ) = L s s † ( — B ) , L e e ( B ) = L e e † ( — B ) , L e s ( B ) = L s e † ( — B ) , where Lss is the heat conduction tensor, Lee the electrical conduction tensor, Les and Lse the tensors describing the cross-effects, and where the sign † indicates transpose matrix. The relation between the cross-effects is a non-trivial example of a reciprocal relation derived from the formulation of microscopic reversibility with variables, which are even and odd functions of particle velocities (Casimirs α- and β-variables).Synopsis The thermodynamic theory of galvanomagnetic and thermomagnetic phenomena in anisotropic metals is developed. With the method of De Groot and Mazur, which allows to treat vectorial phenomena, the following reciprocal relations are derived from microscopic reversibility L s s ( B ) = L s s † ( — B ) , L e e ( B ) = L e e † ( — B ) , L e s ( B ) = L s e † ( — B ) , where Lss is the heat conduction tensor, Lee the electrical conduction tensor, Les and Lse the tensors describing the cross-effects, and where the sign † indicates transpose matrix. The relation between the cross-effects is a non-trivial example of a reciprocal relation derived from the formulation of microscopic reversibility with variables, which are even and odd functions of particle velocities (Casimirs α- and β-variables).

On the statistical basis of Onsager's reciprocal relations

Synopsis A survey and discussion is given of various derivations of the Onsager reciprocal relations between irreversible processes. In particular a derivation is presented dealing explicitly with non-equilibrium situations.

On the derivation of reciprocal relations between irreversible processes

Synopsis The Onsager relations have been derived in a quantum statistical treatment of irreversible processes for scalar phenomena. In this paper the problem of deriving reciprocal relations between the coefficients which describe vectorial and tensorial phenomena is studied. The problem is approached in two different ways. Firstly, it is shown that one can derive from the quantum statistical theory the results concerning fluctuations used in Onsagers theory, which formed the starting point of de Groot and Mazurs proof of reciprocal relations between vectorial or tensorial phenomena. Secondly, a reformulation of the symmetry expressed by the scalar (Onsager) relations is given which can be readily utilised for the phenomenological equations describing vectorial and tensorial processes. In this way the reciprocal relations between the phenomenological coefficients can be derived. The method is applied to heat conduction, diffusion, viscous flow and cross effects, and also to electrical conduction in anisotropic solids.

Transformation properties of the onsager relations

Synopsis The invariance of the Onsager reciprocal relations under simultaneous linear transformations of the fluxes and forces occurring in the expression for the entropy production is investigated for the cases of scalar phenomena and of vectorial phenomena (heat conduction and diffusion). The connection between these transformations follows in a natural way from the invariance of the deviation of the entropy from its equilibrium value. The latter condition rather than the invariance of the entropy production guarantees the invariance of the Onsager relations.