Thermodynamic aspects of nonequilibrium current fluctuations

Starting from a macroscopic nonequilibrium entropy, we obtain an expression for the nonequilibrium fluctuations of the electric current in a metallic resistor. Our method goes further than previous theories of irreversible thermodynamics and, as well as microscopic entropies, it leads to results of the same order of magnitude but not completely coincident with the full nonequilibrium corrections obtained from kinetic methods by Tremblay et al.


I. INTRODUCTION
matic perturbation theory, and by kinetic argu- ments.
In the last years, great attention has been devot- ed to the problem of nonequilibrium fluctuations of thermodynamic systems, ' since this analysis is directly related to experiments (for instance, light scattering, neutron scattering, computer simulation)   and since it allows a deeper insight into the problems of nonequilibrium statistical mechanics.Re- cently, we have analyzed the problem of equilibrium fluctuations of dissipative fluxes in the frame- work of the so-called extended irreversible thermo- dynamics, ' where, starting from a generalized Gibbs equation and the Einstein hypothesis for the probability of the fluctuations, we have obtained expressions for the second moments of the fluctua- tions of heat flux, electric current, and the hydro- dynamic dissipative fluxes, in equilibrium.
On the one hand, our theory unifies the usual expressions of the fluctuation-dissipation formulas for the fluc- tuations of the dissipative fluxes.On the other hand, it places some restrictions on the possible nonequilibrium generalizations of the Gibbs equation, which had not been taken into account up to now.
Recently, we have extended our method to the analysis of nonequilibrium fluctuations of the heat flux in some rigid heat conductors, and we have evaluated the nonequilibrium corrections to the classical Landau-Lifshitz formulas for dielectric solids and metallic conductors.
The purpose of this paper is to apply our method to obtain some information in the influence of an external electric field on the current fluctuations, and to compare our results with the expressions calculated by

D. NONEQUILIBRIUM GIBBS EQUATION
Extended irreversible thermodynamics is, essen- tially, a mesoscopic description of thermodynamic systems.While in the usual macroscopic theory the state of a rigid electrical conductor is described by u, the specific internal energy per unit mass, and c" the electron density, in the formalism of extended irreversible thermodynamics one takes as supplementary, independent variables the dissipative fluxes, in order to obtain a more detailed description than the usual one.In the present problem, we take as an independent variable the electric current J.In order to achieve a maximum simplicity, we assume that the electron density c, is uniform and constant, and we do not consider the effects of the heat flux.In this paper we do not give a complete description of the method used by the extended irreversible thermodynamics, which has been given already in the literature.'   The evolution equation for the internal energy u is given by the well-known balance equation pi = J.E, which expresses that the time derivative of the internal energy is equal to the Joule heating.In order to obtain an evolution equation for J one can start from a microscopic model (a Boltzmann equation for the electrons, for instance) or one can proceed in a macroscopic way starting from phenomenological hypotheses.We take this latter point of view and try to obtain an evolution equa- tion for J by following a procedure parallel to that of irreversible thermodynamics." The difference D. JOU, J. E. LLEBOT, AND J. CASAS-UAZQUEZ with the latter formalism is that, in our case, J is included as a variable in the generalized nonequili- brium entropy in order to obtain some information about its evolution equation from a suitable formu- lation of the second law.If one assumes that the nonequilibrium entropy s is differentiable enough, the corresponding generalized Gibbs equation is given by ds =6 'du -a J. d 3, where a can be shown to be ' and 6 is a generalized nonequilibrium absolute temperature given by the following equation of state:

III. NONEQUILIBRIUM FLUCTUATIONS OF THE ELECTRlC CURRENT
As in the classical theory, we assume that the probability of thc fluctuations at constant tempera- ture and constant electric field is given by (see Ref. &-'( J -oE), 8t with E being the electric field.The equation of state (4} is obtained from the equality of the second-order mixed derivatives of (2).From ( 2) and (3) it may be seen that a nonvanishing relaxa- tion time ~in (5) has some repercussions in the en- tropy and in the temperature, so that it is not coherent to retain the usual local-equilibrium en- tropy if one uses the generahzed Ohm's law (5).Starting from {2},for the second differential of the generalized entropy, one obtains Bu -dJdJ-2 " J. -d Jdu, Bu where e is the specific heat per unit mass and Jp is the mean nonequilibrium value of J given by J p -o. E. + Jp.J .0'T Expanding s(u, J ) around the mean value s (up, Jp) in powers of the deviations 5u, 5 J, and neglecting higher-order terms, it is easily found that (7) may be approximated by (Ref. 12,Chap. 15) 5s g -exp 2k (9) which predicts the second moments correctly al- though it is not accurate in the prediction of the higher-order moments.%'e have shown that in equilibrium, {9)leads in a direct way to the classi- cal Nyquist expression for the current Auctuations.
The use of ( 9) with the local-equilibrium entropy has been criticized by some authors' who have used an expression similar to (9) as a basis for ihe construction of nonequilibrium thermodynamic po- tentials from the knowledge of the Auctuations obtained, for instance, from a master equation.Here, in order to establish a connection with these at- tempts to extend classical thermodynamics to nonequilibrium steady states, we study the consequences of hypothesis (9) with the generalized en- tropy (2), which contains the dissipative fluxes and can therefore account for some noncquilibrium features.
From ( 6) and ( 9) we obtain for the second mo- ments of the fluctuations the following expressions: THERMODYNAMIC ASPECTS OF NONEQUILIBRIUM CURRENT. . .Here, we are mainly interested in the current- current fluctuations.Assuming that the applied electric field is low, we can develop (1+4Jo) ' in (12) obtaining for the one time correlation X(1+co r ) (16) where 8 =I./oA is the electrical resistance.For low frequencies such that cov « 1, ( 16) reduces to It can be shown" that the relaxation time ap- pearing in the electrical conductivity is indeed the same as that appearing in (5).This relaxation time ~is related to the mean-free path I by ~= IvF ' =lm (k~A) ' with k~being the Fermi wave vector of the system.Keeping in mind that the specific heat per unit mass for the electron gas is given by14 c=(n /2)(k T/ezm), with e~the Fermi energy given by eF --(fikF) /2m, we obtain from ( 14), This expression gives the second moments of the fluctuations of the electric current near the con- stant value Jo --oE.However, due to the Joule heating, the system is not a stationary state, and the temperature gradually rises, according to (1).
Since the dominant term in ( 17) is proportional to T it will diverge at long times.In order to elim- inate this divergence, one would have to cool the system.It can be easily sho~n that the inclusion of a heat flux perpendicular to the electric current leaves (17) unchanged, so that in a true steady state (when the heat generated by the electric field and the thermal energy flux produced by a thermal gradient balance each other) the electric current Auctuations are also given by ( 17) with a constant T.
%e may compare our phenomenological result (17) with the autocorrelation function obtained by Tremblay et al. by a cumbersome nonequilibrium diagrammatic perturbation theory.They assume a system of metallic electrons that interact only with dilute, static, and isotropically scattering impuri- ties.Their result is, after removing some diver- gences arising from the Joule heating, (5I5I) = 1+0.156 .

{18 kT
In terms of the total intensity I =AJ, in a resistor of cross section A and length I. , and taking into Recently, Tremblay and Vidal' have given a simpler derivation of the nonequilibrium correction D. JOU, J. E. LLEBOT, AND J. CASAS-VAZQUEZ in (18) by means of kinetic arguments.For the leading term, due to the second-order contribution f to the stationary distribution function, they found a value 0.100(eE//kT), while for the contri- bution arising from electron collisions, due to terms of the form f"'f'", with f"' the first-order term in the stationary distribution function, they calculate 0.056(eEI/kT) .As it may be seen, our result is in good agreement with the leading corrections of the microscopic calculations, but it does not give the collision contribution.In fact, as pointed out by Gantsevich et al. i5 and Lax, ' and as commented on by Tremblay and Vidal, ' it may be seen from detailed microscopic calculations that entropy considerations are accurate only in the case when the stationary distribution function satisfies detailed balance, or when Boltzmann statistics are obeyed.Since in the present problem these condi- tions are not fulfilled, the disagreement between both results is not surprising.
A different problem is that in nonequilibrium the temperature of the metallic resistor, due to Joule heating, is not the same as its equilibrium temperature.
In principle, Eq. ( 17) should be ap- plied with the local value of the temperature deter- mined by the Joule heating and the loss mechan- ism.
deeper insight into the signification and the limits of a nonequilibrium entropy.The method outlined in this paper cannot account for corrections due to particle-particle collisions.As we have pointed out, this limitation was already known from microscopic considerations for statistical entropies."' In spite of this shortcoming, its results for correc- tions due to relaxational effects are in very good agreement with the values obtained from microscopic methods.
It must be noted that, in contrast with previous work on this subject, no microscopic considerations have entered into the definition of our nonequilibrium entropy, which has been obtained from purely macroscopic methods.This gives to the present procedure its particular simplicity.Since it has been obtained by comparison with relaxational constitutive equations for the dissipative fluxes, a possible way to improve the present results could be a comparison with more elaborate models for the evolution equations for the dissipative fluxes.This would lead us too far away from our present purpose: to recover from a phenomenological en- tropy some results of nonequilibrium fluctuations which are clearly beyond the scope of classical ir- reversible thermodynamics.

IV. CONCLUSIONS ACKNO%LEDG MENTS
Our aim in this paper was to explore the applicability of a phenomenological generalized entropy to nonequilibrium fluctuations, in order to obtain a This work has been partially supported by the Comision Asesora de Investigacion Cientifica y J alc thc instantaneous values of u and J, while 6p ' and (m jo.T) J p --{m/T)E, are the fixed values of the corresponding parameters 8 '=As/Bu and (~/o T) J = -Bs/8 J. On the other hand, S[8O ', (rU/o T-) Jo] is the corre- sponding I egendre transform of S, given by 6 these expressions, T is the local equilibrium ab- solute temperature, o the electrical conductivity, s the relaxation time of the electric current, and u the specific volume per unit mass.The equation of state (3) may be obtained by comparing the constitutive equation for the electric current into account the equation of state (3) and the well-known relation' o. =(ne /m)~(with n, e, and m the electron density, the electron charge, and the electron mass, respectively) we get (