Microreversibility and nonequilibrium response theory in magnetic fields

Abstract

For open systems subjected to external magnetic fields, relations between the statistical cumulants of their fluctuating currents and their response coefficients are established at arbitrary orders in the deviations from equilibrium, as a consequence of microreversibility. These relations are systematically deduced from the extension of the fluctuation relation for this class of systems, and analyzed by using methods developed in [M. Barbier and P. Gaspard, J. Phys. A: Math. Theor. 51 (2018) 355001]. We unambiguously identify, among the statistical cumulants and their nonequilibrium responses, which of these quantities are independent and thus left unspecified by the fluctuation relation, i.e. by microreversibility. We also find the explicit expression of the dependent quantities in terms of the independent ones by means of coefficients of Euler polynomials.

Microreversibility, nonequilibrium response, and Euler’s polynomials

“…This result holds for any odd integer m 1 and any n 1. In the case where the index n is even, i.e., n = 2l with l 1, hence making the total number N ≡ m + n odd, the result (44) is equivalent to equation (58) of reference [34]. On the other hand, for n odd, i.e., n = 2l − 1 with l 1, now making the total number N ≡ m + n even, the result (44) is equivalent to equation (59) of reference [34].…”

“…We recently studied the fluctuation relation for open systems, connected to reservoirs of energy and particles, in nonequilibrium steady states both in the absence [33] and the presence [34] of an external magnetic field B. We showed that the fluctuation relation reduces by about half the total number of statistical cumulants, and of their responses to the affinities, that need to be known in order to fully describe the statistical properties of the nonequilibrium currents.…”

Fluctuation Relations for Dissipative Systems in Constant External Magnetic Field: Theory and Molecular Dynamics Simulations

“…This fact originated the idea that these systems require special treatment when discussing properties based on time reversibility. In particular, because the currents (and therefore the magnetic field that they generate) are reversed under , classical text books [ 1 , 2 ] as well as current literature [ 3 ] report statistical relationships in the presence of a magnetic field using pairs of systems with identical interparticle interactions but under magnetic fields of opposite signs. For example, the Onsager reciprocal relations were adapted by Casimir to relate cross-transport coefficients of systems under opposite magnetic fields [ 4 ].…”

“…Similarly, Kubo [ 5 ] derived symmetry properties of time-correlation functions of two such systems. In nonequilibrium statistical mechanics, results for currents, response, and Fluctuation Relations (FRs) are typically presented under the same conditions [ 3 , 6 , 7 , 8 ]. All these results are to be contrasted with their “standard” counterparts, i.e., those derived in the absence of magnetic fields, that refer to a single system.…”

“…For example, within linear response theory, null values of transport coefficients in experiments concerning a single system in a given magnetic field cannot be predicted based on symmetry properties of the time-correlation functions. Similarly, in the context of nonlinear response, which includes the Fluctuation Relations [ 9 , 10 , 11 , 12 , 13 ], null cumulants of the dissipation cannot be identified via symmetry [ 3 ].…”