Onsager-Casimir reciprocal relations Sylvain D. Brechet1 Institute of Physics Station 3 Ecole Polytechnique F ed erale de Lausanne - EPFL CH-1015

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Onsager-Casimir reciprocal relations

Sylvain D. Brechet1

Institute of Physics, Station 3, Ecole Polytechnique F´ed´erale de Lausanne - EPFL, CH-1015

Lausanne, Switzerland

Abstract

The Onsager reciprocal relations are established within the phenomenolog-

ical framework of the thermodynamics of irreversible processes. In order to do

so, the dissipated power densities associated to scalar and vectorial processes

are written as positive semi-deﬁnite quadratic forms of the corresponding gen-

eralised forces, as required by the local expression of the second law in the

neighbourhood of the equilibrium. The antisymmetric part of the scalar and

vectorial Onsager matrices do not contribute to the dissipation, which yields

the scalar and vectorial Onsager reciprocal relations. Furthermore, the positive

semi-deﬁnite quadratic forms of the generalised scalar and vectorial forces are

invariant under time reversal, which yields the scalar and vectorial Casimir-

Onsager reciprocal relations, that are a generalisation of the Onsager reciprocal

relations.

Contents

1 Introduction 2

2 Internal power density for irreversible processes 3

3 Scalar Onsager reciprocal relations 5

4 Scalar Onsager-Casimir reciprocal relations 8

Email address: sylvain.brechet@epfl.ch (Sylvain D. Brechet)

Preprint submitted to Journal of L

X Templates October 11, 2022

arXiv:2210.04289v1 [cond-mat.stat-mech] 9 Oct 2022

5 Vectorial Onsager reciprocal relations 9

6 Vectorial Onsager-Casimir reciprocal relations 12

7 Conclusion 14

1. Introduction

In 1931, Lars Onsager determined the symmetries of the phenomenological co-

eﬃcients of the linear applications mapping the generalised forces to the gen-

eralised forces in thermodynamic systems that are locally at equilibrium but

globally out of equilibrium. [1, 2] These relations, known as the Onsager re-

ciprocal relations, were established within the framework of statistical physics.

The approach taken by Onsager is quite general in the sense that he considered

a mechanical system consisting of particles. His theoretical work is based on the

microscopic reversibility, which requires the symmetry of the equilibrium ﬂuc-

tuation correlations. It widely believed that these relations cannot be derived

within the phenomenological framework of the thermodynamics of irreversible

processes. [3, 4, 5, 6] An attempt was made by Christian Gruber to show that

such an assumption is unwarranted for the very particular case of a system of

particles characterised by a global entropy. [7] In this publication, we will show

that for a thermodynamic system that is locally at equilibrium but globally out

of equilibrium [3], the Onsager reciprocal relations, whether of scalar or vecto-

rial nature, are a direct consequence of the local expression of the second law of

thermodynamics.

In 1945, Hendrick Casimir showed that the Onsager reciprocal relations do

no longer hold in general if diﬀerent generalised forces have not the same sym-

metry under time reversal. [8] He showed in particular that in the presence of

a magnetic ﬁeld that changes sign under time reversal these relations do no

longer hold. [9] The generalisation of the reciprocal relations that take into ac-

count the symmetries under time reversal are known as the Onsager-Casimir

reciprocal relations. In this article, we will also show how these relations can

be derived in a purely phenomenological approach by taking into account the

symmetry under time reversal in addition to the local expression of the second

law of thermodynamics.

This publication is structured as follows : in Sec. 2, we review the mathemat-

ical structure of the scalar and vectorial internal power densities of irreversible

processes in a continuous medium consisting of electrically charged chemical

substances undergoing coupled chemical reactions. The Onsager reciprocal rela-

tions between the scalar generalised forces and currents densities are established

in Sec. 3. The scalar Onsager-Casimir reciprocal relations are derived in Sec. 4.

In Sec. 3, we establish the Onsager reciprocal relations between the vectorial

generalised forces and current densities. The vectorial Onsager-Casimir recip-

rocal relations are derived in Sec. 4. Finally, we conclude our analysis of the

Onsager-Casimir relations in Sec. 7.

2. Internal power density for irreversible processes

In order to establish the Onsager relations entirely within the framework of

the thermodynamics of irreversible processes, we consider a continuous medium

consisting of relectrically charged chemical substances, undergoing ncoupled

chemical reactions in the absence of shear and vorticity. The irreversible ther-

modynamic evolution of the system is described by the internal power density

pint accounting for the dissipation,

pint =

a=1

ωaAa+τfr (∇·v) + js·(−∇T) +

A=1

jA·−∇¯µA>0 (1)

where ωais the rate density and Aais the aﬃnity of the chemical reaction a,

τfr is the isotropic internal friction and (∇·v) is the local volume expansion

rate density, jsis the entropy current density and ∇Tis the temperature gra-

dient, jAis the matter current density and ¯µAis the electrochemical potential

of substance A. The ﬁrst two power density terms are the product of scalar

quantities describing the dissipation due to chemical reactions and to the in-

ternal friction within the continuous medium. They will be referred to as the

scalar internal power density pint

S. The last two power density terms are the

product of vector quantities describing the dissipation due to the transport of

chemical substances. They will be referred to as the vectorial internal power

density pint

V. Thus,

pint =pint

S+pint

V>0 (2)

Since scalar and vectorial terms have diﬀerent symmetries, according to the

Curie principle, the scalar internal power density pint

Sand the vectorial internal

power density pint

Vhave to be separately irreversible,

pint

S>0 and pint

V>0 (3)

Following Onsager’s approach, the scalar internal power density (3) can be for-

mally written in matrix form as the product of a line vector and a column vector

consisting of n+ 1 scalar components,

pint

S= (∇·v,A1,...,An)







τfr

ω1

ωn







>0 (4)

Similarly, the vector internal power density (2) can be formally written as a

matrix product of a line vector and a column vector in consisting of r+ 1 vector

components,

pint

V= (−∇T, −∇¯µ1,...,−∇¯µr)·













>0 (5)

The generalised scalar force column vector F∈Rn+1 and the generalised vec-

torial force column vector F∈Rn+1 are deﬁned as,







∇·v







and F=







−∇T

−∇¯µ1

−∇¯µr







(6)

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Onsager-CasimirreciprocalrelationsSylvainD.Brechet1InstituteofPhysics,Station3,EcolePolytechniqueFederaledeLausanne-EPFL,CH-1015Lausanne,SwitzerlandAbstractTheOnsagerreciprocalrelationsareestablishedwithinthephenomenolog-icalframeworkofthethermodynamicsofirreversibleprocesses.Inordertodoso,thediss...

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Onsager-Casimir reciprocal relations Sylvain D. Brechet1 Institute of Physics Station 3 Ecole Polytechnique F ed erale de Lausanne - EPFL CH-1015

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