Geometrical thermodynamic field theory

Geometrical Thermodynamic FieldTheory

GIORGIO SONNINO, JARAH EVSLINEURATOM, Belgian State Fusion Association and International Solvay InstitutesDepartment of Theoretical Physics and Mathematics, Free University of Brussels, Blvd du Triomphe,Campus de la Plaine, C. P. 231, Building NO, Brussels B-1050, Belgium

Received 01 February 2006; accepted 1 May 2006Published online 5 October 2006 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.21134

ABSTRACT: A manifestly covariant, coordinate independent reformulation of thethermodynamic field theory (TFT) is presented. The TFT is a covariant field theory thatdescribes the evolution of a thermodynamic system, extending the near-equilibriumtheory established by Prigogine in 1954. We introduce the minimum rate of dissipationprinciple, which applies to any system relaxing toward a steady state. We also derivethe thermodynamic field equations, which in the case of �–� and �–� processes havealready appeared in the literature. In more general cases, the equations are notablysimpler than those previously encountered, and they extend beyond the weak-fieldregime. Finally, we derive the equations that determine the steady states as well as thecritical values of the control parameters beyond which a steady state becomes unstable.© 2006 Wiley Periodicals, Inc. Int J Quantum Chem 107: 968–987, 2007

Key words: relaxation; nonequilibrium systems; universal criterion of evolution;entropy production; geometry in physics

1. Introduction

T he thermodynamic field theory (TFT) was pro-posed in 1999, to describe the behavior of

thermodynamic systems beyond the linear (On-sager) regime [1–5]. The characteristic feature of thetheory is its purely macroscopic nature. We mean

by this not a formulation based on the macroscopicevolution equations, but rather a purely thermody-namical formulation starting solely from the en-tropy production and from the transport equations,i.e., the flux–force relations. The latter provide thepossibility of defining an abstract space, whosemetric is given by the transport matrix. The law ofevolution is not the dynamical law of motion ofparticles, or the set of two-fluid macroscopic equa-tions of plasma dynamics. Rather, the evolution inthe thermodynamical space is determined by pos-tulating three purely geometrical principles: theshortest path, the closeness of the thermodynamic

Correspondence to: G. Sonnino; e-mail: [email protected] grant sponsor: IISN.Contract grant number: 4-4505-86.Contract grant sponsor: European Commission RTN.Contract grant number: HPRN-CT-00131.

International Journal of Quantum Chemistry, Vol 107, 968–987 (2007)© 2006 Wiley Periodicals, Inc.

field strength, and the least action. From these prin-ciples, a set of field equations, constraints, andboundary conditions are derived. These equationsdetermine the nonlinear corrections to the linear(Onsager) transport coefficients.

When a thermodynamic system approachesequilibrium, the solution of the field equations re-duces to the Onsager–Casimir tensor and satisfiesthe Prigogine theorem of minimum entropy pro-duction [6]. Far from equilibrium, Glansdorff andPrigogine [7] demonstrated that, for time-indepen-dent boundary conditions, a thermodynamic sys-tem relaxes to a steady state satisfying the universalcriterion of evolution. Glansdorff and Prigogine ob-tained this result in 1954, using a purely thermody-namical approach. As indicated in Refs. [1–5], mak-ing use of the second law of thermodynamics, theshortest path principle ensures the validity of theuniversal criterion of evolution.

The validity of this theory in the weak-field ap-proximation has been tested in many examples of�–� and �–� processes, such as the thermoelectriceffect and the unimolecular triangular chemical re-action (see Refs. [1, 2]). [Here, we adopt the termi-nology of De Groot and Mazur [8]; i.e., when thevelocity distribution function is an even (odd) func-tion of the velocities of the particles, a process issaid to be an �-process (�-process). It is possible toshow that this definition implies that the � pro-cesses only involve the symmetric part of the On-sager tensor, whereas the � processes only involvethe only skew-symmetric one.] In the first case, theTFT provides new predictions when the material issubjected to a strong electric field and, in the sec-ond, the thermodynamic field equations reproducethe well-known De Donder law.

The thermodynamic field equations, in the weak-field approximation, have also been applied to sev-eral � � � processes. For example, the Field–Koros–Noyes model, in which the thermodynamic forcesand flows are related by an asymmetric tensor, wasanalyzed in Ref. [9]. Even in this case, the numericalsolutions of the model are in agreement with thetheoretical predictions of the TFT. Recently, theHall effect [10, 11] and, more generally, the galva-nomagnetic and thermomagnetic effects have beenanalyzed in nonlinear regimes. In each of thesepapers, it was shown that the TFT successfully de-scribes the known physics in the nonlinear regimesand also predicts new interesting effects, such asthe nonlinear Hall effect. The theoretical predic-tions of the nonlinear Hall effect have been con-firmed experimentally. In Ref. [11] it was shown

that, for materials with low thermoelectric powercoefficients and within the temperature range ofavailable experimental data, the theoretical predic-tions of TFT agree with experiment. Another exam-ple of physical application in the technological fieldfor which a large number of experimental data arealready available, without any theoretical under-pinning, is the case of semiconductors submitted tolarge electric fields. This case was analyzed in greatdetail in Ref. [12].

Transport processes in magnetically confinedplasmas, of � � � type, have also been analyzedusing the TFT. In particular, in Ref. [13] the ther-modynamic field equations were solved in theweak-field approximation of the classical and thePfirsch–Schluter regimes. We found that the TFTdoes not correct the expressions for the ionic heatfluxes predicted by the neoclassical theory. In con-trast, the fluxes of matter and electronic energy(heat flow) are enhanced in the nonlinear classicaland Pfirsch–Schluter regimes. This phenomenonwould have been amplified had we used the strong(exact) thermodynamic field equations. These re-sults are in line with experimental observations.

In the present work, using the coordinate-inde-pendent language of Riemannian geometry, we re-formulate the TFT in a generally covariant way.This allows us to extend several weak-field resultsto the strong-field regime. After expressing the con-cepts of thermodynamic forces, conjugate flows,and entropy production in this new language inSection 2, we go on to study the relaxation of athermodynamic system in Section 3. In particular,we prove that if the shortest path principle is valid,the universal criterion of evolution, written in co-variant form, is automatically satisfied. We alsodemonstrate that the term expressing the universalcriterion of evolution, written in a covariant form,has a local minimum at the geodesic line. Physicallythis means that a thermodynamic system evolvestoward a steady state with the least possible dissi-pation. We shall refer to this proposal, togetherwith its corollary, the shortest path principle, as theminimum rate of dissipation principle. We wouldlike to stress that here the term “rate” refers to theaffine parameter, and not to time. This principle iscompletely different from the minimum dissipationprinciple introduced by Onsager in his classicalworks [14–16], which is valid only for systems closeto equilibrium. For a system near equilibrium, On-sager introduced a functional of the thermody-namic fluxes and an associated variational princi-ple, called the minimum dissipation principle.

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Through a variational calculation, Onsager deriveda set of linear equations describing the relaxation ofthe thermodynamic fluxes to equilibrium. The func-tional results to be proportional to the entropy pro-duction along the solution of the relaxation equa-tion. The physical meaning of the minimum rate ofdissipation principle is therefore unrelated to thatenunciated by Onsager.

In Section 4 we obtain the thermodynamic fieldequations. For �–� and �–� processes we obtain thesame equations found in Refs. [1–5] but, for generalprocesses, we propose a new set of the field equa-tions, which apply even beyond the weak-field re-gime. These equations are notably simpler than theones found in Refs. [3–5]. In Section 5, we establishthe equations that determine the steady states andthe critical values of the control parameters of ageneric thermodynamic system at which a steadystate becomes marginally stable. These equationsare also conjectured to hold in the strong-field re-gime and therefore generalize the weak-field equa-tions reported in Refs. [3–5]. Examples of calcula-tions of the geometric quantities used in the newformulation of TFT can be found in Appendixes Aand B.

2. Forces, Flows, and EntropyProduction

The configurations of a thermodynamic systemin TFT are points in the thermodynamic space,which is a smooth, path-connected, geodesicallycomplete, real manifold M equipped with a Ri-emannian metric g. A Riemannian metric is a pos-itive-definite quadratic form on a manifold’s tan-gent spaces TxM, which varies smoothly from apoint x of the manifold M to another. The metric gprovides a smoothly varying inner product

� A, B� � A�g��B�, A, B � TxM, (2.1)

where following the Einstein summation conven-tion repeated indices are summed, as they will be inthe remainder of this note. The inner product al-lows one to define the lengths of curves, angles andvolumes.

M contains one special point named x � 0, whichcorresponds to the unique equilibrium state. Note,however, that in practice the thermodynamicalforces, which provide local coordinates on patchesof M, will be not be independent, but rather will

satisfy a set of constraints that determines a sub-manifold N of M. When the submanifold N of phys-ical configurations does not include equilibrium,the system relaxes to a different steady state.

Given two points x � M and y � M, we mayconstruct a shortest path �(t) such that

� : �ti, tf� 3 M, ��ti� � x, ��tf� � y. (2.2)

Path � is automatically a geodesic. Clearly for somemanifolds and choices of x and y there will bemultiple paths � that minimize the length; thus, it iscrucial that observable quantities be independent ofthe choice of path. The metric g�� allows one todefine the invariant affine parameter �:

d�2 � g��dx�dx�, (2.3)

which is used to define the length L of a path

L � ��1

�2

� g��x�x��1/ 2d�. (2.4)

The positive definiteness of the matrix g�� ensuresthat d�2 � 0 and allows us to choose an affineparameter � that increases monotonically as thethermodynamic system evolves to a stable state.

We now introduce the exponential map, which isa map from the tangent space TxM of a Riemannianmanifold M at the point x to another point expx(V)in M (see, e.g., Ref. [17]). The point expx(V) is thepoint �V(tf) on the geodesic �V that starts at �V(ti) �x, where it is tangent to �V(ti) � V. tf is determinedby the condition that the arc length along �V from xto �V(tf) is the norm �V, V�1/2 of the vector V. Thismap is usually denoted

expx�V� � �V�tf�. (2.5)

We next introduce the generalized thermody-namic force vector Ux � TxM at each point x � M,which is defined for a shortest path �� Ux

by therelation

expx�Ux� � �� Ux�tf� � 0. (2.6)

Therefore, the exponential map Eq. (2.6) associateswith every tangent vector Ux � TxM the equilib-rium point �� Ux

(tf) � 0 in M. We will later proposethat a thermodynamic system at x that relaxes to theequilibrium state will travel along a shortest path

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970 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 107, NO. 4

�� Ux. The norm �Ux� is the geodesic distance from the

point x to the equilibrium point measured along theshortest path �� Ux

. The components Ux� of Ux are

interpreted as the generalized thermodynamicforces.

The exponential map is a one-to-one correspon-dence between a ball in TxM and a neighborhood ofx � M, and so Eq. (2.6) defines a correspondencebetween the generalized thermodynamic forces andthe points of the manifold M. From now on we shallomit the subscript x, leaving implicit the depen-dence of the vector U on point x. In Appendixes Aand B, we provide two examples of calculations ofthe map exp and the vector Ux. In Appendix A, thethermodynamic space M is flat space, so the metrictensor coincides with the Onsager matrix, while inAppendix B M is the two-sphere.

The thermodynamic flows J�, which are conju-gate to the thermodynamic forces U�, are definedby

J� � ��U�, (2.7)

where �� is an asymmetric tensor and, for brevity,we have suppressed the dependence of J, , and Uon the point x and geodesic �� . Any 2-tensor may bedecomposed into a symmetric and an antisymmet-ric piece. The symmetric piece of is identified withthe metric tensor g�� and we shall name the skew-symmetric piece f��. Equation (2.7) can then be re-written as

J� � ��U� � � g�� f��U�. (2.8)

J� is a one form J � T*M which, like U, only van-ishes at the equilibrium point x � 0, although theremay be points at which both U and J depend on achoice of �� . Equation (2.8) allows one to generalizethe inner product �A, B� of two vectors A and B inTxM from Eq. (2.1) to

� A, B� � � g�� f�� A�B�. (2.9)

Finally, we will define the entropy production �at the point x to be

� � �U, U� � � g�� f��U�U� � g��U�U� � 0,

(2.10)

where the inequality corresponds to the second lawof thermodynamics. As the metric tensor g�� is apositive definite matrix, this inequality is always

satisfied. The entropy production � is an observablequantity, and the consistency of the formulationdemands that it be independent of the arbitrarychoice of �� . In fact, � is just the square of the lengthof the shortest geodesic from x to 0, so it is inde-pendent of the choice among geodesics of equallength and thus is globally defined as was required.

In the present study, we will restrict our atten-tion to homogeneous systems. Therefore, a pointx � M will correspond to the thermodynamicalconfiguration at a point in space and equivalentlyto the configuration of the entire system. In partic-ular, the entropy flux through the boundaries willbe held fixed in our experiments. This implies thatany variation in the entropy production � is equalto the variation of the time derivative of the totalentropy. The generalization to inhomogeneous sys-tems is straightforward and follows the strategyemployed in Ref. [5].

3. Relaxation to a Steady State

In the TFT description of a thermodynamic sys-tem, a homogeneous configuration corresponds to apoint x in the thermodynamic space M. The corre-sponding thermodynamic forces, which are com-pletely determined by the configuration x, are as-sembled into the vector U defined in Eq. (2.6). If thepoint x is not a steady state (see Section 5), thesystem will evolve. In this section, we will considerthe process known as “relaxation,” in which theinitial velocity vanishes and the system relaxes to asteady-state y.

The TFT description of relaxation rests upon twopostulates: (i) the manifold (M, g) is a Riemannianmanifold with metric g; and (ii) the minimum rate ofdissipation principle. The minimum rate of dissipa-tion principle is formulated in Section 3.3, wherewe will see that it implies that during the process ofrelaxation the configuration traces out a geodesic �in thermodynamic space. In Refs. [1–4], the geode-sic property was referred to as the shortest pathprinciple. It allows one to write

x � ��ti�, y � ��tf� � expx�V�, (3.1)

where V is a tangent vector to � at x whose norm isthe geodesic distance from x to y. The system beginsat x at time ti and reaches the steady-state y at thetime tf. The process of relaxation begins as soon asthe system is released, that is, immediately after



time ti. In support of this conjecture, we will nowargue that V automatically satisfies a manifestlycovariant form of the minimum entropy productiontheorem (near equilibrium), the Glansdorff–Prigog-ine universal criterion of evolution and the mini-mum rate of dissipation principle.

3.1. MINIMUM ENTROPY PRODUCTIONTHEOREM

In 1945–1947, Prigogine proved the minimumentropy production theorem [7], which concernsthe relaxation of thermodynamic systems nearequilibrium. This theorem states that:

Minimum Entropy Production Theorem. Re-gardless of the type of process considered, a ther-modynamic system, near equilibrium, relaxes to-ward a steady-state y in such a way that theinequality

ddt �V�V��

ddt �V, V� � 0 (3.2)

is satisfied during throughout the evolution and isonly saturated at y.

I. Prigogine’s proof of the minimum entropy pro-duction theorem used a purely thermodynamicalapproach. We shall now demonstrate that this the-orem is a consequence of the postulates of TFT.

Consider a geodesic �(�) parametrized by � � [�i,�f] equipped with a tangent vector field W normal-ized to have unit length

W �

�� *�

�� , �W, W� � 1. (3.3)

Note that � is not necessarily the time t, as thevelocity is not necessarily unity throughout the evo-lution. The vector field W evaluated at any point xon the curve � is a vector W(x) � Tx� � TxM. Wethen have the following identity:

LW��V, V� �dd�

�V�V��, (3.4)

where LW[�(�)] denotes the Lie derivative with re-spect to the vector field W and, since we are nearequilibrium, indices are raised and lowered usingthe Onsager matrix which is a metric for Euclideanspace.

There always exists a neighborhood U � M of y,which physically corresponds to the near-equilib-rium approximation when y is near equilibrium,such that we may uniquely identify the coordinatesx� of a point x with the vectors V(x) in a subset ofa tangent fiber. One example of such a U is a ballwhose radius is the injectivity radius of exp, whichin topologically trivial situations is roughly the in-verse square root of the curvature, but any U suchthat each x and y in U are connected by a uniquegeodesic will suffice. To show that there is a corre-spondence between points and vectors, we need toshow that given x we can determine V(x) and viceversa. We have already seen that we may uniquelydefine V(x) from the unique geodesic � in U con-necting x and y, V(x) is just the tangent vector to �at x whose norm is equal to the length of �.

To choose the coordinates of the point x given achoice of V(x) we must first confront the fact thatV(x) is an element of the tangent space at x, TxM,and so to compare different V(x)’s, we must firstsomehow place them in the same space. This maybe done canonically in U by using the fact that in Uthere is a unique geodesic �x from x to the steady-state y. Thus, we may define the vector Ay(x) � TyMby parallel transporting V(x) from x to y along �.We may now use any choice of coordinates for y towrite x on the same coordinate chart:

y � x � Ay� x� � V� x�, (3.5)

where we have used parallel transport to constructan isomorphism between TxM and TyM that allowsus to identify V(x) and A. Note that each V(x) arisesfrom a unique x because V(x) is parallel to � and thegeodesic equation implies that parallel vectors re-main parallel under parallel transport. Thus, A isalso parallel to �. Thus, the uniqueness of the geo-desic implies the uniqueness of A; thus, with Eq.(3.5), the uniqueness of x. The sign in Eq. (3.5) is aresult of the fact that V(x) points from x to y.

In these coordinates, the metric is the identity aty, the Christoffel symbols vanish at y, the geodesiccurves in U that pass through y are straight lines inU, and the entropy production is given by

� � �x�2 O� x3�. (3.6)

Let us now evaluate the Lie derivative of �V, V�when the thermodynamic system relaxes from thepoint x to the stationary point y � �(tf). According

SONNINO AND EVSLIN


to TFT, this relaxation follows the unique geodesicin U that connects them, which is the straight line:

x � y V��, V�� V� x� f�� (3.7)

in the coordinates (3.5). We then find

LW��V��, V�� 2�V(�),

�V(�)� � 2f�� f ��

�V�x�, V�x�� 0, (3.8)

where the last inequality is a consequence of thefact that �V(x), V(x)� is positive by the positivedefiniteness of the Onsager tensor g, while f(�) ispositive and f(�) is negative because f(�) is thelength of V, which monotonically falls to zero dur-ing the relaxation. Only at the steady-state y isf(�f) � 0, and so the inequality is saturated.

3.2. UNIVERSAL CRITERION OF EVOLUTION

The minimum entropy production theorem isgenerally not satisfied far from equilibrium. How-ever, P. Glansdorff and I. Prigogine demonstratedin 1964 that a similar theorem continues to hold forany relaxation to a steady state. We report the orig-inal version of the Glansdorff and Prigogine theo-rem [6]:

When the thermodynamic forces and conjugateflows are related by a generic asymmetric tensor,regardless of the type of process, for time-indepen-dent boundary conditions a thermodynamic sys-tem, even in strong nonequilibrium conditions, re-laxes to a steady state in such a way that thefollowing universal criterion of evolution is satis-fied:

� g�� f��V�dV�

dt � �V,dVdt

� � 0

��V,dVdt

� � 0 at steady state�. (3.9)

Again, Glansdorff and Prigogine demonstrated thistheorem using a purely thermodynamical ap-proach. In this section we shall prove that the short-est path principle implies that a covariant form ofthe universal criterion of evolution is automaticallysatisfied for relaxation processes in TFT.

Consider a geodesic �(�) parametrized by � � [�i,�f] equipped with a tangent vector field W:

W �

�� *�

�� , �W, W� � 1.

(3.10)

Again, we have parametrized � so that the ve-locity vector W always has unit norm. As the ve-locity of the relaxation is an observable and is notnecessarily equal to one, we must stress that � refersto a particular parametrization, and not necessarilyto time. However, it is monotonically increasingwith respect to time. As usual, we also define asecond vector field V on the curve � such that at anypoint x the vector V(x) satisfies

��f� � expx�V�x��. (3.11)

We will consider curves �, which contain no closedloops, i.e., curves for which �(�) � �(�) implies � ��, so the condition (3.11) uniquely defines the vec-tor field V.

Now, our task is to evaluate the term in Eq. (3.9)when the thermodynamic system relaxes to asteady-state along a geodesic. This term contains atime derivative of the vector V, which a priori isill-defined as V at different times inhabits the tan-gent fibers over different points of M, which are notcanonically isomorphic. The time derivative may bedefined, however, using the Lie derivative withrespect to the vector field W. This leads to thecovariant form of the universal criterion of evolu-tion:

�V, LWV� � 0 �with �V, LWV�

�0 only at the steady state�. (3.12)

From this point on, the term �V, LWV� will be re-ferred to as the Glansdorff–Prigogine quantity.Note that V[�(�f)] � 0 and so in particular theGlansdorff–Prigogine quantity relaxes to zero:

�V��f��, LWV��f�� 0. (3.13)

The quantity defined by the Lie derivative differsfrom the quantity in the minimum entropy produc-tion theorem; in fact, the difference between thesetwo terms is similar to the difference between theminimum entropy production term and the nonco-variant time derivative term. The decompositionfor the noncovariant time derivative is



dd�

�V, V� � � g�� f��V�dV�

d�

V�dd�

�� g�� f��V��

� �V,dVd�

� V�dd�

�� g�� f��V��, (3.14)

while for the Lie derivative it is

dd�

�V, V� � LW�V, V� � �V, LWV�

� V�[LW�g�f�V��. (3.15)

We shall now prove the covariant form of theuniversal criterion of evolution.

Theorem. For all � � �f, we have the strictinequality �V, LWV� � 0.

Proof. For every point x on the curve �, thevectors V(x) and W(x) are elements of the one-dimensional vector space Tx�; therefore, they areparallel, and a function f[�(�)] exists such that

V�� f ��W��. (3.16)

We may now evaluate the Lie derivative

LW��V�� W��, V��

� W��, f ��W��

� ��*�

��, f [�(�)]�*�

�� f ��

�W��.

(3.17)

Combining Eqs. (3.16) and (3.17), we obtain �V,LWV� as a function of f

�V, LWV��fW, f �

W� � f f �

�W, W�. (3.18)

The fact that g is Riemannian, and so positive def-inite implies that �W, W� � 0. We have seen thatV[�(�f)] � 0, and we have imposed that W � 0;therefore, f(�f) � 0. In addition, the fact that W � 0implies that the distance from �(�) to �(�f) has nocritical points, and so f has no critical points. Thus,f is strictly monotonic in � and at the maximal value

� � �f it vanishes. As a result, f and f/ � haveopposite signs

f f �

� 0. (3.19)

Thus, the right-hand side of Eq. (3.18) is the productof a positive and a negative term. Therefore, �V,LWV� is negative as desired.

At this point, one may object that the Glansdorff–Prigogine quantity in Eq. (3.12) is not actually thequantity that appears in (3.9) because t and � are notequal, so W is not actually the velocity of the sys-tem. This implies that the covariant form of theuniversal criterion of evolution is not in generalequivalent to the original form. However, after alittle algebra one can see that they are equivalentwhenever the deceleration �a�, velocity v and affinelength l of the geodesic satisfy the following in-equality:

�a� �v2

l � �(3.20)

at every affine time �. The acceleration is uncon-strained, so this is only a constraint on the dissipa-tion of the system as it approaches the steady stateand slows. However, near the steady state, the de-nominator of the right-hand side of Eq. (3.20)shrinks to zero. Thus, even near the end of therelaxation the constraint on the rate of dissipationappears to be quite weak. This constraint will becompared with experiment in a subsequent publi-cation.

3.3. MINIMUM RATE OF DISSIPATIONPRINCIPLE

In Refs. [1–4], we introduced the shortest pathprinciple. In Section 3.2 we demonstrated that if theshortest path principle is valid then a covariantform of the universal criterion of evolution is auto-matically satisfied. In this section we will show thatthe Glansdorff–Prigogine quantity �V, LWV� has alocal minimum at the geodesic line. We refer to thistheorem, together with the shortest path principle,as the minimum rate of dissipation principle. Theminimum rate of dissipation principle correspondsto the following physical conjecture:

For any type of physical process, even in strongnonequilibrium conditions, a thermodynamicsystem subjected to time-independent boundary

SONNINO AND EVSLIN


conditions relaxes toward a steady-state follow-ing the path with the least possible dissipation.

Let us consider an arbitrary path �(t) parameter-ized by t � [0, tf] equipped with a tangent vectorfield

W �

t ��t� � �*�

t� . (3.21)

This time we fix the velocity of the trajectory, thisis necessary in any minimization of dissipation be-cause otherwise one could always trivially speedup the fall in dissipation by accelerating the pro-cess. In particular, we will parametrize �(t) by im-posing that W is a unit vector

�W, W� � 1. (3.22)

At every time t, there exists at least one geodesic�t(�) such that

�t�0� � ��t�, �t��f� � ��tf�. (3.23)

There also exists a vector Vt in the tangent spaceT�(t)�t � T�(t)M such that

��tf� � exp��t��Vt�. (3.24)

Equation (3.24) defines the vector Vt at everypoint �(t) on the curve �, so it defines a vector fieldV[�(t)] on � with vectors in the tangent space TM.

We will show that, using the above parametriza-tion,

�V, LWV� � ��V�cos�� , (3.25)

where � is the angle between � and �. This quantityis minimized when � � 0, corresponding to the casein which � and � are parallel. Therefore, the Glans-dorff–Prigogine quantity is minimized everywhereif � and � coincide everywhere, in which case � is ageodesic.

To prove this theorem, we shall perform ourcalculations in Fermi coordinates [18] on the geo-desic �t. In particular, we shall work in the Fermicoordinate system such that the first derivatives ofthe components of the metric along the geodesic �t

vanish

x g��t� � 0. (3.26)

Therefore, in Fermi coordinates, a distance � fromthe geodesic, the components of the metric willdiffer by a correction of order �2:

g��t�� g��t� g��t�

x

� � 12

2g��t�

x x� �� O��3� (3.27)

or

�g�� g��t�� g��t�

�12

2g��t�

x x� �� O��3�. (3.28)

In Fermi coordinates, we may interpret the Glans-dorff–Prigogine quantity (3.25) using the fact that�V, V� is the length squared of the geodesic �t, andits Lie derivative with respect to W is

LW�V, V� � 2�V, LWV� � V�V��LWg��. (3.29)

Note that the term LW�V, V� does not correspond tothe Lie derivative of the entropy production, which,as seen in Section 2, is defined as LW�U, U�.

We shall now show that the second term on theright-hand side is of order �, while the other twoterms are of order one:

V�V��LWg�� O��,

LW�V, V� � 2�V, LWV� � O�1�. (3.30)

Until now we have defined the vector W onlyalong the curve �. However, Eq. (3.29) requires adefinition of the derivative of W in directions thatare not parallel to the curve. Such derivatives havenot yet been defined, but they may be defined if weextend the definition of W to a neighborhood U ofthe point �t(0). We define W at a point �t(0) � � inthis neighborhood by parallel transporting W(�t(0))along the unique geodesic �x � � in U that con-nects �t(0) and �t(0) � � [19] (see Fig. 1):

W�� x� � W0

� �� x��W0

� O��2�. (3.31)

Equation (3.31) allows us to generalize the defini-tion of the quantity �V, LWV� to arbitrary curves.The validity of the theorem would not be ruined ifwe parallel transported along paths that were notgeodesics; the only essential property is that the



lengths of the paths connecting points separated bya geodesic distance � should be of order �.

The last term in Eq. (3.29) is the derivative of themetric along the direction parallel to �t. It is easy tocheck that this term is of order � in our coordinatesystem:

V�V��LWg�� V�V�W g��

x � 2V�V� W�

x� (3.32)

or

V�V��LWg�� 2V�V�W�� 2V�V�

��

x� �W0�

� O��2�, (3.33)

where we have used the identity

g��

x � �� g��

� g��. (3.34)

However, in Fermi coordinates, a distance � fromthe curve the components of the affine connectionwill differ by a correction of order �:

�� t��

��t� ��

��t�

x� �� O��2�

� ��

��t�

x� �� O��2�. (3.35)

Observe that from Eq. (3.35), we also have

�� t�

x� � lim��30

�� t��

��t�

�� O�1�. (3.36)

Therefore, the second term on the right-handside of Eq. (3.29) is of order � as claimed. As a check,we note that Eq. (3.36) reveals that the Riemanniancurvature tensor R��

� is of order O(1), as it must be,since our choice of coordinates cannot affect thecurvature.

Now we have reduced the problem to the calcu-lation of half of the left side of Eq. (3.29), which isthe derivative of the length squared of �t along thecurve �. For this, we have to compare the lengths ofthe geodesics �t and �t��. This leads us to thenotion of Jacobi fields (see, e.g., Ref. [20]). Let usconsider a single infinite family of geodesics in themanifold M. Let � be a parameter varying alongeach geodesic of the family, and let t a parameterwhich is constant along each geodesic of the family,but which varies as we pass from one geodesic toanother. A Jacobi field is a vector field on a geodesicthat describes the deformations that interpolate be-tween different geodesics in a one-parameter fam-ily. For example, the family �t yields the Jacobifield Jt:

Jt��

�t��

t (3.37)

on each geodesic �t�(�).

Consider now two nearby geodesics in the �family, �t and �t��. In particular, we will name twopoints

A � �t��1�, A � �t��1�, (3.38)

which are featured in Figure 3. Let t� � � sin �Jt

� bethe infinitesimal Jacobi vector (see Fig. 2) such that[21]

g��t�t

� � 0. (3.39)

FIGURE 1. We define the vector W in a neighborhoodU of the point �(t) � �t(0) by parallel transporting thevector W[�(t)] along an edge �x � �.

SONNINO AND EVSLIN


This equation tells us that the deviation t� is per-

pendicular to �t�. The vector t

� satisfies the Jacobiequation [21]

DD�2 t

�� R��

�t��

�t

�� t

��

�� 0, (3.40)

where D denotes the covariant derivative. Bothterms in the Jacobi equation (3.40) are linear in �and so, in the limit � 3 0, the second derivative ofour Jacobi field goes to zero. The relative angle � atthat point A (see Fig. 3), between �t and �t��,defined using parallel transport along the flow ofthe Jacobi field, can be easily evaluated using therelation [21]

��t��A�Y��A � � � Y�R��

� �t�

�t

�t

�dtd�, (3.41)

where Y� is any vector at A. Intuitively, � is theintegral of the second derivative of the Jacobi field,which we have seen is order �, and so � is also oforder �.

Substituting the definition of the derivative into(3.29) our task becomes the evaluation of the limit

lim�30

��V, V�

�� 2g��V� lim

�30

�V�

� V�V� lim

�30

�g��

�.

(3.42)

We have already seen that the last term on theright-hand side is of order �, and so vanishes. Asshown in Figure 3, to leading order in �, the differ-ence between the vector V� at the geodesic �t andits image on the curve �t�� under exp*�t

(� sin(�)Jt) isof order �2 because the angle � is of order �. There-fore, the difference between the squared length ofthe geodesic �t and its image �t�� exp�(t)(�sin(� )Jt) is of order �2 and so will not contribute tothe derivative defined in (3.42).

So it may seem that the derivative that we aresearching for [Eq. (3.25)] is zero. However, thecurve �t�� is not quite the same as exp�t

[� sin(� )Jt],because we have parameterized �t�� such that (seeFig. 1):

�t�� 0� � ��t �� exp��t�� sin�� Jt�0��.

(3.43)

That is to say that the Jacobi field is not parallel to�, so it does not take the beginning of the curve �t

to the beginning of the curve �t��. Instead, exp ofthe Jacobi field � sin(�)Jt(0) takes the beginning ofthe curve �t to a point �� cos(�) on the curve �t��.Up to corrections of order �2, the curves �t and �t��

are the same length, and so the distance from�t��(0) to �(tf) is the length of �t minus the correc-tion � cos(�) that we have just calculated. Half of thedistance squared then differs by �V�� cos(�). We

FIGURE 2. As one moves along the curve �, the mini-mal distance to �(tf) shrinks at a speed of cos(�). Thisspeed, which is proportional to the negative quantity�V, LWV�, is locally minimized when � � 0, correspond-ing to the case in which � follows a geodesic.

FIGURE 3. When the relative angle � at the point Abetween �t and �t � is of order �, the difference be-

tween length AB� and length AB� is of order �2. Indeed,

we have AB� � AB�� 1/2��CB� O��2�.



have seen in Eq. (3.29) that the Lie derivative is thisquantity divided by �, which yields the claim (3.25).

3.4. EXAMPLE: DIFFUSION

We conclude this section with a simple exampleof the diffusion of two solutes in an isothermalsystem. The evolution of the mass fractions ci isdescribed by the differential equations [22]:

c1

t � D11�c1 D12�c2

c2

t � D12�c1 D22�c2, (3.44)

where � is the Laplacian operator. We will considerthe Onsager regime, in which the diffusion coeffi-cients Dij are uniform. Equations (3.44) can be di-agonalized and rewritten in terms of the solutesthat diffuse independently

c�1

t � D1�c�1

c�2

t � D2�c�2, (3.45)

where the c�i are linear combinations of the massfractions ci.

Let us now suppose that our system is infiniteand submitted to the initial conditions:

c�1�r, t � 0� � N1��r�

c�2�r, t � 0� � N2��r�, (3.46)

where Ni indicate the total number of particles ofspecies i. We will now analyze the relaxation of thissystem and verify the validity of the minimum rateof dissipation principle. It is easy to check that thesolutions of Eq. (3.45) with initial conditions, Eq.(3.46) can be cast into the form

c�1�r, t� �N1

�4�D1t�3/ 2 exp��r2

4D1t�

c�2�r, t� �N2

�4�D2t�3/ 2 exp��r2

4D2t�. (3.47)

Note that at each moment t the mass fractions areGaussian distributions centered at the origin withstandard deviations �i equal to �2Dit. The origin

follows a geodesic in the space of mass fractions,however we are interested in the space of thermo-dynamic forces and we will see that the forcesalways vanish at the origin in this example.

To understand the sense in which the evolutionis a geodesic in a space of thermodynamic forces, itwill be necessary to follow the relaxation of a pointat the time-dependent radius:

r��t� � ��2Dt, D � � D1D2

D2 � D1� , (3.48)

which lies a fixed number of standard deviationsfrom the origin. We will see that the thermody-namic forces at r� follow a geodesic for every fixedvalue of �. One might object that we should insteadbe following a point at a fixed value of r. However,we will see that such points begin already relaxed,then they are excited, and then relax again. Thus theshortest path principal, which applies only to thephenomenon of relaxation, does not apply to thevalue of a field at a fixed value of r in the presentexample.

The thermodynamic forces are defined to be thegradients of the mass fractions. In particular, theirradial components are

U1�r, t� �dc�1

dr � �rN1

16�3/ 2�D1t�5/ 2 exp��r2

4D1t�

U2�r, t� �dc�2

dr � �rN2

16�3/ 2�D2t�5/ 2 exp��r2

4D2t�

(3.49)

and so they are related by

U1�r, t� �N1

N2�D2

D1� 5/ 2

exp��r2

4Dt�U2�r, t�. (3.50)

At r � r�, Eq. (3.50) reduces to

U1�r�, t� �N1

N2�D2

D1� 5/ 2

exp��2

2 �U2�r, t�. (3.51)

In particular, the ratio of the forces U1 and U2 isindependent of t and so, for each value of �, U1(t)and U2(t) lie along a line in the two-dimensionalspace of forces. We are considering the Onsagerregime, so the metric is flat and the line is a geode-sic. Thus, at each r�, the system traces out a geodesicfrom a �-dependent excited state to the stationarystate 0, which in this case is also the equilibrium

SONNINO AND EVSLIN


state. In the Onsager regime, the Glansdorff–Pri-gogine quantity is just the rate at which the forcesapproaches equilibrium with respect to the affineparameter, which is maximized when the velocityvector points toward equilibrium, as it always doesfor a straight line ending at equilibrium. Thus theminimum rate of dissipation principle is satisfied,as it must be for any geodesic relaxation accordingto the general arguments of the previous subsec-tion.

4. Thermodynamic Field Equations

As shown in Eq. (2.10), the second principle ofthermodynamics is satisfied when the metric g ispositive definite. However, the postulates that wehave presented so far are insufficient to calculate g.In this section, we will add a third postulate to TFTstating that the metric is the solution to a set of fieldequations which are in turn equal to the variationsof an action functional. In short, we include thepostulate:

Principle of Least Action. In particular we willassert that the action is invariant with respect togeneral coordinate transformations, and we will notinclude a cosmological constant term, as there hasbeen no experimental evidence for the presence ofsuch a term in TFT.

We will also place the following three restric-tions on our physical systems: (i) internal fluctua-tions are neglected; (ii) boundary conditions aretime-independent; and (iii) the system is not sub-jected to external perturbations.

If these three conditions are satisfied simulta-neously, the equations of motion will be source-free. The internal fluctuations can be analyzed us-ing the Landau–Lifshitz theory [23] and systemssubmitted to time-dependent boundary conditionshave been examined by York [24]. Examples ofsystems subjected to external perturbations, thosemost commonly encountered in practice, can befound in Ref. [25].

4.1. �–� AND �–� PROCESSES

In �–� and �–� processes, the matrix �� thatrelates the thermodynamic forces and flows (2.7) issymmetric and so is equal to the metric g��. Wetherefore expect the action to be constructed en-tirely from g. After the cosmological constant, theinvariant constructed using the least number of

derivatives of g is the Ricci scalar R � g��R��, whereR�� is the Ricci tensor [26]

R�� ,� � ��,

��

� ��

. (4.1)

Of course, one could add higher powers of R and itsderivatives to the action and maintain general co-variance, but so far such terms have not been re-quired to reproduce the experimental data and nu-merical simulations. If such terms are indeedpresent in the action, by dimensional analysis theywill be multiplied by characteristic scales of posi-tive spatial dimension and they will be negligible atlonger distance scales. Thus it may well be thatbelow some threshold length scale, correspondingto very strong fields, higher-order terms in R willbegin to play a role. For example, in gravitationaltheories such terms may play a role deep inside ofblack holes and immediately after the big bang, butare irrelevant at the scales of all observed phenom-ena. By contrast, in general relativity, the nonlin-earity of the R term considered here already pro-vides a significant correction to the clocks of theGPS satellites and its effect on Mercury’s orbit wasobserved a hundred years ago.

We therefore postulate that the action I takes thesimple form:

I � � �g R��g��d�, (4.2)

where d� is a volume element and the domain ofintegration is the entire manifold M. Here g denotesthe determinant of the metric g��. The principle ofleast action implies that the variation of I withrespect to the metric vanishes, which yields thefamiliar thermodynamic field equations:

R�� 12 Rg�� 0. (4.3)

For n � 2, Einstein’s equation (4.3) reduces to theRicci flatness condition

R�� 0. (4.4)

4.2. GENERAL CASE

In Refs. [1–5], the TFT was formulated postulat-ing the closedness of the thermodynamic fieldstrength where the thermodynamic field strength is



determined from the skew symmetric part f of thetensor relating the thermodynamic forces to theconjugate flows. This postulate reflects the observa-tion that, in all cases examined thus far, the sourcesof the thermodynamic fields, i.e., the internal fluc-tuations, time-dependent boundary conditions andexternal perturbations, only affect the balance equa-tion of the dual form of f. This experimental obser-vation allows us to identify the system of thermo-dynamic field equations as an analogue ofMaxwell’s equations. We shall see that this postu-late is the absence of magnetic sources in a higher-dimensional version of Maxwell’s equations. In thissection we derive the simplest thermodynamic fieldequations, which are in line with the experimentalobservations and, at the same time, do not violatethe inequality expressed by the universal criterionof evolution written in a covariant form.

Let us now consider the general case in which�� is not necessarily symmetric

J� � ��U� � � g�� f��U�. (4.5)

Any completely skew-symmetric tensor f of type(0, 2) defines a 2-form f. Given a skew-symmetrictensor f of type (2, 0) with components f��, we candefine an (n � 2)-form �f called the Hodge dualof f. In terms of components, the relation between�f and f is

�fj1j2. . .Jn�2 �12! g1/ 2��j1j2. . .jn�2f

��, (4.6)

where ��j1j2. . .jn�2is the Levi–Civita symbol:

�i, j. . .n � �i, j. . .n � �1 if ij . . . n is an even permutation of 1, 2 . . . , n�1 if ij . . . n is an odd permutation of 1, 2 . . . , n0 otherwise.

(4.7)

The inverse of Eq. (4.6) reads

f�� 1

�n � 2�! g1/ 2�j1j2. . .jn�2��f �j1j2. . .jn�2. (4.8)

To write the kinetic term for the f-field, we willneed to first define its antisymmetric 3-tensor fieldstrength H

H�� f�� f�� f�� (4.9)

or simply

H � df, (4.10)

where d is the exterior derivative. The square of theexterior derivative, d2, contains the contraction oftwo ordinary derivatives with the � tensor, so itvanishes by antisymmetry. Therefore, H is closed

dH � d2f � 0. (4.11)

In principle, the balance equation for H could alsocontain a source term J

dH � J, (4.12)

where J is a closed 4-form, dJ � 0.

Given a third-rank skew-symmetric tensor H, wecan define an (n � 3)-form �H, the Hodge dual of H,with components

�Hj1j2. . .Jn�3 �13! g1/ 2��j1j2. . .jn�3H

��. (4.13)

As we shall see shortly, �H also satisfies a balanceequation with a source term, which we shall denote�J. This source term may be nonzero only if the firsttwo restrictions on our physical system are relaxed.In other words, internal fluctuations and time-de-pendent boundary conditions can both serve as �Jsources for �H. However in all cases examined sofar no potential J sources have been found for H.

We are therefore led to the following postulate[3, 4]:

The 3-form H has no sources or, equivalently, the3-form H is closed.

Note that, from the mathematical point of view, toimpose J � 0 it is sufficient to require that thetensor field f does not possess any singularities, soEq. (4.10) is everywhere well defined.

We will now present an invariant action I for thethermodynamic system. As in the previous subsec-tion, we will restrict our attention to terms thatcontain the least number of derivatives, as higher

SONNINO AND EVSLIN


derivate terms will be suppressed by a characteris-tic length scale and so may only be relevant in thepresence of very strong fields. Again, the metricmay enter via an Einstein–Hilbert term equal to theRicci scalar R. The antisymmetric tensor f may entervia a 2-derivative term H � �H generalizing theMaxwell action and also via a 0-derivative massterm �f � �f where � is a dimensionful parameter.Here � is the wedge product of differential formsand � is the Hodge star defined in Eqs. (4.6) and(4.13). A cosmological constant term, Chern–Si-mons terms, and more complicated functions of themass term, are not necessary at present to explainthe data, so they will not be included. In compo-nents the proposed Lagrangian density is then

Lkin �1

12 H��H��

2 f�� f ��, (4.14)

where indices are raised using g. The mass param-eter � is an inverse of the characteristic lengthsquared of the thermodynamic system.

The source J�� can be included in the theory byadding an interaction term to the Lagrangian den-sity

Lint � Sn�1J�� f ��, (4.15)

where Sn�1 � 2�n/2�(n/2) is the volume of a unit(n � 1)-sphere and �(n/2) is the Gamma function.Note that if we do not add the source J, we couldadd a new term coupling the source J to the dualfield f(dual), where df(dual) � �H. However, f and f(dual)

are not mutually local, since f(dual) at a point cannotbe constructed from f at that point together with afinite number of derivatives, so no local action cansimultaneously contain both J and J. In line withour postulate, we impose J � 0 and so we onlyconsider the source term in Eq. (4.15).

Combining (4.14), (4.15), and the Einstein–Hil-bert action, we arrive at the total action

I � � �g L�� g R�1

12 H � �H��

2 f � �f

�Sn�1J � �f�. (4.16)

The relative normalization of the first two termsin the action (4.16) may be modified by rescaling themetric, here we have chosen the convention knownas the string frame in string theory [27].

�J will be treated as an external source, so thereare two variables whose variations yield the fieldequations. The variation with respect to the metricyields a modified Einstein’s equation:

R�� 12 g��R � �

14 �H��H�

� �16 g��H��H��

� ��f��f��

14 g��f� f �� 2Sn�1�J� f �

�14 g��J� f ��,

(4.17)

while the variation with respect to the f-field yields

d � H � � f � �Sn�1 � J. (4.18)

We have seen that the antisymmetric tensor fmodifies the field equations for the metric. One maywonder whether it also affects the shortest pathprinciple. In general, a p-tensor potential can becovariantly coupled to a p-dimensional extendedobject that is a J-source and to a (d � p � 2)-dimensional J-source. For example, in four-dimen-sional Maxwell theory, the 1-tensor potential cou-ples to electric and magnetic particles, while in fivedimensions the electric sources are still particles butthe magnetic monopoles are strings. In our case p �2, so the J-source is a string and the J-source is (d �4)-dimensional. In particular, the trajectories in thethermodynamic space are, being paths, one-dimen-sional and so they cannot be J-sources, except infive dimensions, they also cannot be J-sources.Thus, there is no covariant coupling of the trajecto-ries to the f-field, as there would have been for anordinary vector potential.

Concretely, this means that the action for a re-laxing configuration is proportional to the properlength of its trajectory

S � � d��dX�

d�

dX�

d� � 1/ 2

, (4.19)

just as in the case with no f-field. The proper lengthof a path, by definition, is extremized by a geodesic.Thus the solutions of the classical equations of mo-tion for the evolution of a state are still geodesics inthermodynamic space, even when we include the ffield. This is not to say that the solutions with an ffield are just the solutions to the system without thef-field, because the new Einstein’s equation (4.17)no longer admits Ricci-flat metrics, so the Rieman-



nian manifold M itself is changed by the inclusionof f.

5. Steady States and Stability Criteria

In the thermodynamic theory of irreversible pro-cesses, a steady state of order �, where � is less thanor equal to the dimension n of our thermodynamicspace M, is a thermodynamic point x with coordi-nates

U1, U2 . . . U� kept constant �� n�Ja � 0 �a � � 1, · · ·n�.

(5.1)

From now on, Latin indices will run from � � 1 ton and Greek indices will run from 1 to n. Thus (5.1)may be written more compactly as

J�dU��st.state � 0, (5.2)

where d denotes the total differential on the sub-manifold N of solutions of the constraint equations.

Steady states may be stable or unstable. Accord-ing to the Layapunov stability theory, the stabilityof a state is determined as follows [6]:

ifddt �2S�U0

� � 0 then U0� is unstable for t � t0

ifddt �2S�U0

� � 0 then U0� is asymptotically

stable for t � t0,

(5.3)

where S denotes the total entropy of the thermody-namic system. In practice, we can modify the valueof �2S by varying a series of control parameters .When these parameters reach a critical value c, thesign of the inequality (5.3) will change, and a steadystate will lose its stability. A steady state in a con-figuration with critical control parameters is said tobe marginally stable:

ddt �2S�U0

��c� � 0 for t � t0. (5.4)

The total entropy S is related to the entropyproduction � considered in this note via the en-tropy balance equation

ddt S � � �S, (5.5)

where �S denotes the entropy flux through theboundaries. Equation (5.5) may be varied twice toobtain

ddt �2S � �2� �2�S, (5.6)

where the variation of the �S term vanishes becausewe consider inhomogeneous configurations inwhich �S is held fixed in our experiments. In Ref.[6], the authors considered inhomogeneous thermo-dynamic systems subject to boundary conditions inwhich all fluxes vanish at infinity. We are insteadconsidering finite inhomogeneous systems forwhich the entropy flux through the boundaries, �S

is held fixed. Note that in the particular case �S � 0,the only steady state is equilibrium. As a resultsimplifications that occur in Ref. [6], which rely ontheir choice of boundary conditions do not applyhere, but they will not be needed. Thus, in our casethe stability condition is

if �2��U0� � 0 then U0

� is unstable for t � t0

if �2��U0� � 0 then U0

� is asymptoticallystable for t � t0.

(5.7)

These conditions may be interpreted geometri-cally as follows. Recall that the point x is in a jdimensional submanifold N � M of configurationsthat satisfy a set of physical constraints that relatethe thermodynamic forces. Then x is a steady stateof order � if j � n � � and if the derivatives of �with respect to all tangent vectors to N vanish.Diagonalizing the matrix of second derivatives, wemay also define the stable directions to be the pos-itive eigenspace, the unstable directions to be thenegative eigenspace, and the marginally stable di-rections to be the null eigenspace.

The steady-state condition (5.2) is not canoni-cally defined as the U’s at various points x live indifferent spaces. However, using the geometric in-terpretation of steady states as critical points of �,one arrives at a unique covariant definition forsteady states, and thus a definition that we conjec-ture will hold beyond the weak-field regime. Thesteady states are critical points of �, which meansthat d� vanishes, where d is the exterior derivative

SONNINO AND EVSLIN


on the submanifold N. As � is a scalar, the ordinaryand covariant derivatives of � are equal. Thus,

0 � d� � D� � D�U�U�g�� 2g��U�DU�, (5.8)

where the f term vanishes because f is antisymmet-ric and UU is symmetric and Dg vanishes becausethe covariant derivative of the metric is zero. Thusthe covariant form of the steady-state condition isjust

g��U�DU� � 0. (5.9)

The stability condition (5.7) is already covariant,as S is a scalar, so ordinary derivatives on S arecanonically defined. But for completeness, we willevaluate the double variation

�2� � 2�

xa xb �xa�xb, (5.10)

where

2�

xa xb � 2g��U ;a�U ;b

� 2f��;aU0�U ;b

� 2� g��

f��U0��U ;a;b

� �ab� U ;�

� U ;��

2x�

xa xb� . (5.11)

We have again used the fact that g is covariantlyconstant. There are marginally stable states whenthe determinant of the symmetric matrix ( 2�)/( xa xb) vanishes

� (2g�� f��)U0��U ;a;b

� �ab� U ;�

� U ;��

2x�

xa xb� 2g��U ;a

�U ;b� f��;aU0

�U ;b� f��;bU0

�U ;a��

�c

� 0.

(5.12)

We expect that, even beyond the weak-field ap-proximation, Eqs. (5.9) and (5.12) will determine,respectively, the steady states U0 and the criticalvalues c of the control parameters .

6. Conclusions

We have presented a manifestly covariant andcoordinate-independent formulation of the TFT.This allows the theory to be extended, for example,

to thermodynamic spaces M that cannot be coveredby a single coordinate patch corresponding to the-ories for which different observable quantities aresuitable in different regimes. Making use of theshortest path principle, we have demonstrated thevalidity of the universal criterion of evolution, ex-pressed in a covariant form, and we have shownthat this term has a local minimum at the geodesicline. We referred to this theorem, together with theshortest path principle, as the minimum rate ofdissipation principle. Physically, the minimum rateof dissipation principle affirms that, for time-inde-pendent boundary conditions, a thermodynamicsystem evolves towards a steady-state with theleast possible dissipation. This principle should notbe confused with the minimum dissipation princi-ple introduced by Onsager, which applies onlywhen the system is close to equilibrium. In theOnsager formalism, the evolution of the thermody-namic fluxes is given by a system of linear equa-tions, which describes the relaxation to equilibrium.The term “minimum” refers to the fact these relax-ation equations have been derived minimizing afunctional of the thermodynamic fluxes. The mini-mum rate of dissipation principle and the Onsagerprinciple are therefore unrelated.

Of course, the test of any proposed formulationis its agreement with experiment. One physical sys-tem for which a large amount of data are alreadyavailable, and a theoretical underpinning is lacking,is the relaxation of a magnetically confined plasmatoward a steady state in the nonlinear classical andPfirsch–Schluter regimes or in the nonlinear Bananaand Plateau regimes. The metrics in these regimescan be found in Ref. [13].

We have also derived the thermodynamic fieldequations. For the �–� and �–� processes, theseequations reduce to those found in Refs. [1–4]. Thevalidity of these equations, in the weak-field ap-proximation, has been tested in several thermody-namic systems. So far we have found that numeri-cal simulations and experiments are in agreementwith the theoretical predictions of the TFT. Whenthe skew-symmetric tensor f�� does not vanish, wehave proposed a new set of field equations, whichare conjectured to hold even beyond the weak-fieldregime. The strong field equations are, however,notably simpler than the ones obtained in Ref. [4].

The equations that determine the steady statesand the critical values of the control parameters ofa generic thermodynamic system have also beenpresented. These equations are also conjectured tohold beyond the weak-field regime and so they



generalize the weak-field equations reported inRefs. [2–4]. We note that this extension of linearthermodynamics is quite different from the Prigog-ine–Glansdorff extension, which is based on hydro-(or plasma-)dynamical stability theory and bifurca-tion analysis [6]. The relation between the two ap-proaches should be made explicit. This will be thesubject of a further publication.

The current approach describes the trajectory fol-lowed by a relaxing system, but does not give thetime dependence of the relaxation process. It maybe possible to include the time direction in thethermodynamic space with a metric such as gtt ��. The geodesic evolution in spacetime wouldthen give a firm prediction for the time dependenceof the spatial trajectory. In addition, the steady-statecondition and the Layapunov stability conditionsmay be derived from the constraint that particlesfollow spacetime geodesics. Of course, in order forthe system to stop at a steady state, somehow dis-sipation will need to be added to the system. Oncethis is done, it may be possible to apply TFT toprocesses other than relaxation.

Appendix A. Example: Flat Space

Let M � � be a plane. We will compute the mapexp : T� � �. In this case, the Riemannian manifold(M, g) coincides with (M, L) where L is the Onsagermatrix. There is a well-defined exponential mapfrom a ball Ux in Tx� to a neighborhood �Ux of x� �:

Ux�Tx� � R23 �Ux�� R2. (A.1)

When the metric is flat, the geodesics �V(�) arestraight lines

xf� � �A� x�, (A.2)

where A� is a constant vector and A � Tx�, �A(ti) �x and �A(tf) � xf. Therefore, the exp map on x issimply

expx�A � �A x; (A.3)

that is, the exponential map exppV (with p � � andV � Tp�) simply performs a translation:

exppV � p V. (A.4)

Let us now consider the case U � Ux � T0M � R2

where 0 � � denotes the origin, corresponding tothe equilibrium state. In this case, U points from xto the origin and, using the canonical identificationof the plane � and its tangent fibers, which areplanes of the same dimension, we find the relationx � �U [see Eq. (3.5)]. If the coordinates of x are x �(a, b), the coordinates of U are then U � (�a, �b)and, from Eq. (A.4), we have

exppU � p U � �a, b� ��a, �b� � 0. (A.5)

This implies that the exponential map sends thetangent vector U at p to the equilibrium state, whichis the definition of U in Eq. (2.6).

The entropy production of a system at the pointx is just its length squared

� � �U�2 � ��x�2 � �x�2. (A.6)

Appendix B. Example: TheTwo-Sphere

Let M be the unit-sphere S2 with the usual rota-tion-invariant metric

ds2 � d�2 sin2�d�2. (B.1)

We will use both spherical and Cartesian coordi-nate systems, which are related by

x � sin � cos �, y � sin � sin �, z � cos �.

(B.2)

The geodesics on the 2-sphere are the great circles,which are the equators corresponding to variouschoices of north pole. We will now fix the northpole to be � � 0, and so every geodesic � willintersect the equator � � �/2 at two points � � ��and � � � � �. Note that the equator itself is thegeodesic that intersects itself an infinite number oftimes, to apply the following derivation to theequator itself it would be necessarily to deform it bya small rotation and then take that rotation to zeroat the end. However, the final result will makesense for every geodesic, including the equator.

The map exp(v�) is easy to calculate in the case inwhich � is the equator, it just increases � by thenorm v of the vector v�. To evaluate exp on anarbitrary point (�0, �0) with an arbitrary tangentvector v�, which yields an arbitrary geodesic �, it

SONNINO AND EVSLIN


will be simplest to rotate the coordinate system sothat � becomes the equator, then exp may be eval-uated and the coordinate system may be rotatedback to yield the final answer.

The rotation of the coordinate system will pro-ceed in two steps. First we use � to define a rotatedset of coordinates:

x � sin � cos�� , y � sin � sin�� ,

z � z � cos �. (B.3)

Then we rotate about the y-axis until � is the equa-tor, which we parametrize with the azimuthal vari-able �

� xyz� � �1 0 0

0 cos � �sin �0 sin � cos �

��cos �sin �

0�. (B.4)

In coordinates, the rotation (B.4) is

sin � cos�� cos �, sin � sin��

� cos � sin �, cos � � sin � sin �. (B.5)

The tangent vector v� will be decomposed in po-lar coordinates (v, �), where v is the norm of v� and� is the angle between v� and the azimuthal direc-tion �. In this section we will calculate exp(�0,�0)

v�.First, we will find expressions satisfied by the un-known angles � and � in terms of the given data �0,�0, v, and �. These will allow us to calculate the zcomponent z of the function exp, which is partic-ularly simple as it is independent of �. Then we willrotate the coordinates to construct the other com-ponents x and y.

v� is a tangent vector to the geodesic � and so therotation (B.4) relates v� to the equator’s tangent vec-tor / �. For example, the last expression in (B.5)yields

�cos � �

�sin � sin �, (B.6)

where the derivative is evaluated at (�0, �0). Theleft-hand side may be evaluated using the chainrule, where we note that the derivative of � withrespect to � along the geodesic is just the � compo-nent sin � of the unit tangent vector v�/v. The right-hand side is evaluated by noting that � is the angleby which the entire geodesic � is rotated, so it isindependent of �. Thus, (B.6) reduces to

sin � sin �0 � sin � cos �. (B.7)

The dependence on the unknown angle � may beremoved using the last term in Eq. (B.4). We thendivide both sides of the resulting equation by cos �0to obtain

cot � � sin � tan �0. (B.8)

Thus, we have determined the unknown angle � interms of the initial point and tangent vector. Whilein principle we could also find �, it will suffice torecall the last term from (B.5):

sin � � cos �0 csc �. (B.9)

In our new coordinates, the exp operation corre-sponds to a simple shift of the equator’s longitudi-nal coordinate �:

� � � v. (B.10)

Finally, we need to reexpress this shift in terms ofthe old coordinates. In fact, it will suffice to find thecorresponding coordinate z. Using the rotation(B.4) at the new coordinate � � v

�xyz� � �1 0 0

0 cos � �sin �0 sin � cos �

��cos(� v)sin(� v)

0� (B.11)

we find

z � sin � sin�� v�. (B.12)

This may be expanded using (B.9) and the trigono-metric identity for the sine of a sum to yield

z � cos �0 csc ��sin � cos v cos � sin v�

� cos �0�cos v cot � sin v�. (B.13)

Finally, we may use (B.8) to obtain

z � cos �0 cos v sin � sin �0 sin v. (B.14)

To find the corresponding formula for x, wemay rotate the coordinates by �/2, yielding a newz coordinate, z, equal to the old x coordinate

z � cos � � sin � cos � � x. (B.15)



This rotation is an isometry, so v is unchanged;however, the angle � is effected. The rotated valueof � may be found similarly to (B.6):

�cos � � sin � sin � �

��sin � cos ��

� sin � cos � cos � � cos � sin �. (B.16)

In the last term of the last expression of (B.16) wehave used

cos � � g��1/ 2

�

�, g�� sin2�. (B.17)

We may now use (B.15) and (B.6) to rotate ourEq. (B.14) for the z component of the exp function toobtain the x component

x � sin �0 cos �0 cos v �sin � cos �0cos �0

� cos � sin �0�sin v. (B.18)

At last we may rotate � to obtain the y component:

y � sin �0 sin �0 cos v �sin � sin �0cos �0

cos � cos �0�sin v. (B.19)

Putting together Eqs. (B.14, B.18, B.19), we obtain

exp��0,�0��v, �� x, y, z�. (B.20)

Taking the north pole to be equilibrium, the ther-modynamic force at any point is calculated usingthe geodesic that runs straight north from any pointto the north pole

U � �v � �, � ��

2� . (B.21)

This geodesic is uniquely determined for allpoints except for the south pole � � �, where thereare infinitely many northward geodesics and thecoordinate � is ill-defined. The entropy productionis then simply

� � �U, U� � �2. (B.22)

Note that at the south pole, where the geodesic isnot unique, � is independent of the geodesic chosenand thus continues to be well defined.

ACKNOWLEDGMENTS

The authors acknowledge our indebtedness to Pro-fessor C. M. Becchi of the University of Genova, Phys-ics Department, and to Professor E. Tirapegui of theUniversity of Chile, Physics Department, for readingthis manuscript and making helpful suggestions. G. S.expresses sincere gratitude to the hierarchy at theEuropean Commission: Professor A. Mitsos, Dr. P.Fernandez Ruiz, and Dr. E. Rille. G. S. is also verygrateful to Dr. U. Finzi and Dr. M. Cosyns, of theEuropean Commission, for their continuing encour-agement. G. S. thanks the members of the EURATOMBelgian State Fusion Association. The work of J. S. ispartially supported by IISN, Belgium, contract 4-4505-86, by the Interuniversity Attraction Poles Pro-gramme, Belgian Science Policy, and by the EuropeanCommission RTN program HPRN-CT-00131, in asso-ciation with K. U. Leuven.

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Geometrical thermodynamic field theory