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Entropy 2015, 17(9), 6150-6168; doi:10.3390/e17096150
Abstract: In this work, we show that the thermodynamic phase space is naturally endowed with a non-integrable connection, defined by all of those processes that annihilate the Gibbs one-form, i.e., reversible processes. We argue that such a connection is invariant under re-scalings of the connection one-form, whilst, as a consequence of the non-integrability of the connection, its curvature is not and, therefore, neither is the associated pseudo-Riemannian geometry. We claim that this is not surprising, since these two objects are associated with irreversible processes. Moreover, we provide the explicit form in which all of the elements of the geometric structure of the thermodynamic phase space change under a re-scaling of the connection one-form. We call this transformation of the geometric structure a conformal gauge transformation. As an example, we revisit the change of the thermodynamic representation and consider the resulting change between the two metrics on the thermodynamic phase space, which induce Weinhold’s energy metric and Ruppeiner’s entropy metric. As a by-product, we obtain a proof of the well-known conformal relation between Weinhold’s and Ruppeiner’s metrics along the equilibrium directions. Finally, we find interesting properties of the almost para-contact structure and of its eigenvectors, which may be of physical interest.
The geometry of equilibrium thermodynamics and thermodynamic fluctuation theory is extremely rich. In particular, equilibrium thermodynamics is based on the first law, which for reversible processes can be written in the internal energy representation as:
A Riemannian metric can be introduced on the Legendre sub-manifold representing a thermodynamic system by means of the Hessian of a thermodynamic potential. Weinhold [9,10,11,12] was the first to realize this fact and proposed the metric defined as the Hessian of the internal energy. For example, for a closed system:
The two metrics are related by a conformal re-scaling :
The study of Metrics (3) and (5) has been very fruitful. It was found in particular that the thermodynamic length corresponding to (respectively, ) implies a lower bound on the dissipated availability (respectively, to the entropy production) during a finite-time thermodynamic process  and that the scalar curvature of these geometries is a measure of the stability of the system, since it diverges over the critical points of continuous phase transitions with the same critical exponents as for the correlation volume [20,21,22]. Moreover, these geometries are related naturally to the Fisher–Rao information metric, and therefore, the investigation of their geometric properties can be extended (mutatis mutandis) to the statistical manifold  and to microscopic systems, which are characterized by working out of equilibrium [24,25,26]. As such, the intrinsic geometric perspective of Legendre sub-manifolds of the thermodynamic phase space has given new physical insights for thermodynamics itself, with direct interest for applications in realistic processes, outside the realm of abstract reversible thermodynamics.
So far, the geometric properties of the thermodynamic phase space itself have remained less investigated. Mrugala et al.  proved that one can endow naturally the TPS with an indefinite metric structure derived from statistical mechanics, which for a closed system can be defined either as:
An additional physical motivation for our study comes from previous results, where it has also been proved, by means of contact Hamiltonian dynamics, that the lengths computed using Metrics (8) and (9) in the thermodynamic phase space give a measure of the entropy production along irreversible processes identified with fluctuations  (see also ).
In this work, we revisit these ideas from a different point of view, namely that of the theory of connections. In this manner, we present a novel aspect of the geometric structure of thermodynamics and thermodynamic fluctuation theory. In particular, we study the transformations preserving the connection defined by reversible processes, which we call the equilibrium connection. In fact, it is known that the physical content of the first law resides in those processes that annihilate the connection one-form , and therefore, at the level of an equilibrium (reversible) description, we are presented with the physical freedom of rescaling such a form through multiplication by any non-vanishing function. This operation, known as contactomorphism [36,37], does not change the results of equilibrium thermodynamics. One usually encounters such transformations as the change of thermodynamic representation, e.g., from the energy to the entropy representation (cf. Equation (7)). Here, we consider the class of contactomorphisms in the thermodynamic phase space and derive the induced transformations for any object defining its geometric structure. We call these conformal gauge transformations . We prove that the equilibrium connection thus defined is necessarily non-integrable, meaning that its associated curvature (not to be confused with the Riemannian curvature associated with the various thermodynamic metrics) is non-vanishing and not invariant under contactomorphisms. Hence, it follows that, albeit that the equilibrium thermodynamics of reversible processes is independent of the representation used, the description of irreversible fluctuations along such processes does change depending on the choice of a particular representation. We show that the metric structure of the TPS is intimately related to the curvature of the equilibrium connection. Since the connection is non-integrable, it follows that the metric changes under contactomorphisms. We further show that this reduces to the well-known relation (6) between Weinhold’s and Ruppeiner’s geometries in the appropriate case, but our result is valid for a general transformation. This shows that the induced thermodynamic lengths are not invariant with respect to using different potentials or representations (see also [41,42,43] for the definition of inequivalent thermodynamic metrics based on the Hessian of other potentials). Finally, we argue that our results can shed light on the physical significance of these geometric objects, highlighting the ones related to a reversible situation and the ones associated with irreversible evolution. Hopefully, this description will help in the identification of geometric properties of potentials that are relevant in irreversible situations.
2. The Equilibrium Connection
In this section, we will recall some formal developments of thermodynamic geometry. The interested reader is referred to  and  for a detailed discussion about the statistical origin of the structures presented here.
Let us consider a thermodynamic system with n degrees of freedom. As we have argued in the Introduction, the TPS, denoted by , is the -dimensional ambient space of possible thermodynamic states of any system.
The laws of thermodynamics are universal statements (applicable to every thermodynamic system) about the nature of the processes that take place when a system evolves from a particular thermodynamic state to another. Thus, we believe that such laws are better identified in a geometric perspective with the properties of the TPS. In order to accommodate such laws, it is convenient to consider the TPS to be a differentiable manifold. This will make the evolution meaningful in terms of vector fields and their corresponding integral curves.
Our central point is that the first law of thermodynamics (1) is equivalent to defining a dimensional connection Γ over the TPS, which we call the equilibrium connection. This is a smooth assignment of horizontal directions for the tangent vectors at each point of . We express this schematically by:
Let us agree that a curve on represents a possible process. We say that a curve joining two points in the TPS is an equilibrium (reversible) process if its tangent vector lies in the horizontal subspace with respect to the first law. This statement acquires a definite meaning with the aid of a connection one-form η. Recall that a one-form is just a linear map acting on tangent vectors. In the case of the first law, the horizontal directions of are given by the vectors annihilated by η, that is,
From Equation (1), we see that the above condition on X is just the requirement that the corresponding process be a reversible process. In fact, from a geometric point of view, since η is a contact form (see the Introduction), then a theorem by Darboux ensures that around each point on the TPS, one can assign a set of local coordinates , where a takes values from one to n, in which η reads:
Note that the horizontal directions in the TPS are uniquely defined by (11), and any particular thermodynamic system at equilibrium has a tangent space, which is a subspace of at every point. Therefore, Definition (11) encodes the universality of the first law of thermodynamics.
Now, let us find a coordinate expression for the equilibrium directions around every point of the TPS. These are simply the tangent vectors satisfying (11). In Darboux coordinates, a direct calculation reveals that the vectors:
An interesting fact is that equilibrium directions are not propagated along equilibrium processes. To see this, note that the change of the ’s along the integral curves of does not vanish identically, that is, for any smooth function f on ,
Let us observe a crucial consequence of Equation (15). Since the set (13) generates at each point in the TPS, then any non-vanishing Lie-bracket of vectors in will be necessarily vertical. This means that the equilibrium connection Γ is non-integrable . We will return to this point in the next section when we discuss its relevance on conformal gauge invariance.
Now, we have a basis for the tangent space , composed by the Reeb vector ξ and the horizontal basis in (13). Notice, however, that we do not have yet a notion of orthogonality for the vector fields ξ, and . The only information available thus far is that every tangent vector to any point in the TPS can be uniquely decomposed into a vertical part and its equilibrium (horizontal) directions, namely:
In order to introduce the notion of orthogonality between the horizontal and vertical directions, one can introduce a metric structure on the TPS. It was found by Mrugala et al.  (see also ) that there is a natural choice for such a metric based on statistical mechanical arguments, that is:
A word of warning is warranted. It can be directly verified that the metric (19) is not positive definite, that is there are non-zero tangent vectors whose norm vanishes identically. To see this, remember that a metric tensor is a bi-linear map (linear in its two arguments), and hence, it is completely determined by its action on a set of basis vectors. Thus, using the decomposition (18) together with the horizontal basis (13), it follows that:
Now, we want to express the metric tensor in (19) in a coordinate-free manner putting into play the role of η and as the connection one-form and the curvature two-form, respectively. In terms of the geometry of contact Riemannian manifolds, the result of this derivation means that the metric (19) is associated and compatible with the contact one-form η (cf. [36,37]). Since the equilibrium connection Γ is non-integrable, the action of the curvature  of the connection one-form (12) on pairs of horizontal vectors ,
Their inner product is given by:
There is an obvious sign difference due to the fact that the metric is a symmetric tensor, whereas is anti-symmetric. However, we can use here the same argument used in Kähler geometry and introduce a linear transformation of the tangent space at each point, namely , such that:
The map Φ is known in para-Sasakian geometry as the almost para-contact structure . Since Φ is a linear map, it is uniquely determined by its action on the basis vectors. Thus, one can quickly verify that the desired transformation has to satisfy:
Thus, and are eigenvectors of Φ with eigenvalues one and , respectively, and a local expression for in this adapted basis is simply:
Now, we can replace the coordinate dependent part in Equation (19) with an equivalent purely geometric (coordinate independent) expression. Furthermore, since “kills” the vertical part of any tangent vector (cf. Equation (16)), our expressions are carried to any tangent vector. Therefore, for any pair of tangent vectors in , their inner product is given by:
Our final expression for the metric poses a compelling geometric structure, expressed as the sum of the equilibrium connection one-form η and its associated field strength , respectively. This was made with the aid of an intermediate quantity Φ, whose role is revealed by means of its “squared” action on any vector ,
Finally, Φ can be independently obtained as the covariant derivative of ξ with respect to the Levi–Civita connection of G, closing the hard-wired geometric circuit associated with the first law of thermodynamics .
Thus far, we have re-formulated the first law as the definition of a connection whose horizontal vector fields are reversible processes (cf. Equations (10) and (11)). This sets up a suitable framework to work out the local symmetries shared by every thermodynamic system, that is the various points of view in which a thermodynamic analysis can be made without changing its physical conclusions. In the present case, such conclusions are restricted to the directions in which a system can evolve and the possible interpretation (not analyzed here) of the thermodynamic length of a generic process, not necessarily an equilibrium one, by means of Metric (31). In the next section, we will analyze an important class of such local symmetries, i.e., conformal gauge symmetries.
3. Conformal Gauge Transformations in Thermodynamics
In the previous section, we presented the first law of thermodynamics as a connection over the TPS, that is the assignment of equilibrium directions at each point of the tangent space. Such directions were explicitly obtained as the ones that annihilate a one-form whose local expression is the same for every thermodynamic system. There is, however, a whole class of one-forms generating exactly the same connection, each obtained from the other through multiplication by a non-vanishing function. This is referred to here as a conformal gauge freedom. Thus, the central point of this section is to present the class of transformations that leave the equilibrium connection Γ invariant (cf. Equation (10)), together with its corresponding effect on the whole intertwined geometric structure of thermodynamic fluctuation theory, namely the para-Sasakian structure .
Consider the thermodynamic connection one-form η. It is easy to see that any re-scaling defines the same equilibrium directions at each point as the original η, that is:
Moreover, the directions annihilated by do not coincide with those of , That is, while , we have:
Let us take the defining properties of the Reeb vector field, Equation (16), as our starting point. We need a new vertical vector field satisfying:
The first condition is easily met if we define the new vertical vector field as:
The second condition in Equation (37) is not as trivial. A direct evaluation yields:
Evaluating the above expression on ξ and recalling that annihilates ξ, we obtain that:
Now, recalling that Ω is fixed by the change of the gauge (34), we have obtained the desired equation for ζ. Moreover, substituting (41) back into (40), we obtain the expression for the derivative of the scaling factor:
From these calculations, we see that the auxiliary equilibrium (horizontal) vector field ζ plays a central geometric role. Note that in the new gauge , the fundamental vertical vector field is tilted with respect to its unprimed counterpart, that is it has a horizontal component. However, the equilibrium directions are unaltered and, therefore, are generated by the same basis vectors (13). Thus, we write the equilibrium split at each point as:
Note that the expression for was obtained by requiring that its geometrical properties be the same as those of ξ in the new gauge (cf. Equations (16) and (37)). From the same reasoning, in analogy to (31), we require the new metric to be given by:
Consider two vector fields and their inner product in terms of the new gauge. This is expressed by the action of (44) as:
We work out each individual term inside the brackets separately. Let us do this in reverse order and start with the last term. One can immediately see that:
To conclude, note that both expressions, in (48) and in (49), correspond to inner products involving at least one equilibrium vector. Thus, we can re-write them as and , respectively. Substituting (47) to (49) back into (46), adding the null term and collecting the various resulting expressions, we obtain:
Now, a straightforward calculation reveals that:
Finally, using (42) to obtain the coordinate independent expression:
Thus, we have completely determined the new structures in terms of the old ones and the scaling factor relating them. Let us summarize the action of a change of gauge , that is:
To close this section, we shall make a few remarks on conformal gauge invariance in equilibrium thermodynamics, that is the mathematical structures that are indistinguishable along equilibrium processes when we make a change of gauge. Firstly, notice that the curvature of the thermodynamic connection one-form is not a conformally gauge-invariant object, as opposed to a standard gauge theory. This is because the equilibrium connection Γ is, by construction, non-integrable (cf. Equations (15) and (16)). This can be interpreted physically by saying that thermodynamic fluctuations are not gauge invariant. Secondly, note that in spite of the rather non-trivial expression for the transformed metric, Equation (60), its action on equilibrium vectors, say , is remarkably simple, that is:
Notoriously, one can immediately see that the null equilibrium directions at each point of the TPS are exactly the same. Thus, the null structure is gauge invariant. Thirdly, the linear transformation Φ that we introduced on the tangent space at each point of to obtain a coordinate-free expression for the metric tensor is also a gauge invariant object with respect to equilibrium processes,
Thus, combining the statistical origin of the metric [27,28] and the fact that its null directions are eigenvectors of Φ suggests that there is a physical role played by this structure. This will be the subject of future investigations. We believe that quantities that can be directly linked to gauge-invariant structures for equilibrium thermodynamics will be of great interest, since, on the one hand, their meaning will have a universal scope (valid for every thermodynamic system) and, on the other, their values are independent of the thermodynamic representation one decides to use.
4. Change of Thermodynamic Representation as a Gauge Transformation
In the previous sections, we explored some of the consequences of the geometrization of the first law as a connection of the TPS. In this section, we will study a particular example and observe that the various thermodynamic representations are all related by conformal gauge transformations. It follows that, albeit the directions in which a state can evolve through an equilibrium path are independent of the thermodynamic representation, the fluctuations associated with the path will be different when using a different gauge.
Consider the conformal gauge transformation defined by:
The transformation for Φ is just a straightforward calculation, whose result is:
Finally, in order to obtain the expression for the transformed metric, note that for this gauge, ζ is a re-scaling of a null vector (cf. Equation (65) together with (21)). Hence, its squared norm, , is identically zero. Thus, it only remains to evaluate the expression:
The relevance of this exercise is that the conformal gauge transformation presented here corresponds to a change of thermodynamic representation. To see this, let us consider a closed system with the change of gauge defined in (7). It is clear that the equilibrium directions for both and are the same, as shown in the previous section. Hence, they both annihilate the vectors in . Moreover, noticing that in this case and that by Equation (29):
Equation (76) means that the metrics and on are related to each other by the precise conformal gauge transformation that corresponds to a change in the thermodynamic representation (cf. Equations (6) and (7)). Moreover, it follows that on the equilibrium connection Γ, we obtain:
5. Closing Remarks
In thermodynamics, equilibrium (i.e., reversible) processes are defined by the first law (1). In this work, we have given a general geometric statement of the first law in terms of a connection on the thermodynamic phase space. Indeed, we have shown that (1) defines the equilibrium connection Γ (cf. Equations (10) and (11)). Note that the connection one-form η defining Γ is not unique. Indeed, any non-vanishing re-scaling shares the same kernel with η and, thus, defines the same equilibrium connection. Therefore, we call a fixing of a particular one-form determining Γ a conformal gauge choice. The name conformal is in place to denote a difference with gauge theories, such as electromagnetism, where one demands gauge invariance on the curvature of the connection, also referred to as field strength. There, a choice of gauge refers to selecting a one-form generating the same field, whereas in our case, a choice of conformal gauge refers to selecting a one-form generating the same connection. An interesting property of the equilibrium connection is that it is always non-integrable, which means that its curvature does not vanish, independently of the choice of the conformal gauge.
To introduce a further notion of orthogonality between the horizontal (i.e., reversible) and vertical (i.e., irreversible) directions with respect to the equilibrium connection Γ, we followed the work of Mrugala et al.  and equipped the thermodynamic phase space with the indefinite metric structure (19). One can justify such a choice by means of the statistical mechanical arguments contained in  and . Interestingly, the null directions of such a metric correspond precisely to the basis elements generating the horizontal directions (13). The physical significance of such directions remains to be explored and will be the subject of future work. Here, we have given a coordinate invariant formulation (31) of the metric (19), which highlights the role played by the connection one-form η, as well as by the curvature in the definition of the distance and explicitly shows that this is an associated metric in the sense of contact Riemannian geometry [36,37].
The main use of presenting equilibrium thermodynamics as a connection theory relies on the notion of gauge invariance, i.e., those geometric objects that are independent of the particular gauge choice. From the mathematical point of view, the conformal gauge transformations presented here are relevant because they preserve the para-Sasakian structure . As we have argued, in thermodynamics, the curvature of the equilibrium connection is not a gauge-invariant object, nor is the metric. Here, we found the explicit transformations relating the various geometric objects defining the thermodynamic phase space under a conformal gauge transformation. The explicit formulas are summarized by Equations (57)–(60). We observed that the null directions of the metric are gauge invariant. Additionally, when restricted to horizontal directions, the tensor field Φ is also gauge invariant, and the metric structures are conformally related. As an example, we have shown that Metrics (8) and (9), which induce Weinhold and Ruppeiner’s metrics on Legendre sub-manifolds, respectively, are precisely related by the conformal gauge transformation that corresponds to the change in the thermodynamic representation from energy to entropy. This in turn implies that the restriction of such metrics to the equilibrium connection Γ yields the well-known conformal relation (6).
Finally, let us close this work with some comments on the geometry of the equilibrium connection, its conformal gauge transformations and their physical relevance in various prospect applications. Firstly, the construction presented here exhibits the principal bundle nature of the thermodynamic phase space. That is, we readily have a -dimensional (symplectic) base manifold together with a one-dimensional fiber isomorphic to the real line. Such a construction might be suitable to make use of the theory of characteristic classes to formulate universal statements about the nature of thermodynamic processes. Secondly, from the fact that the curvature form of the connection is not preserved by a change of thermodynamic representation together with its statistical origin, one can conclude that thermodynamic fluctuations are not gauge invariant. This is interesting, because thermodynamic fluctuations enter the description of irreversible processes. Therefore, our results can provide new geometric insights on the different extremization problems that one encounters in non-equilibrium thermodynamics, e.g., minimizing dissipation versus maximizing work.
Alessandro Bravetti acknowledges the A. della Riccia Foundation (Florence, Italy) for financial support. Cesar S Lopez-Monsalvo was supported by a UNAM-DGAPA Post-doctoral Fellowship. Francisco Nettel acknowledges financial support from CONACYT Grant Nos. 207934 and 250298.
Each author made an equally valuable contribution in preparing this manuscript. All authors have read and approved the final manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
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