Relativisticlike structure of classical thermodynamics

# Relativisticlike structure of classical thermodynamics

Hernando Quevedo, Alberto Sánchez and Alejandro Vázquez Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, AP 70543, México, DF 04510, Mexico Dipartimento di Fisica and ICRA, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma, Italy Departamento de Posgrado, CIIDET, AP 752, Querétaro, QRO 76000, Mexico
###### Abstract

We analyze in the context of geometrothermodynamics a Legendre invariant metric structure in the equilibrium space of an ideal gas. We introduce the concept of thermodynamic geodesic as a succession of points, each corresponding to a state of equilibrium, so that the resulting curve represents a quasi-static process. A rigorous geometric structure is derived in which the thermodynamic geodesics at a given point split the equilibrium space into two disconnected regions separated by adiabatic geodesics. This resembles the causal structure of special relativity, which we use to introduce the concept of adiabatic cone for thermodynamic systems. This result might be interpreted as an alternative indication of the inter-relationship between relativistic physics and classical thermodynamics.

###### pacs:
04.20. -q, 02.40.-k, 05.70.-a

## I Introduction

The not yet understood similarity between the laws of black hole dynamics and classical thermodynamics (for a recent review, see wald01 ()) seems to indicate that there exists a deep relationship between gravitational dynamics and thermodynamics. This relationship was also established in flat spacetime in the sense that thermodynamic variables like temperature can be ascribed to any null surface that acts as a horizon for observers moving along specific accelerated trajectories in flat spacetime (Unruh effect) unruh76 (). In all these examples, the null surface that can be interpreted as a horizon seems to play an essential role. Nevertheless, other examples are known in which no horizon exists, for instance in a spherically symmetric perfect fluid in equilibrium, but certain similarity continues to hold in the sense that Einstein’s equations are a consequence of demanding that the total entropy of the matter be an extremal open01 ().

The inter-relationship between gravity and thermodynamics was further strengthened by the discovery that Einstein’s equations can be interpreted as a first law of thermodynamics jac95 (). Additional studies of the dynamics of gravity have shown that the validity of this relationship is wider than originally suspected and it holds also in other gravity theories (for a review, see pad13 ()). These results suggest that gravity is not a fundamental interaction, but is an emergent phenomenon with its field equations having the same status as the equations of fluid dynamics. If this turns out to be true, far-reaching consequences are to be expected, in particular, regarding the quantization of gravity.

In all the above studies, one stars from a gravitational system and tries to attribute thermodynamic variables in a consistent manner in order to find the analogy between the dynamics of the system and classical thermodynamics. In this work, we explore the connection between relativity and thermodynamics from a different perspective, namely, we start from a thermodynamic system and try to find the analogy between its “dynamics” and relativity. Indeed, we will show that in the space of equilibrium states of the ideal gas, quasi-static processes can be represented as geodesics which induce a causality structure, resembling that of special relativity. To this end, it is necessary to interpret the equilibrium space as a differential manifold, i.e., we need a geometric approach to thermodynamics.

The idea of using differential geometry in thermodynamics is due to Gibbs gibbs () who realized that the first law of thermodynamics can be represented in terms of differential forms. Caratheodory car () interpreted the laws of thermodynamics in an axiomatic way and in terms of Pfaffian forms. Later on, Hermann her () introduced the concept of thermodynamic phase space where the thermodynamic variables play the role of coordinates. In turn, the phase space contains, in principle, an infinite number of subspaces called equilibrium spaces, each of which is determined by a particular fundamental equation. Recall that in classical thermodynamics, the properties of any thermodynamic system are completely described by the corresponding fundamental equation callen (). The points of each equilibrium space are interpreted as equilibrium states at which the system can possibly exist.

In addition, in the phase space one can introduce a contact structure her () which is invariant with respect to Legendre transformations. The connection between the structure of the phase space and the equilibrium space was recently incorporated in the formalism of geometrothermodynamics (GTD) in an invariant manner quev07 (). In classical thermodynamics, different thermodynamic potentials are related by means of Legendre transformations callen (). Consequently, the Legendre invariance of GTD guarantees that the properties of a thermodynamic system do not depend on the thermodynamic potential used for its description. GTD delivers for any thermodynamic system a Legendre invariant metric which describes the geometry of . In this manner, the equilibrium space becomes a Riemannian manifold whose geometric properties should be related to the properties of the corresponding thermodynamic system.

In GTD, we use some intuitive geometric concepts that were first used in general relativity. For instance, the curvature tensor is expected to be a measure of the interaction between the components of the thermodynamic system. In the case of an ideal gas, where the interaction between particles can be neglected, we say that the thermodynamic interaction is zero and, consequently, the curvature must vanish. We will present a metric for the equilibrium space whose curvature satisfies this condition. Moreover, the curvature tensor is not zero for systems with intrinsic thermodynamic interaction. For this reason the curvature of the equilibrium space is called thermodynamic curvature. In general relativity, curvature singularities indicate the break down of the theory. We use the same idea in GTD and associate the singularities of the thermodynamic curvature with the occurrence of phase transitions at which the equilibrium approach of classical thermodynamics is no more valid.

In this work, we introduce the concept of thermodynamic geodesic as a solution of the geodesic equations along which the laws of thermodynamics are fulfilled. We show that thermodynamic geodesics describe quasi-static processes and that the affine parameter can be used as a “time” parameter. Moreover, thermodynamic geodesics have a definite direction that can be interpreted as the “arrow of time”. In the case of an ideal gas, we show that the thermodynamic geodesics split the equilibrium space into two non-connected regions that resemble the causality structure in Minkowski spacetime.

This paper is organized as follows. In Sec. II, we present the fundamentals of GTD in the case of systems with two thermodynamic degrees of freedom. In particular, we present a Legendre invariant metric for the phase space from which it is possible to derive an invariant geometry for the space of equilibrium. In Sec. III, we focus on the geometry of the ideal gas and show that the corresponding thermodynamic curvature vanishes as a result of the lack of thermodynamic interaction. Furthermore, in Secs. IV and V, we perform a detailed analysis of the thermodynamic geodesics of the ideal gas. The equilibrium space presents a very rich and unexpected structure that resembles the causality structure of spacetime in relativistic physics. In particular, we use the entropy as an affine parameter along thermodynamic geodesics to introduce at each equilibrium state an “adiabatic cone” that determines the entropic past and future of that state. Finally, Section VI is devoted to discussions of our results and suggestions for further research.

## Ii Geometrothermodynamics of simple systems

A rigorous formulation of GTD implies the introduction of geometrical concepts like contact manifolds, tangent and cotangent contact manifolds, smooth maps, pullbacks, etc. quev07 (); qq11 (); blnq14 (). The main purpose of this section is to present GTD in a less rigorous manner, using only the essential tools of differential geometry and emphasizing the intuitive physical and geometric aspects.

The simplest thermodynamic systems are those with only two thermodynamic degrees of freedom. To describe this kind of systems, it is necessary to specify two extensive variables, say entropy and volume , two intensive variables, say temperature and pressure , and a thermodynamic potential, say the internal energy . In GTD quev07 (), the thermodynamic variables are used as coordinates for the construction of the thermodynamic phase space , which in this case is 5-dimensional. We assume that is a well-behaved differential manifold so that it allows us to introduce additional geometric structures. In particular, we introduce the so-called fundamental Gibbs form

 Θ=dU−TdS+PdV , (1)

which, as we will see below, contains the information about the first law of thermodynamics. In addition, we introduce a metric structure that, in general, can depend on all the coordinates of , i.e. .

The triplet is called a Riemannian contact manifold and represents an auxiliary structure which is necessary to implement in a consistent manner the properties of classical thermodynamics. In particular, it is known that the thermodynamic properties of ordinary thermodynamic systems do not depend on the thermodynamic potential used for its description callen (). Since different thermodynamic potentials are related by means of Legendre transformations, we demand that the geometric structure of the phase space be Legendre invariant. In the case of a system with two degrees of freedom, a Legendre transformation is defined in as the change of coordinates with the following possibilities arnold ():

 ~U1=U−TS ,S=−~T ,T=~S ,V=~V ,P=~P , (2) ~U2=U+PV ,S=~S ,T=~T ,V=~P ,−P=~V , (3) ~U3=U−TS+PV ,S=−~T ,T=~S ,V=~P ,−P=~V . (4)

Usually, is called the Helmholtz free energy, is the enthalpy, and is the Gibbs potential. If we denote by , , the result of applying any of the particular Legendre transformations (2)-(4) to the fundamental Gibbs form, then it is easy to see that , showing that in fact is a Legendre invariant geometric object.

In GTD, we also demand that the metric be Legendre invariant. It is possible to write down and solve the algebraic conditions that an arbitrary metric must satisfy in order to be Legendre invariant quev07 (). A particular solution was found in vqs08 () that can be written as

 G=(dU−TdS+PdV)2+(ST)2k+1dSdT+(VP)2k+1dVdP , (5)

where is an integer, positive or negative. As in the case of the fundamental Gibbs form, if we denote by the metric resulting from the application of the Legendre transformations (2)-(4) to the metric (5), the Legendre invariance of becomes clear. In fact, the functional dependence of coincides with (5) with replaced by , by , and so on. In this manner, we see that the particular triplet , with and given as in Eqs.(1) and (5), respectively, is invariant with respect to all possible Legendre transformations in the case of systems with two thermodynamic degrees of freedom111It should be noticed that the metric (5) is not unique. One still can multiply by an arbitrary Legendre invariant function which, however, can be associated with a change of representation that should not affect the geometric properties of blnq14 (). Moreover, all mixed terms of the form etc. can be added to the right-hand side of Eq.(5), without changing its symmetry properties. These additional terms, however, do not satisfy the additional condition that the metric must generate a flat metric for the ideal gas gl14 ().. This is an important property which guarantees that our further results are independent of the choice of thermodynamic potential.

The next important element of GTD is the space of equilibrium states which, in quite general terms, is the space where systems in thermodynamic equilibrium can exist and their properties can be investigated. This means that is a 2-dimensional subspace of which we define in the following manner. Let us choose the set of extensive variables as the coordinates of . Then, when evaluated on , the remaining coordinates of must be functions of and only, i. e.

 U=U(S,V) ,T=T(S,V) ,P=P(S,V) . (6)

The first of these equations is known as the fundamental equation in the energy representation. In fact, once is given explicitly, one can derive all the equations of state and thermodynamic properties of the corresponding thermodynamic system. To guarantee the existence of the second and third equations of (6), we demand that the projection of the fundamental Gibbs form on vanishes, i. e.,

 Θ|E=0⟺dU=TdS−PdV , (7)

a relationship that is immediately recognized as the first law of thermodynamics. Furthermore, since , the first law of thermodynamics implies that

 ∂U∂S=T ,∂U∂V=−P , (8)

so that and become functions of and , as stated in (6). In classical thermodynamics, the relationships (8) represent the conditions for thermodynamic equilibrium. As for the metric of , we demand that its projection on , by using (7) and (8), induces a metric on , i. e., . In the particular case of the Legendre invariant metric (5), a straightforward calculation leads to

 g= (S∂U∂S)2k+1∂2U∂S2dS2+(V∂U∂V)2k+1∂2U∂V2dV2 (9) +[(S∂U∂S)2k+1+(V∂U∂V)2k+1]∂2U∂S∂VdSdV .

If we specify a fundamental equation , the above metric is unique. The above description corresponds to the energy representation in classical thermodynamics. One of the advantages of GTD is that the formalism allows us to handle different representations in an invariant way. For later purposes, we present here the entropy representation for a system with two thermodynamic degrees of freedom. A simple rearrangement of Eq.(1) leads to the Gibbs form in the entropy representation

 ΘS=dS−1TdU−PTdV , (10)

so that the coordinates of the phase space are , and the metric as given in (5) can be written as

 GS=(dS−1TdU−PTdV)2+(UT)2k+1dUd(1T)+(VPT)2k+1dVd(PT) . (11)

Applying the corresponding change of coordinates (see Appendix), it is easy to show that the above geometric objects are invariant with respect to Legendre transformations. Furthermore, for the equilibrium subspace we choose the extensive variables and so that the remaining coordinates become functions of and when projected on . In particular, the fundamental equation must now be given as . As before, we demand that the projected Gibbs form and the metric satisfy the relationships

 ΘS|E=0 ,GS|E=gS=gS(S,V) , (12)

so that from the first condition we obtain the first law of thermodynamics and the conditions for thermodynamic equilibrium in the entropy representation:

 dS=1TdU+PTdV ,∂S∂U=1T ,∂S∂V=PT . (13)

Moreover, the metric of the equilibrium space can be calculated in a straightforward manner from the above equations and we obtain

 gS= (U∂S∂U)2k+1∂2S∂U2dU2+(V∂S∂V)2k+1∂2S∂V2dV2 (14) + [(U∂S∂U)2k+1+(V∂S∂V)2k+1]∂2S∂U∂VdUdV  .

Again we see that once the fundamental equation is given, the metric of is uniquely determined. As in classical thermodynamics, the fundamental equation must satisfy the second law which in the case under consideration can be written as callen ()

 ∂2U∂Ea∂Eb≥0 ,∂2S∂Fa∂Fb≤0 , (15)

for the energy and entropy representation, respectively. Here and .

It is worth mentioning that in the above construction we chose the extensive variables as coordinates for the equilibrium space in order to obtain the energy and entropy representation which are the most common approaches used in classical thermodynamics. However, in general it is possible to choose any 2-dimensional subspace of to define the equilibrium space which would correspond to a different representation in classical thermodynamics. The Legendre invariance of the phase space has as a consequence that all possible representations are equivalent and the properties of thermodynamic systems do not depend on the representation.

The above geometric construction can be applied to any thermodynamic system with two degrees of freedom. One only needs to specify the fundamental equation of the thermodynamic system in order to investigate its geometric properties. However, GTD allows a generalization to include any system with a finite number of degrees of freedom, say . In this case, the phase space has the dimension and the subspace of equilibrium states is dimensional. The fundamental Gibbs form and the metrics and can be generalized to the dimensional case in a straightforward manner vqs08 ().

## Iii The equilibrium space of the ideal gas

In the specific case of an ideal gas, the fundamental equation in the entropy representation can be expressed as callen ()

 S(U,V)=S0+NkBcVln(UU0)+NkBln(VV0) , (16)

where is the dimensionless heat capacity at constant volume, is Boltzmann’s constant, is the constant number of particles and , and are constants.

The intensive thermodynamic variables for the ideal gas can be calculated by using the conditions of thermodynamic equilibrium (13). We obtain

 1T=NkBcVU ,PT=NkBV . (17)

Furthermore, from Eq.(14) we obtain the simple metric

 gS=−(NkB)2k+2[c2k+2VdU2U2+dV2V2] . (18)

It is straightforward to show that the curvature of this metric vanishes. In GTD, we interpret the absence of curvature as a manifestation of the absence of thermodynamic interaction. One of the goals of GTD is to interpret the curvature of the space of equilibrium states as a measure of thermodynamic interaction. This goal has been reached here in the case of the ideal gas. Moreover, one can show quev07 (); qr11 () that in the case of the van der Waals gas the curvature tensor of the metric (14) does not vanish in accordance with the fact that the van der Waals gas corresponds to a system with non-vanishing thermodynamic interaction. In addition, the locations of the thermodynamic curvature singularities turn out to coincide with the points where phase transitions occur qr11 (). We interpret this result as a further indication that thermodynamic curvature can be used to measure the thermodynamic interaction.

Since the curvature of the metric (18) vanishes, there must exist coordinates in which the metric takes the simple Euclidean form. In fact, the Cartesian-like coordinates

 ξ=ξint+(NkBcV)k+1lnU ,η=ηint+(NkB)k+1lnV (19)

 gS=dξ2+dη2 , (20)

where for simplicity we have chosen a positive definite signature. Moreover, the additive constants of integration and and can always be chosen such that and . This means that the equilibrium space of the ideal case can be represented as the positive definite quadrant of the Cartesian plane.

## Iv Thermodynamic geodesics

Let us define in the equilibrium space the thermodynamic length as , where is the volume element in quev07 (); vqs08 (). Then, the condition that the thermodynamic length be extremal, , leads to the geodesic equations for the coordinates of the space . In the case of the flat metric (20), the geodesic equations become and , where the dot denotes differentiation with respect to an affine parameter . The solutions are found to be and , where and are constants, i. e., they represent straight lines. For instance, consider all geodesics with initial state and . Then, on the plane the geodesics must be contained within the quadrant determined by and , due to our choice of integration constants for and . Consequently, the geodesics of the ideal gas can be depicted by using the equation , with constants and . For any arbitrary initial state, there is always a straight line that connects that state with any arbitrary point on the plane. This means that the entire space of equilibrium states can be covered by geodesics. This behavior is schematically illustrated in Fig. 1.

However, not all the solutions of the geodesic equations must be physically realistic. Indeed, there could be straight lines connecting equilibrium states that are not compatible with the laws of thermodynamics. In particular, one would expect that the second law of thermodynamics imposes strong requirements on the solutions. In classical thermodynamics, two equilibrium states are related to each other only if they can be connected by means of a quasi-static process. Then, a geodesic that connects two physically meaningful equilibrium states can be interpreted as representing a quasi-static process. Since a geodesic curve is a dense succession of points, we conclude that a quasi-static process can be seen as a dense succession of equilibrium states, a statement which coincides with the definition of quasi-static processes in equilibrium thermodynamics callen (). Accordingly, we define a thermodynamic geodesic as a geodesic along which the laws of thermodynamics are satisfied, i.e., a geodesic that represents a quasi-static process. This implies that a thermodynamic geodesic must have a definite direction associated with the direction in which the entropy increases, in agreement with the second law of classical thermodynamics. Moreover, since the affine parameter is defined up to a linear transformation, it should be possible to choose it in such a way that it increases as the entropy of a quasi-static process increases. This opens the possibility of interpreting the affine parameter as a “time” parameter with a specific direction which coincides with the direction of entropy increase. We will explore in detail this possibility in Sec. V.

In the special case of the ideal gas, the fundamental equation (16) in coordinates and represents a straight line (with a new additive constant which we denote as before as for simplicity)

 S=S0+(NkB)−k(c−kVξ+η) , (21)

so that the entropy increases as and increase. Consequently, any straight line pointing outwards of the initial zero point and contained inside the allowed positive quadrant connect states with increasing entropy. This behavior is schematically illustrated in Fig. 2 where the arrows indicate the direction in which a quasi-static process can take place, i.e., in which the entropy increases. A quasi-static process connecting states in the opposite direction is not allowed by the second law of thermodynamics. Consequently, the affine parameter represents a time parameter and the direction on each geodesic indicates the “arrow of time”.

Since the above description is based upon a rigorous analysis of the geodesic equations and their solutions in the equilibrium space of an ideal gas, we conclude that the above result can be interpreted as a rigorous geometric proof of the intuitive and well-known idea that the direction of time coincides with the direction in which entropy increases. It must be mentioned that our interpretation of “time” is pure classical in the sense that it corresponds to an affine parameter along a macroscopic geodesic, and cannot be extrapolated to the microscopic level where time must be interpreted in a different manner and possibly quantum effects must be considered huang (); landau ().

If the initial state is not at the origin of the plane, the second law permits the existence of geodesics for which one of the coordinates, say , decreases as long as the other coordinate increases in such a way that the entropy increases or remains constant. In fact, the region in the plane available from a given initial equilibrium state is situated within two extreme geodesics which span a maximum angle that can be determined in the following way. Let the initial state be at the point . According to the second law of thermodynamics and Eq.(21), the state characterized by the coordinate values can be reached by a geodesic with origin at if the condition

 cVΔξ+Δη≥0 ,Δξ=ξf−ξi ,Δη=ηf−ηi , (22)

is satisfied. Consider geodesics for which . Hence, only those geodesics are allowed for which . The equal sign determines the extreme geodesic with constant entropy (adiabatic geodesic) which intersects the axis at the point and . This geodesic intersects the axis at an angle such that . Consider now geodesics with . The allowed geodesics must satisfy and the extreme adiabatic geodesic intersects the axis at the point with coordinates and . The adiabatic geodesic intersects the axis at an angle with . Since the intersection angles are complementary, , we conclude that the angle spanned by the two adiabatic geodesics (one with and the second one with ) is .

An alternative derivation of the above geometric construction of adiabatic geodesics consists in considering the corresponding equation in the form , which can be rewritten as

 ξfξi+ηi/cV+ηfηi+ξicV=1 , (23)

and is immediately recognized as the equation of a straight line. This line in the equilibrium space can be occupied only by states belonging to an adiabatic process. Moreover, since the entropy remains constant along this straight line, the “arrow of time” can point in both directions. This is illustrated in Fig. 3.

From the last equation it is then easy to obtain the general relationships

 tanα=ξi+ηi/cVηi+ξicV ,tanα′=ηi+ξicVξi+ηi/cV . (24)

These formulas are valid for any values of the initial state, except the one situated on the origin of coordinates. In fact, for the initial states and , we recover the values of the intersection angles described above.

If the initial state is on the origin, the entropy condition (22) is always satisfied since any arbitrary straight line that starts at the origin is characterized by and so that the allowed geodesics could occupy the entire positive quadrant as illustrated in Fig. 2. However, this result changes drastically if we take into account the third law of thermodynamics which postulates the impossibility of reaching absolute zero of temperature or, equivalently, the minimum value of the entropy callen (). For the ideal gas with fundamental equation (21) the minimum value for the entropy is and corresponds to and . Consequently, the origin of coordinates must be “removed” from the space of equilibrium states.

We conclude that in the case of the ideal gas, a thermodynamic geodesic can be represented as a straight line that never crosses the origin of coordinates and possesses a definite direction which coincides with the direction of entropy increase.

We see that the laws of thermodynamics imply that the geometric structure of the equilibrium space is as illustrated in Fig. 4. For any given initial equilibrium state , there exist two different regions. The first one is determined by all the states than can be reached from the initial state by means of quasi-static processes, i. e., by thermodynamic geodesics. This could be called the region of connectivity of the initial state. If we identify as a time parameter, the connectivity region acquires the characteristics of a causally-connected region, resembling concepts of relativistic physics. The second region is composed of all the equilibrium states that cannot be reached from the initial state by thermodynamic geodesics. We call it the region of non-connectivity. Again, it could be also identified with the non-causally connected region of spacetime in relativistic physics. The boundary between the connectivity and non-connectivity regions is occupied by adiabatic thermodynamic geodesics and this is the only place in the equilibrium space where reversible thermodynamic processes can occur.

As mentioned in the previous section, it is possible to use the entropy as an affine parameter. Indeed, introducing the geodesic solutions and in Eq.(21), we obtain so that the affine parameter can be represented as , where and are constants. This implies that in terms of the logarithmic coordinates can be expressed as

 ξ=~ξ0+~ξ1S ,η=~η0+~η1S , (25)

where

 ~ξ0=ξ0η1−ξ1η0−S0(NkB)kξ1c−kVξ1+η1 ,~ξ1=(NkB)kξ1c−kVξ1+η1 , (26)
 ~η0=c−kV(ξ1η0−η1ξ0)−S0(NkB)kη1c−kVξ1+η1 ,~η1=(NkB)kη1c−kVξ1+η1 . (27)

It is easy to see that the new constants , , , and can always be taken as positive definite by choosing appropriately the original constants , , etc.

In this representation, an initial equilibrium state with coordinates is connected to a final state by a thermodynamic geodesic if the conditions

 Δξ=ξf−ξi=~ξ1ΔS≥0 ,Δη=ηf−ηi=~η1ΔS≥0 , (28)

are satisfied. Since the second law demands that , we conclude that all the thermodynamic geodesics must satisfy the conditions and , simultaneously. Consequently, all the thermodynamic geodesics that initiate at a particular equilibrium state must be contained within the region defined by and . On the other hand, all the thermodynamic geodesics that end at a particular equilibrium state, say must obey the same conditions. Figure 5 illustrates this behavior. We conclude that all the “incoming” and “outgoing” thermodynamic geodesics at a given point must be contained within a “cone” that we will call adiabatic cone.

Notice that the boundaries of the adiabatic cone are determined by adiabatic processes ( which in this representation must obey the condition , i. e., they are confined to one point in the equilibrium space. We designate as entropic future of an equilibrium state the region of the equilibrium space occupied by the thermodynamic geodesics that start at that state. The entropic past is defined in a similar way with all the thermodynamic geodesics that end at that particular equilibrium state.

## Vi Discussions and conclusions

In this work, we presented a detailed analysis of the geometry of the ideal gas equilibrium space. The main property of our analysis is its invariance with respect to Legendre transformations, i. e., it is independent of the choice of thermodynamic potential, a property which is essential in classical thermodynamics. The geometry of the ideal gas turns out to be flat, in accordance with our intuitive expectation that the absence of thermodynamic interaction would imply absence of curvature. This is an indication that thermodynamic curvature can be used as a measure of thermodynamic interaction.

Our analysis of the geodesics in the space of equilibrium space shows that they can be represented as straight lines when logarithmic thermodynamic variables are used. We introduce the concept of thermodynamic geodesics as those solutions of the geodesic equations which satisfy the laws of thermodynamics. Then, the equilibrium space can be represented as a Cartesian-like plane where thermodynamic geodesics correspond to quasi-static processes. The third law of thermodynamics implies that the origin of coordinates must be removed from the space of equilibrium states. This opens the possibility of interpreting the third law as a topological property of the equilibrium space.

For any given initial state, the equilibrium space can be split into two different regions. The connectivity region is occupied by all states which are connected to the initial state by means of thermodynamic geodesics. On the contrary, the region of non-connectivity corresponds to those equilibrium states than cannot be reached from the initial state by using only thermodynamic geodesics. In the boundary between these two regions there exist adiabatic thermodynamic geodesics. It can be shown that this is the only place where adiabatic geodesics can exist so that the boundary determines the only region in the equilibrium space where reversible thermodynamic processes can take place. Moreover, the concept of “arrow of time” becomes well-defined in terms of thermodynamic geodesics.

If we use the entropy as the affine parameter along the thermodynamic geodesics, we have shown that the connectivity region of a given equilibrium state can be represented as an adiabatic cone inside which the thermodynamic geodesics have a definite direction, which coincides with the direction of entropy increase, and can be interpreted as the “arrow of time”. Since our results are based upon a rigorous analysis of the geodesic equations and their solutions, we conclude that we have provided a geometric rigorous proof of the intuitive and well-known idea that the “arrow of time” coincides with the direction in which entropy increases. Of course, we have shown this explicitly in this work only for the case of an ideal gas. Whether this conclusion holds also in the case of more realistic thermodynamic systems remains a task for future investigations. However, from a geometric point of view this seems to be true in general. In fact, since any differential manifold can be interpreted as locally flat, all the results obtained here for the ideal gas should be valid locally for any thermodynamic system.

In this work, we used the entropy representation. For the sake of completeness, we performed the same analysis using the energy representation. The main difference in the analysis follows from the fact that the fundamental equation (16) in the energy representation, , leads to a non-diagonal term in the metric of the space of equilibrium states. This complicates the analysis of the geodesic equations. Nevertheless, the corresponding metrics with all possible thermodynamic potentials were shown to describe a flat space so that in all the cases it is possible to introduce Cartesian-like coordinates and the investigation of the geodesic equations leads to results equivalent to those obtained in the entropy representation. This result corroborates one of the most important properties of the formalism of GTD, i. e., the choice of thermodynamic potential or representation does not affect the results. This is also explicitly shown in the Appendix.

We found that the space of equilibrium states of the ideal gas possesses a very rich geometric structure which resembles the structure of spacetime in relativistic physics. Since this structure is the result of applying the laws of thermodynamics in a geometric context, we expect similar structures in the case of more general thermodynamic systems. For instance, we analyzed the van der Waals gas which is generally accepted as describing the thermodynamics of realistic gases. In this case the metric for the equilibrium space can be obtained immediately from Eq.(14) by introducing the corresponding fundamental equation . The resulting metric, however, is no longer flat. This is due to the fact that the van der Waals gas possesses a certain degree of thermodynamic interaction which, according to the formalism of GTD, generates a non-vanishing thermodynamic curvature. It turns out that the corresponding curvature scalar possesses singular points which coincide with the points of phase transitions of the van der Waals gas. Moreover, as a result of the non-vanishing curvature, the geodesic equations are much more complicated and there are no Cartesian-like coordinates in which the geodesics could be represented as straight lines. The geodesic equations must be analyzed by using numerical methods qr11 ().

In this work, we limit ourselves to systems with only two thermodynamic degrees of freedom. We can increase the number of degrees of freedom, maintaining the flatness of the equilibrium space, by considering, for instance, a multicomponent ideal gas, a paramagnetic ideal gas, etc. In fact, it can be shown explicitly that the formalism of GTD, as described in II, can be generalized to include systems with a larger number of thermodynamic variables. As a result, one obtains a metric, similar to (9), for higher dimensional manifolds from which one can show that the equilibrium spaces for the above mentioned generalizations of the ideal gas are flat. In all these cases, the flatness of the equilibrium space implies that there exist Cartesian-like coordinates in which the thermodynamic geodesics can be represented again as straight lines in higher dimensional spaces. The splitting of the equilibrium space in a region of connectivity separated by adiabatic geodesics from the non-connectivity region is similar; however, those regions are now represented by spaces with dimensions higher than 2. We conclude that the analysis of the corresponding thermodynamic geodesics will require to consider the topological and geometric properties of equilibrium spaces in higher dimensions.

The results presented in this work suggest that the flat equilibrium space of the ideal gas is to GTD what the Minkowski spacetime is to general relativity. We found an analogy between the thermodynamic structure of the equilibrium space as determined by thermodynamic geodesics and the causality structure of the Minkowski spacetime as determined by test particles. This is a new aspect of the inter-relationship between thermodynamics and relativistic physics.

## Acknowledgements

We would like to thank the GTD-UNAM group for helpful comments and discussions. One of us (HQ) thanks F. Schaller for stimulating discussions. This work was supported DGAPA-UNAM, Grant No. 113514, and Conacyt-Mexico, Grant No. 166361.

## Appendix A The entropy representation

As mentioned in Section III, in the entropy representation the coordinates of the phase space are , with and so that the auxiliary metric in can be written as

 GS=(dS−βdU−ϑdV)2+[(Uβ)2k+1dUdβ+(Vϑ)2k+1dVdϑ] . (29)

As before, the structure of this auxiliary metric is such that any extensive variable is multiplied by its corresponding intensive variable. As a result the above metric is invariant with respect to the following Legendre transformations, :

 ~S1=S−Uβ ,U=−~β , β=~U ,V=~V ,ϑ=~ϑ , (30)
 ~S2=S−Vϑ ,U=~U ,β=~β ,V=−~ϑ ,ϑ=~V , (31)
 ~S3=S−Uβ−Vϑ ,U=−~β , β=~U ,V=−~ϑ ,ϑ=~V . (32)

The thermodynamic potentials , , and are known as Massieu functions callen (). For a given fundamental equation they represent the same subspaces of in different coordinates. In the case of the ideal gas, the Massieu functions can be derived explicitly by using the state equations (17) in the form and and the fundamental equation (16). We get (dropping the tildes)

 S1(β,V)=S01−cVNkBlnβ+NkBlnV , (33)
 S2(U,ϑ)=S02+cVNkBlnU−NkBlnϑ , (34)
 S3(β,ϑ)=S03−cVNkBlnβ−NkBlnϑ , (35)

where , , and are constants. Then, the metric of the subspace of equilibrium states in each case corresponds to

 g1=dβ2β2+dV2V2 , (36)
 g2=dU2U2+dϑ2ϑ2 , (37)
 g3=dβ2β2+dϑ2ϑ2 , (38)

where for simplicity we omit the constants and choose a positive definite signature. Clearly, all these metrics represent the same flat space of equilibrium states for the ideal gas. The geodesic equations can be solved and we obtain , where and and are constants. An analogous representation for the geodesics can be obtained from the analysis performed in Sec. IV. This is a concrete example of the invariance of the results obtained by using the formalism of GTD.

## References

• (1) R. Wald, Living Rev. Rel. 4, 6 (2001).
• (2) W. G. Unruh, Phys. Rev. D 14, 870 (1976).
• (3) J. Oppenheim, A thermodynamic sector of quantum gravity, arXiv: gr-qc/0112001
• (4) T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).
• (5) T. Padmanabhan, arXiv: 1312.3253
• (6) J. W. Gibbs, The collected works, Vol. 1, Thermodynamics (Yale University Press, 1948).
• (7) C. Caratheodory, Math. Ann. 67, 355 (1909).
• (8) R. Hermann, Geometry, physics and systems (Marcel Dekker, New York, 1973).
• (9) H. B. Callen, Thermodynamics and an introduction to thermostatics (John Wiley & Sons, Inc., New York, 1985).
• (10) H. Quevedo, J. Math. Phys. 48, 013506 (2007).
• (11) H. Quevedo and M. N. Quevedo, Electr. Jour. Theor. Phys, 2011, 1 (2011); arXiv:1111.5056 [math-ph]
• (12) A. Bravetti, C. S. López-Monsalvo, F. Nettel and H. Quevedo, Jour. Geom. Phys. 81, 1 (2014).
• (13) V. I. Arnold, Mathematical methods of classical mechanics (Springer Verlag, New York, 1980).
• (14) D. García and C. S. López-Monsalvo, Infinitesimal Legendre symmetry in the geometrothermodynamics programme, J. Math. Phys. (2014) in press.
• (15) A. Vázquez, H. Quevedo, and A. Sánchez, J.Geom.Phys. 60, 1942 (2010).
• (16) H. Quevedo and A. Ramírez, A Geometric Approach to the Thermodynamics of the van der Waals System, in Proceedings of Mario Novello’s 70th Anniversary Symposium, edited by N. Pinto-Neto and S. E. Perez-Bergliaffa (2012) pp. 301-313; arXiv:1205.3544 [math-ph]
• (17) K. Huang, Statistical Mechanics (John Wiley & Sons, Inc., New York, 1987).
• (18) L. D. Landau and E. M. Lifshitz, Statistical physics (Pergamon Press, London, UK, 1980).
• (19) W. L. Burke, Applied differential geometry (Cambridge University Press, Cambridge, UK, 1987).
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