 # An eikonal equation approach to thermodynamics and the gradient flows in information geometry

We can consider the equations of states in thermodynamics as the generalized eikonal equations, and incorporate a "time" evolution into thermodynamics as Hamilton-Jacobi dynamics. The gradient flows in information geometry is discussed as this dynamics of a simple thermodynamic system.

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## 1 Introduction

In Ref. Giovanni , Prof. Pistone discussed Lagrangian function on the finite state space statistical bundle, in which statistical bundle is defined by the set of couples

of a probability density

and a random variable

such that , and on a finite state space, a probability density

gives an affine atlas of charts, consequently the resulting manifold is a non-parametric model for Information Geometry (IG). He has derived the Euler Lagrange equations from the Lagrange action integral by computing the ”velocity” and ”acceleration” of a one-dimensional statistical model. The quite remarkable points of his results are as follows.

• introducing the concepts of “velocity” as Fisher’s score, and “acceleration” as a second order geometry, so that a flat structure in IG is characterized as an orbit along the curve with zero-acceleration.

• introducing the dynamics of a density in IG as a gradient flow,

 ddtlnq(t)q2=−lnq(t)q2+D(q(t)∥q2), (1)

 D(q(t)∥q2):=Eq(t)[lnq(t)q2]. (2)

The solution of Eq. (1) was given by

 q(t)=exp(e−tlnq0+(1−e−t)lnq2−Ψ(t)), (3)

where is the normalization functional. This solution evolves in the time parameter smoothly from to . For more technical details see for example MP15 ; Giovanni15 .

As is well-known, Newton’s mechanics describes the dynamics of a point particle in Euclidean geometry, and zero-acceleration of a particle means that there is no external force acting on the particle. The point i) reminds us Einstein’s general relativity Dirac , in which a particle free from all external non-gravitational force always moves along a geodesic in a curved space-time. Indeed, for a curve on a smooth manifold with an affine connection , the geodesic equation is

 ∇dγdtdγ(t)dt=0, (4)

which states that there is no tangent component of the acceleration vector of the geodesic curve

. In addition, there is an analogy between classical mechanics and thermodynamics. For example, Rajeev Rajeev showed that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory. On the point ii), we wonder the functional form of the time -dependency, i.e., the double exponential -dependency of in (3). A natural question is that where does this double exponential -dependency come from? In other words, what is the time parameter in this setting in IG? This is the main motivation of this contribution.

Also for the parametric case in IG, the gradient flows have been studied. The pioneering works were Fujiwara F93 , Fujiwara and Amari FA95 , and Nakamura N94 , in which they showed that the gradient systems on manifolds of even dimensions are completely integrable Hamiltonian systems. Boumuki and Noda BN16 gave a theoretical explanation for this relationship between Hamiltonian flow and gradient flow from the veiw point of symplectic geometries. In this paper we consider a gradient flow in a parametric case, which is given by

 dθidt =−gij(→η)∂∂θjΨ(→θ), (5)

or

 dηidt =−gij(→θ)∂∂ηjΨ⋆(→η), (6)

for the potential functions and for a dually-flat statistical manifold .

In this contribution we take a different approach to the gradient flows in IG. Since the relation between the Legendre transformations in a simple thermodynamical system and those in IG are already known (e.g, section 2 in WMS15 ), our discussion is base on this correspondence between thermodynamic macroscopic variables and affine coordinates in IG. Basically in both (equilibrium) thermodynamics and IG, there is no apparent ”time” parameter. Nevertheless we can introduce a ”time” parameter 111In order to avoid possible confusion, we use as the ”time” parameter to distinguish it from the different time parameter which is used in gradient equations. as Hamilton-Jacobi (HJ) dynamics. By generalizing the eikonal equation in optics, which is equivalent to the square of HJ equation, we can describe a ”time” () evolution of thermodynamic systems.

The rest of the paper consists as follows. The next section briefly reviews the basic of geometrical optics. The eikonal equation describes a real optical path between two different point in a optical media. The generalized eikonal equation is introduced and relations with HJ equation are explained. In section 3, a ”time” parameter is introduced in the standard settings of equilibrium thermodynamics. Einstein’s vielbein formalism (or Cartan’s the method of moving frame) plays a role to give a Riemann metric in an equilibrium thermodynamic system. By studying the gradient flow in the ideal gas model, we will relate the time parameter with the temperature of the thermodynamic system. Section 4 discusses two main issues. In the first subsection a simple canonical probability of a thermal system, and relations with the gradient flow equation (1) and the origin of the double exponential time () dependency of the solution (3). In the next subsection we discuss the relation between the gradient flow and Hamilton flow. Final section is devoted to our conclusion. In Appendix A, HJ equation is derived by applying appropriate canonical transformation. Appendix B explains a constant pressure process in the ideal gas model.

## 2 Geometrical optics and generalized eikonal equation

In geometrical optics, the real path of the ray in a media with reflective index

is characterized by Hamilton’s characteristic function (or point eikonal function)

, which is the length of a real optical path between a point and another point .

 I(→q;→q0) =∫ss0n(→r(s))∣∣∣d→r(s)ds∣∣∣ds=:∫ss0Lop(→r,d→r(s)ds)ds, (7)

where is the so called optical Lagrangian. From Fermat’s principle, the real optical path satisfies . Then the variation of w.r.t the infinitesimal variations of the two end points and is given by

 δI=⎡⎢ ⎢⎣∂Lop∂(d→qds)⋅δ→q⎤⎥ ⎥⎦ss0=[→p⋅δ→q]ss0=→p(→q)⋅δ→q−→p(→q0)⋅δ→q0. (8)

Consequently if we find the point eikonal function , the ray vector at a position and that at another point are obtained by

 →p(→q)=∂I∂→q,→p(→q0)=−∂I∂→q0, (9)

respectively. Since , the point eikonal function satisfies the eikonal equations

 (∂I∂x)2+(∂I∂y)2+(∂I∂z)2 =n2(→q), (10a) (∂I∂x0)2+(∂I∂y0)2+(∂I∂z0)2 =n2(→q0). (10b)

In order to determine the real optimal path between and of the ray the both relations (10) are needed. Recall that the eikonal equation is obtained from the wave equation in the limit of short wavelength , i.e., . Historically Hamilton invented his formalism of classical mechanics from his theory of systems of rays Hamilton .

Now let us consider the following generalized eikonal equation

 gij(→q)(∂W∂qi)(∂W∂qj)=E2, (11)

on a -dimensional smooth manifold. Here is Hamilton’s characteristic function,

is the inverse tensor of a metric

of the manifold, and is a real constant, which is determined later. It is known that is a generating function which relates the original variables and new variables by the relations (17). We use Einstein summation convention through out this paper. It is known that the generalized eikonal equation (11) is related with the Hamilton-Jacobi (HJ) equation

 H(→q,∂W∂→q)−E=0, (12)

for the time-independent Hamiltonian

 H(→q,→p)=√gjk(→q)pjpk. (13)

It is also known that for any time-independent Hamiltonian, by separation of variables, the corresponding action is expressed as

 S(→q,→P,τ)=W(→q,→P)−E(→P)τ, (14)

where is a ”time” parameter, is a total energy of the Hamiltonian as a function of . In addition, the action is a generating function of the (type-2) canonical transformation from the original variables to new variables which are conserved, i.e.,

 dQidτ=0,dPidτ=0,i=1,2,⋯,m. (15)

The transformed new Hamiltonian is given by

 K(→Q,→P)=H(→q,∂W∂→q)−E=0, (16)

which is equivalent to HJ equation (12) in this case, and the relations

 pi=∂W∂qi,Qi=∂W∂Pi,i=1,2,⋯,m, (17)

are satisfied (Appendix A).

It is worth noting that since

 pidqidτ=pi∂H∂pi=gij(→q)pipj√gjk(→q)pjpk=H, (18)

the corresponding Lagrangian , which is the Legendre dual of the Hamiltonian (13), becomes zero,

 L(→q,d→qdτ):=pidqidτ−H(→q,→p)=0. (19)

 dS(→q,→P,τ)=L(→q,d→qdτ)dτ=0, (20)

and consequently, from Eq. (14) we see that

 dW(→q,→P)=E(→P)dτ. (21)

In addition, according to Carathéodry Snow67 , we introduce the generating function

 G(→q,τ;→q0,τ0):=S(→q,→P,τ)−S(→q0,→P,τ0), (22)

which generates the canonical transformation between a set of canonical coordinates and another set of canonical coordinates via the conserved quantities , since is the generating function of the canonical transformation between and and is that of the canonical transformation between and . Later we shall use this generating function in order to obtain the ”time” dependency or dynamics described by the -parameter.

## 3 Thermodynamical systems

In general, equilibrium thermodynamic systems with -independent macroscopic variables are completely described by the -independent equations of states, which can be cast into the following form,

 eiμ(q)pμ=ri,i,μ=1,2,…m, (23)

where are some constants. We assume that the matrix to be invertible so that we have

 pμ=(e−1)μiri. (24)

The following relations are satisfied.

 (e−1)μiejμ(q)=δij,(e−1)νieiμ(q)=δμν, (25)

where and denote Kronecker’s delta. In Eq. (23), if we take to define an orthogonal basis with an invertible constant matrix , then can be thought of as a vielbein. The vielbein or Cartan formalism is used in the field of general relativity and it is known as Einstein-Cartan theory. In the fields of IG, to the best of our knowledge, the first trial of applying vielbein formalism is Ref. W19 . In general the frame of on a manifold are non-orthogonal and characterized by a metric . The vielbein field relates this non-orthogonal frame of with the local orthogonal frame of . In order to distinguish the two different frames, Greek and Latin indices are often used, and we follow this convention here.

The inner product in the orthogonal frame of with a diagonal metric and that in the frame of with a metric are related with

 ηijrirj=gμνpμpν, (26)

and using Eq. (23) it follows the next relation

 gμν=ηijeiμejν, (27)

where is the inverse matrix of the metric on the -dimensional smooth manifold , i.e., they satisfy

 gμρgρν=δμν. (28)

Next, from the Hamiltonian (13) and Hamilton’s equation of motion, we have

 dqμdτ =∂H∂pμ=gμνpμE, (29a) dpμdτ =−∂H∂qμ=−12E∂gνρ∂qμpνpρ. (29b)

Substituting Eq. (29a) into the new Hamiltonian (16), we obtain the relation

 √gμνdqμdτdqνdτ=1, (30)

which implies that

 dτ2=gμνdqμdqν. (31)

Consequently, the ”time” parameter in this setting gives a natural distance between equilibrium states of the thermodynamic systems on the manifold equipped with the metric .

For the sake of simplicity, we only consider thermodynamical gas model with dimensions Vaz . A generalization for case is straightforward. A thermal equilibrium system characterized by the specific entropy as a state function of the internal energy and volume per a molecule of the gas. We use the so-called entropy representation , and the first law of thermodynamics is expressed as

 ds=1Tdu+PTdv, (32)

where is the temperature and the pressure of the gas, respectively. It is well known that they are related with

 1T=(∂s∂u)v,PT=(∂s∂v)u. (33)

Mathematically the above physical explanation is equivalent with that the Pfaff equation

 ds(u,v)−1Tdu−PTdv=0, (34)

is an exact differential, which was originally shown by Carathéodory.

Next, introducing the Planck potential given by

 Ξ(1T,PT):=s(u,v)−1Tu−PTv, (35)

which is the total Legendre transform of the entropy . These thermodynamical potentials and their variables can be corresponded with the potential functions and - and -coordinates in IG as follows.

 the η-coordinates: η1=u,η2=v, (36a) the θ-coordinates: θ1=−1T,θ2=−PT, (36b) the θ-potential: Ψ(→θ)=Ξ(1T,PT), (36c) the η-potential: Ψ⋆(→η)=−s(u,v), (36d)

The two potentials are of course related by the total Legendre transformation

 Ψ(θ1,θ2)=θkηk−Ψ⋆(η1,η2), (37)

and

 ηi=∂∂θiΨ(→θ),θi=∂∂ηiΨ⋆(→η),i=1,2. (38)

An example of Maxwell’s relations in thermodynamics is described by

 ∂∂v(1T)=∂∂u(PT), (39)

which is equivalent to the integrability condition

 ∂2∂v∂u(−s(u,v))=∂2∂u∂v(−s(u,v)), (40)

i.e., the symmetric Hessian matrix of the negentropy . It is known Amari that a metric obtained by the Hessian of a convex potential leads to the dually flat structure in IG. Such a Hessian metric in thermodynamics is proposed by Ruppeiner Ruppeiner . The Ruppeiner metric is written by

 (gR)ij=∂2∂qi∂qj(−s),i,j=u,v, (41)

where and .

### 3.1 Ideal gas model

Ideal gas model is the simplest thermodynamic model of an dilute gas, which consists of a huge number of molecule. It is described by

 u=f2kBT,Pv=kBT, (42)

where stands for the specific internal energy, the specific volume, Boltzmann constant, the temperature, the pressure of the ideal gas. The parameter

stands for the degree of freedom of a molecule of the gas, e.g., for a monatomic molecule

. The first equation of (42) states the equipartition theorem and the second states the equation of state of the ideal gas model. It is known that the corresponding entropy can be expressed as

 s(u,v)=Pulnuu0+Pvlnvv0, (43)

where we introduce the reference state which satisfy , and

 Pu:=f2kB,Pv:=kB. (44)

Note that and are constants (conserved quantities) and canonical transformed momenta in HJ equation.

The equations (42) can be cast into the form (23), i.e.,

 (u00v)(1TPT)=(f2kBkB), (45)

with

 enμ=(u00v),pμ=(1TPT),rn=(PuPv), (46)

Let

 ηmn=⎛⎜⎝1α2001β2⎞⎟⎠,ηmn=(α200β2), (47)

where and are scale factors scaleF . Then, from Eq. (27), the corresponding metric becomes

 gμν=⎛⎜⎝u2α200v2β2⎞⎟⎠. (48)

Next, from the relations (24) and (27), we have

 H =√gμνpμpν=√ηijeiμpμejνpν=√ηijrirj=E, (49)

then we find the explicit expression of constant as

 E(Pu,Pv)=√(Pu)2α2+(Pv)2β2. (50)

The corresponding generalized eikonal equation (11) becomes

 (51)

from which a complete solution can be obtained as

 W(u,v,Pu,Pv)=Pulnu+Pvlnv, (52)

where the constant of integration is set to zero. Consequently the action of the ideal gas model becomes

 S(u,v;Pu,Pv,τ)=Pulnu+Pvlnv−E(Pu,Pv)τ, (53)

and the corresponding generating function (22) is

 G(u,v,τ;u0,v0,τ0)=Pulnuu0+Pvlnvv0−E(Pu,Pv)(τ−τ0), (54)

which satisfies the relations

 ∂G∂Pi=0,i=u,v, (55)

since . Using the expression (53) of the action of the ideal gas model, it follows that

 ∂G∂Pu =lnuu0−Pu(τ−τ0)α2E=0, (56a) ∂G∂Pv =lnvv0−Pv(τ−τ0)β2E=0, (56b)

By choosing and , which corresponds to a constant pressure process as explained in Appendix B, we have

 u(τ) =u0exp[(τ−τ0)E],v(τ)=v0exp[(τ−τ0)E]. (57)

and

 dudτ =uE,dvdτ=vE, (58)

Next let us consider the gradient flow of and . From the correspondence relations (36d), the direct mapping of the gradient flow equations (6) in IG to those in the ideal gas model become

 dudt =guu∂s(u,v)∂u,dvdt=gvv∂s(u,v)∂v. (59)

Substituting the metric (48) and entropy (43) into these equations, we find

 dudt=Puα2u=u,dvdt=Pvβ2v=v. (60)

for a constant pressure process. Comparing the results (58) and (60) we find

 dτ=Edt. (61)

Next, taking the derivative of the specific entropy (43), we have

 ds(u,v)=1Tdu+PTdv, (62)

which corresponds to the relation

 dW(→q,→P)=∂W∂qidqi+∂W∂PidPi=pidqi, (63)

in analytical mechanics, because in (17) and in (15). The relation (21) in analytical mechanics corresponds to

 ds(u,v)=Edτ, (64)

in thermodynamics.

From the equations (42), we have

 1Tdu=−ud(1T),PTdv=−vd(PT). (65)

Substituting these relations into Eq. (62), it follows that

 ds(u,v)=−ud(1T)−vd(PT). (66)

For a constant pressure process, this becomes

 ds(u,v)=−(u+vP)d(1T)=(u+vP)dTT2. (67)

Since and for a constant pressure process, we have

 ds(u,v)=(Pu+Pv)dTT=E2dlnT. (68)

Comparing this relation, (61) and (64), we finally obtain that

 dτ=EdlnT,dt=dlnT. (69)

In this way, we find out the relation between the time parameter in the gradient flows in IG and the temperature in a simple gas model of thermodynamics.

## 4 Discussion

Having found out the time parameter of the gradient flows in IG, let us turn our focus on the gradient flow equation (1).

### 4.1 On the double exponential t-dependency

First we’d like to answer the question concerning on the double exponential -dependency of the solution in (3), in other words, the physical meaning of the time parameter in this solution from the view point of our results obtained until the previous sections.

Let us begin with the well known canonical probability of a thermally equilibrium system,

 pi(β)=1Z(β)exp(−βEi),i=1,2,…,N, (70)

where is coldness or the inverse temperature, is the partition function, is the energy level of -th state. We assume that each discrete energy level is ordered as . Consequently in the high temperature limit (), every probability becomes equal, i.e.,

 ∀i.pi(0)=1N. (71)

Taking the logarithm of the both sides of (70) we have

 lnpi(β)=−βEi−lnZ(β). (72)

Taking the expectation of this with respect to leads to

 n∑i=1pi(β)lnpi(β)=−βN∑i=1pi(β)Ei−lnZ(β). (73)

Differentiating the both sides of (72) with respect to , if follows that

 ddβlnpi(β)=−Ei−ddβlnZ(β)=−Ei+N∑i=1pi(β)Ei, (74)

where we used the well known relation

 N∑i=1pi(β)Ei=−ddβlnZ(β). (75)

Multiplying the both sides of (74) by , we have

 βddβlnpi(β)=−βEi+βN∑i=1pi(β)Ei, (76)

By subtracting (73) from (72), the right hand side of (76) is rewritten as

 βddβlnpi(β) =lnpi(β)−N∑i=1pi(β)lnpi(β) (77)

Since is constant, we can rewrite this relation as

 ddlnβln(pi(β)p0) =ln(pi(β)p0)−N∑i=1pi(β)ln(pi(β)p0). (78)

From the second relation in (69) we see that

 dlnβ=−dln(kBT)=−dt, (79)

from which we can relate the time parameter and the coldness as

 β=exp(−t)+C, (80)

with a constant of integration. Setting for simplicity, and substituting into (78) we finally obtain that

 ddtln(pi(e−t)p0) (81)

This is equivalent to the gradient flow equation in (1) when we make the following associations.

 β⇔e−t,q(t)⇔pi(β)=pi(e−t),q2=q(t→∞)⇔p0, (82a) D(q(t)∥q2):=Eq(t)[lnq(t)q2]⇔N∑i=1pi(e−t)ln(pi(e−t)p0) (82b)

Therefore the double exponential -dependency in the solution in (3) can be explained by these associations. The time parameter characterizes the temperature evolution through the relation .

### 4.2 The gradient flow and Hamilton flow

Next we focus our attention on the relation between the gradient flow (6) and Hamilton flow. In this subsection, for the sake of clarification, we don’t use Einstein’s convention, i.e., the summation over the repeated indices.

Let us consider the case in which the characteristic function is given by

 W(q,P)=m∑μ=1Pμlnqμ, (83)

and the diagonal matrix is

 gμν=(qμαμ)2δμν,μ,ν=1,2,…,m, (84)

which is the inverse matrix of the metric on a smooth manifold and is Kronecker’s delta. We remind that each is a constant of motion, each is a scale factor.

The corresponding generalized momentum is given by

 pμ=∂W∂qμ=Pμqμ,μ=1,2,…,m. (85)

Consequently each constant of motion satisfies that

 Pμ=qμpμ,μ=1,2,…,m. (86)

The corresponding Hamiltonian of Eq. (13) becomes

 H(q,p)=√∑μ,νgμνpμpν= ⎷∑μ(qμαμpμ)2, (87)

which is a constant because this Hamiltonian has no explicit time dependence. Indeed, by using the relations (86), it follows that

 E(P)= ⎷∑μ(Pμαμ)2. (88)

Then Hamilton’s equations of motion leads to

 ddτqμ =∂H∂pμ=Pμqμα2μE, (89a) ddτpμ =−∂H∂qμ=−Pμpμα2μE. (89b)

Now if we choose each scale factor as

 α2μ=Pμ,μ=1,2,…,m, (90)

and using the relation (61), then the equations of motion (89) become

 ddtqμ =qμ, (91a) ddtpμ =−pμ. (91b)

We remind the correspondence between IG and analytical mechanics:

 ημ ⇔qμ, θμ ⇔−pμ, (92a) Ψ⋆(η) ⇔−W(q,P), θμ=∂Ψ⋆(η)∂ημ ⇔−pμ=−∂W(q,P)∂qμ. (92b)

Then we see that the Hamilton flow described by (91) is equivalent to the gradient flow in IG described by the gradient equations (6).

Finally let us comment on the Lemma 3.3 in Ref. BN16 by Boumuki and Noda, where they introduced the potential functions for a dually flat space in IG as

 Ψ⋆(→η)=−∑ilnηi,ηi=−1/θi,i=1,2,…,m. (93)

This mathematical model is a simple but non-trivial, since its gradient flow equation is reformulated as Hamilton equation. For the details see Ref. BN16 . From the above correspondence between IG and analytical mechanics, we note that the first relation in (93) is a special case, in which all are set to , of (83), and the second relation comes from the relation (86). As a result, their model (93) is a special case of the model discussed in this subsection.

## 5 Conclusions

We have studied the gradient flows in IG as the dynamics of a simple thermodynamic system, in which the equations of states in thermodynamics are considered as the generalized eikonal equations, and incorporate a ”time” () evolution into thermodynamics as HJ dynamics. Through this ”time” parameter in the HJ dynamics of the thermodynamics, we have related the time parameter t in the gradient flow equation with the temperature of the thermodynamic system as shown in (69). Based on this fact, we have found the physical origin of the double exponential time () dependency of the solution (3) in subsection 4.1.

In the history in the fields of IG, since Refs. F93 ; FA95 ; N94 found the relations between Hamilton flows and gradient flows, some research works have been done. However, because the gradient flow equations were introduced by hand, the time parameter in the gradient flow equations was introduced as a merely mathematical parameter. To the best of our knowledge, no physical meaning is given to this time parameter . We can consider the equations of states in thermodynamics as the generalized eikonal equations, which enable us introducing a ”time” evolution to thermodynamics as HJ dynamics as shown in section 3. In this way we have described the gradient flows (60) of the HJ dynamics of the simple ideal gas model.

###### Acknowledgements.
The authors thank to Prof. G. Pistone for his interesting talk in the SigmaPhy2017 Conference held in Corfu, Greece, 10–14 July 2017, and for organizing a seminar held at Collegio Carlo Alberto, Moncalieri, on 5 September 2017. The first author (T.W) also thanks to Dr. S. Goto for useful discussion on the early version of this work.

## Appendix A Hamilton-Jacobi equation by canonical transformation

Here we briefly review the Carathéodory derivation Snow67 of HJ equation by canonical transformation. Recall that the non-uniqueness of Lagrangian, i.e., two Lagrangians which differ by a total derivative of some function with respect to time, describe the same system. For example consider the following two Lagrangians and related with

 L(→q,˙→q,t)=L⋆(→Q,˙→Q,t)+df(→q,t)dt. (94)

Both Lagrangians lead to the same Euler-Lagrange equation, and consequently describe the same system.

Now, introducing the Hamiltonians and which are Legendre duals of