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Affine geometric description of thermodynamics

Thermodynamics provides a unified perspective of thermodynamic properties of various substances. To formulate thermodynamics in terms of sophisticated mathematics, thermodynamics is described by a variety of differential geometries, including contact and symplectic geometries. Meanwhile affine geometry is a branch of differential geometry and is compatible with information geometry, where information geometry is known to be compatible with thermodynamics. By combining above, it is expected that thermodynamics is compatible with affine geometry, and is expected that several affine geometric tools can be introduced in the analysis of thermodynamic systems. In this paper affine geometric descriptions of equilibrium and nonequilibrium thermodynamics are proposed. It is then shown that several thermodynamic quantities can be identified with geometric objects in affine geometry, and that several geometric objects can be introduced in thermodynamics. Examples of these include: specific heat is identified with the affine fundamental form, a flat connection is introduced in thermodynamic phase space. In addition, two classes of relaxation processes observed in thermodynamic systems are shown to be described in an extension of affine geometry. To see some links between existing geometric frameworks and the present one, this affine geometric description of thermodynamics is compared with a contact geometric one.


Introduction
Thermodynamics is a branch of physics, provides a unified perspective of thermodynamic properties of various substances, and has been applied to various branches of sciences and technologies [1]. Thus further developments in thermodynamics are beneficial in these branches. One way to develop thermodynamics further is to apply well-developed pure mathematics to thermodynamics [2], and differential geometry is one of such mathematics [3]. By introducing notions developed in differential geometry to thermodynamics, some new views and applications in thermodynamics were expected to be found. Such views and applications are described by contact geometry [4,5], symplectic geometry [6], and so on [7,8]. Here contact geometry is known as an odd-dimensional counterpart of symplectic geometry [9,10,11,12], and is used in describing not only thermodynamics, but also singularities in hyper-surfaces [13]. Note that there have been other geometric formulations of thermodynamics [14,15], and various developments are in progress.
Affine geometry is a branch of differential geometry, and studies invariant properties under affine transforms [16]. It is compatible with information geometry [17], where information geometry is a geometrization of mathematical statistics [18]. There are a variety of remarkable theorems linking affine geometry and information geometry. One of them is that divergence that plays a central role in information geometry has been extended in the framework of affine geometry [19]. Meanwhile information geometry is compatible with thermodynamics [20,21,22]. Note that affine geometry is deeply related to the so-called Hessian geometry [23].
By combining above, it is expected that thermodynamics is compatible with affine geometry, and is expected that several geometric tools can be introduced in the analysis of thermodynamic systems(see the This paper is intended to link between affine geometry and thermodynamics, and to provide the first step towards the materialization of affine geometric descriptions of thermodynamics. To this end, an affine geometric description of thermodynamics and that of thermodynamic processes are proposed and discussed in this paper. More specifically the following are shown: • The point of departure for this paper is to identify the set of equilibrium states in thermodynamics with the image of a graph immersion (see Interpretation 3.1 of this paper).
• It is then shown that several thermodynamic quantities can be identified with geometric objects employed in affine geometry, and that several geometric objects can be introduced in thermodynamics (see Proposition 3.3 of this paper). Such examples include: specific heat is identified with affine fundamental form, a flat connection is introduced in thermodynamic phase space.
• In addition, two classes of relaxation processes observed in thermodynamic systems are shown to be described in an extension of affine geometry (see Theorems 3.9 and 3.14 of this paper). A simple spin model with a unique set of equilibrium states shows how the description is justified.
• To see some links between existing geometric frameworks and the present one, this affine geometric description of thermodynamics is then compared with a contact geometric one. It is then shown that the present geometric formulation of relaxation processes are consistent with the existing theory (see Theorem 4.7 of this paper).
The rest of this paper is organized as follows. In Section 2, some of necessary background of geometries and thermodynamics are summarized. In Section 3 an affine geometric thermodynamics is proposed. In Section 4, the affine geometric description of thermodynamics is compared to the contact geometric thermodynamics. Finally Section 5 summarizes the present geometric formulation of equilibrium and nonequilibrium thermodynamics, and discuss future studies.

Preliminaries
This section is intended to provide a brief summary of the necessary background of geometries and thermodynamics, and is intended to fix notations here. Throughout this paper manifolds are connected, and every object on any manifold is smooth, unless otherwise stated. Given a manifold M, its tangent and cotangent bundles are denoted by T M and T * M, respectively. Various formulae and tools developed in differential geometry are known [2,3]. For example, the Lie derivative of a k-form α ∈ ΓΛ k M on a manifold M along a vector field X ∈ ΓT M can be written as L X α = dı X α + ı X dα, where d is the exterior derivative and ı X the interior product with X. This is known as the Cartan formula. This formula is valid even k = 0, where ΓΛ 0 M is identified with the space of functions on M. If φ is a map from a manifold to another one, then φ * denotes the push-forward of φ, and φ * denotes its pull-back. When a (affine) connection ∇ is equipped on M, the (1, 2)-type tensor field T ∇ such that T ∇ (X, Y ) := ∇ X Y − ∇ Y X − [X, Y ] for all X, Y ∈ ΓT M, is called torsion tensor field, where [X, Y ] = XY − Y X is the Lie bracket. If T ∇ (X, Y ) ≡ 0 for all X, Y ∈ ΓT M, then the ∇ is said to be torsion-free. The (1, 3)-type tensor field R ∇ such that Y ] Z for all vector fields X, Y, Z, is called the curvature tensor field. If R ∇ (X, Y )Z ≡ 0, then ∇ is called curvature-free. If a connection ∇ is torsion-free and curvature-free, then ∇ is said to be flat. If a connection ∇ is flat, then there exists a coordinate system such that all the connection coefficients vanish Γ k ij ≡ 0. These coordinate systems are said to be affine coordinate systems.

Affine geometry
In this subsection necessary background in affine geometry are summarized (see Ref. [16] for more details. In Ref. [16], (co)tangent space at a point is identified with a point due to the property of affine space. Meanwhile in this paper, this identification is not adopted.). Let M be an n-dimensional manifold (n = 1, 2, . . .), f an immersion of M into R n+1 , and ξ a vector field along f . If for arbitrary point p ∈ M the condition is satisfied, then the pair (f, ξ) is referred to as an affine (hyper-surface) immersion, and ξ a transversal vector field. There are various formulae associated with affine immersions. Let (f, ξ) be an affine hyper-surface immersion, and D the standard flat affine connection on R n+1 . Then there exists a torsion-free connection where h is a (0, 2)-type symmetric tensor field called an affine fundamental form associated with ξ. Equation (1) is known as the Gauss formula [16]. Throughout this paper, immersions are always affine hyper-surface immersions unless otherwise stated. For all X ∈ ΓT M, one can write the Weingarten formula: where S is a (1, 1)-type tensor field being referred to as an affine shape operator, and τ a one-form being referred to as a transversal connection form. When τ = 0, the affine immersion (f, ξ) is referred to as being equiaffine. If h is non-degenerate everywhere, then f is said to be non-degenerate. The dual of vector spaces provides various tools in differential geometry in general. In affine geometry, the dual of a vector space also provides useful geometric tools. Given an affine immersion (f, ξ), introduce a map v : If a map v satisfies the conditions v(p) , ξ p = 1, and v(p) , f * X p = 0, then v is referred to as the conormal map.
The graph immersion is a class of affine immersions. This is explained below. Let Ω be an n-dimensional region of R n , and F a function defined on Ω, F : Ω → R. The coordinate system of Ω ⊂ R n is denoted by x = (x 1 , . . . , x n ), and the coordinate of the other R is denoted by z. A graph immersion associated with F is a pair (f, ξ) satisfying the conditions written in coordinates as It follows from the flatness of D that D X ξ = 0 for all X ∈ T p M. This and (2) yield S = 0 and τ = 0. Since τ = 0, any graph immersion is equiaffine. The affine fundamental form h is calculated as follows. First, one calculates From (4), one calculates From (1) and (5), one has and It follows from (6) that if the matrix (∂ 2 F/∂x i ∂x j ) is strictly positive at x, then h is non-degenerate. In addition, it follows from (7) that the coordinate system x is affine with respect to ∇. If h is non-degenerate everywhere on M, then the graph immersion is said to be non-degenerate. The conormal map at p ∈ Ω is expressed as

Information geometry
Information geometry is a geometrization of mathematical statistics [18], and a study on statistical manifolds [17]. The definition of statistical manifold is given from a viewpoint of differential geometry as follows. Let (M, g) be a (pseudo-) Riemannian manifold, and ∇ a torsion-free affine connection. If ∇ satisfies the Codatti equation [16], then the triplet (M, ∇, g) is referred to as a statistical manifold [19]. One pair of connections explained below plays various roles in information geometry. Let g be a pseudo-Riemannian metric tensor field on M, and ∇ an affine connection. If a connection ∇ * satisfies the condition then ∇ * is referred to as the dual connection of ∇ with respect to g. It can be shown that if ∇ is flat then ∇ * is flat. In information geometry, the tetrad (M, g, ∇, ∇ * ) is referred to as a dually flat space, and this class of manifolds has been well-studied [17]. On a dually flat space, let θ = (θ 1 , . . . , θ n ) be an affine coordinate system. Then it can be shown that there exists an affine coordinate system η = (η 1 , . . . , η n ) for ∇ * that satisfy where δ b a is the Kronecker delta giving unity when a = b and zero when a = b. The coordinate system η is said to be the dual coordinate system of θ with respect to g. One basic proposition on dually flat spaces is as follows. Let (M, g, ∇, ∇ * ) be a dually flat space, θ an affine coordinate system, η a dual affine coordinate system, where M is simply connected and has a global coordinate system. Then there exist functions ψ and ϕ, and it follows that ∂ψ ∂θ a = η a , ∂ϕ ∂η a = θ a , ψ(p) + ϕ(p) − n a=1 θ a (p)η a (p) = 0, a, b = 1, . . . , n, ∀ p ∈ M (9) , a, b = 1, . . . , n where g ab = g(∂/∂θ a , ∂/∂θ b ) and (g ab ) is the inverse matrix of (g ab ). In mathematical statistics and information geometry, divergence plays various roles [18]. The function M × M → R, is called the canonical divergence. Several relations in affine geometry and information geometry are known in the literature. One link between affine geometry and information geometry is known as follows: [17]). If (f, ξ) is non-degenerate and equiaffine, then (M, ∇, h) is a statistical manifold.
Another link between affine and information geometries is found on the study of divergence. To define the geometric divergence, consider a graph immersion (f, ξ) associated with ψ : Ω → R, written in coordinates as f : θ → (θ, ψ(θ)), ξ = ∂ ∂z .
If ψ is convex, then the explicit form of the conormal map acting on p ∈ Ω is obtained in coordinates with (3) and (9) as Then, let (f, ξ) be a non-degenerate equiafffine immersion, v its conormal map, and ∆ f : Ω × Ω → T R n+1 a map such that Then the function D G : Ω × Ω → R, called geometric divergence, is introduced in the context of the study of affine geometry. A relation between D and D G is obtained as shown explicitly in Ref. [17]. It is obtained from (10) and In more general case, a relation between the divergence and the geometric divergence has been known as follows: 17]). Let (M, ∇, g) be a simply connected dually flat space. Then the canonical divergence and the geometric divergence on (M, ∇, g) coincides.

Thermodynamics and existing geometric formulations
In this subsection necessary background of thermodynamics for this study is briefly summarized. Symbols introduced here for thermodynamic quantities are duplicated with symbols for geometry introduced in Section 2.1. These duplicated symbols are identified when affine geometric thermodynamics is constructed. In addition, thermodynamic processes in this paper are assumed to be quasi-static for simplicity.
To formulate equilibrium thermodynamics in an abstract manner, thermodynamic variables expressed as a point on a subspace of R n are needed. In addition we demand that R should be needed for the value of a complete thermodynamic function, where complete thermodynamic functions are functions that derive equations of state at equilibrium and response functions [24]. Examples of complete thermodynamic functions are the internal energy, entropy, Helmholtz and Gibbs free-energies with appropriate arguments. Thermodynamic variables and complete thermodynamic functions play fundamental roles, in the sense that these induce equations of state at equilibrium. More details about these variables and functions are explained below. Let x = (x 1 , . . . , x n ) be a set of thermodynamic variables in R n and F a complete thermodynamic function (free-energy or internal energy). If x is a set of arguments or equivalently variables of F, then x 1 , . . . , x n are called primal thermodynamic variables in this paper. Then the set of the thermodynamic conjugate variables y = (y 1 , . . . , y n ) is such that y a = ∂F ∂x a , a = 1, . . . , n.
In addition the value of a complete thermodynamic function F (a free-energy, entropy, or internal energy) at x should be the same as that of F(x), The thermodynamic phase space is where (12) and (13) are satisfied. Hence, the thermodynamic phase space is a subset of R 2n+1 . The fundamental relation of thermodynamics can be written as where U is internal energy, T the absolute temperature, S entropy, P pressure, and V volume. In addition, the heat (one-form) Q and the work (one-form) W are defined as respectively [2]. The first law of thermodynamics states that there exists the function U so that (14) holds with some processes, where processes are integral curves of vector fields. In other words, it states that the sum Q+W is an exact one-form dU . Meanwhile, the second law of thermodynamics states that there exists the function S with some properties. To change variables, the Legendre transform is applied to functions, where the transformed functions are convex if the original functions are convex. Given a function F, it follows from the theory of Legendre transform that there is a function F * such that x a = ∂F * ∂y a , a = 1, . . . , n.
After changing thermodynamic variables, (14) can be written in various forms, such as and so on, where A denotes the Helmholtz free-energy. Response functions, such as heat capacity and specific heat, are obtained by differentiation of (12) as Equilibrium states can be divided into at least three classes. They are unstable, metastable, and most stable equilibrium states. Since equilibrium states are fundamental objects in the study of thermodynamics, a classification of equilibrium states provides further understanding of thermodynamic properties of substances. The three classes above are explained below. The most stable equilibrium states are equilibrium states that are structurally stable within some time-length against some small external perturbation. Metastable equilibrium states are states that exist without very small external perturbation, but states deviate from the metastable states under some strength of external perturbation. Unstable equilibrium are states that are hard to be realized experimentally since the states will not return to the original unstable equilibrium states under very small perturbation.
Nonequilibrium thermodynamics is a developing branch of physics, and several theories have been proposed in the literature [25,26]. To provide a solid foundation for a nonequilibrium theory, we focus on a simple class for clarity. One simple class of nonequilibrium phenomena is that of relaxation processes. In this paper even in a nonequilibrium state, thermodynamic variables are assumed to be defined and described by extending the equilibrium thermodynamic variables. Let y(t) and F(t) be a set of thermodynamic variables and the value of a nonequilibrium free-energy at time t ∈ I ⊂ R, respectively. In addition, y and F denote the corresponding variable set and the function defined at equilibrium, respectively. If a process (or a time-evolution) that satisfies lim t→∞ y(t) = y, and lim t→∞ F(t) = F, then the process is referred to as a relaxation process in this paper.
There are several existing geometric formulations of thermodynamics in the literature. Such existing studies include Hesse geometry and contact geometry: and g a pseudo Riemannian metric tensor field that can be written as g = DdF with F being a function.
• ( [11,10]). A contact manifold is an odd-dimensional manifold M equipped with a distribution ker λ, where λ is a one-form that satisfies the condition that the top-form λ ∧ dλ ∧ · · · ∧ dλ is a volume element on M.

Affine geometric description of thermodynamics
In this section an affine geometric description of thermodynamics is proposed and discussed. This description is divided into two parts, one is equilibrium case and the other nonequilibrium one. The description of the nonequilibrium case can be divided into two, one is the case of a unique set of equilibrium states, and the other the case of two sets of equilibrium states: Equilibrium / Nonequilibrium a unique set of equilibrium states two sets of equilibrium states .

Equilibrium
To describe thermodynamics in terms of affine geometry, the basic idea proposed in this paper is to employ graph immersions of a region Ω ⊂ R n into R n+1 . In this paper the following interpretations are proposed.
1. A point of R n+1 is identified as a set of primal thermodynamic variables and the value of a complete thermodynamic function (a free-energy, entropy or internal energy).
2. The coordinate system x = (x 1 , . . . , x n ) for Ω ⊂ R n represents primal thermodynamic variables in thermodynamic systems. Then the coordinate z for the additional space R represents the value of a complete thermodynamic function (a free-energy, entropy, or internal energy). This is written as z = F (x) when a thermodynamic relation holds, and let ξ = ∂/∂z.
3. The image of a graph immersion (f, ξ) ⊂ R n+1 associated with F is identified with a thermodynamic phase space, where a thermodynamic relation holds. Accordingly, a point of this space is identified with an equilibrium state.
4. The conormal map is identified with one of the expressions of the fundamental relation of thermodynamics, (14), (15), and (16).
Note that as well as item 1 of Interpretation 3.1 where (n + 1)-dimensional manifolds are involved, there are several geometric approaches with (n + 1)-dimensional manifolds in the literature [2,27].
Remark 3.2. The above identifications induce the following.
1. The affine fundamental form h, a (0, 2)-type operator, is identified with a set of response functions at equilibrium (on the thermodynamic phase space) (see (6) and (17)).
2. The flat connection ∇ on Ω is introduced at equilibrium (on the thermodynamic phase space). In addition, x is an affine coordinate system (see (1) and (7)).
3. The z component of the push-forward f * (∂/∂x a ) is identified with the thermodynamic conjugate variable with respect to x a (see (4)). This is written as y a = ∂F/∂x a , where y a is the thermodynamic conjugate variable with respect to a primal thermodynamic variable x a , and the collection is denoted by y = (y 1 , . . . , y n ). Hence thermodynamic primal and conjugate variables are expressed as a point of the tangent bundle T (f Ω).

4.
A succinct information about describing a set of equilibrium states is the image of a graph immersion (f, ξ) into R n+1 , so that the definition of equilibrium state is given by item 3 of Interpretation 3.1.
When thermodynamic conjugate variables are needed, its tangent bundle is considered.
For later purpose of describing nonequilibrium states, a relaxation (or, an extension) of the image of the graph immersion is discussed here. Since the values of a free-energy and thermodynamic variables in a nonequilibrium state are not written F (x) and (x, y(x)) in coordinates, the manifold T (f Ω) is relaxed to (or, extended to) T Ω × R.
A point of T Ω × R can be written in coordinates as (x, y, z), where (x, y) is the coordinates of T Ω and z that of R. Meanwhile the manifold T Ω × R is redundant for the purpose of expressing equilibrium states.
Since T (f Ω) and T Ω × R include redundant components for describing equilibrium states, we have defined equilibrium states as a graph immersion (f, ξ).
6. In the case that F is convex for the graph immersion (f, ξ), it follows from Proposition 2.1 that the equilibrium phase space is identified with a statistical manifold.
7. In the case that F is convex, the geometric divergence (11) is introduced in the thermodynamic phase space.
Some of Remark 3.2 is summarized as follows. This is the main claim in this subsection: (equilibrium states in terms of affine geometry). On a set of equilibrium states that is a thermodynamic phase space in the sense of this paper, an affine fundamental form and flat connection are induced. In addition, thermodynamic conjugate variables are described in the tangent bundle. The geometric divergence is introduced if the complete thermodynamic function is convex or concave.
Examples are given as follows. The first example below shows how to find an affine immersion and its geometric quantities from a given equation of state and a given complete thermodynamic function. where R > 0 is constant, T > 0 temperature, P pressure, and V volume. This equation is written as where A denotes the Helmholtz free-energy. Let T be fixed, and identify Ω = R >0 , and z the value of the free-energy. From the Helmholtz free-energy as a complete thermodynamic function and the equation of state given above, the corresponding affine immersion is shown below. As a transversal vector field, take ξ = ∂/∂z. Then the image of the graph immersion (f, ξ) associated with F is identified with the set of equilibrium states at temperature T . The affine fundamental form h is such that Thus h is non-degenerate on Ω = R >0 , from which this graph immersion is non-degenerate. By applying Proposition 2.1 one can introduce the dual coordinate system x * with respect to h, The example below shows how to find an affine immersion and its geometric quantities from a given complete thermodynamic function and a given fundamental relation of thermodynamics.
where U is internal energy and c a positive constant. Since primal thermodynamic variables are the arguments of S, the primal thermodynamic variables defined on Ω = R 2 >0 := R >0 × R >0 for this model are To specify an immersion, we let z be the value of S and let ξ = ∂/∂z be a transversal vector field. The image of the graph immersion associated with S is identified with the thermodynamic phase space. The conormal map is expressed and specified as The conjugate thermodynamic variables are then These yield the expression of U and the equation of state, respectively. From the first equation above, the specific heat C = dU/dT is derived as C = cR. The explicit form of the affine fundamental form h is shown as Thus h is non-degenerate on Ω = R 2 >0 , from which this graph immersion is non-degenerate. By applying Proposition 2.1 one can introduce the function S * so that This S * (y 1 , y 2 ) is found to be S * (y 1 , y 2 ) = R ln(y c 1 y 2 ) + (constant), and then the geometric divergence D G can be introduced.
There are several complete thermodynamic functions for the ideal gas system, such as A in Example 3.4 and S in Example 3.5. Internal energy as a function of S and V is another choice as a complete thermodynamic function. In this choice, U (S, V ) = c V −1/c exp(S/(cR)) and (14) yield the equation of state P V = RT and the specific heat C = cR, where c is constant. In the examples above, the affine immersions are non-degenerate. Meanwhile in the following example, an affine immersion is shown to be degenerate.
where T denotes temperature, V volume, and P pressure. This equation can be written as where A denotes the Helmholtz free-energy. Let T be fixed, and identify Ω = R >0 , and z the value of the free-energy. Choose ξ = ∂/∂z as a transversal vector field. Then the image of the graph immersion (f, ξ) associated with F is identified with the set of equilibrium states at temperature T . The affine fundamental form h is such that from which this graph immersion is degenerate (not non-degenerate). The z component of f * (∂/∂x), y, is identified with P . Note that there are several existing studies on this model in the language of Riemannian geometry with the Levi-Civita connection in the literature [15,28].

Nonequilibrium
In this subsection, a nonequilibrium geometric theory is proposed. This is based on the equilibrium affine geometric theory, developed in Section 3.1.
To construct a nonequilibrium geometric theory one needs to 1. introduce physical time, 2. extend or relax the equilibrium theory, where the equilibrium theory is based on a graph immersion (f, ξ) into R n+1 , 3. verify that the nonequilibrium theory is consistent with the equilibrium theory in some limits. One of these limits is the time-asymptotic limit.
To develop a nonequilibrium theory, we 1. introduce time, and it is denoted by t ∈ I ⊂ R, 2. introduce the trivial fiber bundle Ω × R, where the restriction of this bundle to the set of equilibrium states is Ω × F (Ω) that is the image of a graph immersion (f, ξ). Hence this trivial bundle is an extension of the graph immersion.
3. focus on the so-called relaxation process. Relaxation processes are time-dependent phenomena in which thermodynamic variables achieve an equilibrium state from a nonequilibrium state through a time evolution. Since there are a variety of classes of nonequilibrium systems and any confusion should be avoided, a simple formulation for a simple nonequilibrium phenomena is proposed in this paper. Then, time-asymptotic limits of the nonequilibrium theory is verified to be consistent with the equilibrium theory.
In what follows, the two cases are considered (see Fig. 1). Relaxation processes for systems with • a unique set of equilibrium states • two sets of equilibrium states x z F x z I II Figure 1: Relaxation processes as integral curves of vector fields. (Left) System with a unique set of equilibrium states. The thick line represents the set of unique equilibrium states that is the graph of F , and the arrows represent the relaxation generating vector field X F (see Section 3.2.1). (Right) System with two sets of equilibrium states. The thick line represents the set of the most stable equilibrium state that is the graph of F I , the dashed line the set of metastable equilibrium states that is the graph of F II , and the arrows represent the relaxation generating vector field X II→I (see Section 3.2.2).

Nonequilibrium, (i) case of unique set of equilibrium states
In the following, a nonequilibrium geometric theory is proposed for the case that there is a unique set of equilibrium states.
The following is the definition of relaxation process in the case that there is a unique set of equilibrium states.
Definition 3.7. (relaxation process). Consider a graph immersion (f, ξ) associated with F from Ω into R n+1 , (dim Ω = n), and let x be a coordinate system of Ω, z ∈ R, ξ = ∂/∂z and y a = f * (∂/∂x a ), (a = 1, . . . , n). Suppose that the image of (f, ξ) is identified with a set of equilibrium states. Then let φ : I t → p(t) ∈ T R n+1 be a curve, where the coordinates of p(t) are denoted by (x(t), z(t), y(t), w(t)), (t ∈ I ⊂ R). For each x kept fixed in t, if a curve satisfies the conditions lim t→∞ y a (t) = ∂F ∂x a (x), and lim t→∞ z(t) = F (x), a = 1, . . . , n, then the curve is said to be a relaxation process towards a point of the set of equilibrium states (f, ξ).
To state an affine geometric description of this class of relaxation processes in terms of vector fields, a set of equilibrium states is placed as follows. Let (f, ξ) be a graph immersion into R n+1 written in coordinates as x → (x, F (x)) ∈ R n+1 and ξ = ∂/∂z, that is, this immersion is associated with a function F : Ω → R. First, recognize that Ω × R ⊂ R n+1 is a trivial fiber bundle, π : Ω × R → Ω, π(q, z) = q. Hence z ∈ R is a point of the fiber whose base point is q. Then, consider the vector field on R over the point q ∈ Ω of the form written in coordinates as with w F (−; x) being a function of z at a given x (see Fig. 1 (Left)). Integral curves of (20) are obtained by solving the ordinary differential equation (ODE) Equation (21) is said to be the system of the ODE associated with X F in this paper. If a solution z(t; x) of (21) satisfies lim t→∞ z(t; x) = F (x), then the system (21) together with an appropriate initial condition is referred to as a relaxation generating system, and the corresponding vector field X F a relaxation generating vector field in this paper.
The system in the following example is a relaxation generating system.
Example 3.8. Consider the dynamical system on R associated with X F : Then the explicit form of the solution z(t; x) is immediately obtained from (22) as From lim t→∞ z(t; x) = F (x) for each x, it follows that the system (22) is a relaxation generating system.
Roughly speaking a relaxation generating system is a seed for generating a relaxation process. This is refined as follows: Theorem 3.9. (induced relaxation process 1). Consider the relaxation generating system (21) with some initial condition. Then, introduce a family of graph immersions {(f t , ξ t )} t∈I into R n+1 written in coordinates as x → (x, F t (x)) ∈ R n+1 , (t ∈ I), that is, f t is associated with a function F t . In addition, 1. Choose the function F t to be F t (x) = z(t; x) with z(t; x) being a solution to (21), and 2. let y a (t; x) be such that y a (t; x) = ∂F t (x) ∂x a , a = 1, . . . , n.
If the limit and differentiation are commute, then the curve t → (x, z(t; x), y(t; x), w(t; x)) is a relaxation process in the sense of Definition 3.7: lim t→∞ z(t; x) = F (x), and lim t→∞ y a (t) = ∂F ∂x a (x), a = 1, . . . , n.
Proof. The statement on lim z(t; x) is nothing but the definition of relaxation generating system. Then a proof for y a (t) is given below. For all x, it follows from lim t→∞ F t (x) = F (x), and commutability of the limit and differentiation that lim t→∞ y a (t) = ∂F ∂x a (x), a = 1, . . . , n.
Remark 3.10. Note the following: 1. Given a relaxation generating system, the induced relaxation process is obtained by applying Theorem 3.9.
2. For each t ∈ I of this family of graph immersions, (f t , ξ t ) can be interpreted as a set of equilibrium states.
3. For each t ∈ I of this family of graph immersions, a flat connection is induced where the connection can be written as ∇ t .
4. For each t ∈ I of this family of graph immersions, the geometric divergence is induced if F t (x) is convex or concave with respect to x, where this geometric divergence can be written as D G t . The following is a physical model explaining Theorem 3.9 in a physical language. The following example also shows how a physical system is translated into its affine geometric description of thermodynamics proposed in this paper.
Example 3.11. (kinetic Ising model without spin-coupling [29,30]). Consider a nonequilibrium system consisting of only one spin σ = ±1 in contact with a heat bath of fixed temperature T = /(k B β) with k B being the Boltzmann constant. The dynamics of the statistical average of this spin is governed by an externally applied static magnetic field H, where this H yields a relaxation process towards a point of the unique set of equilibrium states. To describe this system, introduce a time-independent variable x := βµ B H ∈ R. This x is a dimensionless variable for H with µ B being the Bohr magneton. In the following after the equilibrium case is discussed, the nonequilibrium case is discussed. The equilibrium probability distribution function is assumed to be the canonical distribution, P eq (σ, x) = exp(xσ)/Z(x), where Z(x) = σ=±1 e xσ = 2 cosh x.
At equilibrium, one can introduce the negative of a dimensionless free-energy F : R x → F (x) ∈ R as F (x) := ln Z(x) = ln cosh(x) + ln 2, the magnetization at equilibrium is then written as In the following a dynamical equation for this system is derived. Introduce P t : {±1}×R → R >0 , (t ∈ I ⊂ R), that is a probability distribution function of σ at time t. Then the expectation variable at t is denoted by σ (t; x) := σ=±1 P t (σ; x) σ, (t ∈ I ⊂ R). This variable, σ (t; x), is identified with a nonequilibrium magnetization at t for fixed x. Imposing the simple assumptions for P t , • the equation for P t is the Pauli master equation, and • the detailed balance condition holds for the master equation, one derives the ODE: This system of the ODE is referred to as the kinetic Ising model without spin-coupling in Ref. [29]. Solving this ODE explicitly, one verifies that lim t→∞ σ (t; x) = σ eq (x).
To write this system in affine geometry, identify y(t; x) = σ (t; x), and introduce the new variable z(t; x) ∈ R. The physical meaning of z is the value of a nonequilibrium extension of F , and z(t; x) is denoted by F t (x). A relaxation generating system is introduced by letting w F (t; x) = F (x) − z as in (22), The set {(f t , ξ t )} t∈I is a family of graph immersions, and the curve t → (x, y(t; x), z(t; x), w F (t; x)) is a relaxation process.

Nonequilibrium, (ii) case of two sets of equilibrium states
In the following, a nonequilibrium geometric theory is proposed for the case that there are two sets of equilibrium states. This is based on the geometric theories developed in Sections 3.1 and 3.2.1. To avoid unnecessary confusion and to keep discussions in this paper simple, cases where more than three equilibrium states are not discussed in this paper.
The following is the definition of relaxation process from a point of the set of metastable equilibrium states to a point of the set of the most stable equilibrium states. Definition 3.12. (relaxation process with metastable state). For Ω 0 ⊂ Ω, let x be coordinates of a point q 0 ∈ Ω 0 , z ∈ R, ξ = ∂/∂z, and y a = f * (∂/∂x a ), (a = 1, . . . , n). In addition let F I : Ω 0 → R and F II : Ω 0 → R be functions where the condition F I (q 0 ) < F II (q 0 ) holds for any q 0 ∈ Ω 0 . Consider the two graph immersions given below: • A graph immersion (f I , ξ) associated with F I into R n+1 . Suppose that the image of (f I , ξ) is identified with the most stable equilibrium state set.
• A graph immersion (f II , ξ) associated with F II into R n+1 . Suppose that the image of (f II , ξ) is identified with a metastable equilibrium state set.
In addition, let φ : R t → p(t) ∈ T R n+1 be a curve, where the coordinate of p(t) is denoted by (x(t), z(t), y(t), w(t)), (t ∈ R). If a class of curves satisfies the conditions 0.
z(0) < F II (x), and x(t) = x(0), ∀ t ∈ R, then the curve is said to be a relaxation process from a point of (f II , ξ) towards a point of (f I , ξ).

lim
To state an affine geometric description of this class of relaxation processes, two sets of equilibrium states are placed as follows. Let (f I , ξ) and (f II , ξ) be graph immersions into R n+1 written in coordinates as x → (x, F I (x)) ∈ R n+1 and x → (x, F II (x)) ∈ R n+1 , respectively, where ξ = ∂/∂z. In addition, let Ω 0 ⊂ Ω be a region on which the condition F II (x) > F I (x) is satisfied. First, recognize that Ω 0 × R ⊂ R n+1 is a trivial fiber bundle, π : Ω 0 × R → Ω 0 , π(q 0 , z) = q 0 . Hence z ∈ R is a point of the fiber whose base point is q 0 . Then, consider the vector field on the fiber R over a point q 0 ∈ Ω 0 ⊂ Ω of the form written in coordinates with w II→I (−; x) being a function of z at a given x (see Fig. 1 (Right)). Integral curves of (24) are obtained by solving the ODE dz dt = w II→I (z; x).
If a solution z(t; x) whose initial condition is z(0) < F II (x) satisfies then the system (25) is referred to as a relaxation generating system with f I and f II , and the corresponding vector field X II→I relaxation generating vector field with f I and f II in this paper.
Example 3.13. Consider the dynamical system on R associated with X II→I : where the graph of w II→I (z; x) is drawn in Fig. 2. In the case z(0) < F II (x) in R for a given x, it follows that Hence the system (26) is a relaxation generating system.
There are various variants of (26). One of them is Roughly speaking a relaxation generating system is a seed for generating a relaxation process. This is refined as follows: Theorem 3.14. (induced relaxation process 2). Consider the relaxation generating system (25). Then, introduce a family of graph immersions {(f t , ξ t )} t∈R into R n+1 written in coordinates as x → (x, F t (x)) ∈ R n+1 , (t ∈ R), that is, f t is associated with a function F t .
1. Choose F t to be F t (x) = z(t; x) with z(t; x) being a solution to (25), and 2. let y a (t; x) be y a (t; x) = ∂F t (x) ∂x a , a = 1, . . . , n.
If the limit and differential are commute, then the curve t → (x, y(t; x), z(t; x), w II→I (t; x)) is a relaxation process in the sense of Definition 3.12: The symbols I and II in the graph indecate F I and F II , respectively. ; x), a = 1, . . . , n. Remark 3.15. Given a relaxation generating system, the induced relaxation process is obtained by applying Theorem 3.14. Note the following:

lim
• For each t ∈ R, (f t , ξ t ) can be interpreted as a set of equilibrium states.
• For each t ∈ R, a flat connection is induced where the connection can be written as ∇ t .
• For each t ∈ I, the geometric divergence is induced if F t is convex or concave with respect to x, where this geometric divergence can be written as D G t .

Comparison with contact geometric thermodynamics
In this section the developed theory in Section 3 of this paper is compared with a representative existing theory. As a representative thermodynamic theory a contact geometric thermodynamics is summarized first in Section 4.1. Second a lift of the vector field X F on R is discussed in Section 4.2. The reason why lifted vector fields are considered is that the manifold on which X F is defined is different to the manifold on which contact vector fields are defined. To compare these vector fields on the equal footing, one needs the same dimensional manifolds on which vector fields are defined. One way to realize such is to lift the vector fields on a lower dimensional manifold. For the same reason, a lift of X II→I is discussed. Then, in Section 4.3, the present affine geometric theory is compared with the existing contact geometric theory. For nonequilibrium systems, vector fields are compared by introducing appropriate identifications for different manifolds.

Contact geometric thermodynamics
One of developing geometric theories of thermodynamics employs contact geometry, where contact geometry is known as an odd-dimensional counterpart of symplectic geometry. In the contact geometric thermodynamics, the manifold T * Q × R is often considered as an ambient manifold [31,32], where Q is an n-dimensional manifold. Let x be a coordinate system of Q, y that of T * q Q, and z that of R. In addition, equip the one-form λ = dz − n a=1 y a dx a on T * Q × R, where the top-form λ ∧ dλ ∧ · · · ∧ dλ does not vanish anywhere. The pair (T * Q × R, ker λ) is called a contact manifold, where ker λ denotes the kernel of λ.
A particular class of submanifolds, called a Legendrian submanifold, can be interpreted as a thermodynamic phase space [5], where the equation of state at equilibrium is described. Here a Legendrian submanifold is an n-dimensional submanifold of a contact manifold satisfying the condition that the pull-back of λ vanishes. There are some useful projections [13]. In terms of the coordinate system (x, y, z), the projection of a Legendrian submanifold onto the (x, z)-plane is called a Legendre map, its image is celled a wave front. In addition, the projection of a Legendrian submanifold onto the (x, y)-plane is called a Lagrange map.
A diffeomorphism that preserves the contact structure ker λ is called a contact transform, and its vector field is called a contact vector field. Although there are various viewpoints, this vector field is sometimes employed as a tool for expressing nonequilibrium time-evolution of thermodynamic systems [29,32,33] (see Ref. [34] for another interpretation). A way to provide a contact vector field is to provide a function, called a contact Hamiltonian, and the derived contact vector field is called a contact Hamiltonian vector field. A contact Hamiltonian vector field Y h associated with contact Hamiltonian h is determined by where R is called the Reeb vector field defined such that ı R λ = 1, and ı R dλ = 0.
In the case where the contact form is expressed as above, the contact Hamiltonian vector field Y h on T * Q×R is written in coordinates as Y h =ẋ a ∂ ∂x a +ẏ a ∂ ∂y a +ż ∂ ∂z .
By identifying˙= d/dt, one derives When h does not depend on y, h = h(x, z), one immediately has that d dt The following two contact Hamiltonian systems will be focused in Section 4.3.
• Choose h to be h F , where with F being some function. Substituting this h F into (28), one has dx a dt = 0, dy a dt = −y a + ∂F ∂x a , dz dt = h F , a = 1, . . . , n.
The contact Hamiltonian system (29) has been studied in Refs. [29,35], so that a class of relaxation processes is described on a contact manifold.
• Choose h to be h II→I , where with F I and F II being functions of x. Substituting this h II→I into (28), one has where a = 1, . . . , n. The contact Hamiltonian system (30) has been studied in Ref. [32], so that a class of relaxation processes with sets of multiple equilibrium states is described on a contact manifold. In Ref. [32], the dimension of the contact manifold is three, and suffix a in (30) has been omitted.

Lift of relaxation generating vector field
In this subsection vector fields on a high dimensional manifold are defined based on the developed theories in Section 3.2, so that the vector fields on the higher dimensional manifold will be compared with the existing vector fields defined on a contact manifold. In particular, a vector field lifted from X F in (20) and a vector field lifted from X II→I in (24) are discussed here, where the lifted vector fields are denoted by X F and X II→I , respectively. Recall that the vector fields X F and X F generate relaxation processes for the unique equilibrium state systems, and vector fields X II→I and X II→I generate relaxation processes for non-unique equilibrium state systems. Before discussing lifted vector fields for the both cases, how to discuss stability of vector fields on a manifold M of dimension n is summarized here. Let µ be a volume-element of M, and X a vector field on M. Then the phase space compressibility of X with respect to µ, denoted div µ X ∈ ΓΛ 0 M, is defined such that [36] L X µ = (div µ X)µ, where L X is the Lie derivative along X. This function is sometimes considered when stability of vector fields on manifolds is discussed [37]. If the system associated with X has the property that (div µ X)(p) < 0 at p ∈ M, then this dynamical system is said to be contracting at p ∈ M with respect to µ in this paper. A coordinate expression of div µ X is obtained as follows. Let ξ = (ξ 1 , . . . , ξ n ) be coordinates, and µ be such that µ = dξ 1 ∧ · · · ∧ dξ n .
Note that this ξ is nothing to do with any transversal vector field of an affine immersion. Consider the vector field written in coordinates ξ of the form X = n a=1ξ a (ξ) ∂ ∂ξ a , whereξ = (ξ 1 , . . . ,ξ n ) is a set of functions of ξ. Then the Lie derivative of µ along X is calculated to be L X µ = (dξ 1 ∧ dξ 2 ∧ · · · ∧ dξ n ) + · · · + (dξ 1 ∧ · · · ∧ dξ n−1 ∧ dξ n ) n terms = n a=1 ∂ξ a ∂ξ a µ, from which one can write div µ X as a function of ξ, To see the role of this function more clearly, consider the one-dimensional case, X =ż∂/∂z on M = R with µ = dz andż = −z. Integral curves of X are obtained by solving the ODEż = −z, whereż = dz/dt. It follows from the explicit solution for z that the absolute value |z(t)| is decreasing (contracting) as time develops from t = 0 to any t * > 0. Meanwhile, for this system the phase space compressibility as a function of z is obtained by the calculation ∂(−z)/∂z = −1 as (div dz X)(z) = −1 on R, and one concludes that this system is contracting. This example justifies the terminology "contracting." The both cases, the case of the unique equilibrium set and that of non-unique equilibrium state sets, are discussed below separately.

Unique set of equilibrium states
In Theorem 3.9, the variables y and z express time-evolution. Meanwhile the corresponding vector field X F and the dynamical system are written only in terms of z for a fixed q and x. Hence it is natural to consider a dynamical system involving y and z, and is natural to consider its corresponding vector field denoted by X F . This vector field is studied here. Since there are various classes of ODEs, one focuses on systems where 1. a fixed point (y * a , z * ) is (∂F/∂x a , F (x)), (a = 1, . . . , n) with x being a set of coordinate values of a point q ∈ Ω, and 2. the corresponding vector field is a lift of X F , in the sense that π * X F = X F : Note that the manifold T Ω × R has been discussed in (18) as a redundant manifold for describing equilibrium states.
This X F is written of the form: with some functions u = (u F 1 , . . . , u F n ). The reason why item 1 is needed is as follows. If (∂F/∂x a , F ) is a fixed point and attractive in some sense, then integral curves of X F are relaxation processes, which we seek. Here relaxation process in T (T q Ω × R) is defined as follows.
then the curve is said to be a relaxation process towards a point of the set of equilibrium states (f, ξ).
To compare X F with X F , a property of X F is studied first. In particular, a stability of the dynamical system of the ODE (21) associated with X F on R,ż = w F (z; x), is discussed here. Let div X F ∈ ΓΛ 0 R be the phase space compressibility with respect to dz on R, that is, To study properties of X F , as well as the case of X F , one defines the function div X F ∈ ΓΛ 0 (T q Ω × R) such that L X F µ = (div µ X F )µ, where µ = dy 1 ∧ · · · ∧ dy n ∧ dz.
The following Lemma is about the system of the ODEs associated with X F and that with X F . When (y * a , z * ) = (∂F/∂x a , F (x)) is a fixed point for the system of the ODEs associated with X F and conditions for Theorem 3.9 are satisfied, this fixed point is an attractor for the dynamical system. Lemma 4.2. Consider a dynamical system for (y, z) associated with X F , where u F a (−, −; x) is a function of z and y for a given x. Choose Let z * be a point that satisfies w F (z * ; x) = 0, i.e., the point z * is a fixed point for the ODEż = w F in (33). Then the following hold.
1. The contracting property is preserving under the lift in the sense that with some Z ⊂ R.

If
then y * of a fixed point (y * , z * ; x) is given by Proof. (Proof for 1.) The explicit form of div µ X F is obtained as From this and the condition (div dz X F )(z; x) = ∂w F /∂z < 0, item 1 holds.
To find an explicit form of a fixed point fromẏ a = u F a , lettingẏ * a = u F a (z * , y * ; x) = 0 for each x, one has (37).
• In Example 3.8, the case w F (z; x) = F (x) − z has been considered. In this example it follows that (div dz X F )(z; x) = −1. Hence the condition for item 1 and that for 3 are satisfied. In addition, the subset Z for item 2 is found as Z = {z|F (x) − z ≥ 0}. These calculations are summarized as Tab.1. Table 1: Behavior of the system in Example 3.8 • The system consisting of (33), (34), andẋ a = 0 (a = 1, . . . , n), is formally the same as (29), where (29) is the contact Hamiltonian system with the contact Hamiltonian h(x, z) = F (x) − z. Hence, it is expected that there is a link between the lifted relaxation generating systems and contact Hamiltonian systems. In particular, the function w F (z; x) is expected to play a role of h(x, z). This link will be discussed in Section 4.3.
The following Proposition shows how to describe a relaxation process in terms of X F . Proposition 4.3. Consider the dynamical system (33) with (34). In the case where with an initial condition F (x) > z(0), integral curves of X F connect points on T q Ω × R and the fixed point (∂F/∂x a , F (x)).
Proof. This follows from item 2 of Lemma 4.2.
Lemmata 4.2 and 4.3 are associated with Theorem 3.9 that is the case of a unique set of equilibrium states. Meanwhile the following are associated with Theorem 3.14.

Two sets of equilibrium states
In Theorem 3.14, the variables y and z express time-evolution. Meanwhile the corresponding vector field X II→I and the dynamical system are written only in terms of z for a fixed q 0 and x. Hence it is natural to consider a dynamical system involving y and z, and its corresponding vector field denoted by X II→I . This vector field is studied here. Since there are various classes of ODEs, one focuses on systems where 1. a fixed point (y * a , z * ) is (∂F I /∂x a , F I (x)), (a = 1, . . . , n), with x being the coordinate system for Ω 0 , 2. the corresponding vector field is a lift of X II→I , in the sense that π * X II→I = X II→I : Note that the manifold T Ω × R has been discussed in (18) as a redundant manifold for describing equilibrium states.
This X II→I is written of the form: with some functions u II→I = (u II→I 1 , . . . , u II→I n ). The reason why item 1 is needed is as follows. If (∂F I /∂x a , F I ) is a fixed point and attractive in some sense, then integral curves of X II→I are relaxation processes, which we seek. Here relaxation process in T (T q0 Ω 0 × R) is defined as follows.
Definition 4.4. (relaxation process in the lifted space 2). Let φ : R t → p(t) ∈ T (T q0 Ω 0 × R) be a curve, where the coordinates of p(t) are denoted by (y(t), z(t),ẏ(t),ż(t)), (t ∈ R). For each x kept fixed in t, if a curve satisfies the conditions 2.
then the curve is said to be a relaxation process towards a point of the set of equilibrium states (f, ξ).
To compare X II→I with X II→I , a property of X II→I is studied first. In particular, a stability of the dynamical system of the ODE (25) associated with X II→I on R,ż = w II→I (z; x), is discussed here. Similar to the case of div µ X F ∈ ΓΛ 0 R, let div µ X II→I ∈ ΓΛ 0 R be the phase space compressibility with respect to dz on R, that is, Simple calculations yield (div dz X II→I )(z, y; x) = ∂w II→I ∂z (z; x).
If ∂w II→I /∂z < 0 in some Z ⊂ R, then the system is contracting in Z.
To study properties of X II→I , as well as the case of X II→I , one defines the function div µ X II→I ∈ ΓΛ 0 (T q0 Ω 0 × R) such that L XII→I µ = (div µ X II→I )µ, where µ = dy 1 ∧ · · · ∧ dy n ∧ dz.
The following Lemma is about the system of the ODEs associated with X II→I and that with X II→I . Lemma 4.5. Consider a dynamical system for (y, z) associated with X II→I , where u II→I a (−, −; x) is a function of z and y for a given x. Choose u II→I a (z, y; x) = y a ∂w II→I ∂z + ∂w II→I ∂x a , a = 1, . . . , n.
Then the contracting property is preserving under the lift in the sense that (div dz X II→I )(z; x) < 0 on Z ⇒ (div µ X II→I )(z, y; x) < 0, on T q0 Ω 0 × Z, with some Z ⊂ R.

Proof.
A way to prove this is analogous to the proof of item 1 of Lemma 4.2.
In Lemma 4.5, note the following.
• In Example 3.13, the case w II→I (z; x) = −(z − F I (x))(z − F II (x)) 2 has been considered. In this example it follows that (div dz X II→I )(z; x) < 0 in some Z ⊂ R. Hence there is some non-empty Z so that the condition for Lemma 4.5 is satisfied. More precisely, one has Note that F I (x) − z 0 (x) = (F I (x) − F II (x))/3 < 0 and that F II (x) − z 0 (x) = (2/3)(F II (x) − F I (x)) > 0. These calculations are summarized in Tab. 2. From this summary, it follows that (div dz X II→I )(z; x) < 0 and (div µ X II→I )(z, y; x) < 0 around z = F I (x). Moreover, in this Example the point (y * a , z * ) in the system (41) with (42), (y * a (x), z * (x)) = ∂F I ∂x a (x), F I (x) , a = 1, . . . , n, is verified to be a fixed point.
• The system consisting of (41), (42), andẋ a = 0 (a = 1, . . . , n), is formally the same as (30), where (30) is the contact Hamiltonian system with the contact Hamiltonian h(x, z) = −(z − F I (x))(z − F II (x)) 2 . Hence, it is expected that there is a link between the lifted relaxation generating systems and contact Hamiltonian systems. In particular, the function w F (z; x) is expected to play a role of h(x, z). This link will be discussed in Section 4.3.
The following Proposition shows how to describe a relaxation process in terms of X II→I .
Proposition 4.6. Consider the dynamical system (41) with (42). In the case of Example 3.13, w II→I (z; x) = −(z − F I (x))(z − F II (x)) 2 , with an initial condition z(0) < F II (x), integral curves of X II→I connect points on T q0 Ω 0 × R and the fixed point (∂F I /∂x a , F I (x)).
Proof. The strategy for proving this Lemma is to find Lyapunov functions, ans this proof is similar to the proof of Theorem 3.1 in Ref. [32]. Let V I (−; x) and V II (−; x) be the functions for a given x, V I (z; x) = 1 2 (z − F I (x)) 2 , on Z I (x) = {z|z < F II (x)}, and V II (z; x) = z − F II (x), on Z II (x) = {z|F II (x) ≤ z}.
The equality holds when z = F I (x). Hence V I (−; x) is a Lyapunov function on Z I (x). Next, it follows that V II (z; x) ≥ 0 on Z II (x), and that dV II dt =ż = −(z − F I )(z − F II (x)) 2 ≤ 0, on Z II (x).
The equality holds when z = F II (x). Hence V II (−; x) is a Lyapunov function on Z II (x). Applying the Theorem of Lyapunov, one completes the proof.

Comparisons
In the following, the present affine geometric thermodynamics is compared with a contact geometric thermodynamics for equilibrium and nonequilibrium systems. Note that contact geometric thermodynamics is an developing branch of mathematical physics, and these identifications may differ among theories. image of Lagrange map (unnamed projection) Table 3: Comparisons between affine and contact geometric thermodynamics

Nonequilibrium
Notice that the system (28) is formally the same as (33) with (34), and is also formally the same as (41) with (42). In addition, since x is a coordinate of a point q or q 0 , one recognizes that the equations for y and z in (28) are coordinate expressions for the restricted vector field Y h | q ∈ ΓT (T * q Q × R) at q ∈ Q. One then immediately arrives at the following.
Theorem 4.7. (relation between vector fields in affine and contact geometries). A class of contact Hamiltonian vector fields Y h | q ∈ ΓT (T * q Q × R) is formally the same as the lifted relaxation generating vector field X F ∈ ΓT (T q Ω × R) in (31) with (33) and (34), and is formally the same as X II→I ∈ ΓT (T * q Q × R) in (38) with (41) and (42). How to identify Y h | q with X F and X II→I is to put Q = Ω or Q = Ω 0 and identify T * Q ∼ = T Ω or T * Q ∼ = T Ω 0 . From Theorem 4.7, it follows that the present affine geometric formalism is consistent with a contact geometric formalism [29,32,33]. In addition, this Theorem indicates how a contact vector field is constructed from a given relaxation generating vector field on R. Conversely, a relaxation generating vector field on R is easily obtained from a contact Hamiltonian vector field on a contact manifold by defining an appropriate projection.

Concluding remarks
This paper offers an affine geometric formulation of thermodynamic systems. This formulation covers both equilibrium and simple nonequilibrium systems. The main claims of this paper are as follows: • A set of equilibrium states is identified with the image of a graph immersion into R n+1 (see Interpretation 3.1).
• Several affine geometric objects can be introduced in thermodynamics (see Proposition 3.3).
• A class of relaxation processes in thermodynamic systems are described (see Theorems 3.9 and 3.14).
• The present geometric formulation of relaxation processes are consistent with the existing theory (see Theorem 4.7).
The significance of this study includes, as indicated the diagram below, Affine Geom