Hamiltonian thermodynamics in the extended phase space: a unifying theory for non-linear molecular dynamics and classical thermodynamics

Abstract

The Hamiltonian theory of thermodynamics for reversible and irreversible processes is presented in detail. Particularly, it is demonstrated that non-linear molecular dynamics and thermodynamics of equilibrium and non-equilibrium processes can merge in a single dynamical theory by constructing a homogeneous of first degree in momenta Hamiltonian function on the extended thermodynamic state space and in the entropy representation. The geometrical properties of the physical equilibrium state manifold are discussed and results of numerical experiments with the Hénon–Heiles model that dissipates in a simple thermodynamic system with multiple friction parameters are presented. It is well known that this classical paradigm carries most of the non-linear dynamical characteristics of generic molecular systems. The construction of homogeneous Hamiltonians for composite thermodynamic systems is also outlined in the appendixes, where the theory of Hamiltonian thermodynamics is developed in current differential geometry terms.

Appendices

Appendix A: Thermodynamic phase spaces

In the two appendixes we provide details of the Hamiltonian theory of classical thermodynamics as has been theorized in the last two decades. Classical thermodynamics are presented in accordance to Callen’s book [12] by employing principal state functions, such as the internal energy or the entropy. These are functions of the complete set of extensive variables (entropy, internal energy, number of molecules or moles of the chemical substances, etc) that parametrize the physical state manifold. However, we point out that other state functions can also be used, such as the Legendre transformations of internal energy (enthalpy, Helmholtz and Gibbs free energies), as well as of entropy (Massieu’s functions). Legendre transformations result in functions of mixed extensive and intensive variables of thermodynamic system.

1.1 A.1. Entropy representation and thermodynamic contact space

In the entropy representation of thermodynamics, we consider entropy to be a homogeneous function of first degree in the complete extensive coordinate set [12] $q = (q^1,q^2,\dots ,q^n)^T \equiv (U, V, N^1, \dots , N^r)^T \in Q^S$. With these natural coordinates we describe the configuration manifold $Q^S$ of dimension $n=2+r$ in entropy representation. The entropy as a homogeneous function of first degree is written (Euler’s theorem)

$$\begin{aligned} S(U,V,N)&= \left( \frac{\partial S}{\partial U}\right) _{V,N} U + \left( \frac{\partial S}{\partial V}\right) _{U,N} V + \sum ^r_{i=1}\left( \frac{\partial S}{\partial N^i}\right) _{U,V,N^{j\ne i}} N^i, \;\; j = 1,\dots ,r \\&= \frac{1}{T}U + \frac{P}{T}V - \sum ^r_{i=1} \frac{\mu _i}{T} N^i = \gamma _1 U + \gamma _2 V + \sum ^n_{k=3} \gamma _k N^{k-2}, \end{aligned}$$

(45)

where we have introduced the conjugate intensive variables $\gamma _i$ as the partial derivatives of S

$$\begin{aligned} \gamma _1&= \frac{\partial S}{\partial U} = \frac{1}{T} \end{aligned}$$

(46)

$$\begin{aligned} \gamma _2&= \frac{\partial S}{\partial V} = \frac{P}{T} \end{aligned}$$

(47)

$$\begin{aligned} \gamma _k&= \frac{\partial S}{\partial N^{k-2}} = -\frac{\mu _{k-2}}{T}, \;\; k = 3,\dots ,n. \end{aligned}$$

(48)

T is the absolute temperature, P the pressure and $\mu _1, \dots , \mu _r$ the chemical potentials of r compounds.

We write the vector field [22] $X_q^S$ at a point $q \in Q^S$ as

$$\begin{aligned} X_q^S&= \left( \frac{\partial S}{\partial q^1}\right) \frac{\partial }{\partial q^1} + \left( \frac{\partial S}{\partial q^2}\right) \frac{\partial }{\partial q^2} + \sum _{k=3}^n\left( \frac{\partial S}{\partial q^k}\right) \frac{\partial }{\partial q^k} \\&= \frac{1}{T}\frac{\partial }{\partial q^1} +\frac{P}{T}\frac{\partial }{\partial q^2} - \sum _{k=3}^n\frac{\mu _{k-2}}{T}\frac{\partial }{\partial q^k}, \end{aligned}$$

(49)

where $X_q^S \in T_qQ^{S}$ with $T_qQ^{S}$ labeling the tangent space of configuration manifold $Q^S$ at the point q and $\left( \frac{\partial }{\partial q^1}, \frac{\partial }{\partial q^2}, \frac{\partial }{\partial q^3},\dots ,\frac{\partial }{\partial q^n}\right)$ are the base vector fields. The set of tangent spaces at all q forms the tangent bundle$TQ^S$ dimension 2n of configuration manifold $Q^S$.

Furthermore, in the cotangent space $T_q^*Q^S$ of configuration space at point q the total differential of S is written as

$$\begin{aligned} dS = \frac{1}{T}dU + \frac{P}{T}dV - \sum ^r_{i=1} \frac{\mu _i}{T} dN^i, \end{aligned}$$

(50)

or

$$\begin{aligned} dq^0 = \sum _{k=1}^n \gamma _k(q) dq^k, \end{aligned}$$

(51)

where we have assigned the entropy S to $q^0$. $(dq^1, dq^2, dq^3,\dots , dq^n)$ are the base 1-forms of $T_q^*Q^S$. The set of cotangent spaces at all $q \in Q^S$ forms the cotangent bundle, $P^{2n} \equiv T^*Q^S$, of even dimension 2n, named thermodynamic phase space in the entropy representation of dimension 2n.

From Eqs. (50) and (51) we deduce that

$$\begin{aligned} dS - \frac{1}{T}dU - \frac{P}{T}dV + \sum ^r_{i=1} \frac{\mu _i}{T} dN^i = dq^0 - \sum _{k=1}^n \gamma _k(q) dq^k \equiv 0. \end{aligned}$$

(52)

In $P^{2n}$ thermodynamic phase space the canonical Poincaré 1-form $\theta$ is determined as

$$\begin{aligned} \theta = \sum _{k=1}^n \gamma _k dq^k, \end{aligned}$$

(53)

and the skew-symmetric 2-form

$$\begin{aligned} \omega = - d\theta = \sum _{k=1}^n dq^k \wedge d\gamma _k. \end{aligned}$$

(54)

Since, $\theta = dq^0$ we infer that the canonical symplectic 2-form $\omega$ is

$$\begin{aligned} \omega = -d\theta = - d\cdot dq^0 = 0. \end{aligned}$$

(55)

Equation (50) is the Gibb’s fundamental equation that provides a description of the physical state manifold of the thermodynamic system. Considering the triple $(P^{2n}, \omega , X^S)$ as a Hamiltonian system with Hamiltonian the entropy function, $S: P^{2n} \mapsto {\mathbb {R}}$, and since $\omega (X^S,X^S) = 0$ we conclude that entropy is conserved and the processes are reversible.

A formal way to introduce the physical state manifold is to expand the configuration manifold to the Extended Coordinate Manifold, $Q^{n+1}$, which also includes the entropy as an independent variable, $q^e = (q^0,q^1,q^2,\dots ,q^n)^T \equiv (S, U, V, N^1, \dots , N^r)^T \in Q^{n+1}$. Then, together with the n conjugate intensive variables $\gamma _i$, Eqs. (46–48), we make the contact $(2n+1)D$-manifold, $C^{2n+1}$, endowed with the 1-form

$$\begin{aligned} \theta _c = dq^0 - \sum _{k=1}^n \gamma _k dq^k. \end{aligned}$$

(56)

From Eq. (52) we infer that the Physical Thermodynamic Submanifold is the kernel of $\theta _c$, i.e. the set of hyperplanes $\varDelta = ker(\theta _c) \subset TQ^{n+1}$, for which the vectors $\xi$ lying in $\varDelta$ satisfy the equation

$$\begin{aligned} \theta _c(\xi ) = 0, \;\;\;\; \xi \in \varDelta . \end{aligned}$$

(57)

1.2 A.2. Energy representation and thermodynamic contact space

According to Euler’s theorem for the homogeneous internal energy function of first degree in the extended coordinates, we have

$$\begin{aligned} U(S,V,N) = \left( \frac{\partial U}{\partial S}\right) _{V,N}S + \left( \frac{\partial U}{\partial V}\right) _{S,N}V + \sum ^r_{i=1} \left( \frac{\partial U}{\partial N^i}\right) _{S,V,N^{j \ne i}} N^i, j = 1,\dots ,r. \end{aligned}$$

(58)

The conjugate intensive properties, temperature (T), pressure (P) and chemical potentials $(\mu _i)$ are the partial derivatives of the internal energy

$$\begin{aligned} T&= \left( \frac{\partial U}{\partial S}\right) _{V,N} \end{aligned}$$

(59)

$$\begin{aligned} -P&= \left( \frac{\partial U}{\partial V}\right) _{S,N} \end{aligned}$$

(60)

$$\begin{aligned} \mu _i&= \left( \frac{\partial U}{\partial N^i}\right) _{S,V,N^{j\ne i}}, \;\; i,j = 1,\dots ,r. \end{aligned}$$

(61)

Hence, the internal energy is written as

$$\begin{aligned} U(S,V,N) = T(S,V,N)\; S - P(S,V,N)\; V + \sum _{i=1}^r \mu _i(S,V,N)\; N^i. \end{aligned}$$

(62)

In energy representation the coordinates $q = (q^0,q^2,q^3,\dots ,q^n)^T \equiv (S,V,N^1,\dots , N^r)^T \in Q^U$ define a local chart of the configuration nD-manifold $Q^U$. The partial derivatives of internal energy belong to the tangent space, $T_qQ^U$, of configuration space $Q^U$ at the point q. $T_qQ^U$ being a vector space itself, we can define a coordinate system with base vectors $\left( \frac{\partial }{q^0}, \frac{\partial }{q^2},\frac{\partial }{\partial q^3},\dots ,\frac{\partial }{\partial q^n} \right)$

$$\begin{aligned} X_q^U&= \left( \frac{\partial U}{\partial q^0}\right) \frac{\partial }{\partial q^0} + \left( \frac{\partial U}{\partial q^2}\right) \frac{\partial }{\partial q^2} + \sum _{k=3}^n\left( \frac{\partial U}{\partial q^k}\right) \frac{\partial }{\partial q^k} \\&= T(q)\frac{\partial }{\partial q^0} -P(q)\frac{\partial }{\partial q^2} + \sum _{k=3}^n\mu _{k-2}(q)\frac{\partial }{\partial q^k}, \end{aligned}$$

(63)

to express the vector fields $X_q^U \in T_qQ^U$ in the energy representation of the thermodynamic system. The set of tangent spaces at all q, $TQ^U$, consists of the tangent bundle of configuration manifold. The dimension of the tangent bundle is 2n.

It is accustom to in thermodynamics one to work in the dual (cotangent) space of the tangent space at the point q, $T_q^*Q^U$ of dimension n. Then, the total differential of the internal energy is written as

$$\begin{aligned} dU = TdS - PdV + \sum ^r_{i=1} \mu _i dN^i, \end{aligned}$$

(64)

or

$$\begin{aligned} dq^1 = T(q) dq^0 - P(q) dq^2 + \sum ^n_{k=3} \mu _{k-2}(q) dq^k. \end{aligned}$$

(65)

If we define the variables $\beta _k$ as the partial derivatives of U

$$\begin{aligned} \beta _0&= \frac{\partial U}{\partial S} = T \end{aligned}$$

(66)

$$\begin{aligned} \beta _2&= \frac{\partial U}{\partial V} = -P \end{aligned}$$

(67)

$$\begin{aligned} \beta _k&= \frac{\partial U}{\partial N^{k-2}} = \mu _{k-2}, \;\; k = 3,\dots ,n, \end{aligned}$$

(68)

then, Eq. (65) takes the form

$$\begin{aligned} dq^1 = \beta _0(q) dq^0 + \beta _2(q) dq^2 + \sum ^n_{k=3} \beta _k(q) dq^k. \end{aligned}$$

(69)

$(dq^0, dq^2, dq^3,\dots , dq^n)$ provide the base 1-forms of $T_q^*Q^U$. The set of all cotangent spaces consists of the cotangent bundle, $T^*Q^U$, also named thermodynamic phase space of dimension 2n in energy representation, $P^{2n}\equiv T^*Q^U$.

In analogy with the mechanical phase space we identify the canonical Poincaré 1-form to be

$$\begin{aligned} \theta \equiv dq^1 = \beta _0 dq^0 + \sum ^n_{k=2} \beta _k dq^k, \end{aligned}$$

(70)

and the canonical symplectic 2-form

$$\begin{aligned} \omega = - d\theta = dq^0\wedge d\beta _0 + \sum ^n_{k=2} dq^k\wedge d\beta _k = 0. \end{aligned}$$

(71)

Hence, Gibb’s fundamental equation, Eq. (69), describes the manifold of equilibrium states and thermodynamic processes in energy representation. Similarly to the entropy representation we can construct the thermodynamic contact space, $C^{2n+1}$, to determine the thermodynamic state manifold in the energy representation. Instead, in the following section we introduce the projection technique, which facilitates the production of several representations.

1.3 A.3. Extended thermodynamic phase space

Following Balian and Valentin [11] we introduce the thermodynamic extended phase space by adopting the extended coordinate manifold $Q^{n+1}$ with natural coordinates the set

$$\begin{aligned} q^e = (q^0, q^1, q^2, \dots , q^n)^T \equiv (S, U, V, N^1, \dots , N^r)^T \in Q^{n+1}. \end{aligned}$$

(72)

Then, the entropy canonical conjugate momentum, $p_0$, is treated as a non-vanishing free parameter or a gauge variable. The canonical conjugate momenta of the other coordinates of $Q^{n+1}$ are determined as

$$\begin{aligned} p_1&= -p_0 \gamma _1 \\ p_2&= -p_0 \gamma _2 \\ p_3&= -p_0 \gamma _3 \\ \cdots&= \cdots \cdots \\ p_n&= -p_0 \gamma _n. \end{aligned}$$

(73)

Hence, $p_i$ act as new variables, which replace the physical intensive variables $\gamma _i$.

In the Thermodynamic Extended Phase Space, $T^*Q^{n+1}$ ($P^{2n+2}$), Poincaré 1-form becomes

$$\begin{aligned} \theta _e = \sum _{i=0}^n p_idq^i, \end{aligned}$$

(74)

and the canonical symplectic 2-form

$$\begin{aligned} \omega _e = -d\theta _e = \sum _{i=0}^n dq^i\wedge dp_i. \end{aligned}$$

(75)

From Eq. (56) we infer that

$$\begin{aligned} \theta _c = \theta _e/p_0 = dq^0 - \gamma _1(q) dq^1 - \gamma _2(q) dq^2 - \sum ^n_{k=3} \gamma _k(q) dq^k = 0, \end{aligned}$$

(76)

which specifies the physical thermodynamic submanifold.

From Eqs. (74) and (76) we infer that

$$\begin{aligned} \theta _e =0, \end{aligned}$$

(77)

on the Thermodynamic Extended State Manifold, which also yields

$$\begin{aligned} \omega _e = 0. \end{aligned}$$

(78)

The above can be stated and generalized by considering the Thermodynamic Contact Space $C^{2n+1}$ of dimension $2n+1$ as the Projective Space of thermodynamic extended phase space

$$\begin{aligned} \pi : T^*Q^{n+1} \rightarrow {\mathbb {P}}(T^*Q^{n+1}), \end{aligned}$$

(79)

where $\pi$ is the projection map. In coordinates we write

$$\begin{aligned} \pi ^i \left( \begin{array}{c} q^0 \\ q^1 \\ \dots \\ q^n \\ p_0 \\ p_1 \\ \dots \\ p_{i-1} \\ p_i \\ p_{i+1} \\ \dots \\ p_n \end{array} \right) = \left( \begin{array}{c} q^0 \\ q^1 \\ \dots \\ q^n \\ -{p_0}/{p_i} \\ -{p_1}/{p_i} \\ \dots \\ -{p_{i-1}}/{p_i} \\ -{p_{i+1}}/{p_i} \\ \dots \\ -{p_n}/{p_i} \end{array} \right) . \end{aligned}$$

(80)

A significant outcome of introducing the extended phase space is the selection of a representation of thermodynamics by choosing the projection coordinate. We can see that by taking $p_0$ as projection coordinate we obtain the thermodynamic contact $(2n+1)D$-space [see Eq. (56)] with conjugate momenta the variables $\gamma _i$ in the entropy representation

$$\begin{aligned} \pi ^0 \left( \begin{array}{c} q^0 \\ q^1 \\ q^2 \\ q^3 \\ \dots \\ q^n \\ p_0 \\ p_1 \\ p_2 \\ p_3 \\ \dots \\ p_n \end{array} \right) = \left( \begin{array}{c} q^0 \\ q^1 \\ q^2 \\ q^3 \\ \dots \\ q^n \\ -{p_1}/{p_0} \\ -{p_2}/{p_0} \\ -{p_3}/{p_0} \\ \dots \\ -{p_n}/{p_0} \end{array} \right) = \left( \begin{array}{r} q^0 \\ q^1 \\ q^2 \\ q^3 \\ \dots \\ q^n \\ \gamma _1 \\ \gamma _2 \\ \gamma _3 \\ \dots \\ \gamma _n \end{array} \right) . \end{aligned}$$

(81)

In problems where the variable $q^0$ is also excluded, then, we produce the thermodynamic 2nD phase space in entropy representation.

Instead, taking as projection variable to be the $p_1$, we obtain the thermodynamic contact $(2n+1)D$-space in the energy representation with canonical conjugate momenta the variables $\beta$ and 1-form

$$\begin{aligned} \theta _c = dq^1 - \beta _0(q) dq^0 - \beta _2(q) dq^2 - \sum ^n_{k=3} \beta _k(q) dq^k. \end{aligned}$$

(82)

The projection map $\pi ^1$ is

$$\begin{aligned} \pi ^1 \left( \begin{array}{c} q^0 \\ q^1 \\ q^2 \\ q^3 \\ \dots \\ q^n \\ p_0 \\ p_1 \\ p_2 \\ p_3 \\ \dots \\ p_n \end{array} \right) = \left( \begin{array}{c} q^0 \\ q^1 \\ q^2 \\ q^3 \\ \dots \\ q^n \\ -{p_0}/{p_1} \\ -{p_2}/{p_1} \\ -{p_3}/{p_1} \\ \dots \\ -{p_n}/{p_1} \end{array} \right) = \left( \begin{array}{r} q^0 \\ q^1 \\ q^2 \\ q^3 \\ \dots \\ q^n \\ \beta _0 \\ \beta _2 \\ \beta _3 \\ \dots \\ \beta _n \end{array} \right) . \end{aligned}$$

(83)

In problems where the variable U is excluded, then, we produce the thermodynamic 2nD phase space in energy representation.

Similarly, by choosing $p_2$ the projective space gives the volume representation of the thermodynamic contact space.

1.4 A.4. Legendrian and Lagrangian submanifolds of physical states

In the previous sections we have specified the physical thermodynamic submanifold to be a Legendrian submanifold in the contact space and the thermodynamic extended state manifold to be a Lagrangian submanifold in phase space. Here we give their formal definitions.

In the contact space $C^{2n+1}$ the locally defined 1-form $\theta _c$, which satisfies the non-integrability condition, i.e., there is the volume form

$$\begin{aligned} \theta _c \wedge (d\theta _c)^n \ne 0, \end{aligned}$$

determines the nD-Legendrian submanifold $L_c^n$ by requiring $\theta _c = 0$. In other words, the distribution $\varDelta = ker(\theta _c)$ provides the maximal dimension hyperplanes tangent to $L_c^n$.

In even dimensional phase space $P^{2n+2}$ the symplectic 2-form $\omega$ [Eqs. (54, 78)], which satisfies the non-integrability condition (volume form)

$$\begin{aligned} (d\theta )^{n+1} \ne 0, \end{aligned}$$

determines the $(n+1)D$-Lagrangian submanifold $L_p^{n+1}$ by requiring $\omega = 0$.

The thermodynamic contact space $C^{2n+1}$ has been the projective space of thermodynamic extended phase space of dimension $2n+2$

$$\begin{aligned} \pi : T^*Q^{n+1} \rightarrow {\mathbb {P}}(T^*Q^{n+1}), \end{aligned}$$

(84)

where $\pi$ the projection map.

It can be proved that the submanifold $L_c^n \subset {\mathbb {P}}(T^*Q^{n+1})$ is a Legendrian submanifold if and only if

$$\begin{aligned} L_p^{n+1} := \pi ^{-1}(L_c^n)\subset T^*Q^{n+1} \end{aligned}$$

is a homogeneous (in momentum) Lagrangian submanifold.

Also, every homogeneous Lagrangian submanifold originates from a Legendrian submanifold of the form $\pi ^{-1}(L_c^n)$. In Table 2 we show mappings between contact and extended phase space.

Table 2 Projection mappings $(\pi )$ between thermodynamic extended phase space (TEPS) and thermodynamic contact space (TCS)

The thermodynamic extended state manifold is a $(n+1)D$- Lagrangian submanifold of $P^{2n+2}$. In the entropy representation, the n extensive independent variables $(q^1,\dots , q^n)$ together with the $p_0$ parametrize the Lagrangian submanifold $L_p^{n+1}$. If the entropy $S(q^1, \dots , q^n)$ is the generating function of the Legendrian manifold, then for the Lagrangian manifold the generating function is $- p_0 S(q^1, \dots , q^n)$. With this generating function we extract the remaining $n + 1$ variables as

$$\begin{aligned} q^0 = S(q^1, \dots , q^n), \;\;\;\; p_i = - p_0 \frac{\partial S}{\partial q^i} = - p_0 \gamma _i, \;\; i=1,\dots , n. \end{aligned}$$

(85)

1.5 A.5. Metrics on the physical thermodynamic submanifold

We can define a torsion-free, trivial, pseudo-Riemannian metric [22] on a Lagrangian $(n+1)D$-submanifold, $L_p^{n+1} \subset P^{2n+2}$, with generating function $- p_0 S(q^1, \dots , q^n)$. In canonical coordinates (q, p) it takes the form

$$\begin{aligned} R(q, p) = \sum _{i=0}^n dq^i\otimes dp_i = p_0 R_S, \end{aligned}$$

(86)

where

$$\begin{aligned} R_S(q^1, \dots , q^n, \gamma _1, \dots , \gamma _n)&= - \sum _{i=1}^n dq^i\otimes d\gamma _i(q) \\&= - \sum _{i=1}^n \sum _{j=1}^n dq^i\otimes \frac{\partial ^2 S}{\partial q^i \partial q^j} dq^j \\&= - \sum _{i=1}^n \sum _{j=1}^n \frac{\partial ^2 S}{\partial q^i \partial q^j} dq^i\otimes dq^j. \end{aligned}$$

(87)

In the entropy representation and also excluding the coordinate $q^0=S$ the physical thermodynamic submanifold is the nD-Lagrangian submanifold, $L_p^{n} \subset P^{2n}$. $R_S$ is the metric introduced by Ruppeiner [25, 26] and it is usually called Ruppeiner metric.

Weinhold [27, 28] has extracted a metric, $R_U$, in the energy representation of thermodynamics with generating function $- p_1 U(q^0, q^2,\dots , q^n)$

$$\begin{aligned} R(q, p) = \sum _{i=0}^n dq^i\otimes dp_i = -p_1 R_U, \end{aligned}$$

(88)

where

$$\begin{aligned} R_U(q^0, q^2,\dots , q^n, \beta _0, \beta _2, \dots , \beta _n)&= dq^0\otimes d\beta _0 + \sum _{j=2}^n dq^j\otimes d\beta _j \\&= \sum _{i=2}^n \sum _{j=2}^n \frac{\partial ^2 U}{\partial q^i \partial q^j} dq^i\otimes dq^j \\&\quad + \sum _{j=2}^n \frac{\partial ^2 U}{\partial q^0 \partial q^j} dq^0\otimes dq^j \\&\quad + \sum _{i=2}^n \frac{\partial ^2 U}{\partial q^i \partial q^0} dq^i\otimes dq^0 \\&\quad + \frac{\partial ^2 U}{\partial (q^0)^2} dq^0\otimes dq^0 \end{aligned}$$

(89)

As in the entropy representation and excluding the coordinate $q^1=U$, the Weinhold metric is the metric in the nD-Lagrangian submanifold in the 2nD thermodynamic phase space.

From the above we conclude that Ruppeiner’s and Weinhold’s metrics are related by the equations

$$\begin{aligned} p_0 R_S&= -p_1 R_U \\ R_S&= - \frac{p_1}{p_0}R_U = \gamma _1 R_U = \frac{1}{T}R_U. \end{aligned}$$

(90)

1.6 A.6. Symmetries of thermodynamic manifolds

1.6.1 A.6.1. Gauge transformations

We introduce the gauge transformation $(P_0=\lambda p_0)$ of conjugate momenta and leave coordinates the same

$$\begin{aligned} F: P^{2n+2} \rightarrow P^{2n+2}\;\; ; \;\; (q,p)\rightarrow (Q,P), \end{aligned}$$

(91)

with

$$\begin{aligned} Q^i&= q^i, \\ P_i&= \lambda p_i, \;\; i=0,\dots ,n. \end{aligned}$$

(92)

If $\lambda \ne 0$ is a constant, then, the gauge transformation is of first-kind. If $\lambda (q,p) \ne 0$ is a function of coordinates and momenta, then, the gauge transformation is of second-kind. From Eq. (73) we can see that $\gamma _i$ are invariant under these gauge transformations. Therefore, we may infer that the observable quantities $\gamma _i$ are not affected by the values of $p_0$.

Poincaré 1-form of the thermodynamic extended phase space [Eq. (74)] also remains invariant in the thermodynamic extended state manifold [Eq. (77)], which is a $(n+1)D$ Lagrangian submanifold $(L_p^{n+1})$ in $P^{2n+2}$ phase space, parametrized by $(p_0,q^1,q^2,\dots ,q^n)$.

Since, $\lambda$ can be absorbed by the gauge we conclude that the mappings F act on the Poincaré 1-form as

$$\begin{aligned} F^*(\theta _e) = \theta _e. \end{aligned}$$

(93)

It is also valid

$$\begin{aligned} F^*(\omega _e) = \omega _e. \end{aligned}$$

(94)

Hence, the gauge transformations [Eq. (92)] are canonical transformations on the thermodynamic extended state manifold $(\theta _e=0)$. Furthermore, entropy, a homogeneous function of the extended coordinates, is a generating function. Indeed,

$$\begin{aligned} S = q^0 = \sum _{i=1}^n \frac{\partial S}{\partial q^i}q^i, \end{aligned}$$

(95)

and in the thermodynamic extended phase space becomes

$$\begin{aligned} \sum _{i=0}^n p_iq^i = 0. \end{aligned}$$

(96)

The complete set of the physical canonical coordinates assign the extensivity sheet (E) and the Cartesian product $E\times R$ the thermodynamic extended phase space. If the thermodynamic system is restricted with respect to a variable, the thermodynamic extended state manifold does not belong to E.

1.6.2 A.6.2. Legendre transformations

If ${G}(q^1, \dots , q^n)$ is a thermodynamic potential of n independent extensive variables, which sometimes we want to transform to a new potential $\mathscr{L}$ using as independent variables the $q^i, i=1, \dots , r$ and the $u_j = {\partial {G}}/{\partial q^j}, j=r+1, \dots , n$. This is obtained by Legendre transformations

$$\begin{aligned} {\mathscr{L}}(q^1, \dots , q^r, u_{r+1}, \dots , u_n) = G(q) - \sum _{j=r+1}^n q^j\frac{\partial G}{\partial q^j} = {G}(q) - \sum _{j=r+1}^n q^ju_j, \end{aligned}$$

(97)

and

$$\begin{aligned} d{\mathscr{L}}&= d{G} - \sum _{j=r+1}^n ( q^j du_j + dq^j u_j) \\&= \sum _{i=1}^r u_i dq^i - \sum _{j=r+1}^n q^j du_j. \end{aligned}$$

(98)

The Legendre transformations are diffeomorphisms of the thermodynamic contact space [34,35,36]. However, they keep this property when they act on thermodynamic extended phase space if and only if the Hessian of the thermodynamic potential G(q) is non-degenerate

$$\begin{aligned} \frac{\partial ^2 G(q)}{\partial q^i \partial q^j} \ne 0, \;\;\;\; i,j = 1,\dots , n. \end{aligned}$$

(99)

The metric defined in the thermodynamic contact space may change under a Legendre transformation.

Appendix B: Hamiltonians and Hamilton equations for thermodynamic systems

Considering thermodynamic systems as Hamiltonian dynamical systems we have to take into account the homogeneity property of the thermodynamic functions, which makes thermodynamic Hamiltonians a special category. We point out that a Hamiltonian system is the triple $(P^{2m}, \omega , X_H)$, where $X_H$ the Hamiltonian vector field defined in the phase space $P^{2m}$, and $\omega$ a skew-symmetric, non-degenerate, differential 2-form. Then, we can use the tools of geometric mechanics to write down equilibrium and non-equilibrium equations. Thus, the Hamiltonian vector field can be extracted from the equation

$$\begin{aligned} i_{X_H}\omega (X_H, \bullet ) = dH, \;\;\;\;\;\;\; H : P^{2m} \mapsto {\mathbb {R}}. \end{aligned}$$

(100)

The Hamiltonian vector field can also be expressed by the ordinary Hamilton’s equations

$$\begin{aligned} \dot{q}^i&= \frac{\partial H}{\partial p_i} \end{aligned}$$

(101)

$$\begin{aligned} \dot{p}_i&= -\frac{\partial H}{\partial q^i}, \;\; i=0,\dots ,n. \end{aligned}$$

(102)

Moreover, (q, p) is a canonical coordinate system in the corresponding phase space, and thus, satisfy the Poisson brackets

$$\begin{aligned} \{q^i,q^j\}&= \{p_i,p_j\} = 0, \;\;\;\; \{q^i,p_j\} = \delta ^i_j, \end{aligned}$$

(103)

$$\begin{aligned} \dot{q}^i&= \{q^i,H\}, \;\;\;\; \dot{p}_j = \{p_j,H\}. \end{aligned}$$

(104)

Infinitesimal canonical transformations are generated by smooth Hamiltonian functions, H(q, p). For homogeneous Hamiltonian functions of first degree in p, in general, we write

$$\begin{aligned} H(q,p) = \sum _{i=0}^n p_i \frac{\partial H}{\partial p_i}, \end{aligned}$$

(105)

an equation which can also be put in the form

$$\begin{aligned} \theta (X_H) = H, \end{aligned}$$

(106)

with $\theta$ to be the canonical Poincaré 1-form.

Proof

$$\begin{aligned} X_{H} = \sum _{i=0}^n \frac{\partial H}{\partial p_i} \frac{\partial }{\partial q^i} - \sum _{i=0}^n \frac{\partial H}{\partial q^i} \frac{\partial }{\partial p_i}. \end{aligned}$$

(107)

Thus,

$$\begin{aligned} \theta (X_{H})&= \sum _{i=0}^n p_i dq^i (X_{H}) = \sum _{i=0}^n p_i \frac{\partial H}{\partial p_i} - 0 \\&= \sum _{i=0}^n p_i \frac{\partial H}{\partial p_i} = H. \end{aligned}$$

(108)

$\square$

Also, for Hamiltonian systems the symplectic 2-form satisfies the equation

$$\begin{aligned} \omega (X_H, X_H) = 0. \end{aligned}$$

(109)

For homogeneous Hamiltonians of first degree in p it is also proved that the Lie derivative of $\theta$ with respect to the Hamiltonian vector field is

$$\begin{aligned} {\mathcal{{L}}}_{X_H}\theta = 0. \end{aligned}$$

(110)

Inversely, if Eq. (110) is true, then, the Hamiltonian is a homogeneous function of first degree in p.

Proof

$$\begin{aligned} {\mathcal{{L}}}_{X_{H}}\theta = \frac{d\theta }{dt}&= \sum _{i=0}^n \frac{d}{dt} \left( p_idq^i\right) \\&= \sum _{i=0}^n \left( p_i d\dot{q}^i + \dot{p}_i dq^i \right) \\&= \sum _{i=0}^n \left[ p_i d\left( \frac{\partial H}{\partial p_i}\right) - \left( \frac{\partial H}{\partial q^i}\right) dq^i \right] \\&= \sum _{i=0}^n \left[ d\left( p_i\frac{\partial H}{\partial p_i}\right) - \left( \frac{\partial H}{\partial p_i}\right) dp_i - \left( \frac{\partial H}{\partial q^i}\right) dq^i \right] \\&= \sum _{i=0}^n \left[ d\left( p_i\frac{\partial H}{\partial p_i}\right) \right] - \sum _{i=0}^n \left[ \left( \frac{\partial H}{\partial p_i}\right) dp_i + \left( \frac{\partial H}{\partial q^i}\right) dq^i \right] \\&= \sum _{i=0}^n d\left( p_i\frac{\partial H}{\partial p_i}\right) - dH \\&= d\left[ \sum _{i=0}^n \left( p_i\frac{\partial H}{\partial p_i}\right) - H\right] = 0. \end{aligned}$$

(111)

$\square$

We have shown that the Hamiltonian H is a homogeneous function of first degree in the momenta

$$\begin{aligned} H(q,p) = \sum _{i=0}^n p_i\frac{\partial H}{\partial p_i}, \end{aligned}$$

(112)

which yields

$$\begin{aligned} \lambda H(q^0,q^1,q^2,\dots ,q^n,p_0,p_1,p_2,\dots ,p_n) = H(q^0,q^1,q^2,\dots ,q^n,\lambda p_0,\lambda p_1,\lambda p_2,\dots ,\lambda p_n). \end{aligned}$$

(113)

Hamiltonian functions on thermodynamic extended phase space will be denoted as $H_e$, i.e.,

$$\begin{aligned} H_e: P^{2n + 2} \equiv T^*Q^{n+1} \mapsto {\mathbb {R}}. \end{aligned}$$

For completely extensive charts we can prove that the Hamiltonian is a homogeneous function of first degree in coordinates as well. From Eq. (96)

$$\begin{aligned} \frac{d}{dt}\left( \sum _{i=0}^n p_iq^i\right)&= \sum _{i=0}^n \left( p_i \dot{q}^i + \dot{p}_i q^i \right) \\&= \sum _{i=0}^n \left[ p_i \left( \frac{\partial H_e}{\partial p_i}\right) - \left( \frac{\partial H_e}{\partial q^i}\right) q^i \right] \\&= 0. \end{aligned}$$

(114)

Hence,

$$\begin{aligned} H_e = \sum _{i=0}^n p_i \left( \frac{\partial H_e}{\partial p_i}\right) = \sum _{i=0}^n \left( \frac{\partial H_e}{\partial q^i}\right) q^i. \end{aligned}$$

(115)

In other words, it is valid

$$\begin{aligned} \lambda H_e(q^0,q^1,q^2,\dots ,q^n,p_0,p_1,p_2,\dots ,p_n) = H_e(\lambda q^0,\lambda q^1,\lambda q^2,\dots ,\lambda q^n,p_0,p_1,p_2,\dots ,p_n). \end{aligned}$$

(116)

The thermodynamic extended state manifold in the extended Hamiltonian system is determined by taking $q^0=S(q^1,\dots ,q^n)$. We can then prove $H_e=0$.

Proof

$$\begin{aligned} \dot{q}^0 = \frac{\partial H_e}{\partial p_0}&= \frac{d S}{dt} \\&= \sum _{i=1}^n \frac{\partial S}{\partial q^i}\dot{q}^i \\&= \sum _{i=1}^n \gamma _i\dot{q}^i \\&= -\sum _{i=1}^n \left( \frac{p_i}{p_0}\right) \frac{\partial H_e}{\partial p_i} \\&= -\frac{1}{p_0}\sum _{i=1}^n p_i\frac{\partial H_e}{\partial p_i}. \end{aligned}$$

(117)

Thus, rearranging the above equation and using Eq. (112) we take

$$\begin{aligned} \sum _{i=0}^n p_i\frac{\partial H_e}{\partial p_i} = H_e \equiv 0. \end{aligned}$$

(118)

Hence, the Hamiltonian function in the extended cotangent bundle can be written as

$$\begin{aligned} H_e&= \sum _{i=0}^n p_i\left( \frac{\partial H_e}{\partial p_i}\right) = p_0\dot{q}^0 + \sum _{i=1}^n p_i \dot{q}^i \\&= p_0 \dot{S}(q^1,q^2,\dots , q^n) + \sum _{i=1}^n p_i \dot{q}^i \\&= p_0 \left( \sum _{i=1}^n \left( \frac{\partial S}{\partial q^i}\right) \dot{q}^i \right) + \sum _{i=1}^n p_i \dot{q}^i \\&= \sum _{i=1}^n \left( p_i + p_0 \frac{\partial S}{\partial q^i} \right) \dot{q}^i \\&= \left\langle \left( p + p_0 \frac{\partial S}{\partial q}\right) \Bigg |\dot{q} \right\rangle _n. \end{aligned}$$

(119)

We can also prove that the Lagrangian submanifold $(L_p^{n+1})$ in thermodynamic extended phase space is homogeneous of first degree and gauge invariant. This means that $d\theta _e=0$ as well as

$$\begin{aligned} (q,p) \in L_p^{n+1} \implies (q,\lambda p) \in L_p^{n+1}, \;\; {\mathrm {for\;\; every}}\;\; \lambda \ne 0. \end{aligned}$$

Then, $\theta _e = 0$ for every vector field tangent to $L_p^{n+1}$.

1.1 B.1. General requirements for thermodynamic Hamiltonians

Hamiltonians and Hamilton’s equations must obey the conservation laws of certain quantities, as well as the symmetries of the system.

• 1. Thermodynamic Hamiltonians $H_e: T^*Q^{n+1} \mapsto {\mathbb {R}}$ are homogeneous functions of first degree in conjugate momenta Eq. (119).

• 2. Conservation of the total energy, momenta (linear and angular) for isolated systems.

• 3. Conservation of the entropy for reversible processes and consistency with the additivity, monotonicity and concavity properties of this function.

• 4. Zero heat for adiabatic processes.

• 5. If we define a partition$\{\alpha \}$ of a system the fluxes of the quantities $q^i$ between the subsystems $(\alpha \rightarrow \beta )$, i.e., flux is going out from subsystem $\alpha$ to $\beta$, and $(\beta \rightarrow \alpha )$ from subsystem $\beta$ to $\alpha$, should satisfy

  $$\begin{aligned} \Phi ^{(\alpha ,i)}_\beta = - \Phi ^{(\beta ,i)}_\alpha . \end{aligned}$$

  (120)

  The change of the quantities $q^{(\alpha ,i)}$ with time (velocities) for an isolated system should satisfy the macroscopic balance and are given by the equation

  $$\begin{aligned} \dot{q}^{(\alpha ,i)} + \sum _\beta \Phi ^{(\alpha ,i)}_\beta =0. \end{aligned}$$

  (121)

  The change of the quantities $q^{(\alpha ,i)}$ with time for an opened system are given by the equation

  $$\begin{aligned} \dot{q}^{(\alpha ,i)} + \sum _\beta \Phi ^{(\alpha ,i)}_\beta = F_i({\mathrm {sources}} \rightarrow \alpha ). \end{aligned}$$

  (122)

  $F_i$ are the external sources with which the system interacts.

For a system with partition $\{\alpha \}$ the total entropy is

$$\begin{aligned} S\left( \{q^{(\alpha ,i)}\}\right) = \sum _{\alpha } S_\alpha \left( q^{(\alpha ,1)},\dots ,q^{(\alpha ,n)}\right) , \end{aligned}$$

(123)

and the thermodynamic extended state manifold is described by

$$\begin{aligned} q^0&= S\left( {\{q^{(\alpha ,i)}\}}\right) , \end{aligned}$$

(124)

$$\begin{aligned} p_{(\alpha ,i)}&= -p_0\frac{\partial S_\alpha \left( q^{(\alpha ,1)},\dots ,q^{(\alpha ,n)}\right) }{\partial q^{(\alpha ,i)}}, \end{aligned}$$

(125)

and the homogeneous Hamiltonian as

$$\begin{aligned} H_e&= \sum _{i=1}^n\sum _{\alpha }\sum _{\beta } \Phi _{\beta }^{(\alpha ,i)}\left( -\frac{p_{(\alpha ,i)}}{p_0}, -\frac{p_{(\beta ,i)}}{p_0}\right) \\&\left[ \left( p_{(\alpha ,i)} + p_0\frac{S_\alpha \left( q^{(\alpha ,1)},\dots ,q^{(\alpha ,n)}\right) }{\partial q^{(\alpha ,i)}}\right) \right. \\&\left. - \left( p_{(\beta ,i)} + p_0\frac{S_\beta \left( q^{(\beta ,1)},\dots ,q^{(\beta ,n)}\right) }{\partial q^{(\beta ,i)}}\right) \right] . \end{aligned}$$

(126)

The flux $\Phi _{\beta }^{(\alpha ,i)}\left( -\frac{p_{(\alpha ,i)}}{p_0}, -\frac{p_{(\beta ,i)}}{p_0}\right)$ can be expressed by the response coefficients$L_{\beta }^{(\alpha ,i)}\left( \gamma _{(\alpha ,i)}, \gamma _{(\beta ,i)}\right)$

$$\begin{aligned} \Phi _{\beta }^{(\alpha ,i)}\left( -\frac{p_{(\alpha ,i)}}{p_0}, -\frac{p_{(\beta ,i)}}{p_0}\right) = L_{\beta }^{(\alpha ,i)}\left( \gamma _{(\alpha ,i)}, \gamma _{(\beta ,i)}\right) (\gamma _{(\alpha ,i)} - \gamma _{(\beta ,i)}). \end{aligned}$$

(127)

Then, the Hamiltonian is written as

$$\begin{aligned} H_e&= \sum _{i=1}^n\sum _{\alpha }\sum _{\beta } L\left( \frac{\partial S_\alpha \left( q^{(\alpha ,1)},\dots ,q^{(\alpha ,n)}\right) }{\partial q^{(\alpha ,i)}}, \frac{\partial S_\beta \left( q^{(\beta ,1)},\dots ,q^{(\beta ,n)}\right) }{\partial q^{(\beta ,i)}}\right) \\&\left[ -\frac{1}{2p_0}(p_{(\alpha ,i)} -p_{(\beta ,i)})^2 + \right. \\&\left. \frac{p_0}{2}\left( \frac{\partial S_\alpha \left( q^{(\alpha ,1)},\dots ,q^{(\alpha ,n)}\right) }{\partial q^{(\alpha ,i)}} - \frac{\partial S_\beta \left( q^{(\beta ,1)},\dots ,q^{(\beta ,n)}\right) }{\partial q^{(\beta ,i)}}\right) ^2\right] . \end{aligned}$$

(128)

The change of entropy (for example during the dissipation in isolated systems) is computed by integrating the equation

$$\begin{aligned} \dot{q}^0 = \frac{\partial H_e}{\partial p_0}. \end{aligned}$$

(129)

For systems with Hamiltonians independent of $q^0$, it is valid

$$\begin{aligned} \dot{p}_0 = -\frac{\partial H_e}{\partial q^0} = 0, \end{aligned}$$

(130)

i.e., $p_0$ is constant, which is fixed to $p_0=-1$ in the entropy representation.

$$\begin{aligned} H=\sum _{i=1}^n\sum _{\alpha }\sum _{\beta } \Phi _{\beta }^{(\alpha ,i)}(\gamma _1,\dots ,\gamma _n) \left[ \frac{\partial S(q^1,\dots ,q^n)}{\partial q^i} - \gamma _i\right] , \end{aligned}$$

(131)

or

$$\begin{aligned} H=\sum _{i=1}^n\sum _{\alpha }\sum _{\beta } \Phi _{\beta }^{\alpha ,i}(\gamma _1,\dots ,\gamma _n) \left[ \frac{\partial S(q^1,\dots ,q^n)}{\partial q^i} - \gamma _i\right] + \left[ \frac{\partial S(q^1,\dots ,q^n)}{\partial q^i} - \gamma _i\right] ^2. \end{aligned}$$

(132)

The last term is added to stabilize the Hamiltonian flow, since it does not affect the thermodynamic extended state manifold.