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.
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Notes
With complete we mean an unconstrained thermodynamic system.
\((^T)\) denotes the column vector and generally the transpose of a matrix.
The exact differential of internal energy is written as dU and usually the inexact differentials of heat and work flows as \(d\bar{}q\) and \(d\bar{}w\). However, differential forms (exact or inexact) are mathematical entities by themselves (like vectors), and thus, heat and work can be expressed generically by Q and W differential 1-forms. It is interesting to note that the first law of thermodynamics converts the sum of two inexact differentials to the exact dU and the second law of thermodynamics allows us to write the heat differential as \(Q=TdS\).
The wedge product (\(\wedge\)) of two 1-forms produces a 2D oriented surface element, whereas the wedge product of a 1-form with a 2-form gives an oriented volume element. Obviously, with m-forms and what is called exterior algebra [22] we can describe hypersurfaces or hypervolumes in higher dimensional spaces and generalize geometrical concepts familiar in the 3D world. Equation (2) implies that the volume element is zero, and thus, the thermodynamic processes take place on a (hyper)surface.
The momentum \((p_0)\) is necessary to construct an even dimensional phase space, which is a redundant variable (gauge) that we fix by projecting the extended phase space onto the odd dimensional contact space.
A smooth (differentiable) manifold is approximated locally by a (hyper)plane in a small area around a point, and thus, we can employ Euclidean geometry to calculate distances, angles, derivatives. Moreover, we can extend this property to neighboring points in a continuous way, and thus, integrate Hamilton’s equations. We use smooth manifolds to describe the set of states of a physical system.
Submanifolds are subsets of a manifold, usually with less dimensions, embedded in the manifold. It is convenient from mathematical point of view to consider the state manifold embedded in a manifold with more dimensions. In our case, the physical state manifold of a thermodynamic system, Legendrian submanifold, embedded in the contact space is what we describe with the equation of states (in its most general form [23]) and this submanifold is in one-to-one correspondence with the Lagrangian submanifold embedded in the extended thermodynamic phase space.
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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)
where we have introduced the conjugate intensive variables \(\gamma _i\) as the partial derivatives of S
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
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
or
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
In \(P^{2n}\) thermodynamic phase space the canonical Poincaré 1-form \(\theta\) is determined as
and the skew-symmetric 2-form
Since, \(\theta = dq^0\) we infer that the canonical symplectic 2-form \(\omega\) is
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
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
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
The conjugate intensive properties, temperature (T), pressure (P) and chemical potentials \((\mu _i)\) are the partial derivatives of the internal energy
Hence, the internal energy is written as
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 \(\left( \frac{\partial }{q^0}, \frac{\partial }{q^2},\frac{\partial }{\partial q^3},\dots ,\frac{\partial }{\partial q^n} \right)\)
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
or
If we define the variables \(\beta _k\) as the partial derivatives of U
then, Eq. (65) takes the form
\((dq^0, dq^2, dq^3,\dots , dq^n)\) 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
and the canonical symplectic 2-form
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
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
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
and the canonical symplectic 2-form
From Eq. (56) we infer that
which specifies the physical thermodynamic submanifold.
From Eqs. (74) and (76) we infer that
on the Thermodynamic Extended State Manifold, which also yields
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
where \(\pi\) is the projection map. In coordinates we write
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
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
The projection map \(\pi ^1\) is
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
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)
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 \(2n+2\)
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
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.
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
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
where
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 \(- p_1 U(q^0, q^2,\dots , q^n)\)
where
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
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
with
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
It is also valid
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,
and in the thermodynamic extended phase space becomes
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
and
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
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
The Hamiltonian vector field can also be expressed by the ordinary Hamilton’s equations
Moreover, (q, p) is a canonical coordinate system in the corresponding phase space, and thus, satisfy the Poisson brackets
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
an equation which can also be put in the form
with \(\theta\) to be the canonical Poincaré 1-form.
Proof
Thus,
\(\square\)
Also, for Hamiltonian systems the symplectic 2-form satisfies the equation
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
Inversely, if Eq. (110) is true, then, the Hamiltonian is a homogeneous function of first degree in p.
Proof
\(\square\)
We have shown that the Hamiltonian H is a homogeneous function of first degree in the momenta
which yields
Hamiltonian functions on thermodynamic extended phase space will be denoted as \(H_e\), i.e.,
For completely extensive charts we can prove that the Hamiltonian is a homogeneous function of first degree in coordinates as well. From Eq. (96)
Hence,
In other words, it is valid
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
Thus, rearranging the above equation and using Eq. (112) we take
Hence, the Hamiltonian function in the extended cotangent bundle can be written as
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
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
and the thermodynamic extended state manifold is described by
and the homogeneous Hamiltonian as
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\(L_{\beta }^{(\alpha ,i)}\left( \gamma _{(\alpha ,i)}, \gamma _{(\beta ,i)}\right)\)
Then, the Hamiltonian is written as
The change of entropy (for example during the dissipation in isolated systems) is computed by integrating the equation
For systems with Hamiltonians independent of \(q^0\), it is valid
i.e., \(p_0\) is constant, which is fixed to \(p_0=-1\) in the entropy representation.
or
The last term is added to stabilize the Hamiltonian flow, since it does not affect the thermodynamic extended state manifold.
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Farantos, S.C. Hamiltonian thermodynamics in the extended phase space: a unifying theory for non-linear molecular dynamics and classical thermodynamics. J Math Chem 58, 1247–1280 (2020). https://doi.org/10.1007/s10910-020-01128-z
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DOI: https://doi.org/10.1007/s10910-020-01128-z