**6. Contact GENERIC**

We begin by introducing a space

$$\mathbf{M}^{(cont)} = \mathbf{M} \times \mathbf{M}^\* \times \mathbf{N}\_{eq}^\* \times \mathbf{N}\_{eq} \times \mathbb{R} \tag{78}$$

with coordinates (*x*, *x*∗, *y*∗, *y*, *φ*). The space **M** with elements *x* is the state space, the space **M**∗ with elements *x*∗ is its dual. Similarly, **N***eq* with elements *y* is the state space on the equilibrium level, **N**∗ *eq* with elements *y*<sup>∗</sup> is its dual. We recall that *y* = (*E*, *N*) and *y*<sup>∗</sup> = (*E*∗, *N*∗), where *E*<sup>∗</sup> = <sup>1</sup> *<sup>T</sup>* and *<sup>N</sup>*<sup>∗</sup> <sup>=</sup> <sup>−</sup> *<sup>μ</sup> T* . We moreover introduce the fundamental thermodynamic relation *S* = *S*(*x*), *y* = *y*(*x*) represented in **<sup>M</sup>** <sup>×</sup> **<sup>N</sup>***eq* <sup>×</sup> <sup>R</sup> by the **Gibbs manifold** <sup>M</sup>(*G*) that is the image of the mapping

$$\mathbf{x} \hookrightarrow (\mathbf{x}, \mathbf{y}(\mathbf{x}), \mathbf{S}(\mathbf{x})). \tag{79}$$

Corresponding to the fundamental thermodynamic relation is the thermodynamic potential Φ(*x*, *y*∗) = −*S*(*x*) + *y*∗, *y*(*x*), where , denotes the inner product.

The Gibbs manifold <sup>M</sup>(*G*) can be now extended to the **Gibbs–Legendre manifold** <sup>M</sup>(*GL*) (in the shorthand notation *GL manifold*) that is the image of the mapping

$$(\mathbf{x}, \mathbf{y}^\*) \hookrightarrow (\mathbf{x}, \Phi\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}^\*), \mathbf{y}^\*, \Phi\_{\mathbf{y}^\*}(\mathbf{x}, \mathbf{y}^\*), \Phi(\mathbf{x}, \mathbf{y}^\*)) \tag{80}$$

in M(*cont*).

The thermodynamics in **<sup>M</sup>** is completely expressed in the GL manifold <sup>M</sup>(*GL*). Note that [M(*GL*)]*<sup>y</sup>*∗=<sup>0</sup> (i.e., the image of the mapping

$$S(x,0) \hookrightarrow (x, -S\_x(x), 0, y(x), -S(x))\tag{81}$$

in the space **<sup>M</sup>** <sup>×</sup> **<sup>M</sup>**<sup>∗</sup> <sup>×</sup> **<sup>N</sup>***eq* <sup>×</sup> <sup>R</sup>) is an extension of the Gibbs manifold <sup>M</sup>(*G*) by including the conjugate variable *<sup>x</sup>*∗. Moreover, the manifold [M(*GL*)]*<sup>x</sup>*∗=*Sx*=<sup>0</sup> displays the states *xeq*(*y*∗) that represent in **<sup>M</sup>** the equilibrium states and also the fundamental thermodynamic relation *S*∗(*y*∗), *y*(*y*∗) in **N***eq* implied by the fundamental thermodynamic relation *<sup>S</sup>*(*x*), *<sup>y</sup>*(*x*) in **<sup>M</sup>**. Indeed, [M(*GL*)]*<sup>x</sup>*∗=<sup>0</sup> is the image of the mapping

$$(x, y^\*) \hookrightarrow (x\_{\text{eq}}(y^\*), 0, y^\*, y(x\_{\text{eq}}(y^\*)), S^\*(y^\*)). \tag{82}$$

Let us turn to the time evolution in M(*cont*). We begin by introducing in M(*cont*) a bracket

$$\begin{aligned} \{A,B\}^{\text{(cont)}} &= -\left( \langle A\_{\mathcal{X}}, B\_{\mathcal{X}^\*} \rangle - \langle B\_{\mathcal{X}}, A\_{\mathcal{X}^\*} \rangle \right) \\ &- \left( \langle A\_{\mathcal{Y}}, B\_{\mathcal{Y}^\*} \rangle - \langle B\_{\mathcal{Y}}, A\_{\mathcal{Y}^\*} \rangle \right) \\ &- \left( \langle \mathbf{x}^\*, A\_{\mathcal{X}^\*} \rangle B\_{\boldsymbol{\theta}} - \langle \mathbf{x}^\*, B\_{\mathcal{X}^\*} \rangle A\_{\boldsymbol{\theta}} \right) \\ &+ \left( AB\_{\boldsymbol{\theta}} - BA\_{\boldsymbol{\theta}} \right) \\ &+ \left( \langle A\_{\mathcal{Y}}, \mathbf{y} \rangle B\_{\boldsymbol{\theta}} - \langle B\_{\mathcal{Y}}, \mathbf{y} \rangle A\_{\boldsymbol{\theta}} \right), \end{aligned} \tag{83}$$

where *<sup>A</sup>* and *<sup>B</sup>* are sufficiently regular functions <sup>M</sup>(*cont*) <sup>→</sup> <sup>R</sup>. This bracket consists of two contact brackets (95) of paper [41]. With such bracket, we introduce the time evolution in M(*cont*) by an equation

$$A = \{A, H^{(cont)}\}^{(cont)} - AH\_{\phi}^{(cont)} \tag{84}$$

that is required to hold for all *<sup>A</sup>*. The function *<sup>H</sup>*(*cont*) : <sup>M</sup>(*cont*) <sup>→</sup> <sup>R</sup>, called a contact Hamiltonian, will be specified below. The last term on the right-hand side corresponds to the non-conservation of the phase-space volume [40]. Written explicitly, the time evolution equations (84) take the form

$$\dot{\mathbf{x}} = \begin{array}{c} H\_{\mathbf{x}^\*}^{(\text{cont})} \\ \end{array} \tag{85a}$$

$$\dot{\mathbf{x}}^{\*} = -H\_{\mathbf{x}}^{(cont)} - \mathbf{x}^{\*}H\_{\boldsymbol{\Phi}}^{(cont)} \, , \tag{85b}$$

$$\dot{y}^\* = \quad H\_y^{(cont)},\tag{85c}$$

$$\begin{array}{rcl} \dot{y} &=& -H\_{y^\*}^{(cont)} + yH\_{\phi}^{(cont)} \end{array} \tag{85d}$$

$$
\dot{\Phi}^\* = -H^{(cont)} + \langle \mathbf{x}^\*, H\_{\mathbf{x}^\*}^{(cont)} \rangle - \langle H\_{\mathbf{y}}^{(cont)}, y \rangle. \tag{85e}
$$

These are the evolution equations in M.

Next, we specify the contact Hamiltonian *H*(*cont*)

$$H^{(cont)}({\bf x},{\bf x}^\*,{y}^\*,{y},{\bf y}) = -S^{(cont)}({\bf x},{\bf x}^\*,{y}^\*) + \frac{1}{E^\*}E^{(cont)}({\bf x},{\bf x}^\*,{y}^\*),\tag{86}$$

*Entropy* **2019**, *21*, 715

where

$$\begin{array}{rcl} S^{(cont)}(\mathbf{x}, \mathbf{x}^\*, y^\*) & = & \Xi(\mathbf{x}, \mathbf{x}^\*, y^\*) - [\Xi(\mathbf{x}, \mathbf{x}^\*, y^\*)]\_{\mathbf{x}^\* = \Phi\_{\mathbf{x}^\*}} \\ E^{(cont)}(\mathbf{x}, \mathbf{x}^\*, y^\*) & = & \langle \mathbf{x}^\*, L\Phi\_{\mathbf{x}} \rangle. \end{array} \tag{87}$$

Ξ is the dissipation potential entering GENERIC and *L* is the Poisson bivector also entering GENERIC. Both Ξ and *L* are degenerate in the sense

$$
\langle \mathbf{x}^\*, L\mathbf{S}\_\mathbf{x} \rangle = \langle \mathbf{x}^\*, L\mathbf{N}\_\mathbf{x} \rangle = 0, \quad \forall \mathbf{x}^\*,
$$

$$
\langle E\_\mathbf{x}, \Xi\_{\mathbf{x}^\*} \rangle = \langle N\_\mathbf{x}, \Xi\_{\mathbf{x}^\*} \rangle = 0, \quad \forall \mathbf{x}^\*,
$$

$$
\langle \mathbf{x}^\*, [\Xi\_{\mathbf{x}^\*}]\_{\mathbf{x}^\* = E\_\mathbf{x}} \rangle = \langle \mathbf{x}^\*, [\Xi\_{\mathbf{x}^\*}]\_{\mathbf{x}^\* = M\_\mathbf{x}} \rangle = 0, \quad \forall \mathbf{x}^\*.\tag{88}
$$

We note in particular that the contact Hamiltonian (86) is independent of *y* and *φ*. With (86), the time evolution Equations (85a) become

$$\dot{\mathbf{x}}^{\*} = \underbrace{\frac{1}{E^\*}}\_{\cdot} L \Phi\_{\mathbf{x}} - \Xi\_{\mathbf{x}^\*} \tag{89a}$$

$$\left(\dot{\mathbf{x}}^{\*}\right)\_{\mathbf{x}} = \left(\Phi\_{\mathbf{x}\mathbf{x}}\left(\frac{1}{E^{\*}}L\mathbf{x}^{\*} - [\Xi\_{\mathbf{x}}\,^{\*}]\_{\mathbf{x}}{}^{\*} = \Phi\_{\mathbf{x}}\right)\right) \tag{89b}$$

$$-\frac{1}{E^\*} \left< \mathbf{x}^\*, L\_\mathbf{x} \Phi\_\mathbf{x} \right> + \Xi\_\mathbf{x} - \left[ \Xi\_\mathbf{x} \right]\_{\mathbf{x}^\* = \Phi\_\mathbf{x}}.$$

$$\dot{\Phi} = -\left< \mathbf{x}^\*, \Xi\_{\mathbf{x}^\*} + \Xi - \left[ \Xi \right]\_{\mathbf{x}^\* = \Phi\_\mathbf{x}}.\tag{89c}$$

$$\dot{y}^\* = \begin{array}{c} 0, \\ \end{array} \tag{89d}$$

$$
\dot{\mathbf{y}} = \|\boldsymbol{\Xi}\_{\mathbf{y}^\*} - \boldsymbol{\Xi}\_{\mathbf{y}^\*}\|\_{\mathbf{x}^\* = \boldsymbol{\Phi}\_{\mathbf{x}}}.\tag{89e}
$$

If we now evaluate (89a) on the GL manifold <sup>M</sup>(*GL*) (note that [*H*(*cont*)] <sup>M</sup>(*GL*) <sup>=</sup> 0 ), we arrive at

$$\left. \dot{\mathbf{x}}^{\*} \right|\_{\mathbf{x}} = \left. \frac{1}{E^{\*}} L \Phi\_{\mathbf{x}} - \Xi\_{\mathbf{x}}{}^{\*} \right|\_{(\mathbf{x}^{\*} = \Phi\_{\mathbf{x}}, y = \Phi\_{y^{\*}})^{\*}} \tag{90a}$$

$$\dot{\mathbf{x}}^{\*} \quad = \quad \Phi\_{\text{xx}} \left( \frac{1}{E^{\*}} L \Phi\_{\text{x}} - \left[ \Xi\_{\text{x}^{\*}} \right]\_{\left( \mathbf{x}^{\*} = \Phi\_{\text{x}, \mathbf{y}} = \Phi\_{\text{y}^{\*}} \right)} \right) \,\tag{90b}$$

$$\dot{\Phi}^\* = -\langle \mathbf{x}^\*, \boldsymbol{\Xi}\_{x^\*} \rangle \big|\_{(x^\* = \Phi\_{x,y} = \Phi\_{y^\*})'} \tag{90c}$$

$$\dot{y}^\* = \begin{array}{c} 0, \\ \end{array} \tag{90d}$$

$$\begin{array}{rcl} \dot{y} & = & 0, & & \end{array} \tag{90e}$$

which are the GENERIC evolution equations. See [14] for more details.
